Course Handouts ECON 8101 MICROECONOMIC THEORY. Jan Werner. University of Minnesota

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1 Course Handouts ECON 8101 MICROECONOMIC THEORY Jan Werner University of Minnesota FALL SEMESTER

2 PART I: Producer Theory and Convex Analysis 1. Production Sets Production set is a subset Y of commodity space R 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. Example (Activity analysis): If two activities a 1,a 2 R L can be combined together at arbitrary scale, then the production set is Y = {y R L : y = λ 1 a 1 + λ 2 a 2, λ 1 0,λ 2 0}. Some properties that production sets may have: (i) Y closed; 0 Y. (ii) no free production: Y R+ L = {0}. (iii) free disposal: Y R+ L Y. (iv) Y convex, Only property (i) will be assumed throughout. 1

3 A convenient specification of a production set is in the form Y = {y R L : T(y) 0} (1) for some function T : R L R, called transformation function. Typically, function T is increasing, continuous, and such that T(0) = 0. Such specification permits the use of marginal rates of transformation T y i / T y j. 2

4 2. Production Functions Often in applied work and in examples, production technology is specified by a production function. In the simple case of single output, production function is f : R+ n R + that associates a quantity of single output with a vector of some 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, CES, etc. Some properties of production functions: (i) f(0) = 0; f continuous or differentiable. (ii) f concave or quasi-concave. Complementarity of inputs: Complementarity of inputs is expressed by the property of production function being supermodular. Function f : R n + R is supermodular if f(x y) f(x) f(y) f(x y), (2) for every x,y R n +, where we use x y to denote the supremum and x y to denote the infimum of two vectors x,y R n +. That is, x y = (max{x 1,y 1 },max{x 2,y 2 },...,max{x n,y n }), x y = (min{x 1,y 1 },min{x 2,y 2 },...,min{x n,y n }), 3

5 Figure 1 illustrates definition (2) for n = 2. Supermodularity can be characterized using second-order cross derivatives. Proposition 2.1: Let f : R n + R be twice differentiable. Then f is supermodular if and only if 2 f x i x j (x) 0 (3) for every i,j, i j and every x R n +. Production function f gives rise to production set Y f given by Y f = {(x,z) R n+1 : x 0, 0 z f( x)}. (4) 4

6 3. Returns to Scale in Production Properties of returns to scale for production set are defined as follows: constant if y Y, then λy Y for every λ 0, nonincreasing if y Y, then λy Y for every 0 λ 1, nondecreasing if y Y, then λy Y for every λ 1, Actually, returns to scale can be more crisply defined for production function. These definitions are 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. One can show (Exercise) that constant, decreasing or increasing returns to scale for f imply that the production set Y f of (4) exhibits constant, nonincreasing or nondecreasing returns to scale, respectively. 5

7 4. Profit Maximization Profit maximization at price vector p R L is maximize py over y Y. (5) 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 }. (6) The (maximum) profit is π (p) = sup y Y py. (7) 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 R L. The domain of s is a subset of the domain of π. 6

8 5. Convex Analysis and Duality Profit function π is the support function of production set Y. Extensive discussion of properties of support functions can be found in Rockafellar (1970), ch 13. Here we present the most useful results and definitions. For a closed set K R n, the support function µ K is defined by for every p R 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 R n : K = {x R n : px µ K (p), p} (9) See Corollary in Rockafellar (1970). 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) See Theorem 23.5 in Rockafellar (1970), or MWG, Section 3.F. 7

9 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. Being convex, profit function π is continuous on its domain except possibly for points on the boundary. 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. 8

10 Proof: (ii) follows from (ii) and (iii) of Theorem 6.1 if π is differentiable. (iii) follows from (ii), see Math Appendix. 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. (iii) if Y is compact, then π is a continuous function and s is an upper hemi-continuous correspondence. Proof: (i) and (ii) left as exercises, (iii) follow from the Maximum Theorem (in Math Appendix). 9

11 7. Profit-rationalizability Consider a function π : R L R that assigns profit to each price vector p in R L. (π(p) can take infinite value +, but not for every p.) Call π a profit function, but it is not known whether or not π results from maximizing profit on some production set, that is, whether π is a maximum profit function. Production set Y profit-rationalizes function π if π(p) = max{py : y Y } for every p. Properties (i) and (ii) of Theorem 6.1 together with a continuity condition turn out to be sufficient for profit-rationalizability. Theorem 7.1: If π is (i) homogeneous of deg 1, (ii) convex, and (iii) lower semi-continuous, then there exists a closed and convex set Y that profitrationalizes π. Proof: in class. The set Y that profit-rationalizes function π is Y = {y R L : py π(p), p}. Function π : R L R is lower semi-continuous at p, if π(p) lim n π(p n). for every sequence {p n } such that p n converges to p and the limit of π(p n ) exists (possibly equal to + ). 10

