Absolute Risk Aversion, Stochastic Dominance, and Introduction to General Equilibrium

Transcription

1 Absolute Risk Aversion, Stochastic Dominance, and Introduction to General Equilibrium Econ 2100 Fall 2015 Lecture 15, October 26 Outline 1 Absolute Risk Aversion 2 Stochastic Dominance 3 General Equilibrium Notation

2 From Last Class: Utility of Money A cumulative distribution function (cdf) F : R [0, 1] satisfies: x y implies F (x) F (y); lim y x F (y) = F (x) lim x F (x) = 0 and lim x F (x) = 1. { 0 if z < x δ x yields x with certainty: δ x (z) = 1 if z x. µ F denotes the mean (expected value) of F, i.e. µ F = x df (x). von Neumann Morgenstern preference representation: there exists a utility function U on distributions, and a continuous index v : R R over wealth such that F G U(F ) = v df v dg = U(G) If F is differentiable, the expectation is computed using the density f = F : v df = v(x)f (x) dx. The preference relation is risk averse if, for all F, δ µf F. Given a strictly increasing and continuous vnm index v over wealth, the certainty equivalent of F (c(f, v)) is defined by v(c(f, v)) = v ( ) df. the risk premium of F, denoted r(f, v) is defined by r(f, v) = µ F c(f, v). is risk averse v is concave r(f, v) 0

3 Another Application: Asset Demand An asset is a divisible claim to a financial return in the future. Asset Demand An agent has initial wealth w; she can invest in a safe asset that returns $1 per dollar invested, or in a risky asset that returns $z per dollar invested. The general version has N assets each yielding a return z n per unit invested. The risky return has cdf F (z), and assume z df > 1. Let α and β be the amounts invested in the risky and safe asset respectively. Then, one can think of (α, β) as a portfolio allocation that pays αz + β. The agent solves max v(αz + β) df s. t. α, β 0 and α + β = w The first oder conditions for this optimal portfolio problem is (z 1)v (α(z 1) + w) df = 0 If the decision maker is risk averse, this expression is decreasing (in α) because of the concavity of v. One can use this fact to verify that if DM1 is more risk averse than DM2 then her optimal α 1 is smaller than the corresponding α 2 : the more risk averse consumer invests less in the risky asset.

4 How to Measure Risk Aversion Since concavity of v reflects risk aversion, v is a natural candidate measure of risk aversion. Unfortunately, v is not appropriate since it is not robust to strictly increasing linear transformations. Definition Suppose is a preference relation represented by the twice differentiable vnm index v : R R. The Arrow Pratt measure of absolute risk aversion λ : R R is defined by λ(x) = v (x) v (x). The second derivative is normalized to measure risk aversion properly. Notice that by integrating λ(x) twice one could recover the utility function. How about the constants of integration?

5 Absolute Risk Aversion Proposition Suppose 1 and 2 are preference relations represented by the twice differentiable vnm indices v 1 and v 2. Then 1 is more risk averse than 2 λ 1 (x) λ 2 (x) for all x R This confirms that the Arrow-Pratt coeffi cient is the correct measure of increasing absolute risk aversion. We can add this to the characterizations of more risk averse than that you proved in the homework.

6 Proof. 1 is more risk averse than 2 λ 1 (x) λ 2 (x) for all x R We know v 1 = φ(v 2 ) for some strictly increasing φ (by the homework). Differentiating v 1 (x) = φ (v 2 (x))v 2 (x) and v 1 (x) = φ (v 2 (x))v 2 (x) + φ (v 2 (x))v 2 (x) Dividing by v 1 > 0 we have v 1 (x) v 1 (x) = φ (v 2 (x))v 2 (x) v 1 (x) + φ (v 2 (x))v 2 (x) v 1 (x) Subsitute the first equation v 1 (x) v 2 v 1 = (x) (x) v 2 (x) + φ (v 2 (x))v 2 (x) v 1 (x) or v 1 (x) 2 v 1 = (x) v (x) v 2 (x) φ (v 2 (x))v 2 (x) v 1 (x) using the definition of Arrow-Pratt: since φ is concave and v is increasing. λ 1 (x) = λ 2 (x) + something

