Notes I Classical Demand Theory: Review of Important Concepts

Notes I Classical Demand Theory: Review of Important Concepts The notes for our course are based on: Mas-Colell, A., M.D. Whinston and J.R. Green (1995), Microeconomic Theory, New York and Oxford: Oxford University Press, which I will refer to as MWG. The notes review important concepts, however, in no way can the notes substitute for a thorough reading of MWG s text. 1 Notation Notice, we are working with scalars, vectors and matrices. At every step please make sure you are aware of the respective dimension of a variable (particularly, whether it is a scalar, a vector or a matrix). Basically, a vector in R N is a column vector! We denote the gradient vector of a function f(x) : R N R where x R N, N 1 at point x by f( x) R N. So, the nth row of f( x) equals f( x)/ x n, where x n is the nth element of vector x. The inner product of two vectors, x and y, is written as x y, i.e., x y = x T y, where x T is the transpose of x (i.e., a row vector). As a rule of thumb, an expression like x can always be read as x T. Consider a vector x R N. A statement like x 0 means that all elements of the vector are nonnegative: x i 0, i = 1,..., N. Similarly, x = 0 says: x i = 0, i = 1,..., N. Finally, x 0 says: x i > 0, i = 1,..., N, i.e., all elements of vector x are strictly positive: x 0 x R N ++, and x 0 x R N +. Let s consider two important matrices, the Jacobian and the Hessian. Suppose f is a vector valued function, i.e., f : R N R M. So there are m functions, each of which is defined on R N. The Jacobian matrix of f(x) is denoted by Df(x). Certainly, the Jacobian matrix of f(x) is a matrix of dimension M N whose mnth entry is f m / x n. Now suppose not all of the variables, over which the functions are defined, change. Specifically, consider f(x, y), where only the variables x R K change, but not so the variables y R L. Then, D x f(x, y) denotes the Jacobian matrix obtained by differentiating the M functions with respect to the Ronald Wendner I-1 v1.4

variables x i, i = 1,..., K only (and holding the variables y constant). The dimension of this Jacobian matrix, thus, is M K. Consider a function g : R N R (so we are just having a single function). The Hessian matrix of g(x), D( g(x)), is denoted by D 2 g(x), with mnth element [ g(x)/ x m ]/ x n. As x R N, the dimension of the Hessian matrix is N N. Query: What is the Jacobian matrix of a function g(z 1, z 2 ), i.e., when M = 1? Calculate the Hessian matrix of the function g(z 1, z 2 ). Summary of Differentiation in Matrix Notation. 1.) Consider a single function f : R N R. Then Df(x) is the Jacobian which, in this case, equals the transpose of the gradient: Df(x) = [ f(x)] T. Moreover, D 2 f(x) is the N N Hessian matrix whose mnth entry is 2 f(x)/( x m x n ). 2.) Now, consider a vector-valued function f : R N R M. Then Df(x) is the M N Jacobian whose mnth entry is f m (x)/ x n. 3.) Finally, consider f(g(x)), where f : R K R M, g : R N R K, and x R N. In words: there are K functions g k (x 1,..., x N ), and there are M functions f m (x) of g(x): f 1 (g 1 (x),..., g k (x)) f 2 (g 1 (x),..., g k (x))... f m (g 1 (x),..., g k (x)), or, shortly, f(g(x)). What is D x f(g(x))? Here comes the beauty of matrix notation. The derivative according to the product rule simply is: D x f(g(x)) = Df(g(x)) Dg(x). Just to make clear the dimensions: D x f(g(x)) [M N] = Df(g(x)) [M K] Dg(x) [K N]. 2 Setup We start our course with a review of important definitions (I assume you are familiar with these concepts from the Bakk-VWL courses). A market economy is (... an economy beware! notice we did not yet define what we mean by economy so we rather say) a setting where Ronald Wendner I-2 v1.4