12 8. Cost Minimization The problem of minimizing cost for a producer with production function f : R+ n R + 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 is a constrained optimization problem and could be analyzed using the Kuhn-Tucker method. We can also use Section 5 since C (,z) is (the negative of) the support function of the input requirement set V (z) = {x R n + : f(x) z}. (15) The fundamental properties of factor demand and cost function are: Theorem 8.1: Suppose that f is continuous. Then the following hold: (i) C is homogeneous of degree 1 in factor prices w, (ii) C is a nondecreasing and concave function of w, (iii) If C is differentiable at w (this holds iff x is single-valued) then D w C (w,z) = x (w,z). (16) 11

13 Proof: (ii) and (iii) are properties of support functions, Section 5. Equation (16) is the Shephard s Lemma. Theorem 8.2: Suppose that f is continuous. Then the following hold: (i) x is homogeneous of degree 0 in w; (ii) x is negatively -monotone in w, that is, [x (w,z) x (w,z)][w w ] 0, w,w, z (17) (iii) If x is single-valued and differentiable with respect to prices, then the matrix D w x is negative semi-definite and symmetric. Proof: (ii) Follows from (ii) and (iii) of Theorem 8.1. Theorem 8.2 (iii) implies a comparative statics property of factor demand: x i w i 0 (18) Remark: The matrix D w x is singular. This is so because D w x (w,z)w = 0 as follows from (iii) and the Euler s Theorem (see MWG, Appendix). Other properties: (i) C z 0, (ii) If f is concave, then C is convex function of z. 12

14 PART II: Consumer Theory 9. 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 = R+. 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 (l.n.s.), (iv) increasing, or strictly increasing (also called weakly monotone, or strongly monotone), (v) convex, or strictly convex. Other special properties: homothetic, quasi-linear, etc. 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. (19) Properties that a utility function may have: (i) continuous, differentiable, increasing or strictly increasing, locally nonsatiated, concave or strictly concave, quasi-concave, (ii) supermodular, additively separable, homothetic, quasi-linear, etc. Theorem 9.1: If preference relation on X is complete, reflexive, transitive, and continuous, then it has a (continuous) utility representation. Proof: See Hildenbrand and Kirman (1976). An easy proof is available if two additional assumptions are imposed: X = R L +, and strictly increasing. This proof can be found in MWG and in Varian. 14

16 10. Walrasian Demand The problem of utility maximization for a price vector p R+ L and an income w > 0 is written as maximize u(x) (20) subject to px w and x 0. The solutions (there could be many) are denoted by x (p,w) the Walrasian demand (or Marshallian) at prices p and income w. u (p,w) u(x (p,w)) is the indirect utility function. Problem (20) can be analyzed using the Kuhn-Tucker method (see Math Appendix I). If u is continuous, then demand x (p,w) exists for every p >> 0 and w 0. If u is locally nonsatiated, then the following two properties hold: px (p,w) = w, (21) for every p >> 0,w > 0, and if px (p,w ) w, then p x (p,w) w, (22) for every p >> 0,w > 0,p >> 0,w > 0. (21) is budget equation; (22) is GWARP (see Section 14) for demand function. From (22) we obtain 15

17 Proposition 10.1: Let x be Walrasian demand function of a consumer with continuous and l.n.s. utility function. Then [p p][x (p,w ) x (p,w)] 0, (23) for every p >> 0,w > 0,p >> 0,w > 0 such that w = p x (p,w). Relation (23) is the law of compensated demand. 11. The Slutsky Matrix Slutsky matrix of Walrasian demand function x (assumed differentiable) is an L L matrix S(p,w) = [s kl ] defined by s kl = x k (p,w) p l + x k (p,w) x l (p,w). (24) w Law of compensated demand implies that the Slutsky matrix S(p,w) is negative semi-definite. 16

