Course Handouts ECON 4161/8001 MICROECONOMIC ANALYSIS. Jan Werner. University of Minnesota

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1 Course Handouts ECON 4161/8001 MICROECONOMIC ANALYSIS Jan Werner University of Minnesota FALL SEMESTER

2 PART I: Producer Theory 1. Production Set Production set is a subset Y of commodity space IR L, where L is the number of commodities. Vectors in Y represent production plans that are technologically feasible. Negative coordinates of production plan y = (y 1,...,y L ) Y are understood as input quantities; positive coordinates of y are output quantities. Production plan y Y is efficient if there is no alternative production plan y Y, y y, such that y y. Examples: (1) activity analysis: given two activities a 1,a 2 IR L, the production set is Y = {λ 1 a 1 + λ 2 a 2 : λ 1 0,λ 2 0}. (2) see Varian, Ch. 1. Some properties of production sets: (i) Y closed; 0 Y. (ii) no free production: Y IR + L = {0}. (iii) constant returns to scale: if y Y, then λy Y for every λ 0, (iv) Y convex, (v) free disposal: Y IR + L Y. 1

3 2. Production Function f : IR n + IR This is for a technology with single output and n inputs. We write f(x) = z, where x = (x 1,...,x n ) is a vector of inputs (here with positive sign!). Examples: Cobb-Douglas, Leontief, etc. (see Varian, Ch 1). Some properties of production functions: (i) f(0) = 0; f continuous (or differentiable) function. (ii) returns to scale: constant: f(λx) = λf(x), for every λ 0 and x 0. decreasing: f(λx) < λf(x), for every λ > 1 and x 0 such that f(x) 0. increasing: f(λx) > λf(x), for every λ > 1 and x 0 such that f(x) 0. (iii) f concave (or quasi-concave). Production set corresponding to production function f: Y f = {(x,z) IR n+1 : x 0, 0 z f( x)}. If f is concave function, then the set Y f is convex. 2

4 3. Profit Maximization Profit maximization at price vector p IR L is maximize py over y Y. (1) The solutions (there could be many) are the supply of the firm at p, denoted by s (p). We can write s (p) = {y Y : py py, y Y }. (2) The (maximum) profit is π (p) = sup y Y py. (3) π is a function of p while s is, in general, a correspondence. If supply s is a differentiable function, then the L L-matrix Ds (p) is called the supply substitution matrix. Unless set Y is compact, there may exist positive price vectors for which maximum profit is infinite and supply does not exist. The set of price vectors for which profit function takes finite values is the domain of π. It is a convex set in IR L. The domain of s is a subset of the domain of π. 3

5 5. Support Function and Envelope Theorem Profit function π is the support function of production set Y. Basic properties of support functions are discussed in MWG, Section 3.F. More extensive discussion can be found in Convex Analysis by Rockafellar (1970), ch 13. For a closed set K IR n, the support function µ K is defined by for every p IR n. µ K (p) = sup px. (8) x K Support function µ K is a convex function. A duality property holds for a closed and convex set K IR n : K = {x IR n : px µ K (p), p} (9) Theorem 5.1 (Envelope Property): Suppose that K is nonempty and closed. Support function µ K is differentiable at p if and only if there is a unique maximizer x in (8) at p. Moreover, in this case Dµ K (p) = x. (10) 4

6 6. Supply and Profit Fundamental properties of the profit function of a profit-maximizing firm are: Theorem 6.1: Suppose that Y is closed. Then the following properties hold: (i) π is homogeneous of deg. 1; (ii) π is a convex function; (iii) If π is differentiable at p (this holds iff s is single-valued at p), then Dπ (p) = s (p). (11) Proof: (i - iii) are properties of the support function, Section 5. A convex function must be continuous except possibly for points at the boundary of its domain. So π is continuous except for the boundary of the domain. The properties of the supply function are: Theorem 6.2: Suppose that Y is closed. Then the following properties hold: (i) s is homogeneous of deg. 0; (ii) s is -monotone, that is, [s (p) s (p )][p p ] 0, p,p, (12) (iii) If s is differentiable at p, then Ds (p) is positive semi-definite and symmetric. 5