goods and services are available for purchase at publicly known prices that consumers and firms take as unaffected by their own decisions (price taking assumption). Moreover, consumers trade in the marketplace to maximize well-being, and firms produce and trade to maximize profits. Consumption space: L commodities: l = 1,..., L; commodity bundle: x R L, where R L represents the commodity space. The consumption set, X, is a subset of the commodity space: X R L. Usually, we let X = R L + = {x R L : x l 0 for l = 1,..., L}, which also implies that the consumption set is convex. Budget Set: Let the price vector p R L be strictly positive: p R L ++, or, equivalently, p 0. Also, denote the consumer s wealth level (in euros or dollars, etc.) by w. Then, the Walrasian, or competitive budget set, is B p,w = {x R L + : p x w} Notice: B p,w X. Query: Consider any two points, x, x in the set B 0 = {x R L + : p x = w}. What does p (x x ) amount to? Interpret! Walrasian demand correspondence: 1 The consumer s Walrasian demand correspondence assigns to each price-wealth pair the set of optimal consumption bundles x from the budget set. I.e., x = x(p, w) = {x R L + : x x for all x B p,w }. Make sure you understand that x(p, w) represents a system of L functions (or correspondences) of (L + 1) variables i.e., a vector-valued function (correspondence) with M = L, N = L + 1. 3 Preference Relations and Utility: Basic Properties We denote a preference relation by a binary relation, on the consumption set X (=R L +). 2 Preferences are said to be rational if they are (i) complete: for all x, y X: x y or y x, or both, and 1 A correspondence is a kind of function. Suppose our domain is A R N. A (real valued) function f : A R is a rule that assigns to every x A a single value f(x) R (a singleton). In contrast, a (real valued) correspondence ϕ(x) : A R K is a rule that assigns to every x A a set ϕ(x) R K (which is not necessarily a singleton). Obviously, every function is a correspondence. But a correspondence is a function if and only if for every x A we have that ϕ(x) is a singleton. 2 A binary relation on a set X is a rule such that for each x and y in X, we can determine whether x y, y x, or neither, or both, where x y reads x is weakly preferred to y. Ronald Wendner I-3 v1.4

(ii) transitive: for all x, y, z X: x y and y z x z. A utility function u : X R assigns a numerical value to each element of X. It represents the preference relation if, for all x, y X, x y u(x) u(y). Notice that a utility function representing is not unique. Any strict increasing function g : R R, v(x) = g(u(x)), is a new utility function representing the same preferences. Also, a preference relation can be represented by a utility function only if (a) is rational, and (b) is continuous. Rationality, we already defined. Remains continuity... on X is continuous if it is preserved under limits. I.e., for any sequence of pairs {x n, y n } n=1 with x n y n for all n, and with x = lim n x n, and y = lim n y n we have: x y. Remark 1. By the definition of closedness of a set 3, continuity of implies that for all x X both the upper contour set, {x X : x x}, and the lower contour set, {x X : x x } are closed sets. 4 Remark 2. Lexicographic preferences are not continuous (hence, cannot be represented by a utility function). 5 A preference relation on X is locally nonsatiated if for every x X and every ɛ > 0, within the ɛ-ball about x, B ɛ (x) 6, there is an x B ɛ (x) such that x x. Notice that local nonsatiation rules out thick indifference curves. A stronger version of nonsatiation is monotonicity, where x x if x x. An even stronger version is strong monotonicity for which x x if x x. A preference relation on X is (strictly) convex if for every x X the upper contour set is convex. is convex if and only if u(x) is quasiconcave. Notice that if is strictly convex, then for any two consumption vectors x, y: x z and y z, we have for all α (0, 1): α x + (1 α) y z. 3 Consider all converging sequences {x n i } n=1 in A (i.e., every element of a sequence {x n i } n=1 belongs to the set A), and denote the respective limit point by x i. The set A R N is closed if for every converging sequence in A, x i A. 4 The upper and lower contour sets can be respectively described as no worse than-set and no better than-set. 5 Remember, lexicographic on R 2 are defined in the following way: x x if either x 1 > x 1 or (x 1 = x 1 and x 2 x 2). 6 Consider the consumption space X. An ɛ-ball about x X is defined to be B ɛ (x) {x X : x x < ɛ}. Ronald Wendner I-4 v1.4