18 Proposition 11.1: Let x be Walrasian demand function of a consumer with continuous and l.n.s. utility function. If x is differentiable, then the Slutsky matrix S(p, w) is negative semi-definite for every p >> 0, w > 0. Proof: Fix an arbitrary price-income pair (p 0,w 0 ). Define w(p) px (p 0,w 0 ) and the following function F of prices only: F(p) x (p,w(p)). If x is differentiable, then so is F and the Jacobian DF(p) obtains as F l (p) p k = x l (p,w(p)) p k + x l (p,w(p)) x w k(p 0,w 0 ). For p = p 0 (and only there!) the matrix DF(p 0 ) equals the Slutsky matrix S(p 0,w 0 ). Indeed, F l (p 0 ) = x l (p0,w 0 ) + x l (p0,w 0 ) x p k p k w k(p 0,w 0 ) = s lk (p 0,w 0 ). Law of compensated demand (23) implies that [p p 0 ][F(p) F(p 0 )] 0, (25) for every p. It follows from the proof of Proposition II.2 of Math App II that if (25) holds for every p (and fixed p 0 ), then DF(p 0 ) is negative semi-definite. Consequently, Slutsky matrix S(p 0,w 0 ) is n.s.d. QED Slutsky matrix S(p,w) is symmetric. 17

19 12. Integrability First, we summerize properties of Walrasian demand function x (p,w): Theorem 12.1: Let x be Walrasian demand function of a consumer with continuous and l.n.s. utility function. Then for every (p,w) (i) x is homogeneous of degree 0 in (p,w); (iii) x satisfies budget equation; (iii) If x is single-valued and differentiable, then the Slutsky matrix S(p,w) is negative semi-definite and symmetric. Question: Are these all properties of Walrasian demand functions? The way to answer this question is to verify whether for every demand function d satisfying (i), (ii), and (iii) of 12.1 there exists a utility function u such that function d is the Walrasian demand function of u. The answer is yes. Theorem 12.2: Let d : R++ L R + R+ L be a C 1 demand function such that (1) d is homogeneous of deg. 0, (2) pd(p,w) = w, (3) the Slutsky matrix associated with d is negative semi-definite and symmetric. Then there exists a continuous, strictly increasing, strictly quasi-concave utility function u such that d is the Walrasian demand of utility function u. 18

20 Proof: see MWG, Section 3.H. 13. Slutsky Matrix from Hicksian Demand The expenditure minimization problem for p R L + and utility level ū in the image of u is: minimize px (26) subject to u(x) ū and x 0. The solutions are h(p, ū) Hicksian demand correspondence, or function whenever single-valued. e(p,ū) ph(p,ū) is the expenditure function. The problem of expenditure minimization is exactly the same as cost minimization for producer. The fundamental properties of Hicksian demand and expenditure function are the same as those of conditional factor demand and cost function in Section 8. That is Theorem 13.1: Suppose that u is continuous on X = R+. L Then (i) e is concave and homogeneous of deg. 1 in prices; (ii) if e is differentiable at (p,ū), then D p e(p,ū) = h(p,ū). (iii) h is homogeneous of deg. 0 in prices. (iv) the matrix D p h(p,ū) is negative semi-definite and symmetric, (v) [h(p,ū) h(p,ū)][p p ] 0, for every p,p, and every ū. 19

21 Walrasian Demand and Hicksian Demand Let h(p,ū) be the Hicksian demand and x (p,w) be the Walrasian demand correspondences of utility function u on consumption set X = R L +. Let w > 0, ū > u(0) and p >> 0. Proposition 13.1: If u is continuous and locally non-satiated, then h(p,ū) = x (p,e(p,ū)), (27) and x (p,w) = h(p,u (p,w)). (28) Proof (Outline): We first have the following Lemma 13.2: (1) If u is locally non-satiated, then px (p,w) = w. (2) If u is continuous, then u(h(p,ū)) = ū. Step 1: Next we prove the following two relations: (i ) h(p,ū) x (p,e(p,ū)) (ii ) x (p,w) h(p,u (p,w)) Step 2: From (i ) it follows that u (p,e(p,ū)) = ū. From (ii ) it follows that e(p,u (p,w)) = w. Step 3: Since u (p,e(p,ū)) = ū, relation reverse to (i ) follows from (ii ). Similarly, relation reverse to (ii ) follows from (i ) and e(p,u (p,w)) = w. 20

22 The Slutsky Equation and the Slutsky Matrix Suppose that h and x are single-valued and differentiable. Using (27) it follows that D p h(p,ū) = D p x (p,w) + D w x (p,w) x (p,w) (28) where w = e(p,ū), or equivalently ū = u (p,w) More specifically (and rearranging) where ū = u (p,w). x l (p,w) p k = h l(p,ū) p k x l (p,w) x w k(p,w), (29) Equation (29) is the Slutsky equation. It provides decomposition of the effect of change in price of good k on Walrasian demand for good l into the pure substitution effect and the income effect. Slutsky matrix S = [s kl ] associated with Walrasian demand x is, see (24), s kl = x k (p,w) p l + x k (p,w) x l (p,w). w It follows from (28) that S(p,w) = D p h(p,ū), for ū = u (p,w). Using the properties of D p h(p,ū) we obtain that S is negative semi-definite and symmetric. Furthermore, it satisfies S(p, w)p = 0. 21

