Chapter8 - Chapter 8 Infinite-Period Representative...

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© Sanjay K. Chugh 105 Spring 2008 Chapter 8 Infinite-Period Representative Consumer and Asset Pricing Macroeconomic models, at least those based on micro-foundations, used in applied research and in practice assume that there is an infinite number of periods, rather than just two as we have been for the most part assuming . A two-period model is usually sufficient for the purpose of illustrating intuition about how c o n s um e r s m a k e intertemporal choices, but in order to achieve the higher quantitative precision needed for many research and policy questions, moving to an infinite-period model is desirable. Here we will sketch the problem faced by an infinitely-lived representative consumer, describing preferences, budget constraints, and the general characterization of the solution. In sketching the basic model, we will see that in its natural formulation, it easily lends itself to a study of asset-pricing. Indeed, this model lies at the intersection of macroeconomic theory and finance theory and forms the basis of consumption-based asset-pricing theories. We will touch on some of these macro-finance linkages, but we really will only be able to whet our curiosity about more advanced finance theory. For the most part, we will index time by arbitrary indexes 1, , 1 tt t ± ² , etc., rather than “naming” periods as “period 1,” “period 2,” and so on. That is, we will simply speak of “period t ,” “period t +1,” “period t +2,” etc. Before we begin, we again point out that “the consumer” we are modeling is a stand-in for the economy as a whole. In that sense, we of course do not literally mean that a particular individual considers his intertemporal planning horizon to be infinite when making choices. But to the extent that “the economy” outlives any given individual, an infinitely-lived representative agent is, as usual, a simple representation. Preferences The utility function that is relevant in the infinite-period model in principle is a lifetime utility function just as in our simple two-period model. As before, suppose that time begins in period one but now never ends. The lifetime utility function can thus be written as 12345 (, ,, ,, . . . ) vc c c c c . This function describes total utility as a function of consumption in every period 1, 2, 3, … and is the analog of the utility function 12 (, ) uc c in our two-period model. The function v above is quite intractable mathematically because it takes an infinite number of arguments. Largely for this reason, in practice an instantaneous utility function that
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© Sanjay K. Chugh 106 Spring 2008 describes how utility in a given period depends on consumption in a given period is typically used. The easiest formulation to consider is the additively separable function, 234 12345 1 2 3 4 5 (, ,, ,, . . .
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Chapter8 - Chapter 8 Infinite-Period Representative...

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