7 First Order Stochastic Dominance (FOSD) What kind of relationship must exist between lotteries F and G to ensure that anyone, regardless of her attitude to risk, will prefer F to G so long as she likes more wealth than less? In general, if a consumer s utility of wealth is increasing, but its functional form unknown, we do not have enough information to know her rankings among all distributions...but we know how she ranks some pairs; namely, those comparable with respect to the following transitive, but incomplete, binary relation. Definition F first-order stochastically dominates G, denoted F FOSD G, if v df v dg, for every nondecreasing function v : R R. If F FOSD G, then anyone who prefers more money to less prefers F to G. This follows because F G U(F ) = v df v dg = U(G ). Proposition F FOSD G if and only if F (x) G(x) for all x R.

8 First Order Stochastic Dominance (FOSD) F FOSD G, if v df v dg, for every nondecreasing function v : R R. FOSD is characterized by only looking at distribution functions. Proposition F FOSD G if and only if F (x) G(x) for all x R. This follows from integration by parts Thus b a b v (x) f (x) dx = v (x) F (x) b a v (x) F (x) dx = v (b) 1 v (0) 0 a b b = v (b) v (x) F (x) dx a a v (x) F (x) dx b b b v df v dg = v (x) [G (x) F (x)] dx a a a and you can fill in the blanks for a formal proof.

9 Second Order Stochastic Dominance (SOSD) If one also knows that the decision maker is risk averse (her utility index for wealth is concave), we know how she ranks more pairs. Definition F second-order stochastically dominates G, denoted F SOSD G, if v df v dg, for every nondecreasing concave function v : R R. If F SOSD G, then anyone who prefers more money to less and is risk averse prefers F to G. By construction, the set of distributions ranked by FOSD is a subset of those ranked by SOSD F FOSD G implies F SOSD G. Proposition F SOSD G if and only if x F (t) dt x G(t) dt for all x R. SOSD is also characterized by looking at distribution functions. What if F SOSD G an DM chooses G? Is this a reasonable choice?

10 Preferences and Lotteries Over Money So far, we have looked at expected utility preferences over sums of money. Dollar bills cannot be eaten, so where do these preferences come from? There are L commodities and ranks lotteries on X = R L +. The expected utility axioms hold: the consumer is an expected utility maximzer, and u : X R is her vnm utility function. u ( ) is also a utility function over realized consumption (consumer theory). Fix prices p R L ++; let w [0, ) be income, x (p, w) be the Walrasian demand corresponding to u ( ), and v (p, w) the indirect utility function: v (p, w) = u (x) for x x (p, w). Suppose all uncertainty is resolved so that the consumers learns how much money she has before she goes to the markets to buy x. How does the consumer rank lotteries over income? A lottery π (w) is a probability distributions on [0, ). The expected utility of π is y support(π) π (w) v (p, w); and π ρ π (w) v (p, w) ρ (w) v (p, w) y support(π) y support(π) The indirect utility function v ( ) is a vnm utility function over money. How do properties of transfer to v ( )? Think about the answer.

11 General Equilibirum Theory General Equilibrium Theory of market interactions between anonymous agents who take prices as given (everyone is small relative to the market). Nothing is exogenous, hence general (as opposed to partial) equilibirum. 1 How do competitive markets allocate resources? 1 main assumption: agents are price-taking optimizers ; 2 main outcomes of interest: competitive equilibrium (everyone makes optimal choices and prices are such that demand and supply are equal), and Pareto effi cient allocations (cannot be redistributed to improve consumers welfare). 2 Important questions: 1 when is a competitive equilibirum Pareto effi cient? First Welfare Theorem; 2 when is a Pareto effi cient outcome an equilibirum? Second Welfare Theorem; 3 when does a competitive equilibirum exhist? Arrow-Debreu-McKenzie Theorem; 4 when is a competitive equilibrium unique? 5 is a competitive equilibrium robust to small perturbations of the parameters? 6 how do we get to a competitive equilibrium and what happens away from it? 7 what happens if we add time or uncertainty? 3 Applications: finance, macro, labor, etc.

