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Essays on household consumption and labor supply Jutong, Pan 2017

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Essays on Household Consumptionand Labor SupplybyJutong PanB.A., Peking University, 2009M.Phil., The Chinese University of Hong Kong, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2017c© Jutong Pan 2017AbstractThis dissertation studies how households adjust their consumption and labor supply in re-sponse to idiosyncratic shocks.In the first chapter, I propose an empirical strategy for measuring consumption allocationswithin households over time. The strategy consists of imputing gender-specific consumptiondata from a cross-sectional dataset to a panel. I apply it on two publicly available datasets inthe US: the Consumer Expenditure Survey and the Panel Study of Income Dynamics. Thegenerated panel allows researchers to investigate questions such as how the sharing rule shiftsin response to various shocks.The second chapter studies how households insure themselves against idiosyncratic wageshocks and how this insurance interacts with intra-household bargaining. I set up an intertem-poral household model and examine two channels of insurance, self-insurance and family laborsupply adjustment. I consider two alternative specifications of this model: a unitary versionin which I restrict sharing rules to be fixed within households, and a non-unitary one in whichI allow sharing rules to change. I estimate the model using a panel that has information onconsumption allocations within households. I find that intra-household allocations respondstrongly to fluctuations in individual wages. Removing the restriction of fixed sharing rulesdoes not reduce the extent of consumption smoothing within a household, but it significantlychanges the relative importance of different channels. In particular, the relative contributionof family labor supply to household consumption smoothing decreases from roughly 60% inthe unitary model to 30% in the non-unitary model. This is because the added worker effect– the increase in spousal labor supply following an adverse shock to a partner – is muchmilder in the non-unitary specification.Non-stationary income processes are standard in quantitative life-cycle models, promptedby the observation that within-cohort income inequality increases with age. The last chap-ter generalizes Tauchen’s (1986) and Rouwenhorst’s (1995) discretization methods to non-stationary AR(1) processes. We evaluate the performance of both methods in the context ofa canonical finite-horizon, income-fluctuation problem with a non-stationary income process.We find that the generalized Rouwenhorst’s method performs extremely well even with asmall number of states.iiLay SummaryWhat does a household do to insure against income shocks? What happens when the husbandis shifted from a high-paying job to a low-paying job? The household may borrow moneyand use their savings, the husband may work more, and the wife may also choose to work forlonger hours. This dissertation provides an economic analysis of such household behaviorsin situations where the household members’ income fluctuates. I start by proposing a newmethod for measuring consumption allocations in the first chapter. In the second chapter, Ishow that it is important to take bargaining between couples into the analysis. Accountingfor strategic interactions suggests that the wife does not increase her working hours as muchas previous studies find, which assume couples do not bargain. The last chapter generalizestwo numerical methods for approximating income processes and evaluates their performance.We find that the generalized Rouwenhorst method is more accurate and robust than thegeneralized Tauchen method.iiiPrefaceChapter 3 Markov-Chain Approximations for Life-Cycle Models is a joint work with Pro-fessor Giulio Fella and Professor Giovanni Gallipoli. I was involved throughout each stageof the research: proposing the numerical algorithms, coding the simulations, organizing andpresenting results, and writing several subsections of the manuscript.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Measuring Intra-Household Consumption Allocations over Time . . . . 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Consumption Smoothing and Intra-Household Bargaining . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 The Household Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Wage Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 The Family Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 The Solution to the Family Problem . . . . . . . . . . . . . . . . . . . 212.2.4 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Empirical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Data and Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Consumption Imputation . . . . . . . . . . . . . . . . . . . . . . . . . 25vTable of Contents2.3.3 Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Wage Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.3 Consumption, Labor Supply, and Bargaining Parameters . . . . . . . 292.4.4 Transmission of Wage Shocks to Consumption and Labor supply . . . 312.4.5 Insurance Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Markov-Chain Approximations for Life-Cycle Models . . . . . . . . . . . . 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Discrete Approximations of AR(1) Processes . . . . . . . . . . . . . . . . . . 393.2.1 Tauchen’s (1986) Method . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.2 Rouwenhorst’s (1995) Method . . . . . . . . . . . . . . . . . . . . . . 413.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52AppendicesA Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.1 More on Wage Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.2 Approximation of the First Order Conditions and Intertemporal Budget Con-straint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.2.1 Linearization of the First Order Conditions . . . . . . . . . . . . . . . 57A.2.2 Log-Linearization of the Lifetime Budget Constraint . . . . . . . . . . 61A.3 Moment Condtions in GMM estimation . . . . . . . . . . . . . . . . . . . . . 65A.4 Moment Conditions with Measurement Errors . . . . . . . . . . . . . . . . . 69B Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72B.1 Normalized Problem with Unit-Root Labor Income . . . . . . . . . . . . . . 72viList of Tables1.1 Regression for Gender-Specific Clothing Consumption in the CEX . . . . . . 112.1 Descriptive Statistics: Sample Means . . . . . . . . . . . . . . . . . . . . . . . 282.2 Estimates of Wage Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Average Estimates of Transmission Coefficients . . . . . . . . . . . . . . . . . 322.5 Behavioral Responses to -10% Permanent Wage Shock to Husband . . . . . . 342.6 Behavioral Responses to -10% Permanent Wage Shock to Wife . . . . . . . . 343.1 Ratio of Model Moments Relative to Their Counterpart in the QuadratureBenchmark: (a) Markov Chain Simulation and (b) Continuous Random WalkIncome Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.1 Estimates of Wage Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 56viiList of Figures1.1 Comparison of real versus imputed household-level clothing consumption . . . 131.2 Distribution for imputed husbands’ share of consumption . . . . . . . . . . . 14viiiAcknowledgmentsFirst and foremost I want to thank my advisor, professor Giovanni Gallipoli. He has beenvery supportive throughout my Ph.D. studies, from the day I was only a second-year studenttaking his course to now. I would like to thank him for encouraging my research, for pushingme forward, and for caring about my future. This thesis would not have been possible withouthis guidance, and words are hardly adequate for expressing my gratitude.I am also grateful for my committee members, professor David Green and professor YanivYedid-Levi for their continuous guidance and attention. They also supported me intellectuallyand emotionally whenever I need them: when I was struggling in the middle of my research,when my confidence was getting low. . . You were always there, being tremendous mentors tome.Special thanks to professor Giulio Fella at Queen Mary University of London, whom Iworked with on the third chapter of this dissertation. Though we had to collaborate remotely,it has been a very rewarding and enjoyable experience. Thank you to professor Paul Beaudry,professor Matilde Bombardini, professor Patrick Francois, professor Hiro Kasahara, professorPaul Schrimpf, professor Michal Szkup and all other VSE professors who were ever at mypresentations and offered me valuable suggestions and encouragement.I would also like to give a heartfelt thank you to my parents and my sister for alwaysbelieving in me and encouraging me to follow my dreams. Doing a Ph.D. is a huge investmentand commitment, and my family backed me up through thick and thin. And finally to myloving fiance´e Yawen Liang, whose faithful support during the final stages of my Ph.D. meansthe world to me. Thank you.ixIntroductionSuppose that in a family, the husband’s wage is cut down. How would the household respondto such a shock? There are two key channels for consumption smoothing available to thehousehold. One channel is smoothing through financial markets, which is typically doneby saving and borrowing. The other channel is family labor supply adjustments, that is,by increasing the husband’s labor supply and/or increasing the wife’s labor supply. Howimportant is each channel? How should we measure the effectiveness of each channel? Thisdissertation attempts to address these questions.I start by arguing that the relative importance of these two channels depends on howhouseholds make decisions. The theoretical and empirical literature on household intertem-poral decisions has traditionally assumed that households behave as single agents. Under thisassumption, the identity of the recipient of the shock cannot make a difference in terms ofhousehold behavior. Modeling based on this assumption is called the unitary approach. Re-cently, economists have developed models that address some of the limitations of the unitaryapproach as a framework used to answer policy questions. Those models explicitly recognizethat household members have their own preferences. A popular approach to modeling the al-locations that result from the intra-household decision making process is the collective modelof the household, originally proposed in a static form by Chiappori (1988, 1992). There arenow a substantial number of cross-sectional empirical studies (see, e.g. Browning et al., 1994;Chiappori et al., 2002; Blundell et al., 2007) demonstrating that allocations within house-holds are related to the source of income and other factors such as the sex ratio and divorcelegislation, providing supporting evidence to the non-unitary approach.It is theoretically appealing to extend the collective model of the household to dynamiccontexts for answering questions about intertemporal household decisions like ours, and therehas been some progress (Mazzocco, 2007; Voena, 2015; Gallipoli et al., 2016. However,common household survey data typically does not provide information about the evolution ofallocations within households, which is important for testing and characterizing intertemporalnon-unitary models. To address this data limitation, the first chapter of the dissertationproposes a method for measuring intra-household consumption allocation over time when twotypes of data are available: one has information on intra-household consumption allocationbut lacks time variation; the other is a traditional panel of household, in which information1Introductionabout intra-household allocation is missing. The strategy consists of imputing gender-specificconsumption data from a cross-sectional dataset to a panel. I apply this method on twopublicly available datasets in the US: the Consumer Expenditure Survey (CEX) and the PanelStudy of Income Dynamics (PSID). The generated panel allows researchers to investigatequestions such as how the sharing rule shifts in response to shocks over time.Next, the second chapter systematically studies how households insure themselves againstidiosyncratic wage shocks and how this insurance interacts with intra-household bargaining.I set up a life-cycle model of the household and consider two alternative specifications of themodel: a unitary version in which I restrict sharing rules to be fixed within households, anda non-unitary one in which I allow sharing rules to change. Using the empirical strategydiscussed in the first chapter, I estimate the model using a panel that has information onconsumption allocation within households by the generalized method of moments. I find thatintra-household allocations respond strongly to fluctuations in individual wages. Removingthe restriction of fixed sharing rules does not reduce the extent of consumption smoothingwithin a household, but it significantly changes the relative importance of different channels.In particular, the relative contribution of family labor supply to household consumptionsmoothing decreases from roughly 60% in the unitary model to 30% in the non-unitary model.This is because the added worker effect – the increase in spousal labor supply following anadverse shock to a partner – is much milder in the non-unitary specification. Moreover, theestimated model generates rich heterogeneity in responses of allocations within households.Shocks of the same size have very different implications, depending on whose wage theychange.This study has several important policy implications. First, most families (i.e., pooror young families) do not have the assets that would allow them to smooth consumptioneffectively. Without the labor supply channel, one could conclude that they have little inthe way of maintaining living standards when shocks hit. For a correct design of public andsocial insurance policies, it is important to know whether households can use labor supply asan alternative insurance mechanism and to what extent they do so. My research finds thatit is important to incorporate intra-household bargaining into the analysis, otherwise, theeffectiveness of labor supply mechanism would be over-estimated. Moreover, studying howwell families smooth income shocks, how this changes over the life and over the business cyclein response to changes in the economic environment confronted, and how different householdtypes differ in their smoothing opportunities, is an important complement to understandingthe effect of redistributive policies and antipoverty strategies.Finally, the last chapter focuses on the saving and borrowing channel for consumptionsmoothing. Quantitative studies on this channel (Huggett, 1993; Kaplan and Violante, 2010)often approximate continuous stochastic processes using discrete state-space representations;2Introductione.g. Markov chains. The Tauchen’s (1986) and Rouwenhorst’s (1995) methods have beenwidely used in the context of stationary infinite horizon problems. However, covariance-stationary income processes are not consistent with the empirical fact that within-cohortincome inequality increases with the age of a cohort. The last chapter extends both Tauchen(1986) and Rouwenhorst’s (1995) methods to discretize non-stationary AR(1) processes andcompare their respective performance within the context of a life-cycle, income fluctuationproblem. We find that generalized Rouwenhorst’s method performs extremely well even witha relatively small number of grid-points.3Chapter 1Measuring Intra-HouseholdConsumption Allocations over Time1.1 IntroductionHow would a multi-member household respond to a shock to one of its members? Does itmatter if married individuals are taxed jointly or independently? Does a reform in the di-vorce legislation have an effect on married couples’ behavior even if their marriage is stable?Apparently, a complete answer to any of these questions requires a systematic approach tomodeling intertemporal household behavior. Until recently, the standard approach to mod-eling household decision making was based on versions of the so-called “unitary” approach,which assumes that a household can be represented by a single utility function. In a frame-work of this type, what exclusively matters, as far as household decisions are concerned, isthe total amount of resources at the household’s disposal. It ignores any policy effect onthe spouses’ bargaining positions and consequent decisions. Ever since the seminal work byChiappori (1988, 1992), the non-unitary perspective, in particular the “collective approach”,has been replacing the traditional approach. The collective approach explicitly recognizesthat household members each have their own preferences, and thus factors such as the rela-tive incomes of the household members may affect the final allocation decisions made by thehousehold.There are two types of collective models: a static one and an intertemporal one. Whilethe static collective model has become influential in family economics and received muchempirical support such as Browning et al. (1994), Chiappori et al. (2002), and Blundell et al.