0 for T - > c o . In equilibrium, demand for the final labour service satisfies necessary first order condition ^Mj,tKt,AlLj,)0.^(\\-Tl)^-, (30) where PsY ' V ri J (78) The sensitivity of the cyclical component of investment to changes in the cyclical component of the relative shadow price of capital is decreasing in investment adjustment cost parameter x \u2022 This approximate linear investment demand function is subject to investment technology shocks. The cyclical component of the relative shadow price of capital depends on the expected future cyclical component of the relative shadow price of capital, the contemporaneous cyclical component of the real interest rate, the expected future cyclical component of real marginal cost, and the expected future cyclical component of the marginal product of capital according to approximate linear investment Euler equation: l n 4 | = \/?(l-*-\\ 2(2) 3.334 6 .285\" 1.816 7.823* 11.126\"\" 7.377\"' 1.281 8 , 8 2 1 \" G(4) 15.198\" ' 7.363 8.704' 10.755* 16.416\"*' 12 .753\" 1.441 2 0 . 4 3 7 \" ' g 2 ( 2 ) 29.357*\" 12.149*\" 12.939\"\" 11.448* 69 .938\" \" 5 5 . 5 3 2 \" ' 2 4 . 1 5 4 \" ' 2 8 . 9 5 4 \" ' G 2 ( 4 ) 46.188* \" 18.945*\" 23 .797\" \" 23.569' 122.141\"\" 108 .846\" ' 40 .219\"* 58 .337\"* Skewness 0.209 0 .509 \" -0 .305 -0 .283 - 0 . 4 5 2 \" 0.063 0 . 9 9 1 \" ' 1.170\"* Kurtosis 4 . 0 2 9 \" 6 .060\"\" 4 .606\"* 6.040* 2.474 3.218 4 .927* \" 10.565'* ' JB 6 . 8 4 1 \" 5 7 . 6 4 1 ' \" 16 .360\" ' 52.977' 6 .059\" 0.352 4 2 . 3 5 4 \" ' 347 .477*\" C(0T) = -6605.462 Note: Rejection o f the nul l hypothesis at the 1%, 5 % and 10% levels is indicated by * * * , * * and * , respectively. Fi gure 2.5. Predicted cyclical components of observed endogenous variables 101 Note: Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. Fi gure 2.6. Filtered cyclical components of observed endogenous variables 102 Note: Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. 103 Figure 2.7. Smoothed cyclical components of observed endogenous variables L P G D P L P C O N L R G D P 1975 1930 1965 1990 1995 2000 2006 1975 1960 1985 1990 1995 20O0 2005 1975 I960 1985 1990 1995 2000 2005 1975 1960 1985 1990 1995 2030 2005 0 \u2022y 1 1975 1980 1985 1990 19B5 2000 2005 1975 1930 19B5 1990 1995 2000 2005 18751980 1985 1990 N I N T ( A P R ) ft \\ A '\u20224'%' if fl 'Ii 1975 1980 1985 1900 1995 2000 2005 1975 1980 1985 1990 1995 2000 2006 1975 1990 1985 1990 1985 2000 2005 1975 19B0 1985 1990 1995 2003 2005 1975' ' 1980 ' 1935' ' 1990 1975 19B0 1985 1990 1995 2000 2005 1975 I960 1985 1990 1995 2000 N I N T F ( A P R ) 1975 1980 1985 1990 1995 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 Note; Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. 104 Figure 2.8. Predicted trend components of observed endogenous variables L P G D P L P C O N L R G D P L R C O N Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. Fi gure 2.9. Filtered trend components of observed endogenous variables 105 Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. 106 Figure 2,10. Smoothed trend components of observed endogenous variables 1975 1980 1985 1975 1980 1985 1990 1995 2000 2005 2000 2005 NINT(APR) 1990 1995 2000 2005 N I N T F ( A P R ) 1975 1980 1965 1990 1965 1990 1996 2000 2005 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. 107 Figure 2 . 1 1 . Mean squared prediction error differentials for levels LPGDP LPCON LRGDP L R C 0 1 N B 3 2 20 i 0 1 0 10. 0 % -4. - -10 \u20222 \u2022fl- -2 -3--20--4 1 2 3 4 5 6 7 l 1 2 3 . s e ; 1 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 8 LRINV LREXP LRIUP LNWAGE iCO 400 XO 0-400 200 0 400 200 20 10-200 4(1) 800 \u2022200--400-600--200 -400-10 -20 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 8 LEMP RUNENP NINT LNEXCH 4(1 3. 0 6-20- 0.4 100-0- 0 0.0. 0 20-2. -0 4. 100H 40- -3. -0 8-2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 LPGDPF 20-10. LRGDPF LRCONF LRINVF B-4. 2-500. 4. fl--10-\u202220-2--4. \u20226-0--500. 2 3 4 5 6 7 I 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 LNWAGEF LEMPF RUNEMPF NINTF IC i-60. 40-20-0 3. 2-0-2. 10--20--W--60--2--3-2 3 4 5 6 7 8 2 3 4 5 6 7 ; 2 3 4 5 6 7 ( 2 3 4 5 6 7 8 Note: Mean squared prediction error differentials are defined as the mean squared prediction error for the unobserved components model less that for the AR1MA model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors. 108 Figure 2.12. Mean squared prediction error differentials for ordinary differences D L P G D P D L P C O N D L R G D P . \u00ab .3 2 05 X 1 \u20222 - 05 \u2022.