23 14. Revealed Preference: Algebraic Method 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: (i) the consumer s income in situation t is p t x t, (ii) u(x t ) u(x) for every x such that p t x p t x t, (iii) u(x t ) > u(x) for every x such that p t x < p t x t. Note that local nonsatiation is crucial for (i) and (iii). (ii) and (iii) imply that if p t x s p t x t, then p s x t p s x s (30) for all s,t = 1,...,T. Property (30) is the Generalized Weak Axiom of Revealed Preference. 22

24 Thus GWARP is a necessary condition for rationalizability by l.n.s utility function. Is GWARP also sufficient? 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 any bundle x R+ L as follows: x t Rx, if p t x p t x t, (31) x t Px, if p t x < p t x t. (32) 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 (30) as if x t Rx s, then not x s Px t. (30b) Utility maximization implies more than (30b). 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. (33) Property (29) is called the Generalized Strong Axiom of Revealed Preference, or simply Generalized Axiom of Revealed Preference, GARP. 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. (33b) 23

25 Theorem 14.1 (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. 24

26 Remarks: GWARP and GARP are generalizations of two more standard axioms. The Weak Axiom of Revealed Preference is if x t Rx s and x t x s, then not x s Rx t. The Strong Axiom of Revealed Preference is if x t 1 Rx t 2, x t 2 Rx t 3,..., x t n 1 Rx t n and x t1 x t n, then not x t n Rx t 1. These axioms hold for observations strictly rationalized by utility function u, i.e, if u(x t ) > u(x) for every x R L +,x x t, such that p t x p t x t, for every t. SARP is strictly stronger (as long as L > 2) than WARP. 25

27 PART III: Monotone Comparative Statics 15. The Theorem of Topkis Lattice Operations and Supermodular Functions Recall from Section 2 that x y and x y denote the supremum and the infimum, respectively, of two vectors x,y R n. Operations and are called lattice operations. A set X R n is said to be a lattice if x x X and x x X, (31) for every x,x X. Examples: Interval [a,b] R n is a lattice; R n + is a lattice. A function f : X R, where X R n is a lattice, is supermodular on X if f(x y) f(x) f(y) f(x y), (32) for every x,y X, see Section 2. 26

28 Nondecreasing Maximizers and the Theorem of Topkis. Let the set X be either the entire space R n, or the positive orthant R n +. Let T be a subset of R m. For a function f : X T R and a set S X, consider the following maximization problem max f(x,t) (33) x subject to x S. We denote the set of solutions by ϕ (t). That is, ϕ (t) = argmax x S f(x,t). (34) Monotone comparative statics: Under what conditions on function f and set S is ϕ is nondecreasing in t? Correspondence ϕ is nondecreasing in t if ϕ (t) ϕ (t ), (35) for every t t. Inequality (35) between sets means the strong set order: for every x ϕ (t) and x ϕ (t ), it holds x x ϕ (t) and x x ϕ (t ). If ϕ (t) and ϕ (t ) are singleton sets, then (35) is the usual inequality between two vectors. 27

29 Theorem 15.2 (Topkis): If S is a lattice, f(x,t) is supermodular in x for every t and has nondecreasing differences in (x;t), then ϕ is nondecreasing in t. Function f : X T R has nondecreasing differences in (x;t) if the difference f(x,t) f(x,t) is nondecreasing in t, that is, f(x,t ) f(x,t ) f(x,t) f(x,t) (36) for every x x and t t, An equivalent condition in terms of second-order cross derivatives is: Proposition 15.3: Let f : R n R m R be twice differentiable on an interval (a,b) R n R m. Then f has nondecreasing differences in (x;t) if and only if 2 f x i t k (x,t) 0 (37) for every i,k and every (x,t) in the interval (a,b) Remark: If in addition to the assumptions of 15.2 f is continuous and S is compact, then the infimum and the supremum of the set ϕ (t) belong to that set. Denote those by ϕ (t) and ϕ (t). Then ϕ and ϕ are nondecreasing functions. 28