(2007), the dynamic counterpart is much less so. One crucial empirical challenge is thattesting and characterizing an intertemporal collective model requires that intra-householdallocation is observed (at least partially) over time. However, due to data limitations, inmost cases consumption is often measured at the household level only; and in some caseswhere intra-household consumption allocation can be measured, the data does not follow thesame households over time.11The only exception is the Japanese Panel Survey of Consumers, the one that is used in Lise and Yamada(2015). However, this dataset is not publicly available; and it would be still interesting if one could test andidentify the intertemporal collective model using data in other countries such as the U.S. given the substantial41.1. IntroductionThis paper proposes a method for measuring intra-household consumption allocationover time when two types of data are available: one has information on intra-householdconsumption allocation but lacks time variation; the other is a traditional panel of household,in which information about intra-household allocation is missing. In the US, the example ofthe former type is the Consumer Expenditure Survey (CEX), and that of the latter type isthe Panel Study of Income Dynamics (PSID).To be specific, I estimate the demand functions for men’s clothing and women’s cloth-ing, respectively, within married households using the CEX data. The estimated demandfunctions are then used to predict the expenditure on men’s clothing and women’s clothingfor the married households in the PSID. This measures the evolution of the sharing rule ofclothing. Next, to adjust for the gender difference in preferences towards clothing, I esti-mate the functions for the shares of expenditure on clothing for men and women using thesubsamples of single men and single women, respectively. Under the assumption that thisshare is independent of marital status, I use the estimated functions to predict the sharesfor husbands and wives in married households, which are then, combined with the imputedgender-specific clothing expenditure, to back out the intra-household consumption allocation.This paper is related to several strands of literature. One obviously related literature is theresearch that tries to identify intra-household allocation in a static environment. Browninget al. (1994) derive conditions for identifying the intra-household allocation using assignablegoods (which is also clothing in their empirical implementation). Blundell et al. (2007) useleisure as an assignable good. Both studies rely on some assignable good as this paperdoes. Some other researchers use equilibrium conditions in marriage market to identify thesharing rule based on distribution factors such as sex ratios and the nature of divorce laws(Chiappori et al., 2002). Most recent advance is Cherchye et al. (2012), who use Dutch dataon the allocation of private and public consumption expenditures and individual time useto fully identify and estimate a household sharing rule. The data they use is from a singlecross section, and as a result, they are identifying how allocations relate to differences inrelative wages across households, and not necessarily how allocations would change within ahousehold resulting from unanticipated income shocks. Overall, this literature typically findsstrong evidence that rejects the unitary approach to modeling household behavior, but it isbased on static models, which cannot be used to answer questions and evaluate policies thathave any intertemporal dimension.There is a small but growing literature that extends the collective approach to a dynamiccontext.2 Without a panel of intra-household consumption allocation, researchers typicallycultural differences, and more useful when one could use publicly available data to do so.2The key assumption in the collective approach is Pareto efficiency. Depending on how efficiency is definedwithin households, two types of intertemporal collective models exist, namely, a full-commitment model anda limited-commitment model.51.1. Introductionassume that consumption and leisure enter the individual utility function in a separable way.Under this assumption, individual leisure as intra-household time allocation alone help in theidentification of the evolution of intra-household decision powers, and thus one only needs apanel that has information on leisure (or labor supply), which is much more common thanpanels of (individual) consumption. While this separability assumption has been proven tobe a profitable approach to estimating an intertemporal collective model (see, e.g. Voena,2015; Gallipoli et al., 2016), it is not uncontroversial.3 This paper suggests a method forovercoming the data limitations so that (i) researchers can estimate an intertemporal collec-tive model without imposing the separability assumption; and (ii) even when one is willingto assume separability, the intra-household consumption allocation provides another sourcefor the identification of the sharing rule.This paper is also closely related to the consumption imputation literature. This literaturestems from the lack of household consumption panel until very recently. For example, in theUS, the CEX provides comprehensive data set on the spending habits of US households butit follows households for only four quarters at most. The PSID collects longitudinal annualdata, but until 1999 it collects data only for a subset of consumption items, mainly food athome and food away from home. For a long time, researchers have to use food expendituresas a proxy for consumption, which is clearly a poor proxy given that food is largely a necessityand its consumption is much less elastic than other consumption items. Some economiststhus propose methods to impute total consumption in the PSID. For this kind of imputation,the key original reference is Skinner (1987), who proposes to impute total consumption inthe PSID using the estimated coefficients of a regression of total consumption on a series ofconsumption items (food, utilities, vehicles, etc.) that are present in both the PSID and theCEX. The regression is estimated with CEX data. From a statistical point of view, Skinner’sapproach can be formally justified by the idea of matching based on observed characteristics.This method is used in several articles including, among others, Palumbo (1999), Dynan(2000), and Bernheim, Skinner, and Weinberg (2001). The most notable practice, however,is Blundell et al. (2008). Their approach is slightly different from the original Skinner’s:instead of fitting an equation for the total consumption, they start from fitting a standarddemand function for food (a consumption item available in both surveys), and invert it (undermonotonicity of food demands) to obtain the imputed total consumption in the PSID. Myimputation strategy is a combination of Skinner (1987) and Blundell et al. (2008): similarto Skinner (1987), my dependent variable is not available in the PSID and the independentvariables are common in both datasets; similar to Blundell et al. (2008), I fit a standarddemand function for gender-specific clothing consumption. Finally, as far as I know, this is3As Heckman (1974) first noted, the dynamic response of consumption to wage changes will depend onwhether consumption and hours are complements or substitutes in utility. Separability between consumptionand labor supply is rejected empirically in Browning and Meghir (1991) and Blundell et al. (2015).61.2. Methodologythe first imputation method that generates a panel of intra-household allocation.The rest of the paper is organized as follows. Section 1.2 describes the empirical strategyin details. Section 1.3 describes the data and section 1.4 presents the results of applying themethod on the CEX and the PSID. Section 1.5 concludes.1.2 MethodologyFor any married household i at any time t, the total expenditure can be decomposed intothree components:Cit,m = Ci1t,m + Ci2t,m +Git,m (1.1)where the subscripts i,j,t are household, member, and time indexes, respectively, and mindexes ‘married’. Cijt,m is member j’s (j = 1, 2) private consumption, and Git,m is thehousehold’s public consumption. The distinction between private and public consumptionfollows Browning et al. (1994) and Lise and Seitz (2011).In the micro-level data, each consumption category can be classified as either private orpublic consumption. However, among those private consumption categories, very few areexclusive to a specific member (or gender). We therefore only observe CiP t ≡ Ci1t,m +Ci2t,m(subscript P denotes “private”), but not Ci1t,m or Ci2t,m, the split between the husbandand the wife. There is one exception: the subcategory “men’s clothing” can be consideredexclusive to the husband; the same goes for “women’s clothing” to the wife.4The empirical strategy proceeds as the following:First, using the sample of married households in the CEX, I regress the gender-specificclothing consumption, on the demographic characteristics and the total consumption:clothCEXijt,m = βt +X′CEXit,m β1j + β2jclothCEXit,m + β3jCCEXit,m + uijt,m (1.2)for gender j = 1, 2 separately. The dependent variable clothijt is the clothing consumptionfor gender j, whereas clothit on the right-hand side refers to the household-level clothingconsumption. I control for the year fixed-effect βt. X are socioeconomic variables of thehousehold and the household members, including age, age squared, education, and races ofboth partners, region of living5, number of children, and family size. Finally, Cit is thetotal family-level nondurable consumption. The CEX covers more consumption categories4Children’s clothing is usually another subcategory that is separately counted.5Although information on the state of residence is available in the CEX, it is suppressed for some observa-tions to avoid some small-population areas being identified. On approximately 14% of the records on the CEXfamily files the state variable is blank, and approximately 4% of observations are recoded to states that are notwhere they actually reside. By contrast, the variable that identifies the region of living is never suppressed.71.2. Methodologythan the PSID. For the purpose of imputation, the total consumption is defined as thesum of the expenditure in all consumption categories in the PSID. These include the foodconsumption, household utilities, health-related expenditure, expenditure on home repair andfurnish, clothing expenditure and expenditure on entertainment and trips. All consumptionvariables are in natural logarithms and are so throughout this paper unless explicitly stated inlevels. In practice, I allow the elasticity β3j to vary with time and with observable householdcharacteristics by introducing interactions terms.6 Equation (1.2) can be interpreted as anapproximated demand function that relates the gender-specific clothing expenditure to thetotal expenditure.7Then I predict the gender-specific clothing consumption for the married sample in thePSID using the estimated coefficients from the CEX regression, that is,ĉlothPSIDijt,m = βˆt +X′PSIDit,m βˆ1j + βˆ2jclothPSIDit,m + βˆ3jCPSIDit,m (1.3)where the βˆs are the estimates of βs in equation (1.2).The next step is to calculate the sharing rule for each household in the PSID usingthe imputed gender-specific clothing consumption. The simplest way is to assume that thehusband’s share of private consumption can be approximated by his share of clothing con-sumption, in which scenario the private consumption of partner j would be simply given byCˆijt =ĉlothijtĉlothi1t+ĉlothi2tCiP t, where CiP t is the household-level private consumption, i.e., thesum of all private consumption categories for household i at t. The categories that belong topublic consumption are enjoyed by the couple together and thus no splitting rule is required.However, this assumption might be too restrictive. It is likely that the division for clothingdoes not represent the division for total private consumption, which will be true if, say, thewives on average spend more on clothing than the husbands do, relative to other privategoods. In this case, one would under-estimate the husbands’ shares. To correct for this bias,I exploit the information about how much men (women) on average spend on clothing out oftotal private consumption from the sample of single men (women), and adjust the divisionrule to take into account the preference difference between males and females.Let me illustrate the steps using the women’s case; the estimation for men is analogous.First, for each single female observation in the CEX, I calculate the proportion out of totalconsumption she spends on clothing: ψi2t,s ≡ clothi2t,s/Ci2t,s, where the subscript s indicates6To be consistent with the imputation literature, which typically does not use the income information, Ido not include income of household members as explanatory variables. While they are potentially important,I leave this for future research.7It is worth pointing out that here I do not aim to establish any causal relationship between the dependentand independent variables. Rather, the goal of the regressions, as it is common in the consumption imputationliterature, is to find the best fit that describes the matching between a husband’s expense on clothing andall his family’s observable characteristics and behaviors. Thus endogeneity problems arising from the fact thedependent variable is a subset of the independent variable is not an issue given this objective.81.3. Datathe single sample (note that the subscript 2 in this case only refers to the gender beingfemale, not the second member in household i). This proportion ψi2t,s is regressed on a setof individual observable characteristics that are commonly available in the CEX and in thePSID:ψi2t,s = γt,f +X′CEXi2t,s γf + ui2t,f (1.4)where the subscript f stands for female. Then the proportion that a wife spends on clothingout of her total private consumption in the PSID sample is predicted by:ψˆi2t,m = γˆt,f +X′PSIDi2t,m γˆf (1.5)In fact, equations (1.4) and (1.5) are very similar to (1.2) and (1.3). The idea is to estimatethe average proportion of expenditure on clothing for women (or for men), and I allow thisproportion to vary across ages, races, etc. To justify this imputation, I assume that thepreference for clothing relative to other private goods, once controlled for age, education,etc., does not depend on the marital status.Once we obtain ψˆijt,m for j = 1, 2 for each married household in the PSID sample from(1.5), the private consumption for partner j is given by:Cˆijt =ĉlothijt/ψˆijtĉlothi1t/ψˆi1t + ĉlothi2t/ψˆi2tCiP t (1.6)1.3 DataThe main data I use in this paper is the Panel Study of Income Dynamics (PSID) for the years2005-2013. The PSID data are collected biennially since 1999, so 5 waves are used: 2005,2007, 2009, 2011, and 2013. Starting in 1999, in addition to income data and demograph-ics, the PSID collects data about detailed assets holdings and consumption expenditures (atthe household level), the latter of which cover many nondurable and services consumptioncategories, including utilities, health expenditure, transportation, education, and child care.8A few more consumption categories, including clothing, trips and vacation, and other recre-ation, were added in 2005. As the imputation procedure relies on matching the clothingconsumption, I use the waves since 2005.I focus on the core (non-SEO) sample of the PSID.9 I select the households that are stably8Before 1999, the PSID collected data on very few consumption items, mainly food at home and food awayfrom home.9SEO refers to the Census Bureau’s Survey of Economic Opportunities. The SEO sample in the PSIDconsists of low-income families, which are over-sampled, and it distinguishes from the core sample, which isrepresentative of the US households.91.4. Resultsmarried over the sample period, with the male head aged between 25-60 and both partnersare working. I drop the households with missing information on key demographic variables(age, race, and education) and households with zero consumption. I also drop the householdsin which the hourly wage of the husband or of the wife is lower than half of the minimumwage.10 The remaining sample is an unbalanced panel of 9628 observations.I also use the 1998-2013 Consumer Expenditure Survey (CEX) data. The CEX providesa comprehensive flow of data on the buying habits of American consumers. The data arecollected by the Bureau of Labor Statistics and used primarily for revising the CPI. Thedefinition of the head of the household in the CEX is the person or one of the persons whoowns or rents the unit; this definition is slightly different from the one adopted in the PSID,where the head is always the husband in a couple. I make the two definitions compatible byrestricting the CEX sample to be male-headed households.The CEX is based on two components, the Diary survey and the Interview survey. TheDiary survey is conducted over two consecutive one-week periods, designed to track detailedexpenditures data on small and frequently purchased items, such as food, personal care, andhousehold supplies. The Interview survey is conducted quarterly, and it covers about 95percent of all household expenditure11. My analysis below uses only the Interview sample,because it is more comprehensive and it can be used to construct annual expenditure forhouseholds. I apply the same sampling restrictions on the CEX as on the PSID12: marriedhouseholds with the head being male and of age 25-60; both partners are working; and nomissing information on key demographic characteristics.1.4 ResultsTable 1.1 reports the estimate of coefficients in the regressions of gender-specific clothingconsumption in the CEX (equation 1.2). Column (1) is for the regression of men’s clothingconsumption and Column (2) for women’s. Again, all consumption variables are in loga-rithms. The omitted group for education is the ones with less than high school, the basegroup for races is white, and for number of children I omit the household with no child.10I use the highest minimum wage prescribed by federal law and state law. Before the Great Recession,the number of male workers that earn less than half of the minimum wage is very stable across years. In thepost-crisis waves, 2011 and 2013, this number more than doubles. In fact, the whole wage distribution shiftsto the left after the crisis. (The same thing happens to female workers.) To avoid excess truncation, I donot discard the observations whose wages are lower than half of the minimum wage after the crisis but not sobefore the crisis.11With the exclusion of expenditures for housekeeping supplies, personal care products, and non-prescriptiondrugs.12Strictly speaking, these restrictions cannot be identical, because some sampling requirements imposed onthe PSID rely on the fact that the households are observed for continuous years and thus cannot be appliedon the CEX. For example, the households in the PSID sample here are stably married across years, but in theCEX I cannot observe whether a household is married in the adjacent years or not.101.4. ResultsTable 1.1: Regression for Gender-Specific Clothing Consumption in the CEX(1) (2)Male’s clothing Female’s clothing Test βm = βfHead’s age .0005 .0034(.0088) (.0074)Head’s age squared .0000 .0000(.0001) (.0001)Head’s educ - high school -1.1909 1.9926*(1.2913) (1.1293)Head’s educ - some college or more -.9032 2.4208**(1.1987) (1.0741)Spouse’s educ - high school -.9180 -1.7599(1.6504) (1.4272)Spouse’s educ - some college or more -1.2228 -1.9404(1.6270) (1.4254)Head’s race - black -.2549** -.2771***(.1164) (.1009)Head’s race - others -.0390 -.0194(.0714) (.0688)Spouse’s race - black .3153** .2458**(.1248) (.1069)Spouse’s race - others -.0033 -.0387(.0664) (.0633)Family size -.3848*** -.2198***(.0851) (.0764)One child -.9739 -1.8883***(.6160) (.5431)Two children -1.5244** -2.3802***(.6206) (.5688)Three children or more -1.8814** -2.2193***(.9353) (.8524)Household clothing consumption .8034*** .9166***p=0.000(.0147) (.0129)Household total consumption -.0915 .0618p=0.030(.1315) (.1143)Observations 6510 6715R-squared 0.4656 0.5937Notes: The last column reports the p-value for testing whether the coefficients for men’s regression and forwomen’s are the same. I also control for year fixed effects and region dummies. I include interactions betweenthe household total consumption and observables to allow for flexible budget elasticity. The estimates for thosecoefficients are not shown for preserving space. Standard errors in parentheses are clustered at the householdlevel. *** p < 0.01, ** p < 0.05, * p < ConclusionAlthough not reported in Table 1.1 (for compactness), I also include the year fixed effectsand dummies for region of living, and a set of interaction terms between the household totalconsumption and observables, which captures the idea that the budget elasticity may varywith those characteristics.The last column in Table 1.1 is testing whether the men’s coefficient is different from thewomen’s. In particular, I’m interested in the differential responses between men’s clothingconsumption and women’s when the household-level consumption changes, as they are im-portant for identifying the sharing rule.13 The last test (regarding total consumption) is ajoint test of the total consumption alone and the interaction terms (not shown in the table)being no effect.Figure 1.1 shows the comparison between the actual data on household-level clothingexpenditure and the sum of imputed husband’s clothing and wife’s clothing consumption.The upper panel compares the histogram and the lower panel compares the kernel densityestimates of the distributions. As it can be seen from the graphs, the imputed sum matchesthe actual data very well. In addition, the regression of actual household-level clothingexpenditure on the imputed sum yields an R-squared of 0.8326.The imputed sharing rule is shown in Figure 1.2. I draw both the distribution for themen’s share of (imputed) clothing consumption and the distribution for men’s share of privateconsumption adjusted for the gender preferences as shown in equation (1.6). The mean ofmen’s share in consumption is 0.4067, and standard deviation 0.0815. This is in the range ofwhat the literature found. Dunbar et al. (2013) estimate that men absorb 40-47% of householdresources and a relatively small amount of variation using Malawi data. Using a Dutch cross-section dataset, Cherchye et al. (2012) report a roughly half-half division on average but thenumber is sensitive to the definition of private consumption. Lise and Seitz (2011) finds themean of wife’s share of consumption to be 0.33 in 1970 and 0.40 in 2000 using U.K. data, butthis relatively lower wife’s share on average is because they include all households and someof the wives do not participate in the labor market. My estimation, along with Dunbar et al.