\u00ab 4 - 5 - 2 -.10 -.3 1 2 3 4 5 6 7 1 1 2 3 4 5 8 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 D L R I N V D L R E X P DLRIfvF D L N W A G E 12. 10 8-4 o-s. 0-% 2 \u20224 * -5 -8- -2 -12- -10- \u202210 \\ '* 2 3 4 5 6 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 D L E M P DRUNErVP D N I N T D L N E X ; H 1.2. 10- 03- 10-0 8. 02-05- 5-01-0 0- \u2014 \u2014 ^\u2014~~ rm 00-0.4. -.05. \u2022 \" -.01- \\ _ _____ -0 8- -.02. -5. \u20221 2- - ia -10-2 3 4 5 5 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 D L P G D P F D L R G D P F D L R C O N F DLRINVF 1-0.8- J. 20. 2. i-ii. 0.4-o.a 10-i. \u20220.4. 2. 10--OB. 20-2 3 4 5 6 7 6 2 3 4 5 6 7 l 3 \u2022 \u2022 \u2022 \u2022 ' i 2 3 4 5 6 7 8 D L N W A G E F D L E M P F D R U N E N P F DNINTF ; 2- 08-06-04. '\u2022 04. 02-u- 0- 00 nn 2- i. 0*. -.02. .1. -?- OB- - 04-08. 1 2 3 \u00bb 5 6 7 8 1 2 3 4 5 6 7 2 3 4 5 6 7 8 2 3 4 5 6 7 8 Note: Mean squared prediction error differentials are defined as the mean squared prediction error for the unobserved components model less that for the A R I M A model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors. Figure 2.13. Mean squared prediction error differentials for seasonal differences 109 S D L P C O N S D L R G D P e 4 2 ' 0.8 0.4 0.0 9 3 2 -2 -4 \u2022ft -0.4 \u20220.8. -5 -2 -3 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 t 1 2 3 4 5 6 ? 1 1 2 3 4 5 6 7 8 20U 100 0 S D L R I N V S D L R E X P S D L R I N P S D L N W A G E 100 0 150 100 50 ft. 2. 101) -100 -50--100 -2 -4--200. -150 \u00ab-1 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 B S D L E M P S D R U N E I v P S D N I N T S D L N E X C H 15. 10-5-1.0-0 5- .2. 50-0 - 0.0. -5--10. \u202215. . . . \u2014 \"\u2014-\u2014.... \u2022 \u2014 . 0. \u2014\u2014 \u2022 -0.5--1.0-\u20222. -50-2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 S D L P G D P F S D L R G D P F S O L R C O N F S D L R I N V F 10-4 300-200-U-5-0-< 100-fc -5- >- -100. 4--10- -3--4. -2O0--300-2 3 4 5 6 7 8 2 3 4 5 6 7 6 2 3 4 5 6 7 8 2 3 4 5 6 7 8 S D L N W A G E F S D L E M P F S D R U N E M P F S D N I N T F 2. J-20-0 -15. 1.0-0.5-0.0. 5. z. \" \" \" \" \" \" \u2014 \u2014 10- I I 5-4. 6. -20-1 D-l E -.5-1 2 3 4 5 B 7 8 1 2 3 4 5 6 7 2 3 4 5 6 7 8 2 3 4 5 6 7 8 Note: Mean squared prediction error differentials are defined as the mean squared prediction error for the unobserved components model less that for the A R I M A model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors. Figure 2.14. Dynamic forecasts of levels of observed endogenous variables 110 Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Figure 2.16. Dynamic forecasts of seasonal differences of observed endogenous variables 112 Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts. Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. 113 References Ashley, R. (2003), Statistically significant forecasting improvements: How much out-of-sample data is likely necessary?, International Journal of Forecasting, 19, 229-239. Christiano, L . , M . Eichenbaum and C. Evans (1998), Monetary policy shocks: What have we learned and to what end?, NBER Working Paper, 6400. Christiano, L . , M . Eichenbaum and C. Evans (2005), Nominal rigidities and the dynamic effects of a shock to monetary policy, Journal of Political Economy, 113, 1-45. 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(1989), Recursive solution methods for dynamic linear rational expectations models, Journal of Econometrics, 41, 65-89. Wold, H . (1938), A Study in the Analysis of Stationary Time Series, Almqvist and Wiksell . Woodford, M . (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. 115 C H A P T E R 3 Measuring the Stance of Monetary Policy in a Small Open Economy: A Dynamic Stochastic General Equilibrium Approach 3.1. Introduction Estimated dynamic stochastic general equilibrium or D S G E models have recently emerged as quantitative monetary policy analysis and inflation targeting tools. As extensions of real business cycle models, D S G E models explicitly specify the objectives and constraints faced by optimizing households and firms, which interact in an uncertain environment to determine equilibrium prices and quantities. The existence of short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets permits a cyclical stabilization role for monetary policy, which is generally implemented through control of the short term nominal interest rate according to a monetary policy