30 Proof of Theorem 15.2: Consider t t. Let x ϕ (t) and x ϕ (t ). First, we prove that x x ϕ (t ). Supermodularity in x implies that f(x x,t ) f(x,t ) + f(x,t ) f(x x,t ) (38) Nondecreasing differences (36) imply that f(x,t ) f(x x,t ) f(x,t) f(x x,t) (39) By the lattice property of S, we have x x S. This and x ϕ (t) imply f(x,t) f(x x,t). (40) Combining (38), (39) and (40) we obtain f(x x,t ) f(x,t ) (41) Since x x S and x ϕ (t ), (41) implies that x x ϕ (t ). The argument for x x ϕ (t) is similar: f(x x,t) f(x,t) + f(x,t) f(x x,t) f(x,t) + f(x,t ) f(x x,t ) f(x,t). Since x x S, it follows that x x ϕ (t). 29

31 Nonincreasing Maximizers. ϕ is nonincreasing in t if ϕ (t) ϕ (t ) for every t t. A counterpart of Theorem 15.2 for nonincreasing solutions to maximization problem (39) is Theorem 15.4: If S is a lattice, f(x,t) is supermodular in x for every t and has nonincreasing differences in (x;t), then ϕ is nonincreasing in t. Note that only monotonicity of differences gets reversed. The assumption of supermodularity remains unchanged. Function f : X T R has nonincreasing differences in (x; t) if f(x,t ) f(x,t ) f(x,t) f(x,t), (42) for every x x and t t, For twice differentiable function f, (42) is equivalent to 2 f x i t k (x,t) 0 (43) for every i,k and every (x,t). 30

32 15. Examples of Monotone Comparative Statics 15.1 Profit Maximization Consider the problem of profit maximization for a firm with production function f (see Section 2): max x 0 [qf(x) wx] where q is the price of output and w R n is a vector of prices of n inputs. Function f is assumed strictly increasing. If f is nondecreasing, then the objective function F(x,q) = qf(x) wx has nondecreasing differences in (x;q). If f is supermodular, then F(x,q) is supermodular in x. Theorem 15.2 implies that input demand x (q) is nondecreasing in output price q Matching There are m firms and n types of workers. The quality of a worker is described be a real number. For each type i, let X i R be the finite set of qualities of type-i workers. Each firm j needs to hire one worker of every type. For a vector x j = (x j 1,...,xj n) of qualities of a team of n workers, one from each type, the firm s profit is f(x j,j). The firm has profit function f(,j) : R n R. 31

33 We first consider firm s j optimal hiring problem: max f(x j,j) (44) x j subject to x j n i=1x i. Let ˆx j be the solution. The set n i=1 X i is a lattice. It follows from Theorem 15.2 that if f(x j,j) is supermodular in x j and has nondecreasing differences in (x j ;j), then optimal hiring decisions are nondecreasing, that is, ˆx j ˆx j+1. Matching is an assignment of workers to firms (x 1,...,x m ) such that {x 1 i,...,xm i } is a subset of X i for every type i. A matching (x 1,...,x m ) is nondecreasing if x j x j+1 for every j = 1,...,m 1. The optimal matching problem is stated as follows: max (x 1,...,x m ) m f(x j,j) (45) j=1 subject to {x 1 i,...,x m i } X i. Topkis (1998) shows that if f(x j,j) is supermodular in x j and has nondecreasing differences in (x j ;j), then there exists a nondecreasing optimal matching. 32

34 15.3 Normal Demand for Supermodular Concave Utility Theorem 15.3: Suppose that utility function u : R L + R is supermodular, strictly concave and locally nonsatiated. Then the Walrasian demand function x is a nondecreasing function of income, that is x (p,w ) x (p,w) (46) for every w w > 0 and every p >> 0. In other words, the demand for every good is normal. Theorem 15.3 does not follow from Theorem A proof can be found in Appendix III. A version of Theorem 15.3 was first proved by Professor John Chipman in 1977 under the assumption 2 u x i x j 0, i j, instead of supermodularity. References: Books: Topkis, D. M. (1998), Sundaram (1996) Milgrom, P. and C. Shannon (1994): Monotone Comparative Statics, Econometrica, 62(1), Milgrom, P. and J. Roberts (1992), Games with Strategic Complementarities, Econometrica, 58(6), Chipman, J. (1977), Journal of Economic Theory, 14, Quah, J. K.-H. (2007): The Comparative Statics of Constrained Optimization Problems, Econometrica. 33

35 Intertemporal Choice. Readings: MWG, Chapter 20, Section B. Koopmans, T. (1960), Stationary Ordinal Utility and Impatience Econometrica, 28, pg 287, also Representation of preference orderings over time, in Volume in honor of J. Marschak, (1972). Laibson, D. (1997), Golden Eggs and Hyperbolic Discounting, QJE, 34

