Mathematical Preliminaries for Microeconomics: Exercises

Transcription

1 Mathematical Preliminaries for Microeconomics: Exercises Igor Letina 1 Universität Zürich Fall Based on exercises by Dennis Gärtner, Andreas Hefti and Nick Netzer.

2 How to prove A B Direct proof Find a sequence of accepted statements S i for i = 1,..., n, such that A = S 1 S 2 S 3 S n = B Exercise 1 Let (X, ) be a preference relation and u : X R be a utility function representing (X, ). Using a direct proof, show that if f : R R is a strictly increasing function, then v(x) = f (u(x)) is also a utility function representing (X, ). Utility function A function u : X R is a utility function representing preference relation (X, ) if, for all x, y X, x y u(x) u(y).

3 How to prove A B Proof by contraposition It is a special case of the proof by contradiction. Prove B A. SoW A B A B B A B A 1 T T T 2 F T T 3 F F T 4 T F F Exercise 2 Prove that there exists an infinity of prime numbers.

4 How to prove A B Proof by mathematical induction Only useful for statements that can be indexed by natural numbers, but can be used also if there is an infinite number of statements. How to prove that statement S(n) holds for any n? Need to prove: 1. S(1) is true; 2. whenever S(k) is true for some k, then S(k + 1) is also true. First step is called the basis step, second step is called the inductive step and S(k) is called the inductive hypothesis. Exercise 3 Prove that the sum of first n odd numbers equals n 2.

5 Inverse functions Identity map For a given set X the identity map id X is defined as the function: id X : X X, x x Exercise 4 Prove the following theorem: Theorem Let f : X Y be a function. Then f is bijective iff there exists a function g : Y X with g f = id X and f g = id Y. In addition, the function g is uniquely determined.

6 Derivative of inverse functions Exercise 5 Suppose f : X R is injective and differentiable at x 0 with f (x 0 ) 0. Show that (f 1 ) (y 0 ) = 1 f (x 0 ) at y 0 = f (x 0 ).

7 Mean Value Theorem Let f : U R, be a C 1 function on a (connected) interval U R. For any points a, b U, there is a point c between a and b such that f (b) f (a) = f (c)(b a) Exercise 6: Inverse demand functions Suppose Q : R + R, p Q(p), is a continuously differentiable demand function. 1. Suppose that Q (p) 0 p R +. Show that Q(p) is injective on R + (Hint: MVT). Conclude that the inverse demand function P(q) exists and is uniquely determined. 2. Suppose that Q (p) < 0 p R +. Show that Q(p) and P(q) are strictly decreasing functions. 3. Let f : A R, A R be a strictly decreasing and everywhere differentiable function. Then f (x) < 0. Is this correct?

8 Norms Norm A function : R n R is called a norm if it satisfies the following properties: 1. λx = λ x 2. (Triangle inequality) x + y x + y 3. x = 0 x = 0 Examples are (you should verify that these examples indeed satisfy all properties): x 2 x 2 i x 1 x i x max 1 i n { x i }

9 Norms Exercise 7 Suppose is a norm in R n. 1. Prove that 0 = Prove that the norm is a non-negative function. 3. Prove that the norm is symmetric: x y = y x. 4. Prove the reverse triangle inequality: x y x y 5. Prove that the norm is a continuous function.

10 Equivalent norms and local non-satiation Equivalent norms Two norms A and B on R n are equivalent if there exists K 1 with 1 K x A x B K x A x R n It is known that all norms on R n are equivalent. Exercise 8 Let x R n. 1. Show that 1 n x x 1 n x. 2. A preference relation on X = R n + is called to be locally non-satiated if x X and any ε > 0 x X such that x x < ε and x x. Does this property of depend on the choice of?

11 Sequences and limits Converging sequence Let x 1, x 2, x 3, be a sequence of points in R n. A point x R n is called a limit of the sequence if, given any real number ɛ > 0. there exists a positive integer N such that x x n < ɛ whenever n > N. Exercise 9 Suppose that the sequence (x n ) converges in R n. Show that the limit point of (x n ) is unique. (Hint: Use reverse triangle inequality and argue by contradiction.)

12 Compact and convex sets Exercise 10 Let X R n and Y R k be two compact subsets. 1. Show that the product space X Y R n k is a compact set. 2. Suppose that X and Y are convex sets. Is X Y convex? 3. Consider X R. Give an example of a convex set X that is not compact. Give an example of a compact set that is not convex.

