Birkbeck MSc/Phd Economics Advanced Macro Spring 2006 Consumption and Asset Prices Lecture 1: Optimal Consumption under Uncertainty: (Mainly) Partial Equilibrium Models
1 Introductory 1.1 Overall Course Structure 1. Optimal consumption under uncertainty: partial equilibrium models 2. Consumption and asset prices in an endowment economy 3. Optimal consumption and asset prices in the stochastic growth model
1.2 Exercises There are no classes scheduled for this lecture course, but I believe it is essential that you do some worked exercises. These will be included as we proceed; you’ll be expected to have worked through them by the start of the next lecture, when we’ll go over solutions. 1.3 Reading for Lecture 1 • Deaton’s Understanding Consumption is excellent for anyone who wants to understand the consumption problem in any depth. He does not how- ever deal in any depth with the maths of dynamic programming, dynamic consistency, etc. For this see references given by Yunus Aksoy.
• If you have never come across the introductory material I shall cover in early on in this lecture, you could also look at Chapter 7 of Romer, or, for cover- age of the two-period intertemporal problem, any micro text (Williamson’s macro text also covers well. 1.4 Notation • Capital letters, eg, C, Y will always refer to underlying levels before any transformations • Everything is measured in real terms, including all return series, so, 1 + R t = ( I + S t ) / (1 + Π t ) where S is the nominal return, and Π is the in fl ation rate.
• Stock variables, eg K t , A t are values at the beginning of period t (contrast with most published data - often a source of confusion) • Link with convention that, under uncertainty, any t − dated variable is in the information set I t used to make decisions. • Lower case letters will usually refer to logs, ie c = log C (where log x = ln x ) • Except in case of interest rates, growth rates, etc where r = log(1+ R ) ≈ R ; g = log(1 + G ) ≈ G by the log approximation....
• .... and later on in the course lower case letters may sometimes be used to represent log deviations from steady state values, eg I may de fi ne c t = ³ ´ ≈ C t − b C t C t / b log C t b C t
2 The Consumption Decision with Perfect Fore- sight and a Finite Horizon 2.1 The Objective The representative consumer at time t is assumed to maximise U = U ( C t , C t +1 , .....C T ) (1) For now we shall ignore the posssibility that the utility function might contain other arguments, in particular leisure.
It is very common to restrict this very general formulation to be “additive separable” in per period consumption, so that the maximand becomes the discounted sum of per period utility (or “felicity”) functions: T − t X 1 U = (1 + Θ ) i u ( C t + i ) (2) i =0 • Θ is normally referred to as the “subjective discount rate”, or the “rate of pure time preference”. Rationales? • (nb lower case convention will apply as for interest rates so θ = log(1+ Θ )) • Highly restrictive, but restrictions are useful in modelling terms and (I shall argue) insightful
• Deaton Ch1 also discusses aggregation across goods. 2.2 The Constraint We can represent the constraint in two equivalent ways. First we can look at the evolution of assets: A t +1 = ( A t + Y t − C t ) (1 + R t +1 ) , (3) where Y t is labour income; A t is net assets, and then impose a condition on the consumer’s terminal wealth: A T + Y T − C T ≥ 0
i.e. terminal net assets cannot be negative. If no bequest motive is included in the model, and the constraint is assumed to bind, this means that C T = Y T + A T . Just for now assume, for simplicity, that R is constant in all periods. Then we can use the asset evolution identity and solve backwards recursively from this terminal condition, and thus derive the constraint in present value terms as: T − t T − t X X C t + i Y t + i = A t + (4) (1 + R ) i (1 + R ) i i =0 i =0 = A t + W t It is this form that we shall use initially here; in more complex problems, how- ever, (most notably when uncertainty enters the picture) we shall not be able to do so.