7 Proof: (ii) follows from (ii) and (iii) of Theorem 6.1 if π is differentiable. (iii) will be proved in class. Ds (p) is the substitution matrix. It follows from (iii) that This is a comparative statics property. s i p i 0, i. (13) Some extra properties of supply and profit of profit-maximizing firm: Proposition 6.3: (i) if Y exhibits constant returns to scale, then π (p) = 0 wherever it is well-defined. (ii) if Y is convex, then s (p) is a convex set. Proof: (i) and (ii) left as exercises. 6

8 6. Cost Minimization The problem of minimizing cost for a producer with production function f : IR n + IR + is minimize wx (14) subject to f(x) z and x 0, where w = (w 1,...,w n ) is a vector of input prices. Solution is x (w,z) (conditional) factor demand correspondence, or function whenever single-valued. Also C (w,z) wx (w,z) is the cost function. Cost minimization could be analyzed using methods of Section 5 since C (,z) is (the negative of) the support function of the input requirement set V (z) = {x IR n + : f(x) z}. (15) We shall, however, use the Kuhn-Tucker method to analyze the problem.. 7

9 The Lagrangian for (14) is L(λ,x) = wx + λ[z f(x)], (16) where λ IR + is the Lagrange multiplier. Assume that f is differentiable. Also z > 0, and w >> 0. First-order conditions: FOCs for a solution x and a multiplier λ obtain minimizing L over x 0 and maximizing over λ 0. They are: w i λ f i (x ) 0, and if x i > 0, then = 0, (17) z f(x ) 0, and if λ > 0, then = 0. (18) For an interior solution x >> 0, eq. (17) is w i = λ f i (x ) and consequently w i = f i(x ) w j f j (x ). (19) That is, market rate of substitution = technical rate of substitution. Also λ > 0 if x >> 0 and hence f(x ) = z. These FOCs are necessary for x to solve the cost minimization problem. More precisely, if x 0 is a solution to (14), then there is λ 0 such that x and λ satisfy (17) and (18). 8

10 If production function f is quasi-concave, then FOCs are necessary and sufficient. In particular, if f is concave, then FOCs are necessary and sufficient. Topic omitted: Second-order conditions for interior solutions to the costminimziation problem (see Varian, section 4.2). 7. Comparative statics of cost minimization. The most standard method of comparative statics is to derive partial derivatives such as x i w j or x i z, and determine their sign (if possible). This can be done by differentiating the first-order conditions. Of course, this can be done only if x (w,z) is a differentiable function of its arguments. [Implicit Function Theorem]. 9

11 Example with two inputs: Write FOCs for interior factor demand (x 1(w,z),x 2(w,z)) and the multiplier λ (w,z) as λ (w,z)f 1 (x 1(w,z),x 2(w,z)) = w 1 (20) λ (w,z)f 2 (x 1(w,z),x 2(w,z)) = w 2 (21) f(x 1(w,z),x 2(w,z)) = z (22) Differentiate equations (20-22) with respect to w 1 (or w 2, or z), and solve for the partial derivatives x 1 w 1, x 2 w 1, λ w 1. The results include (see Varian, section 4.4): x 1 w 1 < 0, x 2 w 2 < 0, (23) where the use has been made of SOCs. However, we cannot say much about the sign of x i z. The differentiate-foc method works for any number n of inputs; although with considerably more complicated algebra. 10

12 8. Properties of Cost Function and Conditional Factor Demand We start with properties of the cost function C. They are as follows: Theorem 8.1: Suppose that f is continuous. Then C has the following properties: (i) C is homogeneous of degree 1 in factor prices w, (ii) C is a concave function of w, (ii) C is a continuous function. C is nondecreasing in output level z, (iv) If C is differentiable at (w,z), then C w i = x i, C z = λ. (24) Proof: in class. Part (iv) is a consequence of the envelope theorem for constrained optimization. Equations (35) are the Shephard s Lemma. It follows from (35) that the multiplier λ is the marginal cost. Further, it follows that the cost function, if differentiable, is nondecreasing in each factor price w i, ceteris paribus. 11

