UBC Theses and Dissertations
Essays in comparative dynamics Davidson, Russell
Problems in the theory of economic dynamics are tackled both by theoretical arguments and by use of specific examples. The work is divided into three essays. The first treats optimal control theory from an economic point of view, giving an exposition of the mathematical theory in terms of economic concepts. The idea of the marginal worth of time is introduced and found to be useful in a variety of problems. An interpretation is given of the phase planes of optimal control problems as demand-and-supply diagrams. The second essay makes use of the techniques developed in the first to solve the problem of when and how a firm faced with adverse economic circumstances will choose to go out of business if its operations depend on a stock of some fixed asset that depreciates over time. A straightforward catalogue is presented of different possible outcomes. The third essay deals with a model of urban housing. It contains two main sections. In the first, an equilibrium state is described in which demand by tenants for housing is met by supply from landlords who act as profit maximisers over the whole period of time that their property exists. The rent paid for any particular dwelling is assumed to depend on its state of upkeep, which in turn depends on how much is spent by a landlord on maintenance. The equilibrium is found by a procedure analogous to that regularly used in general equilibrium theory, namely by finding a fixed point of a mapping in a (here infinite-dimensional) vector space. In the second section of the essay, it is assumed that some externality arises which adversely affects urban life and which provokes people to move out to suburbs. The consequences of this are studied and two different kinds of dynamical evolution can be distinguished. One, in which house construction in the suburbs is slow enough to make it necessary for new construction to continue in the city, tends not to be disastrous for the city; the other, in which all urban construction stops when the externality arises, usually leads to complete decay of the city. Throughout the thesis there is an emphasis on the differences in approach between static or quasistatic problems and dynamic ones.
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