(2013)’s and Cherchye et al. (2012)’s, excludes families with non-participation members.1.5 ConclusionTo analyze the intertemporal household behavior in a non-unitary setting, one needs to haveinformation on the evolution of the sharing rule. The lack of panel data on intra-householdconsumption allocation presents an empirical challenge to estimating an intertemporal col-lective model. This paper proposes a method for measuring consumption allocation within a13The differences in the coefficients on the observable characteristics in X, while might be interesting empir-ically, are not useful for the identification here, because in the GMM estimation all variables are “residuals”net of those observables.121.5. Conclusion0. 5 10 15real data imputed data0. 5 10 15xreal data imputed dataFigure 1.1: Comparison of real versus imputed household-level clothing consump-tion131.5. Conclusion0246810Density.2 .3 .4 .5 .6male's share in clothing male's share in totalFigure 1.2: Distribution for imputed husbands’ share of consumptionhousehold over time. I apply it on two publicly available datasets in the US: the ConsumerExpenditure Survey (CEX) and the Panel Study of Income Dynamics (PSID). Using theCEX data, I estimate the gender-specific consumption, clothing, as a function of socioeco-nomic variables that are commonly available in both the CEX and the PSID, and then usethese estimated functions to predict the clothing consumption allocation within the PSIDfamilies. I also estimate the functions for the shares of expenditure on clothing for menand women using the subsamples of single men and single women, respectively. Under theassumption that this share is independent of marital status, I use the estimated functionsto predict the shares for husbands and wives in married households, which are then, com-bined with the imputed gender-specific clothing expenditure, to back out the intra-householdconsumption allocation. Results are consistent with the empirical findings in the literature.The resulting panel provides rich dynamic information inside the “black box” by which ahousehold is taken in a unitary model, and allows researchers to analyze the intertemporaleffect of policies on household behavior based on the more micro-founded collective approach.14Chapter 2Consumption Smoothing andIntra-Household Bargaining2.1 IntroductionMuch past and current literature has examined how households respond to income fluctua-tions. This line of research often attempts to quantify the degree of insurance achieved withinhouseholds (see for example Blundell, Pistaferri, and Preston, 2008; Kaplan and Violante,2010), but it also strives to understand the specific mechanisms used by households to smooththeir consumption (e.g., Low, 2005; Heathcote, Storesletten, and Violante, 2014; Blundell,Pistaferri, and Saporta-Eksten, 2015). Both of these research objectives have crucial pol-icy implications, which justifies the extensive attention they have received so far. Much ofthe existing studies treat the household as a single decision-maker, often ignoring the hetero-geneity in intra-household allocations and, more importantly, how intra-household bargainingover such allocations affects the effectiveness of different insurance channels. These modelingchoices are made in part because of data limitations that reduce the ability to reliably mea-sure changes in intra-household allocations. Nonetheless, growing empirical evidence castsdoubts on the innocuousness of modeling the household decision making as a unitary process(see, among others, Mazzocco, 2007; Lise and Seitz, 2011; Gallipoli, Pan, and Turner, 2016).In this paper, I show both theoretically and empirically that taking a non-unitary approach tohousehold decision-making has non-trivial implications for the analysis of within-householdlabor supply patterns, consumption allocations, and insurance provision.In the presence of wage uncertainty, there are several ways in which households can ac-commodate shocks and smooth consumption. Economists have mostly focused on two keysmoothing channels.14 The first one is self-insurance through borrowing and saving. This isprobably the most studied source of insurance in the macro and labor literature. The otherinsurance channel encompasses all those adjustments to labor supply that members of thehousehold make in response to individual wage fluctuations. Previous studies have tradition-ally looked at these two channels separately, until some more recent work has stressed the14Other insurance channels include progressive taxation, social insurance programs, family transfers, infor-mal networks, default or bankruptcy, et cetera.152.1. Introductionimportance of considering both mechanisms within the same framework.15 In an influentialpaper, Blundell et al. (2015) estimate that following a 10% permanent decline in the hus-band’s wage, roughly 3.9% of consumption is effectively insured. Of this 3.9%, 2.5 percentagepoints (63% of the total insurance effect) come from the family labor supply channel and 0.7percentage point (17% of the total) comes from self-insurance.16 This analysis is based on aunitary household framework, that is, a framework in which spouses pool consumption andmake decisions as if they were one person.In this paper I show that the relative importance of family labor supply as a consumptionsmoothing device decreases when couples can bargain over the intra-household allocation andthat the bargaining power of each member of the household depends on his/her labor incomerelative to the spouse’s. Intuitively, in the case of unitary decision making, when one earnerin the family is hit by a negative permanent wage shock, there is an incentive for the otherearner to work more in order to compensate for the family’s income loss to the family. This issometimes called the “added worker effect”. This effect becomes less obvious in a non-unitarydecision-making household. If an earner’s bargaining power depends on his/her contributionto the total labor income of the family, the earner who experiences an adverse shock may notbe willing to let own labor supply be substituted by the spouse’s, because that would reducetheir bargaining power and likely trigger a renegotiation leading to a new agreement (sharingrule) favoring the spouse.I set up a life-cycle collective household model in which individual wages are subject toidiosyncratic (transitory and permanent) shocks. Individual members derive utility from indi-vidual consumption and labor supply. A household determines the intra-household allocationgiven Pareto weights on individual utilities. These weights are agreed upon by the couple. Ifthe utility weights are fixed, the model assumes full-commitment and results in allocationsconsistent with a unitary specification.17 For comparison with previous studies, I refer to15The literature on the first insurance channel (self-insurance) dates back to at least Deaton (1991). Recentdevelopments include Kaplan and Violante (2010) and Krueger and Perri (2006). This literature typicallyassumes exogenous labor supply. Studies on the responsiveness of individual labor supply to wage changes,as surveyed in Keane (2011), do not consider the joint consumption-labor supply choice and focus on thesingle earner case. There is a parallel, related literature in labor economics (e.g. Lundberg, 1985) askingto what extent a secondary earner’s labor supply increases in response to negative wage shocks faced by theprimary earner, known as the “added worker effect”. However, this literature does not normally provideexplicit measures of the consumption smoothing achieved by households.16The remaining 0.7 percentage point is due to residuals capturing other formal and informal insurancechannels.17As Chiappori and Mazzocco (2015) noted, “There exists situations under which the unitary and collectivemodels generate the same set of household decisions. This is the case, for instance, if the relative decision poweris constant and therefore does not depend on prices, wages, income, and distribution factors.” In addition,Mazzocco (2007) proves that the unitary model is a special case of the full-commitment collective model. Thisimplies that if one rejects the full-commitment model, the unitary model is also rejected. Readers interestedare referred to the technical proof in the appendix of Mazzocco (2007), but the proposition is basically ageneralization of Gorman’s aggregation theorem to an intertemporal framework with public consumption.162.1. Introductionthis specification of the model as “unitary”. Otherwise, if the utility weights are subject torenegotiation, the model is a limited-commitment one, and I refer it to as “non-unitary”.The non-unitary variant of the model encompasses the unitary one as a special case, whenthe response of the utility weights to wage shocks is restricted to be zero.The literature on collective household models typically adopts a fully structural esti-mation approach. That is, one fully parameterizes the household model, making explicitassumptions about preferences, shocks, and the determination of the sharing rule. Examplesof papers using this approach (albeit pursuing different research questions) include Lise andSeitz (2011) and Cherchye et al. (2012). The degree of insurance estimated by this kindof approach is, by construction, reliant on the chosen functional forms.18 In this paper, Iuse a less restrictive approach: while keeping the preferences and bargaining process non-parametric, I approximate the household optimization conditions by log-linearization andderive a system of equations describing the transmission of the wage shocks to consumptionand hours worked, which is then used for evaluating the importance of different channels ofconsumption smoothing.19In the analytical section I show how the transmission coefficients, and the underlyingFrisch elasticities and “bargaining parameter” (the elasticity of utility weights with respectto wages), can be identified using joint moments of individual wages, individual consumption,individual labor supply, and household assets. Importantly, the expression describing theshock transmission coefficients admits a “bargaining effect”, in addition to the traditionalsubstitution and wealth effects. The sign of this effect is unrestricted by the theory - in fact,I show that it is heterogeneous across households and has to be established empirically.I use the 2005-2013 Panel Study of Income Dynamics (PSID) to estimate the model andI focus on continuously married households. To identify the bargaining effect, I need data onindividual-specific consumption. However, while the PSID features a long time-series withrich information about individual labor market activities, it only provides consumption dataat the household level. This is a common disadvantage of most existing consumption datasets - in fact, there is no panel of individual consumption in the US. This presents an em-pirical challenge for incorporating bargaining into the analysis of consumption smoothing.20To overcome this limitation, I use a method developed in Pan (2017), which approximatesindividual consumption expenditures by imputing the gender-specific private consumption18For example, Lise and Seitz (2011) use a utility function that is separable between consumption andleisure. In contrast, Blundell et al. (2015) stress that non-separability may play an important role in thehousehold responses to shocks.19A drawback of this approach is that it is subject to approximation errors. See Blundell et al. (2013) for adiscussion of the accuracy of the approximation approach in the context of unitary household models.20The same problem is also present in data for Canada, the UK, and other countries. As far as I know,such a panel only exists in Japan; see Lise and Yamada (2015) for the details of the Japanese Panel Survey ofConsumers.172.2. The Household Modelbased on estimates from the Consumer Expenditure Survey (CEX). In this way, I obtaina panel of households whose intra-household expenditures are continuously observable. Fi-nally, I construct the data moments of individual consumption, as well as of labor supply andwages that are directly available in the PSID, and estimate the parameters by the generalizedmethod of moments (GMM) using the moment conditions implied by the model structure.I find that the utility weight strongly responds to individual wage shocks, lending supportto the non-unitary specification of the model. The bargaining effect on the shock transmissionis also economically important. In particular, in terms of consumption smoothing, allowingfor intra-household bargaining lowers the relative contribution of family labor supply byroughly 16%-29%.21 Consistent with the intuitive argument proposed above, this is mainlydue to a weaker added worker effect in the non-unitary setting. Moreover, following a positivepermanent shock to the husband’s wage, his own consumption increases more and his spouse’sconsumption increases less than in the unitary case.The remainder of this paper is organized as follows. Section 2.2 introduces the analyticalframework, derives the model solutions, and discusses how the model can be identified andestimated using a panel of individual consumption and labor supply, as well as householdassets. Section 2.3 describes the data and the empirical strategy. Section 2.4 presents anddiscusses the results. Section 2.5 concludes.2.2 The Household ModelIn this section, I develop a life-cycle model for a family consisting of two potential earners.2.2.1 Wage ProcessThe primitive source of exogeneity and uncertainty to the family members are the hourlywages they earn. For each earner within the household, I posit a permanent-transitory wageprocess, assuming that the permanent component evolves as a unit root process. Formally, Iassume that the log of individual j’s real hourly wage in household i at t follows a permanent-transitory process and is given bylnWijt = Xijtζj + wijt (2.1)wijt = wPijt + uijt (2.2)wPijt = wPijt−1 + vijt. (2.3)The vector Xijt contains observed characteristics affecting wages and known to the householdat time t. wijt is the residual wage, which can be decomposed into wPijt, the permanent21This figure varies depending on whether the shock is in the husband’s wage or the wife’s.182.2. The Household Modelcomponent, and uijt, the transitory shocks. The permanent component wPijt follows a random-walk process and vijt is the permanent shock.Deviations from the deterministic path for wages occur because permanent and transitoryshocks, positive or negative, hit the individuals. A permanent shock shifts the value of one’sskills in the market permanently (for example, an accident causing long-term disability, asudden promotion); a transitory shock is mean reverting (for example, a short illness affectingproductivity, a one-time bonus payment). When shocks hit, I assume the partners canperfectly observe and distinguish between them; moreover they hold no advance informationabout the shocks (Et[ui,j,t+1] = 0, Et[vi,j,t+1] = 0; E denotes subjective expectations).I assume that earner j’s permanent and transitory wage shocks are serially uncorrelatedwith variance σ2vj and σ2uj , respectively. I also assume that permanent (transitory) shockscan be contemporaneously correlated within a family, with covariance σv1v2 (σu1u2). Thiscorrelation is theoretically ambiguous. For example, if spouses were to adopt sophisticatedrisk sharing mechanisms, they would select jobs where shocks are negatively correlated. Al-ternatively, assortative mating or other forms of sorting imply that spouses work in similarjobs, similar industries, and sometimes in the same firm - hence their shocks may be poten-tially highly positively correlated.22 In the baseline specification, I impose stationarity forthe variances and covariances of the shocks. Finally, I assume that transitory and permanentshocks are uncorrelated.The properties of the shocks can be summarized as follows:E(vij1tvij2t+s) =σ2vj if j1 = j2 = j, s = 0σv1,v2 if j1 6= j2, s = 00 otherwise(2.4)E(uij1tuij2t+s) =σ2uj if j1 = j2 = j, s = 0σu1,u2 if j1 6= j2, s = 00 otherwise(2.5)E(uij1tvij2t+s) = 0 for all j1, j2, s (2.6)Given the specification of the wage process (2.1)-(2.3) the growth in the residual log wagescan be written as∆wijt = ∆uijt + vijt (2.7)22I do not attempt to model those potential sophisticated risk sharing mechanisms. Labor supply adjustmentin my model only works at the intensive margin — by working longer hours. The sign of correlation betweenspousal wage shocks, which are taken as exogenous, is not restricted in the model and to be determinedempirically, and there might be other possible explanations for non-zero correlation.192.2. The Household Modelwhere ∆ is a first difference operator and ∆wijt is the log change in hourly wages net ofobservables.2.2.2 The