13 Properties of the conditional factor demand x are as follows: Theorem 8.2: Suppose that f is continuous. Conditional factor demand x has the following properties: (i) x is homogeneous of degree 0 in w; (ii) If x is differentiable at (w,z), then the substitution matrix D w x (w,z) is negative semi-definite and symmetric. In particular, Proof: for every i. x i w i 0 (25) (ii) follows from Theorem 8.1. Since C is concave in factor prices, the Hessian D 2 wc is negative semi-definite and symmetric. By 8.1 (iv), D 2 wc (w,z) = D w x (w,z). Hence (ii) follows. 12

14 PART II: Consumer Theory 10. Preferences and Utility Functions Consumption set is a subset X R L. Vectors in X represent consumption bundles that the consumer considers possible for consumption. Often, it is assumed that consumption set X is closed and convex, or more specifically that X = IR +. L The consumer s preferences over commodity bundles in X are specified by a preference relation. Properties that a preference relation may have: (i) reflexive, transitive and complete, (ii) continuous, (iii) nonsatiated, or locally nonsatiated, (iv) increasing, or strictly increasing (also called weakly monotone, or strongly monotone), (v) convex, or strictly convex. Examples of preferences: lexicographic; Leontief; etc. 13

15 Function u : X R is a utility representation of if, for every x,x X, u(x) u(x ) if and only if x x. (26) Existence of a utility representation. Theorem 10.1: If preference relation on IR L + is complete, reflexive, transitive, continuous, and strictly increasing, then it has a (continuous) utility representation. Proof see Varian. 11. Utility Maximization The problem of utility maximization for a price vector p R L and an income m > 0 is written as maximize u(x) (27) subject to px m and x 0. Solution to (27) is denoted by x (p,m) it is the demand at prices p and income m. Demand x (p,m) is often called Walrasian or Marshallian. u (p,m) u(x (p,m)) is the indirect utility function. 14

16 First-order conditions: Assume that u is differentiable. Also m > 0 and p >> 0. Let u i denote u x i. We analyze utility maximization problem (38) using the Kuhn-Tucker approach. We write the Lagrangian as L(λ,x) = u(x) + λ[m px], where λ 0 is the Lagrange multiplier. First-Order Conditions for a solution x and a multiplier λ are: u i (x ) λ p i 0, and if x i > 0, then = 0, (28) m px 0, and if λ > 0, then = 0. (29) Where do these conditions come from? (28) obtains from L x i 0 and (29) from L λ 0. Those are as if we were maximizing Lagrangian L(λ,x) with respect to x and minimizing it with respect to λ over x 0 and λ 0. If x >> 0 and λ > 0, then FOC (28-29) imply that p i = u i(x ) p j u j (x ), that is, ratio of prices = marginal rate of substitution. 15

17 FOCs (28-29) are necessary for x to solve the utility maximization problem. They are necessary and sufficient if utility function u is quasi-concave. 12. Expenditure Minimization and Hicksian Demand The expenditure minimization problem for a price vector p R L and a utility level ū is: minimize px (30) subject to u(x) ū and x 0. Solution is h(p,ū) Hicksian demand function if single-valued. e(p,ū) ph(p,ū) is the expenditure function. The expenditure minimization problem is exactly the same as cost minimization for production function. Results of Theorems 8.1 and 8.2 hold for expenditure function and Hicksian demand. We have (i) e is homogeneous of deg. 1 and h is homogeneous of deg. 0 in prices. (ii) e is a concave function of prices; (iii) e p i = h i, i, or in vector notation D p e(p,ū) = h(p,ū). (iv) the matrix D p h(p,ū) is negative semi-definite and symmetric. In particular, h i p i 0. 16