36 PART V: Choice Under Uncertainty 16. Expected Utility under Uncertainty 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 (reflexive, transitive, and complete) preference relation on the set R S + of state-contingent consumption plans. We say that has state-separable utility representation if there exist utility functions v s : R + R for all s, such that c c iff S v s (c s ) s=1 S v s (c s) (52) s=1 for every c,c R S +. We say that has expected utility representation with respect to probabilities {π s } if there exists function v : R + R such that c c iff S π s v(c s ) s=1 S π s v(c s), (53) s=1 for every c,c R S +. 35

37 Utility function v in the expected utility representation is the von Neumann- Morgenstern (or Bernoulli) utility. Expected utility is written as E[v(c)]. Axiomatization of State-Separable Utility For c R S + and y R +, let c s y denote the consumption plan c with consumption c s in state s replaced by y, that is, (c 1,...,c s 1,y,c s+1,...,c S ). The independence axiom (sure-thing principle): for all c,d R S + and y,w R +. c s y d s y iff c s w d s w (54) Theorem 16.1: Assume that S 3, and that preference relation is strictly increasing and continuous. Then has a state-separable utility representation iff it obeys the independence axiom. Remarks: Proof: see Debreu (1959), Topological methods in cardinal utility theory. Theorem 16.1 does not hold for S = 2. The independence axiom is trivially satisfied by every strictly increasing preference relation of R+. 2 The assumption of strictly increasing can be relaxed to there being at least three essential states. State s is essential if y,w,c such that c s w c s y. 36

38 Axiomatization of Risk-Averse Expected Utility For probabilities {π s } of states such that π s > 0 for each s, let E(c) = s π sc s be the expected value of c = (c 1,...,c S ) and let E(c) denote the deterministic (or risk-free) consumption plan (E(c),...,E(c)). Preference relation is risk averse (with respect to {π s }) if E(c) c (55) for every c. That is, if risk-free consumption plan equal to E(c) is preferred to risky c. Expected utility E[v( )] is risk averse if and only if v is concave. (This will be proved later.) Theorem 16.2: Assume that S 3, and that is strictly increasing and continuous. Then satisfies the independence axiom and is risk averse with respect to probabilities {π s } if and only if it has a concave expected utility representation with respect to {π s }. Proof: Theorem 16.1 implies that has a state-separable representation s v s(c s ). Suppose that each function v s is differentiable. For each x R, consider the problem max c subject to E(c) = x. v s (c s ) (56) s 37

39 By risk aversion, c = (x,...,x) must be a solution to (56). FOCs for this solution are v s(x) = λπ s, s = 1,...S. (57) It follows from (57) that Equation (58) holds for every x R. Therefore v s(x) = π s π 1 v 1(x). (58) v s (x) = π s π 1 v 1 (x) + A s for some A s R. Consequently, s π sv(c s ) with v v 1 represents. Since is risk averse, v is concave (again, this will be proved later). Proof without the extra assumption that functions v s are differentiable can be found in Werner (2005). Remark: There are two interpretations of the probabilities in the above theorem. One is that probabilities {π s } are objectively given. The other is that they are consumer s subjective probabilities revealed by risk aversion in her choice. That is, when faced with a choice between an arbitrary statecontingent plan c and deterministic plan E(c), where E(c) is calculated using {π s }, she prefers risk-free E(c). Then, the theorem says, probabilities {π s } are her subjective probabilities. 38

40 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)], P P (mpeu) 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. 39

41 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. Smooth Ambiguity Aversion Model Another alternative to expected utility motivated by the Ellsberg paradox is the smooth ambiguity aversion model. It takes the form E µ [φ(e P v(c))], (sm) where v : R + R is the vnm utility and φ : R R is a strictly increasing function. The probability measure µ is the second-order prior, that is, a probability distribution on the set of probability measures S on S. Smooth model with strictly concave φ can explain the Ellsberg paradox. It can be argued that concave φ stands for ambiguity aversion. Utility representation (sm) is often called second-order expected utility. 40

42 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. 41

43 17. Risk Aversion and the Pratt s Theorem A consumer with expected utility function E[v( )] on R S + is risk averse if E[v(c)] v(e(c)), (59) for every consumption plan c R S +. This is the definition of risk aversion in Section 16, see (55), specialized to expected utility. The consumer is strictly risk averse if E[v(c)] < v(e(c)) (60) for every consumption plan c R S + such that c E(c). The consumer is risk neutral if E[v(c)] = v(e(c)) (61) for every c R S +. 42

44 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) ). (62) 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), (63) the Arrow-Pratt measure of relative risk aversion R(x) v (x) v (x) x. (63b) 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 17.1 (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. 43