Family ProblemGiven the exogenous wage process described above, a two-earner family maximizes a weightedsum of the husband’s individual utility and the wife’s:maxE0T∑t=0βt (µitU1(Ci1t, Git, Hi1t; di1t) + (1− µit)U2(Ci2t, Git, Hi2t; di2t)) (2.8)subject to the intertemporal budget constraintAi,t+1 = (1 + r)(Ait +Wi1tHi1t +Wi2tHi2t − Ci1t − Ci2t −Git) (2.9)where I denote the household’s asset by Ait, partner j’s individual private consumption byCijt, partner j’s hours worked by Hijt, and the family’s public consumption by Git. Notethat the public consumption enters both partners’ individual utilities. β is the time discountfactor. I assume households have access to a risk-free asset that pays an exogenous interestrate of r, and all assets of a family are shared by the two earners. Finally, dijt are observablepreference shifters, such as the number of children and the age of the earner; I account forthese empirically by using the residual measures of consumption, wages and earnings.I make a distinction between private and public consumption in the model. As publicconsumption is consumed by both members of the household, ignoring the presence of publicconsumption is likely to lead to overestimation of the degree of inequality within households.I follow the strategy of Browning et al. (1994) and partition all expenditures into either publicor private expenditures. Intuitively, a good is deemed private if consumption of one unit of itby one partner implies that this given unit is no longer available to the other partner, whereasa good is deemed public if consumption by one family member does not reduce the availableamount to the other family (i.e. it can be “shared”). More details about the classificationare discussed in Section 2.3.1.The last issue is how the intra-household allocation rule, i.e. the utility weight µit, isdetermined. Depending on whether the utility weight can change over time, there are twoversions of the model.In the first version, the utility weight is fixed so thatµit = µit−1 = · · · = µi0 = µu(zi0). (2.10)In this case, all household decisions are efficient in the sense that they are always on the202.2. The Household Modelex-ante Pareto frontier. In this case, the only thing that matters for determining the Paretoweight is the relative bargaining power at the time of marriage. In other words, the Paretoweight is only a function of information available at the time of marriage (including theforecastable components), zi0 = {E0zit}Tt=0. I refer to this version of the model as a unitaryone.23In the second version, I allow for intra-household bargaining and the utility weight issubject to change when there is news or shock to the household. In this case the Paretoweight depends both on the date-0 forecastable components zi0 and the realized deviationsfrom this forecast εit ≡ zit − E0zit:µit = µn(zi0, εit) (2.11)I call this version of the model a non-unitary one. Since the only exogenous shocks are inwages, εit consists of the accumulated wage shocks for the husband and for the wife. In fact,the non-unitary specification nests the unitary one as a special case: if µit is inelastic withrespect to εit, then equation (2.11) would collapse to equation (2.10). In other words, if thebargaining power doesn’t change in response to wage shocks, the non-unitary variant of themodel would behave as the unitary one. This suggests a simple test of whether couples makedecisions in a unitary way or not, which is equivalent to testing whether the elasticities ofutility weight with respect to wages are zero.2.2.3 The Solution to the Family ProblemThe model outlined above does not have an exact analytical solution unless one imposesstrong restrictions on the functional forms. Keeping the preferences non-parametric, I usean approximation approach to derive the solution. Similar approaches have been employedby Attanasio et al. (2002), Blundell et al. (2008), and Blundell et al. (2015), but only in theunitary framework. I extend the approach to apply in the collective household model.The approximation can be summarized into two parts. First, I approximate the first orderconditions to derive a system that links the endogenous variables (individual consumption,public consumption, etc.) to the exogenous shocks and the change in the marginal utility ofwealth. Second, by log-linearizing the life-time budget constraint, I derive the change in themarginal utility of wealth as a function of the wage shocks. At the end, I derive the followingexpression of how the permanent and transitory shocks of individual wages affect the changes23In fact, this is called the “full commitment” model in Mazzocco (2007). It is different from the purelyunitary model in the sense that there are individual utilities in the full commitment model. However, thedistinction between the pure unitary model and the full commitment model is only useful cross-sectionally.For a married household, whether it is purely unitary or full commitment, there is no time variation in theintra-household allocation.212.2. The Household Modelin the consumption and earnings:∆ci1t∆ci2t∆git∆yi1t∆yi2t ≈κc1u1 κc1u2 κc1v1 κc1v2κc2u1 κc2u2 κc2v1 κc2v2κgu1 κgu2 κgv1 κgv2κy1u1 κy1u2 κy1v1 κy1v2κy2u1 κy2u2 κy2v1 κy2v2∆ui1t∆ui2tvi1tvi2t (2.12)where κlm is the transmission coefficient of shock m into the choice variable l. Note that ingeneral, κ’s are heterogeneous across households and time (i.e., I should write κitc1u1 , etc.).To avoid cluttering, I leave this individual and age-dependence implicit.These transmission coefficients are explicit (but very complicated) functions24 of a set ofFrisch elasticities, household assets, and the husband’s and the wife’s lifetime earnings:κlm = κlm(η, piit, sit) (2.13)where piit ≈ AitAit+HumanWealthit is a “partial insurance” parameter (Blundell et al. (2008)).HumanWealthit = HumanWealthi1t + HumanWealthi2t, and HumanWealthijt is earnerj’s expected discounted lifetime labor income.25 The higher piit the lower the sensitivity ofconsumption to shocks because the household has more financial assets (relative to humanwealth) to smooth the wage shocks. sit ≈ HumanWealthi1tHumanWealthit is the share of husband’s humanwealth over family human wealth. η is the vector of all elasticities, which includes genderj1’s Frisch elasticity of labor supply with respect to j2’s wage (denoted by ηhj1 ,wj2 )26, gen-der j’s consumption elasticity of intertemporal substitution (ηcj ,p)27, and the “bargainingelasticity”(ηµ,wj ), the elasticity of utility weight with respect to changes in the wages.24Full analytical expressions are presented in Appendix A.2.25Formally, HumanWealthijt =∑Ts=tEt−1[Yijs(1+r)s−t ], where T is the retirement age.26Frisch elasticity of labor supply is, by definition, the elasticity of labor supply with respect to wage whenthe marginal utility of wealth is fixed (which is different from the Marshallian elasticity that captures both thesubstitution effect and wealth effect). It can be identified through labor supply response to transitory shocks,which by definition is mean-reverting and does not induce changes in lifetime earnings. However, I follow thecommon notation ηh,w instead of ηh,u.27p is the ”price” of a unit of current consumption relative to future consumption.222.2. The Household Model2.2.4 IdentificationThe parameters of the wage process are identified independently of preferences. The followingmoments of the joint distribution of the individual wages deliver identification:σ2uj = −Et[∆wijt∆wijt+1], j = 1, 2σ2u1u2 = −Et[∆wi1t∆wi2t+1],σ2vj = Et[∆wijt(∆wijt−1 + ∆wijt + ∆wijt+1)], j = 1, 2σ2v1v2 = Et[∆wi1t(∆wi2t−1 + ∆wi2t + ∆wi2t+1)]where ∆wijt is given by (2.7). Identification of the transitory variances rests on the ideathat wage growth rates are autocorrelated due to mean reversion caused by the transitorycomponent (the permanent component is subject to i.i.d. shocks). Identification of σ2u1u2is an extension of this idea: between-period and between-earner wage growth correlationreflects the correlation of the mean-reverting components. Identification of the permanentvariances rests on the idea that the variance of wage growth (Et(∆wijt)2), net of the meanreverting component (Et[∆wijt(∆wijt−1 + ∆wijt+1)]), identifies the variance of innovationsto the permanent component. Identification of σ2v1v2 follows a similar logic.The transmission coefficients of wage shocks into consumption and earnings are iden-tified by the covariances between these outcome variables and wages. Consider for exam-ple the transmission of wage shocks into the wife’s private consumption. The coefficients(κc2u1 , κc2u2 , κc2v1 , κc2v2) can be identified by the following moments:E[∆ci2t∆wi1t+1] = −κc2u1σ2u1 − κc2u2σu1u2E[∆ci2t∆wi2t+1] = −κc2u1σu1u2 − κc2u2σ2u2E[∆ci2t(∆wi1t−1 + ∆wi1t + ∆wi1t+1)] = κc2v1σ2v1 + κc2v2σv1v2E[∆ci2t(∆wi2t−1 + ∆wi2t + ∆wi2t+1)] = κc2v1σv1v2 + κc2v2σ2v2Suppose the wage parameters (σ2u1 , σ2u2 , σu1u2 , σ2v1 , σ2v2 , σv1v2) have been identified. Then thefirst two moments identify the loading factors of transitory shocks (κc2u1 , κc2u2), and the lasttwo moments identify the loading factors of permanent shocks (κc2v1 , κc2v2).Identification of the remaining transmission coefficients follows the same logic. Note thatthe equations above serve to illustrate the idea of identification. More moment conditionsare available and the model is over-identified.Another thing worth mentioning is that, on the one hand, the PSID data are collectedbiennially since 1999; on the other hand, all variables (at least all the ones used in this paper)reported in the PSID are measured at the annual level. One has to be careful about this232.3. Empirical Implementationfeature when deriving the empirical-relevant moment conditions. The full set of momentconditions in this context is presented in Appendix A.3.2.3 Empirical Implementation2.3.1 Data and Sample SelectionThe main data I use in this paper is the Panel Study of Income Dynamics (PSID) for the years2005-2013. The PSID data are collected biennially since 1999, so 5 waves are used: 2005,2007, 2009, 2011, and 2013. Starting in 1999, in addition to income data and demograph-ics, the PSID collects data about detailed assets holdings and consumption expenditures (atthe household level), the latter of which cover many nondurable and services consumptioncategories, including utilities, health expenditure, transportation, education, and child care.A few more consumption categories, including clothing, trips and vacation, and other recre-ation, were added in 2005. As the imputation procedures relies on matching the clothingconsumption, I use the waves since 2005.I focus on the core (non-SEO) sample of the PSID. I select the households that are stablymarried over the sample period, with the male head aged between 25-60 and both partnersare working.28 I drop the households with missing information on key demographic variables(age, race, and education) and households with zero consumption. I also drop the householdsin which the hourly wage of the husband or of the wife is lower than half of the minimumwage. The remaining sample is an unbalanced panel of 9628 observations.29I also use the 1998-2013 Consumer Expenditure Survey (CEX) data. The CEX providesa comprehensive flow of data on the buying habits of American consumers. The data arecollected by the Bureau of Labor Statistics and used primarily for revising the CPI. Thedefinition of the head of the household in the CEX is the person or one of the persons whoowns or rents the unit; this definition is slightly different from the one adopted in the PSID,where the head is always the husband in a couple. I make the two definitions compatible byrestricting the CEX sample to be male-head households.28That is, I select the people who report positive annual working hours. People who are temporarilyunemployed for less than a year remain in the sample. I focus on the couples who have at least someattachment to the labor market. While the extensive margin of labor supply adjustment is also importantfor family insurance, the non-parametric method for solving the household problem that I use in this paperassumes interior solutions only and is thus not suitable for analyzing the entering-and-exiting labor marketchoice. Incorporating the extensive margin requires one to impose more structure on the household behaviorand is beyond the scope of the current paper.29When calculating the relevant consumption, hourly wage and earnings moments, I do not use data display-ing extreme “jumps” from one year to the next (most likely due to measurement error). A “jump” is definedas an extremely positive (negative) change from t− 2 to t, followed by an extreme negative (positive) changefrom t to t+ 2. Formally, for each variable (say x), I construct the biennial log difference ∆2 log(xt), and dropthe relevant variables for observation in the bottom 0.1 percent of the product ∆2 log(xt)∆2 log(xt−2).242.3. Empirical ImplementationThe CEX is based on two components, the Diary survey and the Interview survey. TheDiary survey is conducted over two consecutive one-week periods, designed to track detailedexpenditures data on small and frequently purchased items, such as food, personal care, andhousehold supplies. The Interview survey is conducted quarterly, and it covers about 95percent of all household expenditure. My analysis below uses only the Interview sample,because it is more comprehensive and it can be used to construct annual expenditure forhouseholds. I apply the same sampling restrictions on the CEX as on the PSID: marriedhouseholds with the head being male and of age 25-60; both partners are working; and nomissing information on key demographic characteristics.To meet the data requirement of the model outlined in Section 2.2, I categorize andaggregate the consumption items in the PSID into private and public goods. The baselinecategorization is that public consumption comprises food at home, rent, home insurance,health insurance, utilities, and child care, and all other items are considered private goods,such as clothing and apparel30, telecommunications, food out, transportation, recreationalgoods, etc.Finally, I also use the 1998-2013 series of Consumer Price Indexes, also collected by theU.S. Bureau of Labor Statistics, to deflate the nominal income, consumption, and assets.2.3.2 Consumption ImputationTo identify the relative changes of bargaining power within families, I need information abouthow couples divide the resources, i.e. Cijt, the private consumption of each partner acrosstime. The PSID has rich dynamic information but only provides household-level consumptiondata. The CEX, on the other hand, has partial information on gender-specific consumptionbut is not a longitudinal data (follows the same family for only up to four quarters). I combinethe strengths of these two datasets by imputing the gender-specific consumption in the CEXto the PSID.Specifically, the CEX collects data on consumption of men’s clothing and consumption ofwomen’s clothing.31 The imputation utilizes the household and individual information thatis commonly available in both the CEX and the PSID, and predicts the husband’s and thewife’s clothing expenditure separately for the married households in the PSID. Readers arereferred to Pan (2017) for details about the imputation procedure.32 I assume that the error30Expenditure on children’s clothing is categorized as public consumption. There might be concern thatmothers care more (or less) about children’s welfare than fathers do. In the current model, this difference canbe (partially) accommodated by the different elasticities of public consumption between men and women. Amore fundamental treatment of this issue is left for future study.31Both are referred to adults clothing explicitly. There are additional categories for boys’ (aged 2-15)clothing, for girls’ (aged 2-15) clothing, and for clothing for children aged less than 2, separately.32Browning et al., 1994 also makes use of data on clothing consumption to identify the sharing rule. Aproblem with using clothing is that some clothing expenditure is work-related and has nothing to do with252.3. Empirical Implementationterms in the imputation regressions are not related to the intra-household allocations. Thatis, changes in intra-household allocations are mean-zero after controlling for the householdand individual characteristics and household consumption.2.3.3 Measurement ErrorEarnings data and consumption data are subject to measurement errors. Ignoring the vari-ance of measurement error in wages or earnings is problematic since it has a direct effect onthe estimates of the structural parameters. Following Meghir and Pistaferri (2004), I use find-ings from validation studies to set a priori amount of wage or earning variability that can beattributed to error. I use the estimates of Bound et al. (1994), who estimate the share of vari-ance associated with measurement error using a validation study for the PSID. Specifically,denoting the measurement error in variable x (in logs) by ξx, I set: V ar(ξw) = 0.13V ar(w),V ar(ξy) = 0.04V ar(y), V ar(ξh) = 0.23V ar(h). These estimates can be used to correct all“own” moments (such as E((∆yijt)2), E (∆yijt∆yijt+1), etc.) with the only assumption (notentirely uncontroversial, see Bound and Krueger, 1991) that measurement error is not cor-related over time. Cross moments (such as E(∆wijt∆yijt)) involve the covariance betweenmeasurement errors in wages and in earnings. This covariance is non-zero by construction,since our wage measure is annual earnings divided by annual hours. This is the so-called“division bias”. To correct for this, I write the relationship between measurement errors inlog earnings, hours and wages asV ar(ξw) = V ar(ξy) + V ar(ξh)− 2Cov(ξy, ξh)and given the variance of measurement errors in earnings, hours and wages, I can back outthe covariance between measurement errors in wages and in earnings. Finally, for separableutility, log consumption is a martingale. Hence, the variance of the measurement error inconsumption is directly identified from the moment V ar(ξc) = −E(∆cit∆cit+1). I keep thisexact identification also for the non-separable case.33Using these estimates of measurement errors, I estimate the parameters using the momentconditions that are properly adjusted (See Appendix A.4).2.3.4 Estimation ProcedureHere is a summary of my estimation procedure:preference or bargaining. A man who gets promoted (a positive permanent shock) is likely to spend moremoney on suits, even if there is no bargaining. I make no attempt to address this issue in the current paper.33In the case of non-separability between hours and consumption, −E(∆cit∆cit+1) gives the upper boundon the measurement error as long as the signs of κcu1 and κcu2 are the same, i.e. a transitory shock to thehusband’s wage affects consumption in the same direction as a transitory shock to the wife’s wage does.262.4. ResultsFirst, I impute the private consumption (Cˆijt) of each member in each household