12 The Economy An economy consists of I individuals and J firms who consume and produce L perfectly divisible commodities. Each firm j = 1,..., J is described by a production set Y j R L. Each consumer i = 1,..., I is described by four objects: a consumption set X i R L ; preferences i ; an individual endowment given by a vector of commodities ω i X i ; and a non-negative share θ ij of the profits of each firm j, with θ ij [0, 1] for all j. Definition An economy (with private ownership) is a tuple { } {X i, i, ω i, (θ i1,..., θ ij )} I i=1, {Y j } J j=1 The resources available to the economy are given by the aggregate endowment: I ω = If we do not need to track who owns what, we can } define an economy as {{X i, i } Ii=1, {Y j } Jj=1, ω i=1 where ω R L is the aggregate endowment. ω i

13 Exchange Economies and Edgeworth Boxes Exchange Economy In an exchange economy there is no production; all consumers can do is trade with each other. This is described by assuming that there is one firm with a single technology: free disposal. Formally, J = 1, and Y j = R L +. This is a simple environment in which many results are easier to prove, and that most of the time has the same intuition as an economy with production. Edgeworth Box An Edgeworth Box is an exchange economy with two consumers and two goods. Formally L = 2, I = 2, X 1 = X 2 = R 2 +, J = 1, and Y j = R 2 +. Draw an Edgeworth box.

14 Definition Allocations and Feasibility An allocation specifies a consumption vector x i X i for each consumer i = 1,..., I, and a production vector y j Y j for each firm j = 1,..., J; an allocation is L(I +J) (x, y) = (x 1,..., x I, y 1,..., y J ) R This lists what everyone consumes and what every firm produces. Definition An allocation (x, y) is feasible if i x i ω + j y j Consumption and production are compatible with endowment. Definition The set of feasible allocations is denoted by {(x, y) X 1... X I Y 1... Y J : i x i ω + j L(I +J) y j } R

15 Feasible Allocations: Examples Example In an exchange economy the set of feasible allocations is {x X 1... X I : x i ω} R LI i Example In an Edgeworth Box, any point in the box represents a feasible allocation.

16 Proposition Feasible Allocations: Conditions If the following conditions are satisfied, the set of feasible allocations is closed and bounded. a 1 X i is closed and bounded below for each i = 1,..., I. b 2 Y j is closed for each j = 1,..., J. 3 Y = j Y j is convex and satisfies 1 0 L Y (inaction), 2 Y R L + {0 L } (no free-lunch), and 3 if y Y and y 0 L then y / Y (irreversibility). Moreover, if R L + Y j (free-disposal) and there are x i X i for each i such that i x i ω, then the set of feasible allocations is also non empty. a Closed and bounded means there exists an r > 0 such that for each l = 1,..., L, i = 1,..., I, and j = 1,..., J, we have x l,i < r and y l,j < r for each (x, y ) A. b Bounded below means there exists an r > 0 such that x l,i > r for each l = 1,..., L. When these conditions are not satisfied, the latter in particular, we may have nothing to talk about. So we take them for granted.

17 Problem Set 9 (Incomplete) Due Monday 2 November 1 Suppose there are two goods and the consumer has a vnm utility index over consumption bundles of v(x) = f (x 1 + x 2 ) where f : R R is a strictly increasing function. Let p = (1, 3) and p = (3, 1). Let q =.5p +.5p = (2, 2). In Regime 1, there is a 1/2 probability that the realized price will be p and a 1/2 probability that the realized price will be p, so her ex-ante utility is.5 max x B p,w f (x 1 + x 2 ) +.5 max x B p,w f (x 1 + x 2 ). In Regime 2, the price is q with certainty, so her ex-ante utility is max x B q,w f (x 1 + x 2 ). Prove that, for any strictly increasing f, the consumer is ex-ante better off in Regime 1. 2 Kreps Kreps Suppose a consumer s utility for CDFs is U(F ) = µ F αvar(f ) where Var(F ) = (x µ F ) 2 df is the variance of F. 1 Prove that if α > α then the preference corresponding to α is more risk averse than the one corresponding to α. 2 Prove that U violates expected utility.