18 Comments: Hicksian demand is not an observable demand function, and therefore may appear as not of much interest. However, its relation to the Walrasian demand and its relatively easy comparative statics will turn out very helpful in deriving some results about Walrasian demand. 13. Walrasian Demand and Hicksian Demand Proposition 13.1: Suppose that p >> 0, and u is continuous and strictly increasing. Then: (i) If x solves the utility maximization problem at p and m, then x also solves the expenditure minimization problem at p and u (p,m). (ii) If x solves the expenditure minimization problem at p and ū, then x solves the utility maximization problem at p and e(p,ū). Remark: The proof is a somewhat different from the one in Varian, Ch 7, pg Proof (i): It is a proof by contradiction. Suppose that x solves the utility maximization problem at p and m, but is not a solution to the expenditure minimization at p and u (p,m). The latter means that there is some x with u( x) u (p,m) and p x < px. Since u (p,m) = u(x ) and px m, we have that u( x) u(x ) and p x < m. We show that this contradicts x being utility 17

19 maximizing at p and m. Let α = m p x p 1 > 0 and define ˆx = x + (α,0,...,0). Then u(ˆx) > u(x ) because u is strictly increasing, and pˆx = m. This ˆx contradicts x being utility maximizing. Proof (ii): It is a proof by contradiction. Suppose that x solves the expenditure minimization problem at p and ū, but is not a solution to the utility maximization problem at p and e(p,ū). It follows that there is x with px e(p,ū) and u(x ) > u(x ) ū. Since utility function is continuous and u(x ) > ū u(0), there exist t with 0 t < 1 such that u(tx ) = ū. It also holds that p[tx ] < px e(p,ū). Such tx contradicts x being expenditure minimizing at p and and ū. If the Walrasian and Hicksian demands are single-valued, we can express (i) of Proposition 13.1 as x (p,m) = h(p,u (p,m)), (31) and (ii) as h(p,ū) = x (p,e(p,ū)). (32) 18

20 14. Slutsky Equation and Slutsky Matrix Consider equation (32) for good i: h i (p,ū) = x i (p,e(p,ū)). (33) Differentiate (33) with respect to price p j (assuming differentiability of all involved functions). We obtain h i (p,ū) p j = x i (p,e(p,ū)) p j + x i (p,e(p,ū)) m e(p, ū) p j. (34) We know that e(p,ū) p j = h j (p,ū) = x j (p,e(p,ū)). Also, let us use m to denote e(p,ū). Then we have h i (p,ū) p j = x i (p,m) p j + x i (p,m) m x j(p,m). (35) Rearranging terms, we obtain where m = e(p,ū). x i (p,m) p j = h i(p,ū) p j x i (p,m) m x j(p,m), (36) Equation (36) is the Slutsky equation. Slutsky equation provides a decomposition of the effect of change in price of good i on Walrasian demand for good j into the pure substitution effect and the income effect. 19

21 In particular, we have x i (p,m) p i = h i(p,ū) p i x i (p,m) m x i (p,m). (37) We know that the first term on the right-hand side of (37) is negative. But this does not imply negative sign of x i p i, since x i m may be negative! Good i may be an inferior good. If x i p i > 0, we have Giffen good effect. Define the L L matrix S(p,m) = [s ij ] j=1,...,l i=1,...,l by s ij = x i (p,m) p j + x i (p,m) m x j(p,m). (38) S is the Slutsky matrix associated with Walrasian demand x. Of course, the Slutsky equation (36)implies that or, in matrix notation, s ij = h i(p,ū) p j, (39) S(p,m) = D p h(p,ū), (39a) where ū = u (p,m). It follows from (39) and section 12 that S is negative semi-definite and symmetric. In particular, s ii 0, that is x i (p,m) p i + x i (p,m) m x i (p,m) 0. 20

