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 find 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 final 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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