13 Compact and convex sets Bounded, Compact Set A set A R N is (i) bounded if there exists r R such that x < r for all x A. (ii) compact if it is bounded and closed relative to R N. Convex Set A set A R N is convex if αx + (1 α)x A whenever x, x A and α [0, 1]. It is strictly convex if αx + (1 α)x Int X A whenever α (0, 1), for given X.

14 Differentiability Exercise 11 Let f : R R 2 such that x 2 y f (x, y) = x 2 + y 2 (x, y) (0, 0) 0 else. 1. Show that f is everywhere directionally differentiable. 2. Is f everywhere continuously differentiable? 3. Is f everywhere differentiable?

15 The Implicit Function Theorem Exercise 12 Consider a two-player supermodular game and let Π i (x) denote the profits of player i. Let D 2 i Πi (x) be negative definite. Assume that the game is two-dimensional, i.e. x i = (x i1, x i2 ), and let S i, the strategy space of player i, be a product of two compact intervals. Assuming that the best-response function of player i is in the interior of S i, show that it is non-decreasing in x i. Supermodular games A differentiable game is called supermodular if for every player 1 i n the payoff function Π i (x i, x i ) C 2 has only non-negative cross-partial derivatives 0 for k h and Π i x ih x jk 0 for all j i and any h, k. Π i x ih x ik

16 The Implicit Function Theorem Theorem M.E.1 Suppose every f i is continuously differentiable (with respect to both x and q). If f 1 ( x, q)/ x 1... f 1 ( x, q)/ x N... 0, then f N ( x, q)/ x 1... f N ( x, q)/ x N (i) we can locally solve the system at ( x, q), (ii) by continuously differentiable functions η = (η 1,..., η N ), (iii) with derivative Dη( q) = [D x f ( x, q)] 1 D q f ( x, q).

17 Jensen s inequality Exercise 13 Prove Jensen s inequality in the finite form. Jensen s inequality (finite form) The function f : A R is concave if and only if f (αx α K x K ) α 1 f (x 1 ) α K f (x K ) for any (finite) collection of vectors x 1,..., x K A and numbers α 1,..., α K [0, 1] with K k=1 α k = 1.

18 Jensen s inequality Another useful formulation of Jensen s inequality Jensen s inequality (probabilistic form) If the function f : A R is concave and X an integrable real-valued random variable then f (E(X )) E(f (X )) Observe that Jensen s inequality delivers risk aversion. But, is concavity a good assumption for a utility function?

19 Quasiconcavity Figure: John von Neumann and Bruno de Finetti

20 Quasiconcavity Exercise 14 Consider the following functions: f = x 3, g = x Suppose f, g : R R. Are these functions quasiconcave or quasiconvex? Are they concave or convex? 2. Consider g : [0, ) R. Does your answer of 1) change? What about strict concavity (convexity)?

21 Definition M.C.3 (i) The function f : A R is quasiconcave if f (x) t and f (x ) t implies f (αx + (1 α)x ) t (1) for any t R, x, x A and α [0, 1]. (ii) If the concluding inequality in (1) is strict whenever x x and α (0, 1), then f is strictly quasiconcave.

22 Theorem M.C.3 (i) A continuously differentiable function f : A R is quasiconcave if and only if f (x) (x x) 0 whenever f (x ) f (x) (2) for all x, x A. (ii) If the first inequality in (2) is strict whenever x x, then f is strictly quasiconcave. (iii) In the other direction, if f is strictly quasiconcave and if f (x) 0 for all x A, then the first inequality in (2) is strict whenever x x.

23 Quasiconcavity Exercise 15 Prove that a function f : A R is quasiconcave if and only if f (αx + (1 α)x ) min{f (x), f (x )} (3) for all x, x A and α [0, 1].

24 Exercise 16 Let f : R 2 R be twice continuously differentiable and denote partial derivatives by indices, i.e. f 12 (x 1, x 2 ) = 2 f (x 1, x 2 )/ x 1 x 2. Using Theorems M.C.2 and M.D.2, show that f is (i) strictly concave if f 11 (x 1, x 2 ) < 0 and f 11 (x 1, x 2 )f 22 (x 1, x 2 ) [f 12 (x 1, x 2 )] 2 > 0 for all (x 1, x 2 ) R 2. (ii) concave if and only if f 11 (x 1, x 2 ) 0, f 22 (x 1, x 2 ) 0 and f 11 (x 1, x 2 )f 22 (x 1, x 2 ) [f 12 (x 1, x 2 )] 2 0 for all (x 1, x 2 ) R 2.