22 Example: Consider utility function on IR 2 + given by u(x 1,x 2 ) = x 1 e x 2. Solving the utility maximization problem, we obtain the Walrasian demand for p 2 < m (interior demand), and for p 2 m. x (p,m) = ( p 2 p 1, m p 2 p 2 ), (40) x (p,m) = ( m p 1,0), (41) In order to calculate the Slutsky matrix of x when p 2 < m, we use (40) to find and x 1 p 1 = p 2 (p 1 ) 2, x 1 p 2 = 1 p 1, x 2 p 1 = 0, x 1 m = 0, x 2 = m p 2 (p 2 ) 2, x 2 m = 1. p 2 Using s ij = x i p j + x x i j m, we find s 11 = p 2 (p 1 ) 2, s 12 = 1 p 1 s 21 = 1 p 1 s 22 = 1 p 2 The Slutsky matrix S = [s ij ] is negative semi-definite and symmetric for p 2 < m. The same holds for p 2 > m but it requires separate calculations using (41). 21

23 15. Properties of Walrasian Demand Functions We found that Walrasian demand function x (p,m) of a consumer whose utility function u is strictly increasing has the following three properties for p >> 0 and m > 0: (i) x is homogeneous of degree 0 in (p,m), (ii) budget equation holds, i.e. px (p,m) = m, (iii) If x is differentiable, then the Slutsky matrix of x is negative semidefinite and symmetric. Question: Are these all properties of Walrasian demand functions? Yes! See MWG, or Varian (section 8.5) There are some useful properties of the indirect utility function u (p,m). The envelope theorem for constrained optimization implies that u p i = λ x i, u m = λ. (42) if x i > 0 for every i and λ > 0. (42) implies that x i (p,m) = Equation (43) is the Roy s Identity. u p i (p,m) u m (p,m) (43) Also, it follows that u p i < 0, and u m > 0. 22

24 16. Revealed Preference Suppose that we have several observations of price vectors and consumption plans of a consumer. They are x 1 at p 1,......, x T at p T, where x t R L + and p t R L ++ for all t. Utility function u on R L + rationalizes observations {(p 1,x 1 ),...,(p T,x T )} if, for every t, u(x t ) u(x) for every x R L + such that p t x p t x t. If observations {(p 1,x 1 ),...,(p T,x T )} are rationalized by locally non-satiated utility function u, then the following must hold: (1) the consumer s income in situation t is p t x t, (2) u(x t ) u(x) for every x such that p t x p t x t, (3) u(x t ) > u(x) for every x such that p t x < p t x t. Note that local nonsatiation is crucial for (1) and (3). (2) and (3) imply that if p t x s p t x t, then p s x t p s x s (44) for all s,t = 1,...,T. Property (44) is the Generalized Weak Axiom of Revealed Preference. 23

25 We have just shown that GWARP necessarily holds for a set of observations rationalized by locally nonsatiated utility function. Is GWARP also a sufficient condition for rationalizability? The answer is no. To understand why, we take another look at what follows from utility maximization. Define relations R and P between an observation x t and a bundle x R L + as follows: x t Rx, if p t x p t x t, (45) x t Px, if p t x < p t x t. (46) If x t Rx, we say that x t is (directly) weakly revealed preferred to x. If x t Px, we say that x t is (directly) strictly revealed preferred to x. We can write GWARP (44) as if x t Rx s, then not x s Px t. (44b) Utility maximization implies more than (44b). For every subset of observations (p t 1,x t 1 ),...,(p t n,x t n ), if x t 1 Rx t 2, x t 2 Rx t 3,..., x t n 1 Rx t n, then not x t n Px t 1. (47) Property (47) is called the Generalized Strong Axiom of Revealed Preference, or simply Generalized Axiom of Revealed Preference, GARP. 24

26 Without using relations R and P, GARP is written as if p t 1 x t 2 p t 1 x t 1,..., p t n 1 x t n p t n 1 x t n 1, then p t n x t 1 p t n x t n. (47a) Theorem 16.1 (Afriat): Observations (p 1,x 1 ),...,(p T,x T ) satisfy GARP if and only if there exists a locally nonsatiated utility function u that rationalizes these observations. Proof: See Varian, Ch. 8, also Varian (1982). The utility function u is defined as follows: First, it is proved that the system of inequalities u t u s + λ s p s (x t x s ), t,s. has solution u t,λ t with λ t > 0 for all t. Then, function u is defined by u(x) = min t {u t + λ t p t (x x t )}. It holds u(x t ) = u t. This function u is continuous, concave, and increasing. 25

