Ec 3310 Advanced Mathematical Economics Fall 2022 Problem Set #5
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Ec 3310
Advanced Mathematical Economics
Problem Set #5
Fall 2022
Due Friday, December 2
Problem 1 (Simon and Blume, problem 21.2)
Which of the following functions are concave or convex? At least attempt the first-order test before settling down with the second-order test:
(a) f (x, y) = -3x2 + 2xy - y2 + 3x - 4y on R2 ;
(b) f (x, y, z) = 3e北 + 5y4 - ln z on R
;
(c) f (x, y, z) = Axayb zc , where A, a, b, c > 0, on R
.
Problem 2
Prove that if V C Rn is a convex set, f : V → R is quasiconcave/quasiconvex, and g : R → R is weakly increasing, then h(x) = g(f (x)) is quasiconcave/quasiconvex . (This implies that if f is concave/convex, then h is quasiconcave/quasiconvex).
Problem 3 (Simon and Blume, problem 21.23)
Use the definitions, the second-order test, or Problem 2 to try to determine whether the following functions are quasiconcave, quasiconvex, both, or neither:
(a) f (x, y) = ye–北 on R2 ;
(b) f (x, y) = ye–北 on R
;
(c) f (x, y) = (2x + 3y)3 on R2 ;
(d) f (x, y, z) = (e北 + 5y4 + lzl)1/2 on R3 .
Problem 4
Consider a cost-minimization problem of a firm that uses n inputs:
． min北 c(x, w) = 
wi xi
( s.t. f(x) > q,
where x = (x1 , . . . , xn ) is the input bundle, f : R2 → R is a strictly concave production function, q is the target output level, and w = (w1 , . . . , wn ) > 0 is the vector of input prices. Let x* (w, q) e Rn be a solution to this problem, and λ* (w, q) e R be the associated value of the Lagrange multiplier. Assume that x* (w, q) and λ* (w, q) are unique for every (w, q)
and are C1 . Let c* (w, q) = c(x* (w, q), w) be the value function.
Use the appropriate envelope theorem to evaluate 
(i = 1, . . . , n) and 
.
2022-12-03