2.3 First Order Conditions If the consumer maximises (2), by choice of C t , C t +1 , ...C T subject to (4), the fi rst order conditions for any two adjacent periods are: µ 1 + R ¶ i 0 ( C t + i ) = λ C t + i : (5) u 1 + Θ µ 1 + R ¶ i +1 0 ( C t + i +1 ) = λ C t + i +1 : u (6) 1 + Θ
where λ is the Lagrange multiplier on the constraint (4). ∗ Hence the optimal consumption path between any two periods will satisfy the “Euler Equation”: µ 1 + R ¶ 0 ( C t + i ) = 0 ( C t + i +1 ) u u (7) 1 + Θ Note what is in, and is not in this expression. Bear in mind the inverse relationship between marginal utility and consumption. Implications of R = Θ ; R > Θ ; R < Θ ? The assumption that R = Θ is particularly popular because it gives a straight- forward result whatever the form of the felicity function. ∗ Alternative is to have T − t Lagrange multipliers, one for each period’s constraint but this alternative also disappears under uncertainty so won’t be pursued here (cf Yunus’s lectures)
2.4 A Popular Special Case: Quadratic Utility If per period utility is quadratic, eg, of form ³ ´ 2 u bliss − β C t − C bliss t 2 marginal utility is linear. Exercise 1: Why should we worry about assuming quadratic utility? Linearity of marginal utlility is a very useful property once uncertainty is allowed into the problem, since (as future lectures will show) quadratic utility implies “ certainty equivalence”. If in addition we assume that R = Θ , we get a special case which Deaton refers to as the “Permanent Income Consumption” representation, whereby optimal consumption is constant, and equal to the perfect foresight solution, even under uncertainty. More on this later
2.5 Another Popular Special Case: Power Utility If per period utility function is of the “constant relative risk aversion” (or “power utility”) form: C 1 − γ u ( C ) = (8) 1 − γ = ln( C ) for γ = 1 u 0 ( C ) = C − γ (9) the Euler equation can be manipulated to yield: µ 1 + R ¶ 1 C t + i +1 γ = (10) C t + i 1 + Θ
Hence this speci fi cation implies that the elasticity of intertemporal substitution is equal to the reciprocal of the coe ffi cient of relative risk aversion. Some economists have objected to this linkage, and have produced formulations which separate the two; my personal view is that the link is insightful. Exercise 2: The fi rst line of (8) is unde fi ned for γ = 1 . Show however, using L’Hopital’s Rule, that C 1 − γ lim 1 − γ = ln C γ → 1 Exercise 3: Show that with power utility the coe ffi cient of relative risk aversion c.u 00 /u 0 is constant and equal to γ
2.6 The “Closed Form Solution” when R = Θ . The Euler Equation only tells us about the time pro fi le of consumption, not its absolute level. A simple closed form solution can however be derived if R = Θ ,.(A possible, but quite restrictive general equilibrium rationale for this assumption is given below). If this is the case optimal consumption is constant in all period, ie C t = C ∀ t and we can use the budget constraint to derive the level of C , thus: T − t X 1 C (1 + R ) i = A t + W t (11) i =0
which can be manipulated to yield µ ¶ R C = κ ( A t + W t ) (12) 1 + R Ã ! 1 where κ = > 1 (13) 1 − (1 + R ) − ( T − t +1) Exercise 4: Show how! Note that as T goes to in fi nity, κ goes to unity, and the consumer simply consumes the annuity value of their total wealth ( fi nancial and human capital) - hence the link with Friedman’s “Permanent Income” model. How much does it matter (in modelling terms) that people do not live for ever?
Do not forget the dependence of this solution on the assumption that R = Θ . For more general cases, the solution is nastier (and can only be arrived at by making explicit assumptions on the nature of the felicity function). It is though relatively easy to understand conceptually. 2.7 Comparitive Statics for An Atomistic Consumer Why do we need to assume the consumer is atomistic? 2.7.1 Changes in Labour Income In the special case above, with R = Θ , it is easy to see that if current labour income changes with income in all future periods constant:
µ ¶ R ∂C/∂Y t = κ ≈ R (14) 1 + R What however if the change in Y t persists for all periods? ie, is “permanent”: think of a shock dY t = dY t + i ... = dY T . In this case: µ ¶ T − t X R dC/dY t = κ ∂W t /∂Y t + i (15) 1 + R i =0 µ ¶ T − t X R 1 = κ (1 + R ) i 1 + R i =0 = 1
2.7.2 Changes in the Interest Rate The Euler Equation tells us that the growth of consumption is positively related to the interest rate.But this does not tell us what the closed form response will be except under certain more restrictive assumptions. Exercise 5 Show geometrically in a 2 period model that for a consumer with initial assets the impact of a permanent change in the interest rate on the level of consumption is ambiguous. Exercise 6 In a multi-period model, if the interest rate changes only temporarily, describe a restriction on the nature of the path of interest rates that will ensure that consumption unambiguously falls when the interest rate rises.
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