M.Tolotti - Mathematics (Preparatory) - September Exercises. Maximize p(g(x))g(x) q x subject to x R +

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1 M.Tolotti - Mathematics (Preparatory) - September Exercises EXERCISE 1. where Maximize p(g(x))g(x) q x subject to x R + p : R R is constant, i.e. p(g(x)) = p = 1 for all x. g(x) = 35x x 2. q = Can we rely on the Weierstrass Theorem in this case? 2. Find the optimal production level. EXERCISE 2. Let f(x) = { x, if x 1 1, if x > 1 Find (if any): global maxima and global minima and the corresponding maximizers/minimizers. Find local maxima and local minima and the corresponding (local) maximizers/minimizers. EXERCISE 3. Prove that [1, 2) R is neither closed nor open. Is [ 2, + ) R closed? is it compact? Is R n closed? is it open? Find sup, inf, max and min for the following sets: (0, 1), [0, 1],(0, 1], [0, + ). EXERCISE 4. Let B R 2 be defined as B = {(x 1, x 2 ) R 2 x 1 0, x 2 0, x 2 + x 1 3 0} Is B open? bounded? compact? Let A R 2 be defined as Is A open? bounded? compact? Compute A B. Is it closed? A = {(x, y) R 2 1 < x < 2, y = x}

2 M.Tolotti - Mathematics (Preparatory) - September EXERCISE 5. f : R 3 R 2, f(x) = (f 1 (x), f 2 (x)), f(x 1, x 2, x 3 ) = (x x 2, e x3 ). Is f C 1? Write the differential of f in x = (1, 1, 0). EXERCISE 6. f : R 2 R, f(x 1, x 2 ) = (x x 2 2). Is f C 1? Write df(x 1, x 2 ) and df(0, 0). EXERCISE 7. f : R 2 R, f(x 1, x 2 ) = (x 1 ln x 2 ). Find the domain of f (S R n where f is well defined). Write the differential of f in x = (1, 1). Compute df(1, 1) h, where h = (0.1, 0.1). Compute now f(1.1, 1.1) f(1, 1) and compare it with df(1, 1) h. EXERCISE 8. f(x 1, x 2 ) = 10 x 2 1 x 2 2, D = R 2. Find all the critical points for f on D. (Sol: (0, 0) ) EXERCISE 9. f(x 1, x 2, x 3 ) = x 2 1 2x 2 2 3x x 1x x 2x 3, D = R 3. Find all the critical points for f on D. (Sol: (0, 0, 0) ) EXERCISE 10. f(x 1, x 2, x 3 ) = x 1 ln(x 2 ) e x3, D = {x R 3 x 2 (0, + )}. Find all the critical points for f on D. (Sol: (0, 1, α), α R) EXERCISE 11. f : R 2 R, f(x 1, x 2 ) = (x x 2 2). Is f C 2? Write H(x 1, x 2 ) and H(0, 0). Is H symmetric? EXERCISE 12. f : R 2 R, f(x 1, x 2 ) = (x 1 ln x 2 ). Compute H = d 2 f(x 1, x 2 ). EXERCISE 13. f(x 1, x 2 ) = 10 x 2 1 x 2 2, D = R 2. Solve EXERCISE 14. f(x 1, x 2, x 3 ) = x 2 1 2x 2 2 3x x 1x x 2x 3, D = R 3. Solve EXERCISE 15. f(x 1, x 2, x 3 ) = x 1 ln(x 2 ) e x3, D = {x R 3 x 2 (0, + )}. Solve