followingPan (2017).Second, separately for the husbands and the wives, I regress the log of hourly wage,the log of annual labor income, and the log of imputed private consumption on observablecharacteristics and work with the residuals (the empirical counterparts of wijt, yijt, and cijt)in the following steps. The residual public consumption gijt is also obtained.Third, I estimate the variances and covariances of the wage shocks (σ2uj , σ2vj , σu1u2 , andσv1v2) by GMM using the wage moments.Fourth, I estimate the smoothing parameters piit and sit using asset and (current andprojected) earnings data.Finally, given the estimates of the wage parameters and smoothing parameters, I estimatethe transmission coefficients (κ’s) and the underlying preference and bargaining parametersusing the restrictions that the model imposes on the second order moments of ∆ci1t, ∆ci2t,∆git, ∆yi1t and ∆yi2t.2.4 Results2.4.1 Descriptive StatisticsTable 2.1 presents summary statistics for the PSID sample. The first panel reports the wages,hours worked, and earnings of males and females. Note that my sample is conditioning onworking, so these are the conditional means of wages, hours, and earnings. Real wages forboth males and females exhibit an inverse-U shape pattern across years, peaking at 2008(the 2009 PSID actually reflects the situation in 2008 due to the retrospective nature of thesurvey) and then shrinking afterward. The average female labor supply changes very little(on the intensive margin); there is a slight drop in the male’s hours worked during 2008-2010but the change is minor. The earnings follow the same time pattern as the wages.The second panel reports the average expenditure on different consumption categories.The public consumption items and the private consumption items are listed separately. Usingthis baseline categorization, the total public consumption accounts for roughly 59.2% of thehousehold consumption on average. The time pattern of aggregate consumption is also inter-esting. Aggregate household consumption, whether private or public, starts to shrink from2008, although wages and earnings have not decreased until 2010, suggesting that householdshave some forward-looking behaviors.272.4. ResultsTable 2.1: Descriptive Statistics: Sample Means(1) (2) (3) (4) (5) (6)2005-2013 2005 2007 2009 2011 2013Panel A: Labor supply variablesMale’s hourly wage 30.13 29.4 29.47 31.63 30.58 29.54Female’s hourly wage 20.49 20.13 21.28 21.04 20.01 19.91Male’s hours worked 2245 2322 2317 2192 2174 2217Female’s hours worked 1671 1708 1650 1648 1665 1682Male’s earnings 66669 66206 67870 69118 64228 65760Female’s earnings 33859 33419 33320 34771 33890 33897Panel B: Consumption variablesPublic consumption itemsFood at home expenditure 6044 6171 6119 5796 6112 6030Rent or rent equivalent 13813 14963 15828 13295 12749 12105Home insurance expenditure 734 723 750 712 749 738Utility expenditure 2797 2655 2757 2860 2946 2766Health insurance expenditure 2139 1899 1987 2033 2034 2770Childcare expenditure 905 800 876 878 979 998Total public consumption 26432 27211 28316 25574 25569 25407Private consumption itemsFood out expenditure 2388 2581 2474 2203 2341 2342Gasoline expenditure 2884 2609 3069 2327 3271 3169Transportation (exc. gas) 3650 3891 3758 3474 3586 3538Clothing expenditure 1808 2170 2053 1710 1619 1465Education expenditure 2445 2525 2524 2289 2379 2511Health care expenditure 1597 1445 1567 1614 1562 1806Trips expenditure 2260 2297 2393 2218 2226 2160Other recreation expenditure 1168 1257 1260 1172 1118 1022Total private consumption 18200 18776 19097 17007 18103 18013Observations 9628 1920 1983 1979 1895 1851Notes: Data from the 2005-2013 PSID. Sample means of the variables are reported. Allvariables are annual (except for hourly wages). All wage, earnings, and consumptionvariables are expressed in dollars and deflated by the CPI index (base year is 2005).282.4. ResultsTable 2.2: Estimates of Wage ParametersEstimateMalesTrans. σ2u1.0266***(.0087)Perm. σ2v1.0391***(.0061)FemalesTrans. σ2u2.0180***(.0070)Perm. σ2v2.0503***(.0060)Spousal CovarianceTrans. σu1,u2.0052(.0038)Perm. σv1,v2.0015(.0027)Notes: Standard errors in parentheses are clustered at thehousehold level. *** p < 0.01, ** p < 0.05, * p < Wage ParametersTable 2.2 reports the estimates of the wage variances and covariances. A few things are worthnoting.First, there is some evidence of “wage instability” (see Gottschalk and Moffitt, 2008) bothfor males and for females, as can be seen from the variances of the transitory components, andit is larger for males. Second, the variance of the more structural component (the varianceof permanent shocks), in contrast, is larger for females, perhaps reflecting greater dispersionin the returns to unobserved skills, etc. Finally, neither the transitory components or thepermanent components of the two spouses are significantly correlated.In Appendix A.1, I also compare these estimates for 2005-2013 with those for 1999-2009(which is the sample period used in BPS). On the one hand, for both men and women, thevariances of the transitory components increase slightly in the later period, if any. On theother hand, the variances of the permanent shocks increase, from 0.032 to 0.039 for malesand from 0.039 to 0.050 for females, perhaps reflecting the greater risks due to the GreatRecession.2.4.3 Consumption, Labor Supply, and Bargaining ParametersTable 2.3 reports the estimates of gender-specific consumption and labor supply Frisch elas-ticities and bargaining parameters.Some results are worth noting. First, I find an estimate of the consumption Frisch elas-ticity of ηc1,p around 0.41-0.46 for males, implying a relative risk aversion of around 2.3,292.4. ResultsTable 2.3: Parameter Estimates(1) (2)Unitary Non-UnitaryFrisch elasticitiesηc1,p .4139*** .4610***(.0722) (.0519)ηc2,p .3313*** .3622***(.0415) (.0266)ηc1,w1 -.0082 -.0233(.0209) (.0340)ηc2,w2 .0286 -.0126(.0174) (.0084)ηg,p .1307*** .1622***(.0270) (.0113)ηg,w1 -.0131 -.0189(.0481) (.0261)ηg,w2 .0207 .0131*(.0131) (.0060)ηh1,p .0114** .0120***(.0042) (.0022)ηh2,p .0342 .0421(.0384) (.0271)ηh1,w1 .9122*** .8630***(.0688) (.0544)ηh2,w2 1.1203*** 1.0010***(.0743) (.1025)ηh1,w2 .2320*** .1072***(.0189) (.0030)ηh2,w1 .3912*** .1413***(.0455) (.0202)bargaining parametersηµ,w1 0 .2301***(n/a) (.0067)ηµ,w2 0 -.3130***(n/a) (.0101)Notes: Standard errors in parentheses are clustered at thehousehold level. *** p < 0.01, ** p < 0.05, * p < 0.1.302.4. Resultswhich is in the plausible range of this parameter. The female consumption Frisch elasticityηc2,p is lower, implying a higher relative risk aversion for females. Second, the Frisch laborsupply elasticity of males is smaller than that for females, which is consistent with intuitionand previous findings in the literature. In particular, I estimate ηh1,w1 being around 0.86-0.91 for males. Keane (2011) surveys 12 influential studies and reports a range of 0.03-2.75and an average estimate of 0.85.34 For females, I estimate ηh2,w2 ≈ 1, which is similar tothe estimate reported by Blundell et al. (2015) and Heckman and Macurdy (1980). Third,the elasticities of Pareto weights with respect to individual wages are significantly differentfrom zero, which rejects the assumption of full commitment or unitary decision making. Em-pirically, ceteris paribus, if the husband’s wage increase by 10%, the Pareto weight on thehusband would increase by 2.3%; if the wife’s wage increase by 10%, the Pareto weight onthe husband would decrease by 3.1%. Finally, estimates of cross elasticities of labor supplyare smaller in the non-unitary model, both for the husband and the wife. This suggests aweaker “added worker effect” once I allow for intra-household bargaining, which is consistentwith the intuitive argument in the introduction of the paper.2.4.4 Transmission of Wage Shocks to Consumption and Labor supplyTable 2.4 reports the estimates of transmission coefficients in (2.12). Note that in general,these transmission coefficients are heterogeneous across households and time. The valuesreported here are the sample averages.A few results are worth noting. First, the signs of the transmission coefficients are mostlythe same as in the unitary model and in the non-unitary model. Second, for the transmissionto consumption, a positive permanent wage shock to the husband increases the husband’sconsumption (reflected by κc1v1 in the table) in both models, but more so in the non-unitarycase; the same shock leads to an increase in the wife’s consumption (κc2v1) as well, but lessso in the non-unitary case. A symmetric pattern is found for a permanent shock to the wife’swage (κc1v2 , κc2v2). This suggests that the permanent shocks to individual wages impact theintra-household allocation, consistent with the hypothesis that bargaining power changes asthe spouses experience individual shocks. Third, for the labor supply response, I find someevidence for “added worker effects” in both cases (κh1v2 < 0 and κh2v1 < 0 in both columns),and it is stronger when the shocks come from the husband and the wife increases her laborsupply (|κh2v1 | > |κh2v1 |). However, these effects become much weaker in the non-unitarycase, that is, the spousal labor supply does not increase as much as in the unitary case whenan adverse shock hits. Fourth, a negative shock to the wife’s wage induces a decrease in herown labor supply (κh2v2) in both models, but the decrease is smaller in the non-unitary case,34Studies in the 1980s and 1990s typically find a Frisch elasticity close to zero. Studies after 2000 typicallyfind larger estimates.312.4. ResultsTable 2.4: Average Estimates of Transmission Coefficients(1) (2)Unitary Non-Unitaryκc1u1 -.0083 .0024κc1u2 -.0026 -.0018κc1v1 .2957 .3216κc1v2 .1560 .1245κc2u1 -.0103 -.0076κc2u2 -.0091 -.0001κc2v1 .3226 .2693κc2v2 .1833 .2470κgu1 .0315 .0412κgu2 .0726 .0504κgv1 .3163 .3738κgv2 .1967 .2126κh1u1 .9122 .8401κh1u2 .2320 .2177κh1v1 -.0043 .0323κh1v2 -.2880 -.1792κh2u1 .3912 .2500κh2u2 1.1203 1.0329κh2v1 -.7312 -.5715κh2v2 .4205 .3683Notes: κ’s are heterogeneous acrosshouseholds. The numbers reported hereare the average values in the sample.322.4. Resultssuggesting that the bargaining power shifts to the husband following such a shock. Fifth,the response of labor supply to transitory shocks is larger than the response to permanentshocks (e.g. κh1u1 > κh1v1), for both husband and wife, which is consistent with the findingin the literature and intuition. Finally, the transmissions of transitory shocks to consumptionand to labor supply are different between the unitary case and the non-unitary case, but thediscrepancies are not as large as those of permanent shocks.We can use a numerical example to illustrate how these transmission coefficients reflectthe behavioral responses of an average American household to various shocks. Considera hypothetical family in which the couple earn the US average wages, work the averagehours, and have the average consumption. (See the descriptive statistics in Table 2.1) Inthe PSID sample, the average hourly wage of married men is 30.13 dollars. Consider thescenario in which the husband suffers a permanent wage loss of 3 dollars per hour, whichis a roughly 10% negative shock. According to the estimated transmission coefficients forthe unitary model, this would induce 7.3% increase in the wife’s hours work, whereas thechange would be only 5.7% under the estimation for the non-unitary model. For the familyconsidered, these translate into an increase of 122 hours per year if it is unitary and 94 ifnon-unitary. In both cases, however, the husband barely adjusts his labor supply. The coupleadjusts their individual consumption differently between the two settings as well. The 10%permanent wage loss lowers the husband’s consumption by 2.9%, or $215 per year, if weuse the estimates obtained under the unitary specification and 3.2% ($234 per year) underthe non-unitary one. Wife’s consumption moves even more differently: she cuts down herown expenditure by $352 based on the unitary estimates, but only $293 based on the non-unitary estimates. These suggest that bargaining impact the intra-household allocation inthe direction that the permanent wage loss to the husband hurts the husband’s bargainingposition, and thus relatively increase the wife’s consumption and leisure. The behavioralresponses in this scenario can be summarized in Table 2.5.Next, I consider the same size of wage shock hits the wife in this hypothetical family:the wife’s wage drops by 3 dollars per week permanently, which is roughly a 15% shock.(Married women’s mean wage is $20.49 a week.) By similar calculations, the family adjuststhe individual labor supply and consumption as in Table 2.6. Again, the added-worker effectis much weaker in the non-unitary case: the husband works 97 hours more a year in theunitary case but only 60 hours more in the non-unitary case. The difference is roughly onefull-time week per year. When it comes to consumption, again the one who suffers the wageloss — in this case, the wife — suffers a greater loss of consumption in the non-unitarysetting, and the spouse’s welfare is hurt relatively less.332.4. ResultsTable 2.5: Behavioral Responses to -10% Permanent Wage Shock to HusbandUnitary Non-UnitaryLabor Supply Response (in hours/year)Husband ≈ 0 −7Wife +122 +94Consumption Response (in dollars/year)Husband -215 -234Wife -352 -293Table 2.6: Behavioral Responses to -10% Permanent Wage Shock to WifeUnitary Non-UnitaryLabor Supply Response (in hours/year)Husband +97 +60Wife -105 -92Consumption Response (in dollars/year)Husband -170 -130Wife -299 -405Note: An numerical example of an “average” American household’s response to wage shocks.Numbers are calculated based on the estimated transmission coefficients reported in Table Insurance AccountingI now use the estimates of the transmission coefficients to understand the importance ofvarious sources of insurance available to households.First of all, to be consistent with the discussion in the literature, it is useful to calculate theshock transmission to household consumption, which in my framework can be approximatedby∆c ≈ C1C∆c1 +C2C∆c2 +GC∆gTherefore, the transmission of, say, a permanent wage shock to the husband, to household-level consumption κcv1 can be approximated byC1C κc1v1 +C2C κc2v1 +GCκgv1 , a weighted sumof the private consumption parameters and public consumption parameters. In the imputeddata, the male’s share in consumption is 16.8%, the female’s share is 22.1%, and the publicshare 62.0%. Thus, in the non-unitary model, the transmission is κcv1 ≈ 0.3453, which isslightly greater than the estimate in the unitary model (0.3171). For a permanent wage shockto the wife, the transmission to the household-level consumption κcv2 ≈ 0.2073 in the non-unitary case and 0.1887 in the unitary one. In either case, the consumption is very smoothedeven to permanent shocks, and even more so in the unitary case, i.e., the households achievemore risk sharing in the unitary case.342.4. ResultsHow about the relative contribution of different sources of insurance? Starting from theintertemporal budget constraint,C = Y − S, (2.14)I decompose the response of household consumption growth to a permanent wage shock facedby the primary earner as: 35∂∆c∂v1≈ ∂∆y∂v1− ∂∆(S/Y )∂v1, (2.15)where S/Y is the average propensity to save out of family earnings. Thus, the first term onthe right-hand side represents the extent of insurance achieved via family labor supply andthe second term represents the insurance achieved through asset accumulation.The first term, the response of household earnings to a permanent shock to the maleshourly wage, can be decomposed as follows:∂∆y∂v1≈ ω∂∆y1∂v1+ (1− ω)∂∆y2∂v1(2.16)where ω = Y1/(Y1 + Y2) is the male’s share in household total earnings.From the previous calculation based on the non-unitary estimates, a 10% permanentdecrease in the husband’s wage rate (v1 = −0.1) induces a −3.5% change in household totalconsumption.The response of consumption can be decomposed into several steps. Consider a casein which there is only one earner (ω = 1), labor supply is fixed (∂∆h1∂v1 = 0), and there isno self-insurance through savings. Then ∂∆c∂v1 = 1 and consumption responds one-to-one topermanent shocks in hourly wages. Now if we bring in the second earner, the wife, but stillassuming fixed labor supply and no savings, household earnings would fall by 7.0% (ω = 0.7in the data) and the fall in consumption is of the same magnitude given the absence ofself-insurance through savings and labor supply behavioral responses.The introduction of behavioral responses changes the picture slightly further. Assume,for example, that males can vary their labor supply (while keeping female labor supplyexogenous). Since the husband’s Marshallian elasticity is almost zero (κˆh1v1 = 0.0323),∂∆c∂v1= ∂∆y∂v1 = ωκˆy1v1 = 0.72, almost the same as the case above. Allowing for added workereffects reduces the impact of a 10% decline in male permanent shock on consumption to only6.3% (∂∆c∂v1 =∂∆y∂v1= ωκˆy1v1 +(1−ω)κˆy2v1 = 0.54). Finally, with all insurance channels active,the fall in household earnings is still 5.4%, but the fall in consumption is greatly attenuated35To derive this, first I take logs of both sides of (2.14), and then take first difference: ∆ log(C) = ∆ log(Y −S) = ∆ log(Y ) + ∆ log(1− S/Y ). And then use the approximation: log(1− S/Y ) ≈ −S/Y .352.5. Conclusionto 3.5%. In other words, I use (2.15) to calculate that, of the 35 percentage points (p.p.) ofconsumption “insured” against the shock to the males wage36, 16 p.p. (45.7% of the totalinsurance effect) come from family labor supply adjustment and 19 p.p. (54.3%) come fromself-insurance through borrowing and saving.And how do the estimates from the unitary model and from the non-unitary model implydifferently for insurance? Using my estimates of the unitary model, following the samedecomposition approach as above, I calculate that 38% of consumption is insured whenthere is a permanent shock to the husband’s wage. Out of 38 p.p of consumption insuredagainst the shock to the male’s wage, 23 p.p. (60.5% of the total insurance effect) come fromfamily labor supply adjustment, and the remaining 15 p.p. (39.5%) come from self-insurancethrough credit markets. Therefore, although the total insurance achieved by the householdestimated does not change much with and without intra-household bargaining, the relativeimportance of different channels of insurance does change. Allowing for intra-householdbargaining lowers contribution of insurance from family labor supply (45.7% versus 60.5%)to consumption smoothing.Finally, I discuss the consumption smoothing against a permanent wage shock to the wife.Based on the estimates for the unitary case, 12% of consumption is insured when there is apermanent shock to the wife’s wage. Out of the 12 p.p., 7 p.p. (58.3% of the total insuranceeffect) come from family labor supply adjustment, and the remaining 5 p.p. (41.7%) comefrom self-insurance through credit