**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

max_{x}_{,y}*x*^{2}*y*^{2} subject to 2*x* + *y* ≤ 2, *x* ≥ 0, and *y* ≥ 0.

[You may use without proof the fact that *x*^{2}*y*^{2} 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* + λ*y*) ≥ *f* (*y*).

5. The output of a good is *x ^{a}y*, where

- 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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