27 PART III: Choice under Uncertainty 19. Expected Utility. Uncertainty is described by a set S = {1,...,S} of states of nature. Statecontingent consumption plan specifies consumption conditional on each state. We assume that there is a single commodity. Consumption plan is a vector c = (c 1,...,c S ) R S +. We consider a preference relation on the set R S + of state-contingent consumption plans. Assume that is strictly increasing and continuous. We say that has expected utility representation if there exist probabilities of states {π s } and function v : R + R such that c c iff S π s v(c s ) s=1 S π s v(c s), (64) s=1 for every c,c R S +. Utility function v in the expected utility representation (64) is the von Neumann- Morgenstern (or Bernoulli) utility. Expected utility is written as E[v(c)] 26

28 Some properties of expected utility. Marginal rate of substitution between consumption in any two states does not depend on consumption in any other state. For any deterministic (i.e., risk-free) consumption plan, the marginal rate of substitution between consumption in states s and s is π s π s. If v is continuous, strictly increasing and concave, then expected utility E[v( )] is continuous, strictly increasing and concave. Topic omitted: Axiomatizations of expected utility. State-dependent expected utility is S π s v s (c s ) s=1 for S functions v s : R R. Probabilities don t matter. This is stateseparable utility function. 27

29 Portfolio Choice under Expected Utility. There are S states of nature. Consider a risky asset with return r that can take any of S values r 1 through r S. That is, r = (r 1,...,r S ). There is risk-free asset with state-independent return r. Agent s portfolio choice problem is max a S π s [v((w a)r + ar s )], (65) s=1 where a is the amount of wealth invested in the risky asset and w is the initial wealth. Investment a is unconstrained in this problem. A more convenient way to write maximization (65) is max E[v((w a)r + a r)]. (66) a The first-order condition for an interior solution a to (66) is E[v (wr + a ( r r))( r r)] = 0. (67) If E( r) r = 0, that is, the risk premium on the risky asset is zero and v is concave, then the optimal investment in the risky asset is zero. Example: For quadratic utility v(x) = (α x) 2, we have a = (α wr)(µ r) σ 2 + (µ ˆr) 2, where µ = E( r) and σ 2 = var( r) is the variance of return. 28

30 Ellsberg paradox. An urn has 90 balls of which 30 are red and the rest are blue and yellow. Exact numbers of blue balls and yellow balls are not known. Consider bets of $ 1 on a ball of a certain color (or colors) drawn from the urn. Denote bets by 1 R, 1 B, 1 R Y, etc. Typical preferences over bets are 1 R 1 B, 1 B Y 1 R Y This pattern of preferences is incompatible with expected utility: it cannot be that π(r) > π(b) and π(b Y ) > π(r Y ), because π(b Y ) = π(b)+π(y ) holds for any probability measure π. Multiple-Prior Expected Utility. An alternative to expected utility and one that can explain the Ellsberg paradox is the multiple-prior expected utility. It takes the form min E P[v(c)], (67) P P where v : R + R is von Neumann-Morgenstern utility (with no date-0 consumption) and P is a convex and closed set of probability measures on S. Set of probability measures (priors) P reflects agent s ambiguous beliefs. 29

31 Examples of sets of priors: The set of all probabilities on S. Then min E P[v(c)] = min v(c s ). P s This is the maxmin utility of Hurwicz (1952). Bounds on probabilities: P = {P : λ s P(s) γ s, s}, where λ s,γ s [0,1] are lower and upper bounds on probability of state s, respectively, and such that s λ s 1 and s γ s 1. If c is risk-free, then MPEU is non-differentiable at c, i.e., indifference curve is kinked at risk-free c. Multiple-prior expected utility is often called maxmin utility. 30