45 Risk Aversion and Concavity Let v be twice-differentiable and strictly increasing. Corollary 17.2: (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(2014). 44

46 Decreasing, Constant and Increasing Risk Aversion Corollary 17.3: 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. 45

47 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. Linear Risk Tolerance The risk tolerance: T(x) 1 A(x). The negative exponential utility function, the quadratic utility function, the logarithmic utility function, the power utility function all have linear risk tolerance (LRT or HARA). 46

48 Proof of Pratt s Theorem 17.1: (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)) > 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, (P1) (x)]2 where we used x = v2 1 (t). Equation (P1) 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 (62)) E[v 1 (x + z)] = v 1 ( x ρ1 (x, z) ). (P2) 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)]). (P3) The right-hand side of (P3) equals f ( v 2 (x ρ 2 (x, z)) ) or v 1 (x ρ 2 (x, z)). Using (P2) and (P3) we obtain v 1 ( x ρ1 (x, z) ) v 1 ( x ρ2 (x, z) ). (P4) Since v 1 is strictly increasing, (P4) implies that ρ 1 (x, z) ρ 2 (x, z). (ii) implies (i): (... in class) 47

49 18. Stochastic Dominance and Risk For a consumer whose preferences over state-contingent consumption plans in R+ S have an expected utility representation, it is only the probability distribution of consumption that matters. That is, any two consumption plans that have the same probability distribution have the same expected utility. For instance, if there are two states with equal probabilities, then the expected utility of consumption plans (1,2) and (2,1) is the same. Stochastic dominance is a ranking of random variables based on their distributions. Random variables, such as ỹ and z, could be two state-contingent consumption plans on a finite set of states S equipped with probabilities {π s }, or random variables with continuous distributions on an infinite probability space. All that matters are the cumulative distribution functions of ỹ and z. For simplicity, we assume that ỹ and z take values in a bounded interval [a, b]. Let F z and F y be their cumulative distribution functions. That is, F z (t) = Prob( z t) for t [a,b]. The expected utility of z and the expected value of z can be written as E( z) = b a tdf z (t) and E[v( z)] = b a v(t)df z (t). 48

50 First-Order Stochastic Dominance Definition 18.1: z first-order stochastically dominates iỹ if F z (t) F y (t), t [a,b]. (64) We have Theorem 18.2: z first-order stochastically dominates ỹ if and only if E[v( z)] E[v(ỹ)] for every nondecreasing continuous v. That is, z FSD ỹ if and only if every expected-utility maximizing agent with nondecreasing utility prefers z to ỹ. Example 18.3: Let ỹ take values 1 and 3 with probabilities 1/2, and z take value 1 with probability 1/4, value 3 with probability 1/4, and value 4 with probability 1/2. Then z FSD ỹ. 49

51 Second-Order Stochastic Dominance and Risk Definition 18.4: z second-order stochastically dominates ỹ if w a F z (t)dt w a F y (t)dt, w [a,b]. (65) Since E( z) = b b a F z(t)dt, (65) for w = b implies that E( z) E(ỹ). Thus, if z SSD ỹ, then E( z) E(ỹ). Also, if z FSD ỹ, then z SSD ỹ. Theorem 18.5: z second-order stochastically dominates ỹ if and only if E[v( z)] E[v(ỹ)] for every nondecreasing concave continuous v. That is, z SSD ỹ if and only if every agent with risk-averse nondecreasing expected utility prefers z to ỹ. If z SSD ỹ and z and ỹ have the same expectation E( z) = E(ỹ), then we say that ỹ is more risky than z. Proposition 18.6: If E( z) = 0, then 2 z is more risky than z. Proof: in class 50

52 Risk and Variance For z and ỹ with E( z) = E(ỹ), if ỹ is more risky than z, then var(ỹ) var( z). [This follows from E[v( z)] E[v(ỹ)] applied to the quadratic utility v(x) = (α x) 2.] The converse is not true! Example 18.7: Let z take on the values 1,3,4,6 with equal probabilities, and let ỹ take value 2 with probability 1/2 and values 3 and 7, each with probability 1/4. We have E( z) = E(ỹ) = 3.5, and var(ỹ) = 4.25, var( z) = Thus var(ỹ) > var( z). For the logarithmic utility v(x) = ln(x), we have E[v( z)] = 1 4 ln(72), E[v(ỹ)] = 1 4 ln(84). Thus, E[v( z)] < E[v(ỹ)]. Since v is concave, it follows that ỹ is not more risky than z. [In fact, neither ỹ is more risky than z, nor z is more risky than ỹ.] 51