4 Utility Maximization and Walrasian Demand The utility maximization problem (UMP) is defined as: max x 0 u(x) such that p x w. A solution to the UMP exists, if p 0, x 0 and are continuous. The solution of the UMP, x(p, w), is the Walrasian demand correspondence. Query: Interpret the Lagrangian multiplier of the UMP. Proposition 1 Properties of the Walrasian demand correspondence when preferences are continuous, and locally nonsatiated on the consumption set X: (1) Homogeneity of degree zero: x(p, w) = x(α p, α w), α > 0, (2) Walras law: p x = w for all x x(p, w), (3) Convexity/uniqueness: if preferences are convex (i.e., the utility function is quasiconcave), x(p, w) is a convex set. If preferences are strictly convex (i.e., the utility function is strictly quasiconcave), x(p, w) is a singleton (i.e., the Walrasian demand correspondence is in fact a Walrasian demand function. Given the Walrasian demand correspondence, we can define the indirect utility function as follows: v(p, w) u(x(p, w)). Proposition 2 If on X are continuous and locally nonsatiated, v(p, w) has the following properties: (1) v(p, w) = v(α p, α w), α > 0, (2) v w (p, w) > 0, and v pl (p, w) 0 for all l, 7 (3) v(p, w) is quasiconvex, i.e., the set {(p, w) : v(p, w) v} is convex for any v, (4) v(p, w) is continuous in p and w. 8 7 v w (p, w) v(p, w)/ w, and v pl (p, w) v(p, w)/ p l, etc. 8 Let Z R L+1 denote: Z = {(p, w) : p 0 & w = wealth}. Recall that v(p, w) : R L+1 R is continuous at ( p, w) if for every ɛ R ++, there exists a δ R ++ such that for all (p, w) B δ ( p, w) Z : v(p, w) v( p, w) < ɛ. v(p, w) is continuous on Z if v(p, w) is continuous at each (p, w) Z. Ronald Wendner I-5 v1.4

5 Expenditure Minimization and Hicksian Demand The expenditure minimization problem (EMP) is defined as: min x 0 p x such that u(x) u 0. The EMP is the dual problem to the UMP. The solution of the EMP, h(p, u 0 ), is the Hicksian demand correspondence. Query: Under which conditions is there a solution to the EMP? Query: Interpret the Lagrangian multiplier of the EMP. The Hicksian demand correspondence (or utility-compensated demand correspondence) represents the set of least-cost consumption bundles (at prices p) for which u(x) = u 0. Proposition 3 Suppose preferences are continuous, and locally nonsatiated on the consumption set X, and p 0. Then h(p, u 0 ) is: (1) Homogeneous of degree zero (HD 0 ) in p, (2) For any x h(p, u 0 ): u(x) = u 0, (3) Convexity/uniqueness: if are convex, h(p, u 0 ) is a convex set. If preferences are strictly convex, h(p, u 0 ) is a singleton. The value function, e(p, u 0 ), is called expenditure function. e(p, u 0 ) p h(p, u 0 ) represents the minimum expenditures necessary (at prices p) to purchase a commodity bundle that yields utility level u 0. Proposition 4 Suppose preferences are continuous, and locally nonsatiated on the consumption set X: Then e(p, u 0 ) is: (1) Homogeneous of degree one in p, (2) Strictly increasing in u 0 and nondecreasing in p l, (3) Concave in p, (4) Continuous in p and u 0. 6 True Facts (1) u(h(p, u 0 )) = u 0 (2) e(p, u 0 ) = p h(p, u 0 ) (3) v(p, e(p, u 0 )) = u 0 (4) e(p, v(p, w)) = w Ronald Wendner I-6 v1.4

(5) x(p, w) = h(p, v(p, w)) (6) h(p, u 0 ) = p e(p, u 0 ) (7) D p h(p, u 0 ) = D p x(p, w) + D w x(p, w) x(p, w) T (8) x l (p, w) = [ v(p, w)/ p l ]/[ v(p, w)/ w], for all l = 1,... L Relation (7) are the L 2 Slutsky equations. According to (7), the change in compensated demand of good l due to a change in p k is given by: h l (p, u 0 ) = x l(p, w) + x l(p, w) x k (p, w). p k p k w Finally, relationship (8) is called Roy s Identity. 7 Bonus Stuff Proposition 5 Suppose preferences are continuous, locally nonsatiated, and convex on the consumption set X, and h(., u 0 ) is continuously differentiable at (p, w). Denote its L L derivative matrix by D p h(p, u 0 ). Then: (1) D p h(p, u 0 ) = D 2 p e(p, u 0 ), (2) D p h(p, u 0 ) is a negative semidefinite matrix, (3) D p h(p, u 0 ) is a symmetric matrix, (4) D p h(p, u 0 ) p = 0. (1) follows directly from property (6) above and shows that D p h(p, u 0 ) is the Hessian matrix of e(p, u 0 ). (2) represents the fact that e(p, u 0 ) is a concave function. Moreover it implies non-positive own price effects. (3) is implied by Young s Theorem and implies symmetric compensated responses to price changes. (4) comes from the fact that h(p, u 0 ) is HD 0 in p. Ronald Wendner I-7 v1.4