rule. The persistence of the effects of monetary policy shocks on output and inflation is often enhanced with other features such as habit persistence in consumption, adjustment costs in investment, and variable capital utilization. Early examples of closed economy D S G E models incorporating some of these features include those of Yun (1996), Goodfriend and King (1997), Rotemberg and Woodford (1995, 1997), and McCallum and Nelson (1999), while recent examples of closed economy D S G E models incorporating all of these features include those of Christiano, Eichenbaum and Evans (2005), Altig, Christiano, Eichenbaum and Linde (2005), Smets and Wouters (2003, 2005), and Vitek (2006c). Open economy D S G E models extend their closed economy counterparts to allow for international trade and financial linkages, implying that the monetary transmission mechanism features both interest rate and exchange rate channels. Building on the seminal work of Obstfeld and Rogoff (1995, 1996), these open economy D S G E models determine trade and current account balances through both intratemporal and intertemporal optimization, while the nominal A version o f this chapter has been submitted for publication. V i tek, F., Measuring the stance of monetary pol icy in a closed economy: A dynamic stochastic general equi l ibr ium approach, International Journal of Central Banking. 116 exchange rate is determined by an uncovered interest parity condition. Existing open economy DSGE models differ primarily with respect to the degree of exchange rate pass through. Models in which exchange rate pass through is complete include those of Benigno and Benigno (2002), McCallum and Nelson (2000), Clarida, Gali and Gertler (2001, 2002), and Gertler, Gilchrist and Natalucci (2001), while models in which exchange rate pass through is incomplete include those of Adolfson (2001), Betts and Devereux (2000), Kollman (2001), Corsetti and Pesenti (2002), Monacelli (2005), and Vitek (2006d). Recent research has emphasized the implications of developments in the housing market for the conduct of monetary policy. Existing DSGE models incorporating a housing market include those of Aoki , Proudman and Vlieghe (2004) and Iacoviello (2005), both of which focus on the implications of financial market frictions for the monetary transmission mechanism. In addition to abstracting from open economy elements of the monetary transmission mechanism, these papers do not consider the implications of developments in the housing market for the measurement of the stance of monetary policy. Existing D S G E models featuring long run balanced growth driven by trend inflation, productivity growth, and population growth generally predict the existence of common deterministic or stochastic trends. Estimated D S G E models incorporating common deterministic trends include those of Ireland (1997) and Smets and Wouters (2005), while estimated D S G E models incorporating common stochastic trends include those of Altig, Christiano, Eichenbaum and Linde (2005) and Del Negro, Schorfheide, Smets and Wouters (2005). However, as discussed in Clements and Hendry (1999) and Maddala and K i m (1998), intermittent structural breaks render such common deterministic or stochastic trends empirically inadequate representations of low frequency variation in observed macroeconomic variables. For this reason, it is common to remove trend components from observed macroeconomic variables with deterministic polynomial functions or linear filters, such as the difference filter or the low pass filter described in Hodrick and Prescott (1997), prior to the conduct of estimation, inference and forecasting. As an alternative, Vitek (2006c, 2006d) proposes jointly modeling cyclical and trend components as unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation of a D S G E model. This merging of modeling paradigms drawn from the theoretical and empirical macroeconomics literatures confers a number of important benefits. First, the joint estimation of parameters and trend components ensures their mutual consistency, as estimates of parameters appropriately reflect estimates of trend components, and vice versa. As shown by Nelson and Kang (1981) and Harvey and Jaeger (1993), decomposing integrated observed endogenous variables into cyclical and trend components with atheoretic deterministic polynomial functions or low pass filters may induce spurious cyclical dynamics, invalidating subsequent estimation, 117 inference and forecasting. Second, basing