25 Theorem M.C.2 (i) A twice continuously differentiable function f : A R is concave if and only if its Hessian D 2 f (x) is negative semidefinite for every x A. (ii) If D 2 f (x) is negative definite for every x A, then f is strictly concave. Theorem M.D.2 (i) Suppose M is symmetric. Then M is negative definite if and only if ( 1) r r M r > 0 for every r {1,..., N}. (ii) Suppose M is symmetric. Then M is negative semidefinite if and only if ( 1) r r M π r 0 for every r {1,..., N} and every permutation π of {1,..., N}. (iii) Suppose M is negative definite. Then ( 1) r r M π r > 0 for every r {1,..., N} and every permutation π of {1,..., N}.

26 Homogeneity and homotheticity Exercise 17 Consider the Cobb-Douglas (utility) function u(x 1,..., x n ) = n a i ln(x i ) i=1 where a i > 0, n 1 and 1. Is it homogeneous? 2. Is it homothetic? n i=1 a i = 1.

27 Homogeneity and homotheticity Homogeneity The function f (x 1,..., x N ) is homogeneous of degree r (for r =..., 1, 0, 1,...) if for every t > 0 we have f (tx 1,..., tx N ) = t r f (x 1,..., x N ). Homotheticity A function f : R N + R is homothetic if there exist functions g and h such that g : R N + R is homogeneous of some degree r, h : R R is increasing, and f = h g.

28 Euler s formula Exercise 18 Verify Euler s formula for the following functions: 1. f (x 1, x 2 ) = x 1 /x 2 2. f (x 1, x 2 ) = (x 1 x 2 ) 1/2 Theorem M.B.2 (Euler s Formula) Suppose f (x 1,..., x N ) is differentiable and homogeneous of degree r. Then, at any x = ( x 1,..., x N ) we have N n=1 f ( x 1,..., x N ) x n x n = rf ( x 1,..., x N ), or, in matrix notation, f ( x) x = rf ( x).

29 Correspondences Exercise 19 Sketch the graphs of the following correspondences: 1. F : R R, F (x) = [0, x 2 + 1] 2. G : (0, ) R, G(x) = [ 1/x, 1/x] 3. H : R R, H(x) = {3, 2x 1}

30 Correspondences and hemicontinuity Exercise 20 Verify whether the two following correspondences are uhc or lhc: F (x) = G(x) = H(x) = { 1 0 x /x 1 < x 2 { [0, 1] 0 x 1 [0, 1 + 1/x] 1 < x 2 { [0, 2] 0 x 1 [0, 1/x] 1 < x 2

31 Closed Graph Given A R N and the closed set Y R K, the correspondence f : A Y has a closed graph if for any two sequences x m x A and y m y, with x m A and y m f (x m ) for every m, we have y f (x). Upper Hemicontinuity Given A R N and the closed set Y R K, the correspondence f : A Y is upper hemicontinuous (uhc) if it has a closed graph and the images of compact sets are bounded, that is, for every compact set B A the set f (B) is bounded. Lower Hemicontinuity Given A R N and a compact set Y R K, the correspondence f : A Y is lower hemicontinuous (lhc) if for every sequence x m x A with x m A for all m, and every y f (x), we can find a sequence y m y and an integer M such that y m f (x m ) for m > M.

32 Hemicontinuity as an Extension of Continuity Exercise 21 Prove the following claim: Given A R N and the closed set Y R K, suppose that f : A Y is a single-valued correspondence (i.e., a function). Then f ( ) is an upper hemicontinuous correspondence if and only if it is continuous as a function.

33 Kakutani s FPT Exercise 22 Let F : [0, 1] [0, 1] be defined by {1}, for 0 x < 1 2, F (x) = {0, 1}, for x = 1 2, {0}, for 1 2 < x Verify that F is upper hemicontinuous, and that F (x) is nonempty and compact for each x [0, 1]. 2. Show that F does not have a fixed point. Does this contradict Kakutani s Fixed Point Theorem? Why?

34 Kakutani s Fixed Point Theorem Suppose that A R N is a nonempty, compact, convex set, and that f : A A is an uhc correspondence from A into itself with the property that the set f (x) A is nonempty and convex for every x A. Then f ( ) has a fixed point; that is, there is an x A such that x f (x).

