# Project #69259 - Kuhn-Tucker Economics

ECONOMICS 1935

PROBLEM SET 3

1. Consider the function f : [1,5] →R where f(x) = ln(x)−0.25x .

(a) Can you use the Intermediate Value Theorem to establish whether or not there

is a max/min in the interior of the domain? Explain.

(b) Use Newton’s method to ï¬nd a maximum or minimum (feel free to use a computer). Iterate until |xk −xk−1|≤ 10−6 Do you get the same result regardless of where you start from? Explain. List your initial guess, number of iterations,

and ï¬nal stopping point.

2. Solve the problem

maxx,yx2y2 subject to 2x + y ≤ 2, x ≥ 0, and y ≥ 0.

[You may use without proof the fact that x2y2 is quasiconcave for x ≥ 0 and y ≥ 0.]

3. The function  f  is concave and the function g is quasiconcave; neither is necessarily differentiable. Is the function h defined by h(x) =  f (x) + g(x) necessarily quasiconcave? (Either show it is, or show it isn't.)

4. Show that a concave function is quasiconcave by using the fact that a function  f  is quasiconcave if and only if for all x S, all y S, and all λ [0,1] we have

if  f (x) ≥  f (y) then  f ((1−λ)x + λy) ≥  f (y).

5. The output of a good is xay, where x and y are the amounts of two inputs and a > 1 is a parameter. A government-controlled firm is directed to maximize output subject to meeting the constraint 2x + y = 12.

• Solve the firm's problem.
• Use the envelope theorem to find how the maximal output changes as the parameter a varies.

 Subject Business Due By (Pacific Time) 05/01/2015 02:00 pm
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