estimation on the levels as opposed to differences of observed endogenous variables may be expected to yield efficiency gains. A central result of the voluminous cointegration literature surveyed by Maddala and K i m (1998) is that, i f there exist cointegrating relationships, then differencing all integrated observed endogenous variables prior to the conduct of estimation, inference and forecasting results in the loss of information. Third, the proposed unobserved components framework ensures stochastic nonsingularity of the resulting approximate linear state space representation of the D S G E model, as associated with each observed endogenous variable is at least one exogenous stochastic process. As discussed in Ruge-Murcia (2003), stochastic nonsingularity requires that the number of observed endogenous variables used to construct the loglikelihood function associated with the approximate linear state space representation of a D S G E model not exceed the number of exogenous stochastic processes, with efficiency losses incurred i f this constraint binds. Fourth, the proposed unobserved components framework facilitates the direct generation of forecasts of the levels of endogenous variables as opposed to their cyclical components together with confidence intervals, while ensuring that these forecasts satisfy the stability restrictions associated with balanced growth. These stability restrictions are necessary but not sufficient for full cointegration, as along a balanced growth path, great ratios and trend growth rates are time independent but state dependent, robustifying forecasts to intermittent structural breaks that occur within sample. The primary contribution of this paper is the development of a procedure to estimate the levels of the flexible price and wage equilibrium components of endogenous variables while imposing relatively weak, and hence relatively credible, identifying restrictions on their trend components. Based on an extension and refinement of the unobserved components framework proposed by Vitek (2006c, 2006d), this estimation procedure confers a number of benefits of particular importance to the conduct of monetary policy. First, as discussed in Woodford (2003), the levels of the flexible price and wage equilibrium components of various observed and unobserved endogenous variables are important inputs into the optimal conduct of monetary policy. In particular, the level of the natural rate of interest, defined as that short term real interest rate consistent with price and wage flexibility, provides a measure of the neutral stance of monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure. The proposed unobserved components framework facilitates estimation of the levels as opposed to cyclical components of the flexible price and wage equilibrium components of endogenous variables, while ensuring that they satisfy the stability restrictions associated with balanced growth. Second, given an interest rate smoothing objective derived from a concern with financial market stability, variation in the natural rate of interest caused by shocks having permanent effects may call for larger monetary policy responses than variation caused by shocks having temporary effects. The proposed unobserved components 118 framework yields a decomposition of the levels of the flexible price and wage equilibrium components of endogenous variables into cyclical and trend components, together with confidence intervals which account for uncertainty associated with the detrending procedure. Third, as discussed in Clements and Hendry (1999) and Maddala and K i m (1998), accommodating the existence of intermittent structural breaks requires flexible trend component specifications. However, the joint derivation of empirically adequate cyclical and trend component specifications from microeconomic foundations is a formidable task. The proposed unobserved components framework facilitates estimation of the levels of the flexible price and wage equilibrium components of endogenous variables while allowing for the possibility that the determinants of their trend components are unknown but persistent. The secondary contribution of this paper is the estimation of the levels of the flexible price and wage equilibrium components of various observed and unobserved endogenous variables while imposing relatively weak identifying restrictions on their trend components, with an emphasis on the levels of the natural rate of interest and natural