32 Expected Utility on Lotteries with Objective Probabilities. Let Z be a (finite) set of outcomes, say Z = {z 1,...,z K }. A lottery is a probability distribution on Z, that is, an assignment of probabilities {p i } K i=1 to outcomes so that p i is the probability of winning outcome z i. Lottery with probabilities {p i } K i=1 is denoted by L. Let L be the set of all lotteries on Z. Since probabilities add up to one and are positive, the set L can be identified with the unit simplex in R K. Preference relation on the set of lotteries L has an expected utility representation if there exists function v : Z R such that L L if and only if K p i v(z i ) i=1 K p iv(z i ). i=1 Axiomatization of expected utility on lotteries is due to von Neumann and Morgenstern (1954). See MWG, Chapter 6. 31

33 20. Risk Aversion and Risk Neutrality A consumer with expected utility function E[v( )] on R S + is risk averse if E[v(c)] v(e(c)), (68) for every consumption plan c R S +. The consumer is strictly risk averse if E[v(c)] < v(e(c)) (69) for every consumption plan c R S + such that c E(c). The consumer is risk neutral if E[v(c)] = v(e(c)) (70) for every c R S +. 32

34 Measures of Risk Aversion The risk compensation for additional state-contingent consumption plan z R S with E( z) = 0 at deterministic initial consumption x R is ρ(x, z) that solves E[v(x + z)] = v ( x ρ(x, z) ). (71) If v is twice-differentiable and strictly increasing (so that v (x) > 0 for every x), we also have: the Arrow-Pratt measure of absolute risk-aversion A(x) v (x) v (x), (72) the Arrow-Pratt measure of relative risk aversion R(x) v (x) v x. (73) (x) The Theorem of Pratt Let v 1,v 2 be two C 2, strictly increasing vn-m. utility functions with ρ 1, ρ 2, and A 1 and A 2, respectively. Theorem 20.1 (Pratt): The following conditions are equivalent: (i) A 1 (x) A 2 (x) for every x R. (ii) ρ 1 (x, z) ρ 2 (x, z) for every x R and every z R S with E( z) = 0. (iii) v 1 is a concave transformation of v 2, i.e. v 1 (x) = f(v 2 (x)) for every x, for f concave and strictly increasing. 33

35 Risk Aversion and Concavity Let v be twice-differentiable and strictly increasing. Corollary 20.2: (i) A consumer is risk averse iff his von Neumann-Morgenstern utility function v is concave. (ii) A consumer is risk neutral iff his von Neumann-Morgenstern utility function v is linear. (iii) A consumer is strictly risk averse iff his von Neumann-Morgenstern utility function v is strictly concave. Note: iff means if and only if. This corollary holds true even without the assumption of differentiability of v, see LeRoy and Werner(2001). 34

36 Decreasing, Constant and Increasing Risk Aversion Corollary 20.3: Let v be C 2 and strictly increasing. Then (i) ρ(x, z) is increasing in x for every z with E( z) = 0, iff A(x) is increasing in x. (ii) ρ(x, z) is constant in x for every z with E( z) = 0, iff A(x) is constant in x. (iii) ρ(x, z) is decreasing in x for every z with E( z) = 0, iff A(x) is decreasing in x. Some Common Utility Functions The functions most often used as von Neumann-Morgenstern utility functions in applied work and as examples are: Linear utility: v(x) = x has zero absolute risk aversion, so the consumer is risk-neutral. Negative Exponential Utility: v(x) = e αx, where α > 0, has constant absolute risk-aversion (CARA) equal to α. 35

37 Quadratic utility: v(x) = (α x) 2, for x < α, has absolute risk aversion equal to 1/(α x). Logarithmic utility: v(x) = ln(x + α), for x > α. If α = 0, then relative risk-aversion is constant (CRRA). Power utility: v(x) = x1 γ, for x 0, 1 γ where γ 0,γ 1, has constant relative risk-aversion equal (CRRA) to γ. 36