53 Proof of Theorem 18.2 on First-Order Stochastic Dominance: First, let E[v( z)] E[v(ỹ)] for every nondecreasing continuous v. We want to show that F z (t) F y (t), t [a,b]. Suppose, by contradiction, that F z (t 0 ) > F y (t 0 ) for some t 0 [a,b]. Define the following utility function { 0, if t t0 v(t) = 1, if t > t 0 We have E[v( z)] E[v(ỹ)] = F y (t 0 ) F z (t 0 ) < 0. Function v is nondecreasing, but it is not continuous. However, it can be approximated by a nondecreasing continuous function so that the expression E[v( z)] E[v(ỹ)] remains strictly negative. This is a contradiction. Second, let F z (t) F y (t), t [a,b]. We want to show that E[v( z)] E[v(ỹ)] for every nondecreasing continuous v. Suppose first that v is differentiable. We use integration by parts: E[v( z)] E[v(ỹ)] = [v(b)f z (b) v(a)f z (a)] b b b + F y (t)v (t)dt = a a a v(t)df z (t) b a v(t)df y (t) = F z (t)v (t)dt [v(b)f y (b) v(a)f y (a)] b a (F y (t) F z (t))v (t)dt 0. The same argument holds without differentiability: see Tesfatsion (1976). 52

54 Proof of Theorem 18.5 on Second-Order Stochastic Dominance: First, let E[v( z)] E[v(ỹ)] for every nondecreasing continuous and concave v. We want to show that w a F z(t)dt w a F y(t)dt for all w [a,b]. Suppose, by contradiction, that w 0 F a z (t)dt > w 0 F a y (t)dt for some w 0. Define the following utility function We have E[v( z)] E[v(ỹ)] = { t w0, if t w v(t) = 0 0, if t > w 0 w0 a w0 F z (t)dt + a (t w 0 )df z (t) w0 a w0 a F y (t)dt < 0, (t w 0 )df z (t) = where we used integration by parts. Function v is nondecreasing, continuous and concave. This is a contradiction. Second, let w a F z(t)dt w a F y(t)dt for all w [a,b]. We want to show that E[v( z)] E[v(ỹ)] for every nondecreasing continuous and concave v. Suppose first that v is twice-differentiable. We use the derivation from the proof of FSD and apply integration by parts one more time: v (b)[ b a E[v( z)] E[v(ỹ)] = (F y (t) F z (t))dt] b a b w a [ (F y (t) F z (t))v (t)dt = a (F y (t) F z (t))dt] v (w)dw 0. The same argument holds without differentiability: see Tesfatsion (1976). 53

55 Proof of Proposition 18.6: It suffices to prove that E[v( z)] E[v(2 z)] for every nondecreasing concave v. Let z take S values z s with respective probabilities π s. Since z s = 1 2 (2z s)+ 1 2 (0) and v is concave, we have 1 2 v(2z s) v(0) v(z s). (66) Taking expectations on both sides of (66) (that is, multiplying (66) by π s and summing over s), we obtain 1 2 E[v(2 z)] + 1 v(0) E[v( z)]. (67) 2 Concavity of v and E( z) = 0 imply that E[v( z)] v(0). Substituting this in (67), we obtain E[v( z)] E[v(2 z)]. 54

56 19. Optimal Portfolios under Risk Aversion There are S states of nature. and J assets. The (gross) return on asset j is r j = (r j1,...,r js ), that is, it can take any of S values r j1 through r js. Asset j = 1 is risk-free with state-independent return r The portfolio choice problem of an agent with vn-m utility function v, assumed strictly increasing and differentiable, and wealth w > 0 is max E [ v ( J )] a j r j a 1,...,a J j=1 subject to J a j = w. j=1 The optimal investment is a. The return on the optimal investment is r = J j=1 a j r j. w Some results on optimal portfolios: Theorem 19.1: Assume that v is differentiable and that asset 1 is risk free with return r. Then the payoff of an optimal portfolio of a strictly risk-averse agent is risk free if and only if E(r j ) = r for every risky asset j 2. 55

57 Theorem 19.2: If r is the return on an optimal portfolio of a risk-averse agent and if r E(r ) is more risky than r j E(r j ) for any asset j, then E(r ) E(r j ). If there is single risky asset (with return r), then the portfolio choice problem is max E[v((w a) r + ar)]. a The first-order condition for an interior solution a is E[v (w r + a (r r))(r r)] = 0. Theorem 19.3: If an agent is strictly risk averse and has differentiable vnm utility function, then the optimal investment in the risky security is strictly positive, zero or strictly negative iff the risk premium on the risky security (i.e., E(r) r) is strictly positive, zero or strictly negative. 56