markets. By contract, using the transmission coefficientsestimated in non-unitary model, 10% of consumption in insured; And within this 10 p.p.,only 3 p.p. (30% of the total insurance effect) come from family labor supply adjustment andthe 7 p.p. (70%) come from self-insurance through credit markets. Again, the contributionfrom family labor supply channel decreases once intra-household bargaining is allowed.2.5 ConclusionThis paper investigates how households insure themselves against idiosyncratic wage shocksand how this insurance interacts with intra-household bargaining. I merge information fromthe CEX and the PSID using an imputation procedure to obtain a panel data of house-holds on individual consumption expenditures, income, and labor supply. Using a collectivehousehold model, I derive analytical equations describing how wage shocks are transmittedto consumption and labor supply, and how the transmission mechanism depends on prefer-ence parameters and bargaining parameters. The model is identified and estimated usingthe merged panel data. I find that intra-household allocations of expenditures and leisure36The 36 p.p. figure is derived from the difference between the response of consumption with savings andfamily labor supply responses (a 3.5% decline) and without these (a 7.0% decline).362.5. Conclusionrespond strongly to individual wage shocks, and the same shocks can have very differenteffects depending on whose income they perturb within a household.The non-unitary approach has several interesting implications for the household members’behavioral responses. A permanent decline in the husband’s wage induces an increase in thewife’s labor supply in both the unitary and non-unitary specifications, but the increase issmaller in the non-unitary case. In particular, a 10% permanent decline in the husband’swage increases the wife’s labor supply by 7.3% in the unitary model, but only 5.7% in thenon-unitary model. Moreover, the husband’s labor supply also increases following a negativepermanent shock to the wife’s wage but the increase is smaller when allowing for intra-household bargaining. A 10% permanent decline in the wife’s wage increases the husband’slabor supply by 2.9% in the unitary model versus 1.8% in the non-unitary model.Individual consumption expenditures also respond to the individual wage shocks differ-ently in the two model specifications. For example, a negative permanent shock to thehusband’s wage decreases his own consumption more and decreases his wife’s consumptionless in the non-unitary model relative to the unitary one. In terms of consumption smooth-ing, the overall insurance is not significantly altered, but the contribution of the family laborsupply channel decreases from 60.5% in the unitary model to 45.7% in the non-unitary modelwhen the shock hits the husband’s wage; and from 58.3% to 30% when the shock hits thewife’s wage.These differences are fairly substantial and suggest that removing the restrictions implicitin the unitary model is of critical importance to make sense of observed changes in households’behavior, and to quantify the extent to which different channels contribute to consumptionsmoothing in the face of wage uncertainty.37Chapter 3Markov-Chain Approximations forLife-Cycle Models3.1 IntroductionIn quantitative macroeconomic studies it is often necessary to approximate continuous stochas-tic processes using discrete state-space representations; e.g. Markov chains. Different meth-ods are available to perform such approximations.37 The properties of alternative discretiza-tion methods to approximate covariance-stationary AR(1) processes in the context of station-ary infinite horizon problems have been studied in some detail by Kopecky and Suen (2010).They find that: (a) the choice of discretization method may have a significant impact onthe model simulated moments; (b) the performance of Rouwenhorst’s (1995) method is morerobust, particularly for highly persistent processes.While a covariance-stationary income process is convenient, it is not consistent with thefact, first highlighted by Deaton and Paxson (1994), that within-cohort income inequalityincreases with the age of a cohort. For this reason, most quantitative life-cycle analysesof consumption and income dynamics assume a non-stationary labor income process whosevariance increases with age.38 As a result, the difficulty of accurately approximating theincome process with a small number of discrete states increases with age.We show how to extend both Tauchen’s (1986) and Rouwenhorst’s (1995) methods todiscretize non-stationary AR(1) processes and compare their respective performance withinthe context of a life-cycle, income-fluctuation problem. Both extensions keep the number ofstates in each time period constant, but they allow the state vector and transition matrixto change over time. In both cases, some property of the original stationary counterpart are37The seminal contributions are Tauchen (1986), Tauchen and Hussey (1991) and Rouwenhorst (1995). Addaand Cooper (2003), Flode´n (2008) and Kopecky and Suen (2010) introduce improvements for stationary,univariate, AR(1) processes. Markov-chain approximations for stationary, vector autoregressive processeshave been proposed by Galindev and Lkhagvasuren (2010), Terry and Knotek (2011) and Gospodinov andLkhagvasuren (2014). Farmer and Toda (2016) propose a method that can be applied to stationary, non-linear,multivariate processes.38Non-stationarity in the income process can take the form of distributional assumptions on the initialconditions as in Huggett (1996), a unit root component as in Storesletten et al. (2004), or heteroskedasticityof the innovations as in Kaplan (2012).383.2. Discrete Approximations of AR(1) Processespreserved: Tauchen’s method matches the transition probabilities implied by the normalityassumption, while Rouwenhorst’s method matches the conditional and unconditional firstand second moments of the original process.We evaluate the performance of both methods in the context of a finite-horizon income-fluctuation problem with a unit-root income process with normal innovations.39 We findthat Rouwenhorst’s method performs extremely well even with a relatively small number ofgrid-points.Our paper is related to several studies (see, among others, those listed in footnote 1).However, to the best of our knowledge, it is the first one to formally study the approxi-mation of non-stationary AR(1) processes. Papers studying quantitative life-cycle problemswith non-stationary stochastic processes have typically approximated those processes usinga variety of intuitively appealing approaches. Storesletten et al. (2004) use a binomial tree,Huggett (1996) uses a variant of Tauchen discretization with a different conditional distri-bution at the initial age, Kaplan (2012) uses an age-varying, equally-spaced grid with rangeand transition probabilities chosen to match some moments of the original continuous pro-cess. In most cases these methods are only partially documented, hence we know very littleabout their performance. Our work is meant to provide a more systematic treatment of thisapproximation problem.The remainder of this paper is structured as follows. Section 3.2 discusses how to extendTauchen’s (1986) and Rouwenhorst’s (1995) methods to non-stationary AR(1) processes.Section 3.3 compares the accuracy of the two methods. Section 3.4 concludes.3.2 Discrete Approximations of AR(1) ProcessesConsider an AR(1) process of the following form,yt = ρtyt−1 + εt, εtid∼ N(0, σεt) (3.1)with initial condition y0, where y0 can be deterministic or a random draw from some distri-bution. Let σt denote the unconditional standard deviation of yt. It follows from equation(3.1) thatσ2t = ρ2tσ2t−1 + σ2εt. (3.2)In general the above process is not covariance-stationary. Sufficient conditions for sta-39As we discuss in the main text, the advantage of using such a process for our benchmark is that theassociated optimization problem can be solved using extremely accurate numerical techniques.393.2. Discrete Approximations of AR(1) Processestionarity are that the process in equation (3.1) is restricted toyt = ρyt−1 + εt, |ρ| < 1, εt iid∼ N(0, σε) (3.3)with constant persistence ρ, constant innovation variance σε and y0 randomly drawn fromthe asymptotic distribution of yt; namely, N(0, σ) where σ = σε/√1− ρ2. We call this casethe stationary case in what follows, to distinguish it from the general, unrestricted process40in equation (3.1).The aim of these notes is to show how to adapt both Tauchen (1986) and Rouwenhorst(1995) methods to discretize a non-stationary AR(1) of the general form in equation (3.1).3.2.1 Tauchen’s (1986) MethodStationary caseTauchen (1986) proposes the following method to discretize a stationary AR(1) process. Con-struct a Markov chain with a time-independent, uniformly-spaced state spaceY N = {y¯1, . . . , y¯N} withy¯N = −y¯1 = Ωσ (3.4)where Ω is a positive constant.41 If Φ denotes the cumulative distribution function for thestandard normal distribution and h = 2Ωσ/(N − 1) the step size between grid points, theelements of the transition matrix ΠN satisfypiij =Φ(y¯j−ρy¯i+h/2σε)if j = 1,Φ(y¯j−ρy¯i−h/2σε)if j = N,Φ(y¯j−ρy¯i+h/2σε)− Φ(y¯j−ρy¯i−h/2σε)otherwise.Basically, the method constructs the transition probabilities piij to equal the probability(truncated at the extremes) that yt falls in the interval (y¯j − h/2, y¯j + h/2) conditionally onyt−1 = y¯i.40Note that the general process does not restrict ρt to lie inside the unit circle.41Tauchen (1986) sets Ω = 3. Kopecky and Suen (2010) calibrate it so that the standard deviation of theMarkov chain coincides with that of the original AR(1) process.403.2. Discrete Approximations of AR(1) ProcessesNon-stationary caseOur non-stationary extension of Tauchen (1986) constructs a state space Y Nt = {y¯1t , . . . , y¯Nt }with constant size N , but time-varying grid-points withy¯Nt = −y¯1t = Ωσt (3.5)and step size ht = 2Ωσt/(N − 1). The associated transition probabilities arepiijt =Φ(y¯jt−ρy¯it−1+ht/2σεt)if j = 1,Φ(y¯jt−ρy¯it−1−ht/2σεt)if j = N,Φ(y¯jt−ρy¯it−1+ht/2σεt)− Φ(y¯jt−ρy¯it−1−ht/2σεt)otherwise.The main difference between our extension and its stationary counterpart is that therange of the equidistant state space in equation (3.5) is time varying and, as a result, so arethe transition probabilities.3.2.2 Rouwenhorst’s (1995) MethodThe Rouwenhorst method is best understood as determining the parameters of a two-stateMarkov chain, with equally-spaced state space, in such a way that the conditional first andsecond moments of the Markov chain coincide with the same moments of the original AR(1)process.42Stationary caseIn the case of the stationary AR(1) process in equation (3.3), the state space for the two-stateMarkov chain is y¯2 = −y¯1 and the transition matrix is written asΠ2 =[pi11 1− pi111− pi22 pi22]. (3.6)The moment condition for the expectation conditional on yt−1 = y¯2 isE(yt|yt−1 = y¯2) = −(1− pi22)y¯2 + pi22y¯2 = ρy¯2, (3.7)42In general, a Markov chain of order K is characterized by K2 parameters (K states plus (K2−K) linearly-independent transition probabilities) and can be uniquely identified by K2 linearly-independent moment con-ditions. The Rouwenhorst method is, therefore, a special case of a general moment-matching procedure.413.2. Discrete Approximations of AR(1) Processeswhere the left hand side is the conditional expectation of the Markov chain and the righthand side its counterpart for the AR(1) process for yt−1 evaluated at the grid point y¯2. Itfollows thatpi22 =1 + ρ2= pi11, (3.8)where the second equality follows from imposing the same condition for yt−1 = y¯1 = −y¯2.The moment condition for the variance conditional on yt−1 = y¯2 is43Var(yt|yt−1 = y¯2) = (1− pi22)(−y¯2 − ρy¯2)2 + pi22 (y¯2 − ρy¯2)2 = σ2ε , (3.9)which, after replacing for pi22 from equation (3.8), impliesy¯2 = σ. (3.10)Having determined Π2, the method scales to an arbitrary number of grid points N in thefollowing way.44 The state space Y N = {y¯1, . . . , y¯N} is equally-spaced withy¯N = −y¯1 = σ√N − 1. (3.11)For N ≥ 3, the transition matrix satisfies the recursionΠN = pi[ΠN−1 00′ 0]+ (1− pi)[0 ΠN−10 0′]+ pi[0 0′0 ΠN−1]+ (1− pi)[0′ 0ΠN−1 0],(3.12)where pi = pi11 = pi22 and 0 is an (N − 1) column vector of zeros.The main difference between Rouwenhorst’s and Tauchen’s methods is that in the formerthe transition probabilities do not embody the normality assumption about the distributionof the shocks. Rather, Rouwenhorst matches exactly, by construction, the first and secondconditional and, by the law of iterated expectations, unconditional moments of the continuousprocess independently from the shock distribution.Non-stationary caseAs for Tauchen, our non-stationary extension of Rouwenhorst (1995) constructs an equally-spaced, symmetric, state space Y Nt = {y¯1t , . . . , y¯Nt } with constant size N but time varyinggrid points and transition matrix ΠNt . If N = 2, it follows that y¯2t = −y¯1t and the counterpart43Symmetry implies that the second conditional-variance condition is linearly dependent with equation (3.9)and, therefore, satisfied.44We refer the reader to Rouwenhorst (1995) and Kopecky and Suen (2010) for a rigorous derivation.423.3. Evaluationof the first-moment condition (3.7) becomesE(yt|yt−1 = y¯2t−1) = −(1− pi22t )y¯2t + pi22t y¯2t = ρty¯2t−1,with unique solutionpi22t =12(1 + ρty¯2t−1y¯2t)=12(1 + ρtσt−1σt)= pi11t , (3.13)where the second equality follows from the counterpart of the second moment condition(3.9) which impliesy¯2t = −y¯1t = σt. (3.14)The third equality in equation (3.13) follows from the expression for the conditional firstmoment for yt−1 = y¯t−1.As in the non-stationary version of Tauchen, the points of the state-space are a functionof the time-dependent unconditional variance of yt. Comparing equations (3.8) and (3.13)reveals that, relative to the stationary case, the probability of transiting from y¯2t−1 to y¯2tdepends on the rate of growth of the unconditional variance of yt.Equation (3.13) implies that the condition for the Markov chain to be well defined, andhave no absorbing states, namely 0 < pi11t = pi22t < 1, is equivalent toρ2tσ2t−1σ2t< 1. (3.15)It follows from equation (3.2) that this condition always holds. Therefore Rouwenhorst’sapproximation can be applied to any process of the type defined in equation (3.1).45As in the stationary case, the approach scales to an N -dimensional, evenly-spaced statespace Y Nt by settingy¯Nt = −y¯1t = σt√N − 1 (3.16)and ΠNt to satisfy the recursion (3.12) with the transition matrices and the probability pit =pi11t = pi22t indexed by t.3.3 EvaluationThis section assesses the performance of the two discretization methods above in solving afinite-horizon, income-fluctuation problem with a non-stationary labor income process.45This is also trivially true for Tauchen’s method.433.3. EvaluationConsider the following optimization problem in recursive form46Vt(zt,yt) = maxct,atlog(ct) + βEtVt+1(zt+1, yt+1) (3.17)s.t. zt = (1 + r)at−1 + ytat = zt − ctyt+1 = ytt, log ti.i.d.∼ N(0, σε),at ≥ 0, at given.Individuals start life at age 1, with initial wealth a0 = 0 and y0 = 1, and live until age40. Each model period is a year. In the computation we set the discount rate β to 0.96and the interest rate r to .04 which are standard values. We set the variance of the laborincome process σ2ε = .0161, as in Storesletten et al. (2004). The parameterization implies anaggregate wealth-income ratio of about 0.6, in line with the baseline calibration in Carroll(2009) for a similar model with no retirement and deterministic lifetime.Since the above problem does not have a closed-form solution, we evaluate the accuracyof the two discretization methods by comparing simulated moments under the two methodsto those generated by a very accurate benchmark solution.The advantage of problem (3.17) is that, as first shown in Carroll (2004), the combinationof unit-root (in logs) income process and CRRA felicity function implies that the problemcan be normalized using (permanent) labor income yt, thereby reducing the effective statespace to the single variable zˆt = zt/yt.47. It follows that, under the assumptions that incomeinnovations are log-normally distributed, one can approximate the expectation in equation(3.17) using Gaussian-Hermite quadrature.This allows one to solve the model using a very accurate procedure—the endogenous grid-point method—for the optimization step48 and Gaussian-Hermite quadrature to approximatethe expectation in (3.17). In particular, we compute the policy functions using an exponentialgrid Gz with 1,000 points for the normalized state variable zˆ and 100 quadrature nodes forthe shock log εt. Given the well-known properties of quadrature,49 the model solution usingthe endogenous gridpoint method and quadrature is extremely accurate.We simulate the model by generating 2,000,00050 individual histories for yt using Monte46The lower bound of zero for the choice of next period’s assets is without loss of generality. It is alwayspossible to rewrite the problem so that the lower bound on, the appropriately translated, asset space is zero.47The Appendix reports the derivation48See Barillas and Ferna´ndez-Villaverde (2007) for an assessment of the accuracy of the endogenous gridmethod.49Given n quadrature nodes, Gaussian quadrature approximates exactly the integral of any polynomialfunction of degree up to 2n− 1.50Increasing the number of individuals histories to 20,000,000 does not affect the results in any meaningful443.3. EvaluationCarlo simulation of the continuous AR(1) process and linearly interpolating the policy func-tions for points off the discretized state space Gz. Since, by construction, the non-normalizedpolicy function at(zt, yt) = aˆt(zˆt)yt is linear in labor income, our benchmark simulation doesnot require any approximation with respect to labor income. Therefore, the simulated mo-ments generated by our benchmark method constitute a highly accurate approximation tothe true model moments.Next, we compute the same set of moments by applying the same optimization method asin the benchmark but using either Tauchen or Rouwenhorst’s methods to discretize the laborincome process. To be precise, in each case we solve the (non-normalized) decision problem(3.17) by replacing the continuous income process with the appropriate Markov chain withage-dependent grids Y Nt and transition matrices ΠNt and using a common exponential gridGz with 1,000 points for zt. We consider three different values for the income grid size N ;namely 5, 10 and 25.Given the policy functions thus obtained, we compute the model moments using a MonteCarlo simulation which again generates 2,000,000 income histories. This is done in twodifferent ways. In the first case, we generate the income histories using the discrete Markovchain approximation. The simulation involves interpolating the policy functions linearly onlywith respect to z. In the second case, as in the benchmark quadrature case, we