38 Proof of Pratt s Theorem 20.1: (i) implies (iii): Define f by f(t) = v 1 (v2 1 (t) for every t. The first derivative of f is and is strictly positive since v i f (t) = v 1(v 1 2 (t)) v 2 (v 1 2 (t)) (74) > 0 for i = 1,2. The second derivative is f (t) = v 1(x) (v 2(x)v 1(x))/v 2(x) [v 2, (75) (x)]2 where we used x = v2 1 (t). Equation (75) can be rewritten as f (t) = ( A 2 (x) A 1 (x) ) v 1(x) [v 2 (x)]2. Thus f (t) 0 for every t, and hence f is concave. (iii) implies (ii): By the definition of ρ 1 (see (71)) E[v 1 (x + z)] = v 1 ( x ρ1 (x, z) ). (76) Since v 1 = f(v 2 ) and f is concave, Jensen s inequality yields E[v 1 (x + z)] = E[f(v 2 (x + z))] f(e[v 2 (x + z)]). (77) The right-hand side of (77) equals f ( v 2 (x ρ 2 (x, z)) ) or v 1 (x ρ 2 (x, z)). Using (76) we obtain ( v 1 x ρ1 (x, z) ) ( v 1 x ρ2 (x, z) ). (78) Since v 1 is strictly increasing, (78) implies that ρ 1 (x, z) ρ 2 (x, z). (ii) implies (i): (... in class) 37

39 Portfolio Choice and Risk Aversion Recall the portfolio choice problem with one risky and a risk-free assets: max E[v((w a)ˆr + a r)]. a where a is the amount of wealth invested in the risky asset. Theorem, 20.4: If an agent is strictly risk averse and has differentiable von Neuman-Morgenstern utility function, then the optimal investment in the risky asset is strictly positive, zero or strictly negative iff the risk premium on the risky asset (i.e., E( r) ˆr) is strictly positive, zero or strictly negative. More results on optimal investment: Theorem, 20.5: If an agent is strictly risk averse, if his absolute risk aversion is decreasing, and if the risk premium on the risky asset is strictly positive, then the optimal investment a is increasing in wealth. Consider two agents, one with utility function v 1, the other with v 2, both differentiable. Let a 1 be the optimal investment of agent 1 and a 2 be the optimal investment of agent 2. Theorem, 20.6: If agent 1 is more risk averse than agent 2 in the sense of the Pratt s Theorem 20.1, and if the risk premium on the risky asset is strictly positive, then a 1 a 2. 38

40 21. Increasing Risk Let ỹ and z be two gambles - for example, two state-contingent consumption plans - on the set of states S equipped with probabilities {π s }. Definition 21.1: Let E( z) = E(ỹ). We say that z is more risky than ỹ if only if E[v( z)] E[v(ỹ)] for every nondecreasing concave continuous utility function v. That is, for z and ỹ with E( z) = E(ỹ), z is more risky than ỹ if and only if every risk-averse agent prefers ỹ to z. Note the following: z is more risky than risk-free E( z), for every z. 2 z is more risky than z, for every z with E( z) = 0. More risky is an incomplete ordering. For an arbitrary pair z and ỹ it may well be that neither z is more risky than ỹ nor the other way round. There is a simple characterization of more risky. It is stated in terms of cumulative distribution functions of z and ỹ. Let F z and F y be the cumulative distribution functions of z and ỹ. That is, F z (t) = Prob( z t) and F y (t) = Prob(ỹ t) 39

41 for t IR. For simplicity, assume that ỹ and z take values in a bounded interval [a,b]. That is, F y (a) = F z (a) = 0, and F y (b) = F z (b) = 1. Theorem 18.2: Let E( z) = E(ỹ). z is more risky than ỹ if and only if w a F y (t)dt w a F z (t)dt, w [a,b]. (77) Risk and Variance For z and ỹ with E( z) = E(ỹ), if z is more risky than ỹ, then var( z) var(ỹ). [This follows from E[v(ỹ)] E[v( z)] applied to the quadratic utility v(x) = (α x) 2.] The converse is not true! 40