exchange rate. Definitions of indicators of inflationary pressure such as the natural rate of interest and natural exchange rate vary, and numerous alternative procedures for estimating the natural rate of interest have been proposed, several of which are discussed in a survey paper by Giammarioli and Valla (2004). Within the framework of a calibrated DSGE model of a closed economy, Neiss and Nelson (2003) find that estimates of the deviation of the real interest rate from the natural rate of interest exhibit economically significant high frequency variation. Within the framework of an estimated D S G E model of a closed economy, Smets and Wouters (2003) find that estimates of the deviation of the real interest rate from the natural rate of interest exhibit economically significant high frequency variation and are relatively imprecise, as evidenced by relatively wide confidence intervals. In addition to abstracting from open economy elements of the monetary transmission mechanism, these papers abstract from the trend component of the natural rate of interest, as they employ estimation procedures which involve the preliminary removal of trend components from observed macroeconomic variables with atheoretic deterministic polynomial functions. This paper develops and estimates a D S G E model of a small open economy for purposes of monetary policy analysis and inflation targeting. This estimated D S G E model provides a quantitative description of the monetary transmission mechanism in a small open economy, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts. The model features short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets. The resultant inertia in inflation and persistence in output is enhanced with other features such as habit persistence in consumption and labour supply, adjustment costs in housing and capital investment, and variable capital utilization. 119 Incomplete exchange rate pass through is generated by short run nominal price rigidities in the import market, with monopolistically competitive importers setting the domestic currency prices of differentiated intermediate import goods subject to randomly arriving reoptimization opportunities. Cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. Parameters and unobserved components are jointly estimated with a novel Bayesian procedure, conditional on prior information concerning the values of parameters and trend components. The organization of this paper is as follows. The next section develops a DSGE model of a small open economy. Estimation, inference and forecasting within the framework of a linear state space representation of an approximate unobserved components representation of this DSGE model are the subjects of section three. Finally, section four offers conclusions and recommendations for further research. 3.2. Model Development Consider two open economies which are asymmetric in size, but are otherwise identical. The domestic economy is of negligible size relative to the foreign economy. 3.2.1. The Utility Maximization Problem of the Representative Household There exists a continuum of households indexed by z'e[0,l]. Households supply differentiated intermediate labour services, but are otherwise identical. 3.2.1.1. Consumption, Saving and Investment Behaviour The representative infinitely lived household has preferences defined over consumption C(. , housing Hjs, and labour supply Li s represented by intertemporal utility function U\u201e=ElfjPs-u{C,s,His,L.s), s=l 1 (1) 120 where subjective discount factor \/? satisfies 0 < \/3 < 1. The intratemporal utility function is additively separable and represents external habit formation preferences in consumption, housing, and labour supply, (C,, -a'C^r* | y H (His -aHH^r\" y L (Ljs \u2014aLLsl)M\/n l-i\/o- l-i\/o- v \\+\\in (2) where 0 < \u00ab C < 1 , 0 < \u00ab W < 1 and 0 < a t < l . This intratemporal utility function is strictly increasing with respect to consumption if and only if vf > 0 , and given this parameter restriction is strictly increasing with respect to housing if and only if v\" > 0 , and is strictly decreasing with respect to labour supply if and only if vL > 0 . Given these parameter restrictions, this intratemporal utility function is strictly concave if a > 0 and r\/ > 0 . The representative household enters period 5 in possession of previously purchased domestic currency denominated bonds Bff which yield interest at risk free rate is_x, and foreign currency denominated bonds Bf\/ which yield interest at risk free rate if. It also holds a diversified portfolio of shares {xfjsyj=0 in domestic intermediate good firms which pay dividends {I7fs}f0, and a diversified portfolio of shares {xfk s}[=0 in domestic intermediate good importers which pay dividends {I7kMs}\\=0. The representative household supplies differentiated intermediate labour service Ls ( , earning labour income at nominal wage Wis. Households pool their labour income, and the government levies a tax on pooled labour income at rate TS . These sources of private wealth are summed in household dynamic budget constraint: )KA,,^n- j \u00ab , ^ = ( i H - 1 ) C ^ a + e , ) ^ \/ ;=o *=o ( 3 ) + J (K + K>KA + J \u00ab + KMsK,A + (1 -1 , ) ) WuLlsdl - PfCLs - Pflf. j=0 k=0 1=0 According to this dynamic budget constraint, at the end of period s, the representative household purchases domestic bonds Bff+], and foreign bonds Bf\/ at price \u00a3 s . It also purchases a diversified portfolio of shares [x]'.S+]YJ=0 in intermediate good firms at prices {VfsYj=o> a n Q a diversified portfolio of shares {xfk v + i } [ = 0 in intermediate good importers at prices {VkMs }[=0. Finally, the representative household purchases final consumption good C, s. at price Pf, and final housing investment good lfs at price Pf\" . The representative household enters period 5 in possession of previously accumulated housing stock Hj s, which subsequently evolves according to accumulation function % r ( l - ^ ) ^ ^ \" ( C C ) . (4) 121 where depreciation rate parameter 8H satisfies 0 < 5\" < 1. Effective housing investment function HH'(\/\/l>^\"-i) incorporates convex adjustment costs, n ( ^ . V i ) - 1 ' . 1-X f jH _ j H 1 i,s 1 i,s-\\ I\" V i\" (5) where %H > 0 and v's > 0. In deterministic steady state equilibrium, these adjustment costs equal zero, and effective investment equals actual investment. In period t, the representative household chooses state contingent sequences for consumption {Cis}^,, investment in housing {l\",}\u2122=,, the stock of housing {HiiS+l}\u2122=l, domestic bond holdings {Bff+X}\u2122=l, foreign bond holdings , share holdings in intermediate good firms {{xjjs+l}[j=0}%,, and share holdings in intermediate good importers {{xfk s+x}]k=0}\u2122=, to maximize intertemporal utility function (1) subject to dynamic budget constraint (3), housing accumulation function (4), and terminal nonnegativity ^constraints H. T+l > 0, , > 0 , Bfj+X > 0 , xjjj+l > 0 and xfkT+] > 0 for T \u2014\u00bb oo . In equilibrium, selected necessary first order conditions associated with this utility maximization problem may be stated as uc(C\u201eH\u201eL.l) = PfAl, (6) (7) + Q.-S )QM A,=\/J(1 + \/,)E,A, + 1 , SlA, = B(l + if)E!\u00a3l+lAl+l, (8) (9) (10) (11) (12) where A ( > denotes the Lagrange multiplier associated with the period s household dynamic budget constraint, and ALsQ\" denotes the Lagrange multiplier associated with the period s housing accumulation function. In equilibrium, necessary complementary slackness conditions associated with the terminal nonnegativity constraints may be stated as: 122 A. lim B,;T+X = 0, lim PLTA\u00b1L\u00a3 B P \/ =0 T-Ko 1 Hm PT*>+T yr r = f J r \" i X yj,t+TXj,,+T+l U> r->co 1 (13) (14) (15) (16) (17) Provided that the intertemporal utility function is bounded and strictly concave, together with all necessary first order conditions, these transversality conditions are sufficient for the unique utility maximizing state contingent intertemporal household allocation. Combination of necessary first order conditions (6) and (9) yields intertemporal optimality condition M c ( C , , \/ 7 , , L , , ) = ^ E , ( l + \/ , ) ^ W c ( C ( + l , \/ Y , + 1 , L , , + 1 ) , (18) which ensures that at a utility maximum, the representative household cannot benefit from feasible intertemporal consumption reallocations. Combination of necessary first order conditions (6) and (7) yields intertemporal optimality condition nHT-\/H(TH f \" U C ^Uc(^i+\\'^t+\\'^i,t+]) Pf \/ - , \/ 7 i_\/H \/ jH rH>. Dl\" U, \u00ab l (A + ,~ I T T s ~^C~^H2 ( V l ' 7 \/ > = P > ' uc(cnH\u201eh<) p: (19) which equates the expected present discounted value of an additional unit of investment in housing to its price. Combination of necessary first order conditions (6) and (8) yields intertemporal optimality condition Buc{CM,Hl+],Lil+x) uc(C\u201eH,,Lu) tf. pC \" H ( Q | ^ I + I ' A . H - I ) _ rH ^nH (20) 123 which equates the shadow price of housing to the expected present discounted value of the sum of the future marginal cost of housing, and the future shadow price of housing net of depreciation. Finally, combination of necessary first order conditions (6), (9) and (10) yields intratemporal optimality condition uc(C\u201eH\u201eL,,) J * ' uc{C\u201eH\u201eLu) ftS, ' ' which equates the expected present discounted values of the gross real returns on domestic and foreign bonds. 