generateincome histories using the continuous AR(1) process. We then interpolate linearly over bothz and labor income y.The key difference between these two approaches has to do with the sources of the errorsthat they introduce. Both cases suffer from approximation errors for the policy functionrelative to quadrature due to: (a) the suboptimal approximation of the expectation in (3.17);(b) the fact that the policy functions solve the Euler equations exactly only at a relativelysmall number of grid points for labor income. Compared to the continuous AR(1) simulation,the Markov chain simulation introduces an additional approximation error as the simulatedpolicy functions are step, rather than piecewise-linear, functions along the income dimension.3.3.1 ResultsWe evaluate the accuracy of Tauchen’s and Rouwenhorst’s discretization methods by com-paring simulated moments obtained under the benchmark quadrature approach to thoseobtained under either of the two discretization methods. The moments are: (i) the uncon-ditional mean; (ii) the unconditional standard deviation; and (iii) the Gini coefficient. Eachset of moments is reported for the distributions of labor income, consumption, wealth andtotal income. Given the increasing interest in wealth concentration, at the bottom of eachway.453.3. EvaluationTable 3.1: Ratio of Model Moments Relative to Their Counterpart in the Quadra-ture Benchmark: (a) Markov Chain Simulation and (b) Continuous RandomWalk Income ProcessN = 5 N = 10 N = 25R TΩ∗ TΩ=3 R TΩ∗ TΩ=3 R TΩ∗ TΩ=3(A) Markov chain simulationLabor income (yt) Mean 0.9960 0.9939 1.0880 0.9975 0.9952 1.0543 0.9983 0.9969 1.0070SD 0.9208 0.8798 1.3993 0.9618 0.9085 1.2329 0.9842 0.9490 1.0159Gini 0.9574 0.9608 1.1255 0.9815 0.9817 1.1069 0.9928 0.9942 1.0169Consumption (ct) Mean 0.9966 0.9882 1.0755 0.9978 1.0006 1.0546 0.9984 0.9988 1.0093SD 0.9253 0.8606 1.3517 0.9640 0.9242 1.2140 0.9850 0.9558 1.0213Gini 0.9630 0.9634 1.1392 0.9851 0.9750 1.1045 0.9949 0.9926 1.0148Assets (at) Mean 1.0186 0.7385 0.5370 1.0083 1.2330 1.0706 1.0026 1.0795 1.1079SD 1.0611 0.7689 1.1376 1.0296 1.6243 0.9059 1.0110 1.2334 1.1987Gini 1.1088 1.3562 3.0492 1.0521 1.5599 0.6049 1.0198 1.2017 1.0961Tot. inc. (rat−1 + yt) Mean 0.9966 0.9882 1.0755 0.9978 1.0006 1.0546 0.9984 0.9988 1.0093SD 0.9232 0.8656 1.3723 0.9630 0.9190 1.2181 0.9846 0.9536 1.0190Gini 0.9607 0.9549 1.1071 0.9834 0.9885 1.0966 0.9938 0.9964 1.0174Top 5% wealth share 1.0367 0.8449 1.6747 1.0217 1.3214 0.7299 1.0090 1.1451 1.0785(B) Random walk simulationLabor income (yt) Mean 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000SD 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Gini 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Consumption (ct) Mean 1.0005 0.9941 0.9886 1.0002 1.0065 1.0807 1.0001 1.0023 1.0025SD 1.0027 0.9802 0.9694 1.0013 1.0339 0.8926 1.0005 1.0176 1.0094Gini 0.9994 1.0020 1.0200 0.9998 0.9939 0.5290 0.9999 0.9986 0.9974Assets (at) Mean 1.0232 0.7409 0.4950 1.0106 1.2889 1.0018 1.0039 1.1037 1.1111SD 1.0870 0.7862 0.9270 1.0418 2.2010 0.9888 1.0158 1.6298 1.3198Gini 1.1116 1.3533 2.8598 1.0527 1.7360 0.9890 1.0201 1.2945 1.1070Tot. inc. (rat−1 + yt) Mean 1.0005 0.9941 0.9886 1.0002 1.0065 1.0018 1.0001 1.0023 1.0025SD 1.0018 0.9861 0.9792 1.0009 1.0203 0.9877 1.0003 1.0104 1.0053Gini 1.0012 0.9944 0.9832 1.0006 1.0086 0.9942 1.0002 1.0030 1.0004Top 5% wealth share 1.0670 0.8994 1.5462 1.0338 1.6214 0.7129 1.0130 1.2944 1.1086Note: Parameter values: β = 0.96, r = 0.04, σ2ε = 0.0161.For columns TΩ∗ , Ω = 1.6919 when N = 5, Ω = 2.0513 when N = 10 and Ω = 2.5996 when N = 25.463.3. Evaluationpanel we also report the share of wealth held by the households in the top 5% of the wealthdistribution.Panel (A) and (B) in Table 3.1 report the ratio of the moments obtained from simulatingthe income process using Rouwenhorst and Tauchen’s discretization methods to those com-puted for the Gaussian Hermite benchmark solution. In the table a value of one indicates thatthe approximation entails no error, relative to the benchmark solution. As shown in Flode´n(2008) and Kopecky and Suen (2010), Tauchen’s method is very sensitive to the choice of Ω.Tauchen (1986) originally sets Ω = 3, while Kopecky and Suen (2010) calibrate Ω to matchthe variance of log income. The counterpart of the latter strategy for a non-stationary incomeprocess is not obvious. Hence we choose Ω to match the variance of log income over the wholepopulation, and we report results both for this parametrization (columns TΩ∗) and for thecase in which Ω = 3 (columns TΩ=3).Case 1: Markov chain simulation. Panel (A) shows results for the case in which the dis-cretized income process is used both to compute the expectation in the decision problem andto simulate the model. In this case the Rouwenhorst method and the Tauchen method with“optimal” choice of Ω perform quite similarly in approximating the labor income momentsand the first moment of the consumption distribution. As expected the Tauchen methodwith Ω = 3 performs much worse. The Rouwenhorst method, though, is more accurate withrespect to the standard deviation of consumption, and substantially more so with respect tothe distribution of assets. In the latter case, the Rouwenhorst approximation has a maxi-mum error (for any of the moments) of at most 11 per cent for N = 5 and of only 2 per centfor N = 25. In contrast, the Tauchen approximation is off by anywhere between 1/4 and 2times relative to the benchmark quadrature method. Things are particularly worrying forthe variance of assets, which is very poorly approximated under all Tauchen approximations.The top 5% shares of wealth are very badly approximated, even with a large number ofpoints. Moreover, it is apparent that the approximation error does not necessarily shrink asthe number of grid points increases. Intuitively, when comparing the range of the incomegrid for the Tauchen (equation (3.5)) and Rouwenhorst (equation (3.11)) methods, the rangeof the income grid increases faster with N for the latter method. This implies that, in thecase of Tauchen, a larger number of simulated observations get piled onto the bounds relativeto the benchmark method, reducing accuracy. This problem appears to be quite importantwhen approximating the standard deviation of asset holdings. This conjecture is confirmedby the fact that the Tauchen method with Ω = 3, hence with a larger labor income range,performs better than the one with the “optimal choice” of Ω in this respect.Case 2: Random walk simulation. Panel (B) in Table 3.1 reports the approximationerrors obtained through Monte Carlo simulation using the continuous income process. By473.4. Conclusionconstruction, there is no approximation error for the income process in this case. As expected,the accuracy of both the Tauchen and Rouwenhorst methods generally improves relative toresults for the Markov chain simulation. In fact, the accuracy of the Rouwenhorst method isextremely high even when N = 5.Concerning the asset moments, the performance of the Rouwenhorst method is similar tothat obtained for the Markov chain simulation. The performance of the Tauchen method is,if anything, worse suggesting that, given the narrower income grid relative to Rouwenhorst,extrapolation along the income dimension increases the overall error relative to the Markovchain simulation. In fact, for N larger than 5 the Tauchen method with Ω = 3, hence with alarger labor income range, performs better than the one with “optimal choice” of Ω.In sum, the Rouwenhorst method exhibits considerable accuracy even when using a smallnumber of grid points, and its performance is substantially more robust across all momentsconsidered and for all numbers of grid points. In addition, the accuracy of the Tauchenmethod, especially in terms of asset distributions, does not improve when adding more gridpoints.3.4 ConclusionApproximating non-stationary processes is commonplace in quantitative studies of life-cyclebehavior and inequality. In such studies it is important to reliably model the fanning out overage of the cross-sectional distribution of consumption, income and wealth. Large approxi-mation errors may result in misleading inference and the problem appears to be especiallysevere when approximating the distribution of wealth.In this paper we provide the first systematic examination of the performance of alternativemethods to approximate non-stationary (time-dependent) income processes within a life-cycle setting. We begin by explicitly deriving new generalizations of the Tauchen’s andRouwenhorst’s approximation methods for the case of history-dependent state spaces, like theones commonly employed in life-cycle economies. We then compare the relative performanceof these approximation methods. For each method, we numerically solve a finite-lifetime,income-fluctuation problem, and compute a set of moments for the implied cross-sectionaldistributions of income, consumption and wealth. Next, we gauge the relative performanceof the two methods by comparing these moments to the ones obtained from a quasi-exactsolution of the same income-fluctuation problem.The results of this comparison are quite clear and suggest that, in a life-cycle setting,Tauchen’s method is generally much less precise than Rouwenhorst’s. This discrepancy ismost severe when considering the distribution of wealth. Perhaps more worrying is the factthat adding grid points to the income approximation does not seem to significantly improve483.4. Conclusionthe performance of Tauchen’s method. In contrast, increasing the number of grid points doesimprove the accuracy of the Rouwenhorst approximation. However, we find that the lattermethod offers a very reliable approximation even with just 5 grid points.49ConclusionThis dissertation studies the households’ intertemporal decision making, especially how theyinsure themselves facing idiosyncratic income risks. In order to analyze formally how differentinsurance mechanisms work and how bargaining impacts them, I develop a life-cycle collectivemodel of the household. I show how the model can be identified using joint moments ofindividual wage changes, consumption changes, and hours changes. However, there is a lackof data that continuously measure the intra-household allocations within the same households,an empirical challenge faced by the researchers who want to test and estimate intertemporalnon-unitary household models. The first chapter thus provides a new method for measuringthe evolution of allocations over time using the existing data. Using the CEX data, I estimatethe gender-specific consumption as a function of socioeconomic variables that are commonlyavailable in both the CEX and the PSID, and then use these estimated functions to predictthe consumption allocations within the PSID families. In the second chapter, I estimate thedynamic collective model using the imputed PSID panel. I find the Pareto weights changewith wage shocks. This is a clear rejection of the unitary model. Second, the added workereffects are significantly weaker in the non-unitary model; roughly 25% weaker when comparedto the estimates obtained for a unitary specification. Finally, this implies the contributionof the family labor supply channel to consumption smoothing is significantly lower in thenon-unitary case. The last chapter presents a systematic treatment of numerical methods forapproximating non-stationary income processes. We find that the generalized Rouwenhorstmethod is more efficient and accurate than the Tauchen method.I believe that several contributions of this research are worth highlighting. First, this dis-sertation brings in the non-unitary approach, which has been proved fruitful in other contexts,into the analysis of the household response to wage shocks in a manageable way, without re-quiring strong functional form assumptions. Second, I suggest a new method for combiningthe information from the CEX and the PSID to get a continuous measure of consumptionallocations within households. Third, when I bring the model and the data together, I do findevidence of a substantial bargaining effect, which cannot be captured in the unitary modeland changes the inference we draw about the importance of alternative insurance channels.In particular, this implies that the traditional unitary approach leads to an upward bias inthe estimates of the added worker effect and that the effect of family labor supply channel50Conclusionfor consumption smoothing may be over-estimated when not accounting for renegotiationwithin the household. Fourth, we generalize the Tauchen’s (1986) and Rouwenhorst’s (1995)methods to approximating non-stationary income processes and compare their performance.A good approximation method for the process is important for obtaining accurate approx-imations for the statistics generated from the models and for understanding the householdinsurance mechanisms. 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(2015): “Yours, Mine, and Ours: Do Divorce Laws Affect the IntertemporalBehavior of Married Couples?” American Economic Review, 105, 2295–2332.55Appendix AAppendix for Chapter 2A.1 More on Wage EstimatesTable A.1: Estimates of Wage Parameters(1) (2) (3) (4)2005-2013 1999-2009 1999-2009 1999-2009 (BPS)sample selection I sample selection IIMalesTrans. σ2u1.0266*** .0226*** .0246*** .0275***(.0087) (.0060) (.0062) (.0063)Perm. σ2v1.0391*** .0322*** .0299*** .0303***(.0061) (.0040) (.0049) (.0049)FemalesTrans. σ2u2.0180*** .0169*** .0136** .0125***(.0070) (.0060) (.0058) (.0057)Perm. σ2v2.0503*** .0389*** .0383*** .0382***(.0060) (.0046) (.0045) (.0044)Spousal CovarianceTrans. σu1,u2.0052 .0062** .0052* .0058**(.0038) (.0029) (.0028) (.0027)Perm. σv1,v2.0015 .0029 .0026 .0027(.0027) (.0024) (.0024) (.0023)Notes: Standard errors in parentheses are clustered at the household level. *** p < 0.01, ** p < 0.05, * p < 0.1.Table A.1 presents a few sets of estimates of wage variances and covariances, for differentperiods and different sampling restrictions. Column (1) is the wage estimates used through-out in this paper, the same as reported in Table 2.2. Column (2) uses the same samplingrestriction (labeled as “sample selection I” in the table; see Section 2.3.1 for details) andestimation procedure as (1), but uses the 1999-2009 PSID data instead.Column (4) is the estimates reported by Blundell et al. (2015). The discrepancies betweenmy main estimates (column 1) and Blundell et al. (2015)’s (column 4) can be attributed totwo factors. One is that we use different waves of the PSID: I use 2005-2013 and they use1999-2009. The other factor is that we apply slightly different sampling restrictions: first,I use the sample of male head aged 25-60 and they use 30-57; second, we both drop theobservations with wages lower than half of the minimum wages, but they only consider the56A.2. Approximation of the First Order Conditions and Intertemporal Budget Constraintstate-level minimum wages and I consider the combination of the federal level and the statelevels (whichever is higher in each state); third, whenever a household has a head or wifechange (due to divorce, death, or other reasons) in any year, I drop all year observations ofthis household, whereas they only drop the year of the change and treat the household unit asa new family starting with the observation following the change. In addition, the raw incomedata for 1999 in the PSID has been recalculated and updated (See the PSID documentationfor the 1999 data). The data BPS uses is actually the old version, in which some incomedata do not match the data that are currently available from the PSID.In Table A.1, I label their sampling criterion as “sample selection II”. To show that Ican replicate their wage estimates using my estimation procedure, I apply the same sampleselection criterion as theirs and re-estimate the wage parameters using the 1999-2009 PSID, asreported in column (3). The replication is fairly close, as can be seen by comparing estimatesin columns (3) and (4).A.2 Approximation of the First Order Conditions andIntertemporal Budget ConstraintHousehold i maximizes:Ui =T∑t=0βtEtUit,Uit = µitUm(Ci1t, Git, Hi1t) + (1− µit)Uf (Ci2t, Git, Hi2t),Household period budget constraint:∑j=1,2Cijt +Git +Ait+1 =∑j=1,2WijtHijt + (1 + r)Ait.A.2.1 Linearization of the First Order ConditionsThe first order condition for assets givesλit = β(1 + r)E[λit+1]where λ is the Lagrangian multiplier on the budget constraint. Define eρ = 1β(1+r) , then wehaveE[λit+1] = eρλit. (A.1)Write λit+1 = f(lnλit+1) ≡ exp(lnλit+1) and apply a second-order Taylor expansion of57A.2. Approximation of the First Order Conditions and Intertemporal Budget Constraintf(lnλit+1) around lnλit + ρ:λit+1 ≈ f(lnλit + ρ) + f ′(lnλit + ρ)(lnλit+1 − lnλit − ρ) + f′′(lnλit + ρ)2(lnλit+1 − lnλit − ρ)2= λiteρ + λiteρ(∆ lnλit+1 − ρ) + λiteρ2(∆ lnλit+1 − ρ)2= λiteρ(1 + (∆ lnλit+1 − ρ) + 12(∆ lnλit+1 − ρ)2)Taking expectation at time t yieldsEt[lnλit+1] = λiteρ(1 + Et(∆ lnλit+1 − ρ) + 12Et(∆ lnλit+1 − ρ)2)Substituting for Et[lnλit+1] from (A.1) givesEt(∆ lnλit+1 − ρ) + 12Et(∆ lnλit+1 − ρ)2 = 0and thusEt(∆ lnλit+1) = ρ− 12Et(∆ lnλit+1 − ρ)2which implies that we can write ∆ lnλit+1 as∆ lnλit+1 = $t + ιit+1 (A.2)where$t ≡ ρ− 12Et(∆ lnλit+1−ρ)2 is assumed to be fixed in the cross section and Et(ιit+1) = 0by definition of a prediction error.The first order conditions for Ci1t, Ci2t, Git, Hi1t, and Hi2t are:µitUmc1 (it) = λit(1− µit)Ufc2(it) = λitµitUmg (it) + (1− µit)Ufg (it) = λitµitUmh1(it) = λitWi1t(1− µit)Ufh2(it) = λitWi2twhere Umc1 is the marginal utility of the husband with respect to his private consumption,etc. And I write Umc1 (Ci1t, Git, Hi1t) as Umc1 (it), etc., for compactness.58A.2. Approximation of the First Order Conditions and Intertemporal Budget ConstraintTaking log of both sides and then taking the time difference yields51∆ lnUmc1 (it) = ∆ lnλit −∆ lnµit (A.3)∆ lnUfc2(it) = ∆ lnλit −∆ ln(1− µit) (A.4)ψit∆ lnUmg (it) + (1− ψit)∆ lnUfg (it) ≈ ∆ lnλit − ψit∆ lnµit − (1− ψit)∆ ln(1− µit) (A.5)∆ lnUmh1(it) = ∆ lnλit + ∆wi1t −∆ lnµit (A.6)∆ lnUfh2(it) = ∆ lnλit + ∆wi2t −∆ ln(1− µit) (A.7)where ψit ≡ µitUmg (it)µitUmg (it)+(1−µit)Ufg (it)is the husband’s share of marginal utility of public goods.Write lnUmc1 (C1t+1, Gt+1, H1t+1) = lnUmc1 (ec1t+1 , egt+1 , eh1t+1) and apply a first order Tay-lor expansion around c1t, gt, h1t (omitting the household index i for simplicity):∆ lnUmc1 (t+ 1) ≈Umc1c1(t)Umc1 (t)C1t∆c1t+1 +Umc1g(t)Umc1 (t)Gt∆gt+1 +Umc1h1(t)Umc1 (t)H1t∆h1t+1. (A.8)Similarly for Umg , Umc2 , Ufc2 , Ufg , and Ufh2we have∆ lnUmg (t+ 1) ≈Umgc1(t)Umg (t)C1t∆c1t+1 +Umgg(t)Umg (t)Gt∆gt+1 +Umgh1(t)Umg (t)H1t∆h1t+1 (A.9)∆ lnUmh1(t+ 1) ≈Umh1c1(t)Umh1(t)C1t∆c1t+1 +Umh1g(t)Umh1(t)Gt∆gt+1 +Umh1h1(t)Umh1(t)H1t∆h1t+1 (A.10)∆ lnUfc2(t+ 1) ≈Ufc2c2(t)Ufc2(t)C2t∆c2t+1 +Ufc2g(t)Ufc2(t)Gt∆gt+1 +Ufc2h2(t)Ufc2(t)H2t∆h2t+1 (A.11)∆ lnUfg (t+ 1) ≈Ufgc2(t)Ufg (t)C2t∆c2t+1 +Ufgg(t)Ufg (t)Gt∆gt+1 +Ufgh2(t)Ufg (t)H2t∆h2t+1 (A.12)∆ lnUfh2(t+ 1) ≈Ufh2c2(t)Ufh2(t)C2t∆c2t+1 +Ufh2g(t)Ufh2(t)Gt∆gt+1 +Ufh2h2(t)Ufh2(t)H2t∆h2t+1. (A.13)51The first order condition for public goods Git needs to be log-linearized as follows:∆ ln(µitUmg (it) + (1 − µit)Ufg (it)) ≈ ∆(µitUmg (it)+(1−µit)Ufg (it))µitUmg (it)+(1−µit)Ufg (it)=µitUmg (it)µitUmg (it)+(1−µit)Ufg (it)∆ ln(µitUmg (it)) +(1−µit)Ufg (it)µitUmg (it)+(1−µit)Ufg (it)∆ ln((1− µit)Ufg (it)).59A.2. Approximation of the First Order Conditions and Intertemporal Budget ConstraintSubstituting (A.8)-(A.13) into (A.3)-(A.7) and rearranging them yields:∆c1t+1∆c2t+1∆gt+1∆h1t+1∆h2t+1 ≈γc1w1 γc1w2 γc1λ γc1µγc2w1 γc2w2 γc2λ γc2µγgw1 γgw2 γgλ γgµγh1w1 γh1w2 γh1λ γh1µγh2w1 γh2w2 γh2λ γh2µ︸ ︷︷ ︸Γ∆w1t+1∆w2t+1∆ lnλt+1∆ lnµt+1 (A.14)whereΓ = A−1BA =Umc1c1 (t)C1tUmc1 (t)0Umc1g(t)GtUmc1 (t)Umc1h1(t)H1tUmc1 (t)00Ufc2c2 (t)C2tUfc2 (t)Ufc2g(t)GtUfc2 (t)0Ufc2h2(t)H2tUfc2 (t)ψtUmgc1 (t)C1tUmg (t)(1− ψt)Ufgc2(t)C2tUfg (t)ψtUmgg(t)GtUmg (t)+ (1− ψt)Ufgg(t)GtUfg (t)ψtUmgh1(t)H1tUmg (t)(1− ψt)Ufgh2(t)H2tUfg (t)Umh1c1(t)C1tUmh1(t) 0Umh1g(t)GtUmh1(t)Umh1h1(t)H1tUmh1(t) 00Ufh2c2(t)C2tUfh2(t)Ufh2g(t)GtUfh2(t)0Ufh2h2(t)H2tUfh2(t)B =0 0 1 −10 0 1 µt1−µt0 0 1 −ψt + (1− ψt) µt1−µt1 0 1 −10 1 1 µt1−µt .Apply a first order Taylor expansion on µit = µ(zi0, εit):∆ lnµit ≈ µ2(zi0, εit)µitεit∆ ln εitwhere I utilize the fact that ∆ ln zi0 = 0 since zi0 is predetermined.Recall that εit = zit − E0zit is the deviation of zit from its expected value and that theonly exogenous shock is in individual wages; thus εit is a vector of accumulated individualwage shocks since time 0: ln εit = (∑ts=0(∆ui1s + vi1s),∑ts=0(∆ui2s + vi2s)). This implies∆ ln εit = (∆ui1t + vi1t,∆ui2t + vi2t). Therefore, the equation above can be rewritten as∆ lnµit ≈ ηµ,w1(∆ui1t + vi1t) + ηµ,w2(∆ui2t + vi2t) (A.15)where ηµ,wj ≡µwjwjµ is the elasticity of µ with respect to changes in partner j’s residual wage.60A.2. Approximation of the First Order Conditions and Intertemporal Budget ConstraintNow with (A.2), (A.14), and (A.15), I can write the changes in consumption and hoursas functions of wage shocks and ιit, the latter of which is the deviation of the marginal utilityof wealth from its expectation. In the next section, I will derive ιit as a function of wageshocks.A.2.2 Log-Linearization of the Lifetime Budget ConstraintThe lifetime budget constraint isT∑s=tCi1s + Ci2s +Gis(1 + r)s−t= Ait +T∑s=tWi1sHi1s +Wi2sHi2,s(1 + r)s−t. (A.16)First, write ln∑Ts=tCis(1+r)s−t = ln∑Ts=t exp(cis − (s− t) ln(1 + r)) and apply a first orderTaylor expansion around {Et−1cis − (s− t) ln(1 + r)}Ts=t:lnT∑s=tCis(1 + r)s−t= lnT∑s=texp(cis − (s− t) ln(1 + r))≈ lnT∑s=texp(Et−1cis − (s− t) ln(1 + r)) +T∑s=tθs(cis − Et−1cis) (A.17)where θs ≡ exp(Et−1cis−(s−t) ln(1+r))∑Ts=t exp(Et−1cis−(s−t) ln(1+r)). Taking expectation of (A.17) with respect to timet− 1 and t, respectively, and noting that θs is known (with no uncertainty) at time t− 1 orat time t:Et−1 lnT∑s=tCis(1 + r)s−t≈ lnT∑s=texp(Et−1cis − (s− t) ln(1 + r)) +T∑s=tθs(Et−1cis − Et−1cis)Et lnT∑s=tCis(1 + r)s−t≈ lnT∑s=texp(Et−1cis − (s− t) ln(1 + r)) +T∑s=tθs(Etcis − Et−1cis)Subtracting the first equation from the other:Et lnT∑s=tCis(1 + r)s−t− Et−1 lnT∑s=tCis(1 + r)s−t≈T∑s=tθs(Etcis − Et−1cis). (A.18)By (A.2), (A.14), and (A.15), I have∆ci1t ≈ (γc1w1 + γc1µηµ,w1)(∆ui1t + vi1t) + (γc1w2 + γc1µηµ,w2)(∆ui2t + vi2t) + γc1λ($t + ιit)∆ci2t ≈ (γc2w1 + γc2µηµ,w1)(∆ui1t + vi1t) + (γc2w2 + γc2µηµ,w2)(∆ui2t + vi2t) + γc2λ($t + ιit)∆cgt ≈ (γgw1 + γgµηµ,w1)(∆ui1t + vi1t) + (γgw2 + γgµηµ,w2)(∆ui2t + vi2t) + γgλ($t + ιit)61A.2. Approximation of the First Order Conditions and Intertemporal Budget ConstraintNext, using the approximation that ∆cit ≈ ψi1t−1∆ci1t +ψi2t−1∆ci2t +ψigt−1∆git, whereψi1t−1 = Ci1t−1/Cit−1, ψi2t−1 = Ci2t−1/Cit−1, and ψi1t−1 = Git−1/Cit−1, I writecit ≈ cit−1 + (γcw1 + γcµηµ,w1)(∆ui1t + vi1t) + (γcw2 + γcµηµ,w2)(∆ui2t + vi2t) + γcλ($t + ιit)where γcw1 ≡ ψi1t−1γc1w1 + ψi2t−1γc2w1 + ψigt−1γgw1 , etc. This impliesEtcit − Et−1cit ≈ (γcw1 + γcµηµ,w1)(ui1t + vi1t) + (γcw2 + γcµηµ,w2)(ui2t + vi2t) + γcλ(ιit)Etcit+1 − Et−1cit+1 ≈ Etcit − Et−1cit − (γcw1 + γcµηµ,w1)ui1t − (γcw2 + γcµηµ,w2)ui2tEtcit+2 − Et−1cit+2 ≈ Etcit+1 − Et−1cit+1≈ Etcit − Et−1cit − (γcw1 + γcµηµ,w1)ui1t − (γcw2 + γcµηµ,w2)ui2t· · ·EtciT − Et−1ciT ≈ Etcit − Et−1cit − (γcw1 + γcµηµ,w1)ui1t − (γcw2 + γcµηµ,w2)ui2tSubstituting these into (A.18) yieldsEt lnT∑s=tCis(1 + r)s−t− Et−1 lnT∑s=tCis(1 + r)s−t≈T∑s=tθs(Etcit − Et−1cit)−T∑s=t+1θs((γcw1 + γcµηµ,w1)ui1t + (γcw2 + γcµηµ,w2)ui2t)=(Etcit − Et−1cit)T∑s=tθs − ((γcw1 + γcµηµ,w1)ui1t + (γcw2 + γcµηµ,w2)ui2t)T∑s=t+1θs=(Etcit − Et−1cit)− ((γcw1 + γcµηµ,w1)ui1t + (γcw2 + γcµηµ,w2)ui2t)(1− θt)=(γcw1 + γcµηµ,w1)vi1t + (γcw2 + γcµηµ,w2)vi2t + γcλιit+ θt((γcw1 + γcµηµ,w1)ui1t + (γcw2 + γcµηµ,w2)ui2t)where the last equality comes from the identity∑Ts=t θs = 1. Now assume that θt (consump-tion today as a share of remaining lifetime consumption) is small and can be neglected. Thenthe result of the log-linearization of the left hand side of the lifetime budget constraint (A.16)isEt lnT∑s=tCis(1 + r)s−t− Et−1 lnT∑s=tCis(1 + r)s−t= (γcw1 + γcµηµ,w1)vi1t + (γcw2 + γcµηµ,w2)vi2t + γcλιit(A.19)62A.2. Approximation of the First Order Conditions and Intertemporal Budget ConstraintSecond, the log of the RHS of (A.16) isln(Ait +T∑s=tWi1sHi1s +Wi2sHi2s(1 + r)s−t)= ln(exp(ait) +T∑s=texp(wi1s + hi1s − (s− t) ln(1 + r)) +T∑s=texp(wi2s + hi2s − (s− t) ln(1 + r)))(applying a first-order Taylor expansion around Et−1ait, Et−1wijs, and Et−1hijs)≈ ln (D0 +D1 +D2)+D0D0 +D1 +D2(ait − Et−1ait)+D1D0 +D1 +D2T∑s=tD1sD1(wi1s + hi1s − Et−1wi1s − Et−1hi1s)+D2D0 +D1 +D2T∑s=tD2sD2(wi2s + hi2s − Et−1wi2s − Et−1hi2s)whereD0 = exp(Et−1ait)D1s = exp(Et−1wi1s + Et−1hi1s − (s− t) ln(1 + r))D2s = exp(Et−1wi2s + Et−1hi2s − (s− t) ln(1 + r))D1 =T∑s=tD1sD2 =T∑s=tD2s.Then the time difference in expectation of the log of the RHS of the lifetime budgetconstraint is (noting that Etait − Et−1ait = 0 because ait is determined at t− 1)Et lnRHS − Et−1 lnRHS ≈ D1D0 +D1 +D2T∑s=tD1sD1(Etyi1s − Et−1yi1s)+D2D0 +D1 +D2T∑s=tD2sD2(Etyi2s − Et−1yi2s) (A.20)63A.2. Approximation of the First Order Conditions and Intertemporal Budget ConstraintNote that ∆yijt = ∆wijt + ∆hijt (j = 1, 2). By (A.2), (A.14), and (A.15), I have∆yi1t ≈ (1 + γh1w1 + γh1µηµ,w1)(∆ui1t + vi1t) + (γh1w2 + γh1µηµ,w2)(∆ui2t + vi2t) + γh1λ($t + ιit)∆yi2t ≈ (γh2w1 + γh2µηµ,w1)(∆ui1t + vi1t) + (1 + γh2w2 + γh2µηµ,w2)(∆ui2t + vi2t) + γh2λ($t + ιit)The items being sum in the first term of (A.20) then areEtyi1t − Et−1yi1t ≈ (1 + γh1w1 + γh1µηµ,w1)(ui1t + vi1t) + (γh1w2 + γh1µηµ,w2)(ui2t + vi2t) + γh1λιitEtyi1t+1 − Et−1yi1t+1 ≈ Etyi1t − Et−1yi1t − (1 + γh1w1 + γh1µηµ,w1)ui1t − (γh1w2 + γh1µηµ,w2)ui2tEtyi1t+2 − Et−1yi1t+2 ≈ Etyi1t+1 − Et−1yi1t+1· · ·Etyi1T − Et−1yi1T ≈ Etyi1t+1 − Et−1yi1t+1so the first summation in (A.20) is equal toT∑s=tD1sD1(Etyi1s − Et−1yi1s)≈(Etyi1t − Et−1yi1t)T∑s=tD1sD1− ((1 + γh1w1 + γh1µηµ,w1)ui1t − (γh1w2 + γh1µηµ,w2)ui2t)T∑s=t+1D1sD1=(Etyi1t − Et−1yi1t)− ((1 + γh1w1 + γh1µηµ,w1)ui1t − (γh1w2 + γh1µηµ,w2)ui2t)(1−D1tD1)=(1 + γh1w1 + γh1µηµ,w1)vi1t + (γh1w2 + γh1µηµ,w2)vi2t + γh1λιitwhere I utilize the assumption that D1tD1 (labor income today as a share of remaining lifetimelabor income) is small and can be neglected .Similarly, the second summation in (A.20) isT∑s=tD2sD2(Etyi2s − Et−1yi2s) ≈ (γh2w1 + γh2µηµ,w1)vi1t + (1 + γh2w2 + γh2µηµ,w2)vi2t + γh2λιitTherefore,Et lnRHS − Et−1 lnRHS≈ D1D0 +D1 +D2((1 + γh1w1 + γh1µηµ,w1)vi1t + (γh1w2 + γh1µηµ,w2)vi2t + γh1λιit)+D2D0 +D1 +D2((γh2w1 + γh2µηµ,w1)vi1t + (1 + γh2w2 + γh2µηµ,w2)vi2t + γh2λιit)Define piit ≡ D0D0+D1+D2 , financial wealth as a share of total (financial and human) wealth,64A.3. Moment Condtions in GMM estimationand sit ≡ D1D1+D2 (not to be confused with the time index s), the husband’s share of humanwealth. By “human wealth” I mean the remaining discounted lifetime labor income.Now I combine the results of log-linearization of the left hand side and the right handside of the lifetime budget constraint:(γcw1 + γcµηµ,w1)vi1t + (γcw2 + γcµηµ,w2)vi2t + γcλιit=(1− piit)sit ((1 + γh1w1 + γh1µηµ,w1)vi1t + (γh1w2 + γh1µηµ,w2)vi2t + γh1λιit)+ (1− piit)(1− sit) ((γh2w1 + γh2µηµ,w1)vi1t + (1 + γh2w2 + γh2µηµ,w2)vi2t + γh2λιit)which implies ιit can be written asιi,t = γιv1vi,1,t + γιv2vi,2,t (A.21)whereγιv1 ≡(1− piit)sit(1 + γh1w1 + γh1µηµ,w1) + (1− piit)(1− sit)(γh2w1 + γh2µηµ,w1)− (γcw1 + γcµηµ,w1)γcλ − (1− piit)sitγh1λ − (1− piit)(1− sit)γh2λγιv2 ≡(1− piit)sit(γh2w1 + γh2µηµ,w1) + (1− piit)(1− sit)(1 + γh2w2 + γh2µηµ,w2)− (γcw2 + γcµηµ,w2)γcλ − (1− piit)sitγh1λ − (1− piit)(1− sit)γh2λFinally, plugging (A.2), (A.15), and (A.21) into (A.14) I get the transmission system inthe main text. For example, the transmission equation for ∆ci1t is: ∆ci1t = κc1u1∆ui1t +κc1v1vi1t + κc1u2∆ui2t + κc1v2vi2t, whereκc1u1 = γc1w1 + γc1µηµ,w1κc1v1 = γc1w1 + γc1λγιv1 + γc1µηµ,w1κc1u2 = γc1w2 + γc1µηµ,w2κc1v2 = γc1w2 + γc1λγιv2 + γc1µηµ,w2 .A.3 Moment Condtions in GMM estimationFor simplicity, this section abstracts away measurement errors. The moment conditions withmeasurement errors are derived in Appendix A.4.The PSID is biennial. The difference between year t and t− 2 is actually ∆2wt: ∆2wt ≡wt − wt−2 = wt − wt−1 − (wt−1 − wt−2) = ∆wt − ∆wt−1 = ∆ut + vt − ∆ut−1 − vt−1 =ut − ut−2 + vt − vt−1 = ∆2ut + ∆vt. Keeping this in mind, the moment conditions will be65A.3. Moment Condtions in GMM estimationslightly different from the case of annual data. For example,E[(∆2wt)2] = E[(∆2ut + ∆vt)2]= E[(∆2ut)2] + E[(∆vt)2]= 2σ2u + 2σ2v .Note that for the first difference (annual data):E[(∆wt)2] = E[(∆ut + vt)2]= E[(∆ut)2] + E[(vt)2]= 2σ2u + σ2v .The following formulas are used repeatedly in deriving the moment conditions:E[(∆2ujt)2] = 2σ2ujE[(∆vjt)2] = 2σ2vjE[∆2u1t∆2u2t] = 2σu1u2E[∆v1t∆v2t] = 2σv1v2E[∆2ujt∆2ujt−2] = −σ2ujE[∆vjt∆vjt−2] = 0E[∆2u1t∆2u2t−2] = −σu1,u2 = E[∆2u2t∆2u1t−2]E[∆v1t∆v2t−2] = 0 = E[∆v2t∆v1t−2].They are easy to prove. For example, the first equation:E[(∆2ujt)2] = E[(ujt − ujt−2)(ujt − ujt−2)] = E[(ujt)2] + E[(ujt−2)2] = 2σ2ujOther equations are proved similarly.For a general variable x, where x can be c1, c2, y1, y2, h1, h2, or g, write the transmissionequation from wage shocks {u1t, u2t, v1t, v2t} to variable x as∆xt = κx,u1∆u1t + κx,u2∆u2t + κx,v1v1t + κx,v2v2tAgain, as the PSID is biennial, what I measure is actually ∆2xt, the transmission equation66A.3. Moment Condtions in GMM estimationof which is∆2xt = ∆xt −∆xt−1= κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2tFirst, I have the following moments of x itself:E[(∆2xt)2] = E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)2]= κ2x,u1E[(∆2u1t)2] + κ2x,u2E[(∆2u2t)2] + κ2x,v1E[(∆v1t)2] + κ2x,v2E[(∆v2t)2]+ 2κx,u1κx,u2E[∆2u1t∆2u2t] + 2κx,v1κx,v2E[∆v1t∆v2t]= 2κ2x,u1σ2u1 + 2κ2x,u2σ2u2 + 2κ2x,v1σ2v1 + 2κ2x,v2σ2v2+ 4κx,u1κx,u2σu1u2 + 4κx,v1κx,v2σv1v2E[(∆2xt)(∆2xt−2)] = E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)∗(κx,u1∆2u1t−2 + κx,u2∆2u2t−2 + κx,v1∆v1t−2 + κx,v2∆v2t−2)]= κ2x,u1E[∆2u1t∆2u1t−2] + κ2x,u2E[∆2u2t∆2u2t−2]+ κx,u1κx,u2E[∆2u1t∆2u2t−2 + ∆2u2t∆2u1t−2]= −κ2x,u1σ2u1 − κ2x,u2σ2u2 − 2κx,u1κx,u2σu1u2 ,and the cross moments between the variable x and the wages:E[(∆2w1t)(∆2xt)] = E[(∆2u1t + ∆v1t)(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)]= κx,u1E[(∆2u1t)2] + κx,u2E[∆2u1t∆2u2t] + κx,v1E[(∆v1t)2] + κx,v2E[∆v1t∆v2t]= 2κx,u1σ2u1 + 2κx,u2σu1u2 + 2κx,v1σ2v1 + 2κx,v2σv1v2E[(∆2w2t)(∆2xt)] = E[(∆2u2t + v2t)(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)]= κx,u1E[∆2u1t∆2u2t] + κx,u2E[(∆2u2t)2] + κx,v1E[∆v1t∆v2t] + κx,v2E[(∆v2t)2]= 2κx,u1σu1u2 + 2κx,u2σ2u2 + 2κx,v1σv1v2 + 2κx,v2σ2v267A.3. Moment Condtions in GMM estimationE[(∆2xt)(∆2w1t−2)] = E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)(∆2u1t−2 + ∆v1t−2)]= κx,u1E[∆2u1t∆2u1t−2] + κx,u2E[∆2u2t∆2u1t−2]= −κx,u1σ2u1 − κx,u2σu1u2E[(∆2wt)(∆2x1t−2)] = −κx,u1σ2u1 − κx,u2σu1u2E[(∆2xt)(∆2w2t−2)] = −κx,u1σu1u2 − κx,u2σ2u2E[(∆2wt)(∆2x2t−2)] = −κx,u1σu1u2 − κx,u2σ2u2 .Let zt be another endogenous variable different from xt. The covariance between x andz can also be used for identifying the transmission parameters of x and z:E[(∆2xt)(∆2zt)]= E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)∗(κz,u1∆2u1t + κz,u2∆2u2t + κz,v1∆v1t + κz,v2∆v2t)]= κx,u1κz,u1E[(∆2u1t)2] + κx,u2κz,u2E[(∆2u2t)2] + (κx,u1κz,u2 + κx,u2κz,u1)E[∆2u1t∆2u2t]+ κx,v1κz,v1E[(∆v1t)2] + κx,v2κz,v2E[(∆v2t)2] + (κx,v1κz,v2 + κx,v2κz,v1)E[∆v1t∆v2t]= 2κx,u1κz,u1σ2u1 + 2κx,u2κz,u2σ2u2 + 2(κx,u1κz,u2 + κx,u2κz,u1)σu1u2+ 2κx,v1κz,v1σ2v1 + 2κx,v2κz,v2σ2v2 + 2(κx,v1κz,v2 + κx,v2κz,v1)σv1v2E[(∆2xt)(∆2zt−2)]= E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t)∗(κz,u1∆2u1t−2 + κz,u2∆2u2t−2 + κz,v1∆v1t−2 + κz,v2∆v2t−2)]= κx,u1κz,u1E[∆2u1t∆2u1t−2] + κx,u2κz,u2E[∆2u2t∆2u2t−2]+ κx,u1κz,u2E[∆2u1t∆2u2t−2] + κx,u2κz,u1E[∆2u2t∆2u1t−2]+ κx,v1κz,v1E[∆v1t∆v1t−2] + κx,v2κz,v2E[∆v2t∆v2t−2]+ κx,v1κz,v2E[∆v1t∆v2t−2] + κx,v2κz,v1E[∆v1t−2∆v2t]= −κx,u1κz,u1σ2u1 − κx,u2κz,u2σ2u2 − (κx,u1κz,u2 + κx,u2κz,u1)σu1u2E[(∆2zt)(∆2xt−2)]= −κx,u1κz,u1σ2u1 − κx,u2κz,u2σ2u2 − (κx,u1κz,u2 + κx,u2κz,u1)σu1u2To summarize:The own moments of xt provide 2 moment conditions: E[(∆2xt)2] and E[(∆2xt)(∆2xt−2)].The cross moments of xt and wjt provide 6 moment conditions: E[(∆2xt)(∆2w1t)],E[(∆2xt)(∆2w2t)], E[(∆2xt)(∆2w1t−2)], E[(∆2wt)(∆2x1t−2)], E[(∆2xt)(∆2w2t−2)],and E[(∆2wt)(∆2x2t−2)].These together provide 8 moment conditions, which already over-identify the 4 transmis-68A.4. Moment Conditions with Measurement Errorssion parameters from wage shocks to xt.If the tranmission parameters of xt and zt are estimated jointly, in addition to those8 × 2 moment conditions, we can also use 3 cross moments of xt and zt: E[(∆2xt)(∆2zt)],E[(∆2xt)(∆2zt−2)], and E[(∆2zt)(∆2xt−2)]. In total, we have 19 moment conditions to iden-tify 8 parameters.In general, suppose we have n endogenous variables and we estimate the transmissionparameters of them jointly. The moment conditions are:1. 2n own moments;2. 6n cross moments with wjt;3. 3× Cn2 cross moments between any two endogenous variables.In total this gives 8n+3×Cn2 moment conditions for identifying 4n transmission parameters,as illustrated in the following table.Number of endogeous variables Number of parameters Number of momentsn 4n 8n+ 3× Cn21 4 82 8 193 12 334 16 505 20 70For my application, there are 5 endogenous variables {c1, c2, g, y1, y2}, and thus I use 70moments in the GMM estimation.A.4 Moment Conditions with Measurement ErrorsWith measurement errors taken into account, the moment conditions presented in AppendixA.3 are no longer precise. Denote ξx be the measurement error of variable x. Then forj = 1, 2,∆2wjt = ∆2ujt + ∆vjt + ∆2ξwjtand∆2xt = κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t + ∆2ξxt69A.4. Moment Conditions with Measurement ErrorsSome moment conditions need to be adjusted for measurement errors. For example, thewage moments: for j = 1, 2,E[(∆2wjt)2] = E[(∆2ujt + ∆vjt + ∆2ξwjt)2]= E[(∆2ujt)2] + E[(∆vjt)2] + E[(∆2ξwjt)2]= 2σ2uj + 2σ2vj + 2σ2ξwjE[(∆2wjt)(∆2wjt−2)] = E[(∆2ujt + ∆vjt + ∆2ξwjt)(∆2ujt−2 + ∆vjt−2 + ∆2ξwjt−2)]= −σ2uj − σ2ξwj .Own moments of x:E[(∆2xt)2] = E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t + ∆2ξxt)2]= κ2x,u1E[(∆2u1t)2] + κ2x,u2E[(∆2u2t)2] + κ2x,v1E[(∆v1t)2] + κ2x,v2E[(∆v2t)2]+ 2κx,u1κx,u2E[∆2u1t∆2u2t] + 2κx,v1κx,v2E[∆v1t∆v2t] + E[(∆2ξxt)2]= 2κ2x,u1σ2u1 + 2κ2x,u2σ2u2 + 2κ2x,v1σ2v1 + 2κ2x,v2σ2v2+ 4κx,u1κx,u2σu1u2 + 4κx,v1κx,v2σv1v2 + 2σ2ξxE[(∆2xt)(∆2xt−2)] = E[(κx,u1∆2u1t + κx,u2∆2u2t + κx,v1∆v1t + κx,v2∆v2t + ∆2ξxt)∗(κx,u1∆2u1t−2 + κx,u2∆2u2t−2 + κx,v1∆v1t−2 + κx,v2∆v2t−2 + ∆2ξxt−2)]= κ2x,u1E[∆2u1t∆2u1t−2] + κ2x,u2E[∆2u2t∆2u2t−2]+ κx,u1κx,u2E[∆2u1t∆2u2t−2 + ∆2u2t∆2u1t−2]+ E[∆2ξxt∆2ξxt−2]= −κ2x,u1σ2u1 − κ2x,u2σ2u2 − 2κx,u1κx,u2σu1u2 − σ2ξxI assume that the measurement errors of consumption are uncorrelated with the measure-ment errors of earnings or those of wages. This assumption implies that the cross momentsbetween consumption and earnings, and the cross moments between consumption and wages,have the same expressions as in the case of no measurement errors. I also assume that themeasurement errors of the husband’s variable (wage, earnings, ...) and the measurement er-rors of the wife’s variable are uncorrelated. Thus, the cross moments between the husband’svariable and the wife’s variable are also unchanged.The cross moments between wages and earnings change. Due to the fact that w = y − hand thus ξw = ξy − ξh, we haveV ar(ξw) = V ar(ξy) + V ar(ξh)− 2Cov(ξy, ξh),V ar(ξh) = V ar(ξy) + V ar(ξw)− 2Cov(ξy, ξw).70A.4. Moment Conditions with Measurement ErrorsThe cross moments between wages and earnings become:E[(∆2wjt)(∆2yjt)] =2κyj ,u1σ2u1 + 2κyj ,u2σu1u2 + 2κyj ,v1σ2v1 + 2κyj ,v2σv1v2 + E[∆2ξwjt∆2ξyjt]=2κyj ,u1σ2u1 + 2κyj ,u2σu1u2 + 2κyj ,v1σ2v1 + 2κyj ,v2σv1v2 + 2Cov(ξwj , ξyj )=2κyj ,u1σ2u1 + 2κyj ,u2σu1u2 + 2κyj ,v1σ2v1 + 2κyj ,v2σv1v2+ V ar(ξwj ) + V ar(ξyj )− V ar(ξhj )=2κyj ,u1σ2u1 + 2κyj ,u2σu1u2 + 2κyj ,v1σ2v1 + 2κyj ,v2σv1v2+ 2V ar(ξyj )− 2Cov(ξyj , ξhj )E[(∆2wjt)(∆2yjt−2)] = −κyj ,ujσ2uj − κyj ,u−jσuj ,u−j + E[∆2ξwjt∆2ξyjt−2]= −κyj ,ujσ2uj − κyj ,u−jσuj ,u−j − Cov(ξwj , ξyj )= −κyj ,ujσ2uj − κyj ,u−jσuj ,u−j − V ar(ξyj ) + Cov(ξyj , ξhj )where the variances of ξw, ξy, ξh and their covariances are obtained by the the method de-scribed in Section BAppendix for Chapter 3B.1 Normalized Problem with Unit-Root Labor IncomeIn the case in which the (log) income process has a unit root and the felicity function hasthe CRRA form u(c) = c1−γ/(1− γ), it is well known from Carroll (2004) that it is possibleto normalize problem (3.17) by (permanent) labor income yt, thereby reducing the effectivestate space to zt.To see this, replace for ct = zt−at in (3.17) and consider the problem in the second-to-lastperiodVT−1(zT−1, yT−1) = maxaT−1u(zT−1 − aT−1) + βET−1u(zT ) (B.1)If one defines the state variables zˆt = zt/yt and aˆt = at/yt, equation (B.1) can be rewrittenasVT−1(zT−1, yT−1) = maxaˆT−1u(yT−1(zˆT−1 − aˆT−1)) + βET−1u(yT zˆT )= y1−γT−1{maxaˆT−1u(zˆT−1 − aˆT−1) + βET−11−γT u(zˆT )}(B.2)Note that by definitionzˆt = (1 + r)at−1yt−1t+ 1 = (1 + r)aˆt−1t+ 1, (B.3)which implies that the curly bracket in (B.2) is equal to VT−1(zˆT−1) where the latter satisfiesthe Bellman equationVT−1(zˆT−1) = maxaˆT−1u(zˆT−1 − aˆT−1) + βET−11−γT VT (zˆT ) (B.4)with VT (zˆT ) = u(zˆT ).Equations (B.2) and (B.4) imply that VT−1(zT−1, yT−1) = y1−γT−1V (zˆT−1). The same logicimplies that this holds also for any t < T − 1.72B.1. Normalized Problem with Unit-Root Labor IncomeTherefore the Bellman equation for the problem in normalized form satisfiesVt(zˆt) = maxaˆtu(zˆt − aˆt) + βEt1−γt+1 Vt+1(zˆt+1), (B.5)for all t. It follows from (B.3) and the envelope condition that the associated Euler equationisu′(cˆt) = βRE[−ρt+1u′(cˆt+1)](B.6)The advantage of the normalized problem (B.4) is that one can solve for the saving functionaˆt(zˆt) which is independent of the income realization yt and use at(zt, yt) = aˆt(zˆt)yt to recoverthe policy function for at.Under the assumption that t is i.i.d. and log-normally distributed the expectation inequation (B.4) can be computed using Gaussian Hermite quadrature.73


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