3 M.Tolotti - Mathematics (Preparatory) - September EXERCISE 16. f(x, y) = x 3 + y 3 and g(x, y) = x y. D = {(x, y) R 2 g(x, y) = 0}. Can we conclude that they are maxima or minima? EXERCISE 17. f(x, y) = y and g(x, y) = y 3 x 2. D = {(x, y) R 2 g(x, y) = 0}. What can we say about the maxima? EXERCISE 18. f(x, y) = 10 x 2 y 2 and g(x, y) = x + y, b = 10. D = {(x, y) R 2 g(x, y) = b}. Can we conclude that they are maxima or minima? EXERCISE 19. f(x, y) = 10 x 2 y 2 and g(x, y) = x + y, b = 10. Find local max/min. EXERCISE 20. Given f(x, y) = x 2 y 2, g(x, y) = x 2 y 2. Solve max(min){f(x, y) g(x, y) = 1}. EXERCISE 21. Given f(x, y) = 2x 2 + y, g(x, y) = 2x + 2y and b = 3. D = {(x, y) R 2 x 0, y 0, g(x, y) b}. Solve max{f(x, y) (x, y) D}. EXERCISE 22. One agent may consume two goods in quantities (x 1, x 2 ). Initial income I > 0. Prices of commodities (p 1, p 2 ). u(x 1, x 2 ) = x 1 x 2 max{x 1 x 2 I p 1 x 1 p 2 x 2 0, x 1 0, x 2 0}. EXERCISE 23. f(x, y) = x 2 y, g(x, y) = x 2 + y 2. max{f(x, y) g(x, y) 0}. EXERCISE 24. f(x, y) = x + y, g(x, y) = xy. max{f(x, y) g(x, y) 1} EXERCISE 25. f(x) = 2x 3 3x 2 and g(x) = (3 x) 3. max{f(x, y) g(x, y) 0} EXERCISE 26. Consider the consumption/investment problem with the following data: I = 10. m = 2. Possible states of the world: (ω 1, ω 2 ). Utility u(c 1, c 2 ) = 2 3 ln(c 1) ln(c 2). 2 risky securities on the market: (S 1, S 2 ).

4 M.Tolotti - Mathematics (Preparatory) - September Price vector: p = (1, 1). ( 6 0 Payoff matrix A = EXERCISE Maximize u(c) ). subject to c 0, p h I, c Ah. One good may be produced in two different countries. The cost in country A is C A (x) = ln(1 + 3x 100 ), x 0. The cost in country B is C B (y) = 2 ln(1 + y 100 ), x 0. The firm is aiming at minimizing the global cost of production. The firm wants to produce at least q > 0 units of output. min{c(x, y) = C A (x) + C B (y) x + y q, x 0, y 0}. EXERCISE 28. Prove that a function is affine iff it is simultaneously concave and convex. EXERCISE 29. Prove that the set of concave functions on a set D is a cone in the set of all the functions on D. (Is it a linear space?) EXERCISE { 30. e Let f(x) = x, 0 < x < 1;. 3, x = 0, 1. Prove that f is concave but non continuous on [0, 1]. EXERCISE 31. Prove that h(x) = min{f(x), g(x)} is concave if f and g are too. EXERCISE 32. f(x 1,..., x n ) = x a1 1 xa xan n, (a i > 0), is the Cobb-Douglas function. Prove that it is strictly concave if i a i < 1; concave if i a i 1; quasiconcave for any a i > 0. EXERCISE 33. Suppose g(x) is a concave (convex) function of one variable in the interval I and let us consider f(x, y) = g(x) for all x I and all y in some interval J. Prove that f(x, y) is concave (convex) for x I and y J. Suppose now that g(x) is strictly concave (convex): will f(x, y) also be strictly concave (convex)? EXERCISE 34. Given the function f(x, y) = (ln(x)) a (ln(y)) b defined for x > 1 and y > 1, where a > 0, b > 0 and a + b < 1, prove that f(x, y) is strictly concave.

5 M.Tolotti - Mathematics (Preparatory) - September EXERCISE 35. The profit function Π(p) = sup y Y i p iy i gives the measure of the maximum profit which the firm can earn given prices p and technology Y. Prove that Π is a convex function of p. EXERCISE 36. Let f : R R be an invertible function. Study the concavity/convexity of f in relations with that of its inverse f 1. EXERCISE 37. Let f : R n + R be defined by f(x 1,..., x n ) = log(x a 1... x a n) where a > 0. Study the concavity of f. EXERCISE 38. Any monotone function on D R is both quasiconcave and quasiconvex. EXERCISE 39. If f is quasiconcave and g is increasing then g f is quasiconcave. EXERCISES: 1, 2, 3, 4, 6, 7, 8 at pp. 88, 89 of the book. EXERCISES: 9, 10, 13, 14 at pp. 97, 98 of the book. EXERCISES: 1, 2, 4, 5, 10 pp. 110, 111 of the book. EXERCISES: 1, 2, 3, 4, 5, 10, 11 pp of the book. EXERCISES: 1, 3, 4, 5, 7, 9, 12 pp. 169, 171 of the book. EXERCISES 1 : 16-22, 26-33, 39-42, 44-50, at pp of the book. 1 Exercises on the topics of Chapter 1 of the book that are considered prerequisites for this course.