3.2.1.2. Labour Supply and Wage Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate labour services Lit supplied by households in a monopolistically competitive labour market to produce final labour service Lt according to constant elasticity of substitution production function L. = \\(k,)9' di 9,-1 (22) where df > 1. The representative final labour service firm maximizes profits derived from production of the final labour service nf=WtL- \\WitL.,di, (23) with respect to inputs of intermediate labour services, subject to production function (22). The necessary first order conditions associated with this profit maximization problem yield intermediate labour service demand functions: L... = A- (24) Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final labour service firm earns zero profit, implying aggregate wage index: 124 W. \\(wJ~Ui (25) As the wage elasticity of demand for intermediate labour services Of increases, they become closer substitutes, and individual households have less market power. In an extension of the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction 1 - mL of households adjust their wage optimally. The remaining fraction coL of households adjust their wage to account for past consumption price inflation according to partial indexation rule r P c \\ rt-\\ pC pC W,. (26) where 0 < yL < 1. Under this specification, although households adjust their wage every period, they infrequently adjust their wage optimally, and the interval between optimal wage adjustments is a random variable. If the representative household can adjust its wage optimally in period t, then it does so to maximize intertemporal utility function (1) subject to dynamic budget constraint (3), housing accumulation function (4), intermediate labour service demand function (24), and the assumed form of nominal wage rigidity. Since all households that adjust their wage optimally in period t solve an identical utility maximization problem, in equilibrium they all choose a common wage Wf given by necessary first order condition: w. P'-'uc{Cs,Hs,Lis) uc(C\u201eH\u201eLu) uL(Cs,Hs,Lis) '( pC ' r r - l 11-1 w 1\"'' 4 pC BC V rs-\\ . w ( Pc \\ rl-\\ pc 'pc\\ pc -rL 1 w, (wf -oi (27) This necessary first order condition equates the expected present discounted value of the consumption benefit generated by an additional unit of labour supply to the expected present discounted value of its leisure cost. Aggregate wage index (25) equals an average of the wage set by the fraction 1 - coL of households that adjust their wage optimally in period t, and the average of the wages set by the remaining fraction a>L of households that adjust their wage according to partial indexation rule (26): 125 i\\-coLw;te' +coL ( Pc ^ rt-\\ y- ( Pc \\ ri-i pC V ri-2 ) pC w. (28) Since those households able to adjust their wage optimally in period \/ are selected randomly from among all households, the average wage set by the remaining households equals the value of the aggregate wage index that prevailed during period t -1, rescaled to account for past consumption price inflation. If all households were able to adjust their wage optimally every period, then coL = 0 and necessary first order condition (27) would reduce to: ( 1 f ) f , _ uL{C,,H,,L,) ef-\\uc{CnHt,L^ (29) In flexible price and wage equilibrium, each household sets its after tax real wage equal to a time varying markup over the marginal rate of substitution between leisure and consumption, and labour supply is inefficiently low. 3.2.2. The Value Maximization Problem of the Representative Firm There exists a continuum of intermediate good firms indexed by j e [0,1]. Intermediate good firms supply differentiated intermediate output goods, but are otherwise identical. Entry into and exit from the monopolistically competitive intermediate output good sector is prohibited. 