35 Real-Life Fixed Points Exercise 23 Discuss the following two situations in the context of fixed-point theorems (assumptions and implications): 1. You place a map of Zürich on the ground (in Zürich). 2. You stir your cup of coffee. Brouwer s Fixed Point Theorem Suppose that A R N is a nonempty, compact, convex set, and that f : A A is a continuous function from A into itself. Then f ( ) has a fixed point; that is, there is an x A such that x = f (x).

36 Optimization Exercise 24 Show that max x X y Y f (x, y) = max x X ( max y Y ) f (x, y) Exercise 25 Suppose U R n is a convex subset and f : U R is a quasiconcave function. Let B U denote the nonempty set of maximizers of the problem max f (x). Show that B is a convex subset of U. x U

37 Optimization Exercise 26 Properties of linear objective functions Consider the maximization problem max a x, where C is some x C constraint set, a A and A is a convex set of parameters. Let V (a) denote the maximal value of f for a given a. Show that V is convex and homogeneous of degree one in a. Exercise 27 Properties of the cost function Consider the cost minimization problem: c(w, y) = min {w x : F (x) y}. Show that c(w, y) is concave and homogeneous of degree one in w.

38 Constrained Maximization Exercise 28 An Aggregate Cost Function A firm must produce a total quantity q > 0 of some good. It can spread production over J plants, where producing q j 0 in plant j {1,..., J} costs C j (q j ), C j, C j > 0. Determine the firm s optimal allocation of production over plants. 1. Set up the Lagrangian for this problem. 2. Argue that the constraint q q J q must bind. 3. Write down the Kuhn-Tucker conditions and characterize the optimum. 4. Is the constraint qualification condition met?

39 Constrained Maximization Exercise 29 Non-Negativity Constraints In applications, it is common that a constraint in an optimization problem takes the form of some nonnegativity requirement on some variable x n ; that is, x n 0. The usual first-order conditions (i.e., those without this constraint) require only a small modification to include this. In particular, we need only change the first-order condition for x n to f (x) x n M m=1 λ m g m (x) x n + K k=m+1 λ k h k (x) x n, with equality if x n > 0. Show that this is true by explicitly including x n 0 as a (K + 1)st constraint and showing that the resulting Kuhn-Tucker conditions are equivalent to this modification.

40 Constrained Maximization Exercise 30 Convex Constraint Set Consider the constraint set C R N given by C = {x R N : h k (x) c k for all k = 1,..., K}. Show that if the functions h 1 ( ),..., h K ( ) are quasiconvex, then the constraint set C is convex.

41 Exercise 31 On Non-Binding Constraints Consider a generic constrained optimization problem of the form max x R N s.t. h 1(x) c 1. (4) h K (x) c M, where only inequality constraints are present. Let C k denote the relaxed constraint set which arises if the kth constraint is dropped. Formally establish the following to claims, made in the lecture: 1. If f (x) f (x) for all x C k and if h k (x) c k, then x is a global constrained maximizer in problem (4). 2. Suppose all of the constraint functions h 1 ( ),..., h K ( ) are continuous and quasiconvex and that f ( ) satisfies f (x) (x x) > 0 for any x and x with f (x ) > f (x). Then if x is a solution to problem (4) in which the kth constraint is not binding (i.e., if h k (x) < c k ), we have f (x) f (x) for all x C k.

42 The Envelope Theorem Exercise 32 Envelope Theorem without Constraints A monopolist produces a good q using the linear cost function C(q) = c q (c 0). Facing a demand function q = D(p) (D < 0), the monopolist sets the price p so as to maximize profits π(c, p) (p c) D(p). For any level of variable costs c, let π(c) denote the monopolist s profits given optimal pricing. 1. Set up the monopolist s problem of choosing the profit-maximizing price p. 2. What does the envelope theorem say about π(c)/ c (assume an interior solution)? 3. Derive this result without resorting to the Envelope theorem (i.e., reproduce the proof of the Envelope theorem for the problem at hand).

43 Exercise 33 Envelope Theorem with a Constraint A firm uses a set of N inputs x (x 1,..., x N ) R N to produce output q R (one good) according to the production function q = f (x), where f / x N > 0. It buys each input n at a market price of w n. Given any input price vector w (w 1,..., w N ), any input vector x therefore costs C(x; w) w x = w 1 x w N x N. For any level of output q and input prices w, let C(q; w) denote the minimal costs the firm must bear to achieve output level q. 1. Set up the firm s problem of cost-minimizing factor choice for any output level q. 2. What does the envelope theorem say about C/ w n (assume an interior solution)? 3. Derive this result without resorting to the Envelope theorem (i.e., reproduce the proof of the Envelope theorem for the problem at hand).