3.2.2.1. Employment and Investment Behaviour The representative intermediate good firm sells shares {xfjl+l})=0 to domestic households at price Vjt. Recursive forward substitution for Vjl+s with s > 0 in necessary first order condition (11) applying the law of iterated expectations reveals that the post-dividend stock market value of the representative intermediate good firm equals the expected present discounted value of future dividend payments: .9=\/+! At 126 Acting in the interests of its shareholders, the representative intermediate good firm maximizes its pre-dividend stock market value, equal to the expected present discounted value of current and future dividend payments: (31) The derivation of result (30) imposes transversality condition (16), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to net profits IJYjs, defined as after tax earnings less expenditures on investment in capital: \/ \/ ; . v = ( l - r j ( - ^ ^ . J - P ' \/ \u00ab . (32) Earnings are defined as revenues derived from sales of differentiated intermediate output good Yjs at price Pjs less expenditures on final labour service Z,. s . The government levies a tax on earnings at rate ts, and negative dividend payments are a theoretical possibility. The representative intermediate good firm utilizes capital Ks at rate ujs and rents final labour service Lj s given labour augmenting technology coefficient As to produce differentiated intermediate output good Yj s according to constant elasticity of substitution production function Huj,sKs,AsL.s)-9-1 9-\\ ( +{\\-(pY{AsLjs) 9-1 (33) where 0 < *

0 and As > 0. This constant elasticity of substitution production function exhibits constant returns to scale, and nests the production function proposed by Cobb and Douglas (1928) under constant returns to scale for 9 = 1.1 In utilizing capital to produce output, the representative intermediate good firm incurs a cost G(uJs,Ks) denominated in terms of output: Y.s=HuhsKs,AsLjs)-Q{uhs,Ks). (34) Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the rate of capital utilization at an increasing rate, ' Invoking L 'Hospi ta l 's rule yields l im I n F ( u j J K \u201e A i , L , . J = (oln(\u00bby__KJ + (1 -

\\n(p-(\\-

0 and K> 0. In deterministic steady state equilibrium, the rate of capital utilization is normalized to one, and the cost of utilizing capital equals zero. Capital is endogenous but not firm-specific, and the representative intermediate good firm enters period 5 with access to previously accumulated capital stock K ^ , which subsequently evolves according to accumulation function Ks+l=(l-SK)Ks+HK(lf,d (36) where depreciation rate parameter SK satisfies 0 < SK < 1. Following Christiano, Eichenbaum and Evans (2005), effective capital investment function HK(I*,I*_}) incorporates convex adjustment costs, 2 IK 1s-\\ IK (37) where >0 a n d v's >0- I n deterministic steady state equilibrium,'these adjustment costs equal zero, and effective investment equals actual investment. In period \/ , the representative intermediate good firm chooses state contingent sequences for employment {LJS}^,, capital utilization }\"=\/, investment in capital {I*}\u2122=l, and the capital stock {KS+I}\u2122=L to maximize pre-dividend stock market value (31) subject to net production function (34), capital accumulation function (36), and terminal nonnegativity constraint K T + L > 0 for T \u2014> oo . In equilibrium, demand for the final labour service satisfies necessary first order condition FAL(U.,KI,A1L.,)0.i=(\\-T,) (38) where Pj 1 and ^ > 0. As specified, the deviation of the nominal interest rate from its flexible price equilibrium value is a linear increasing function of the contemporaneous deviation of consumption price inflation from its target value nf - nf, and the contemporaneous proportional deviation of output from its flexible price equilibrium value. Persistent departures from this monetary policy rule are captured by serially correlated monetary policy shock v,. 208 4.2.5. Market Clearing Conditions A rational expectations equilibrium in this D S G E model of a small open economy consists of state contingent intertemporal allocations for domestic and foreign households and firms which solve their constrained optimization problems given prices and policy, together with state contingent intertemporal allocations for domestic and foreign governments which satisfy their policy rules, with supporting prices such that all markets clear. Clearing of the final output good market requires that production of the final output good equal the cumulative demands of domestic and foreign households: Y,=ChJ + CffJ. (38) The assumption that the domestic economy is of negligible size relative to the foreign economy is represented by parameter restriction