Math camp PS_2 _Troyan_

Math Camp Questions - Section 2
The questions below are designed to illustrate three important concepts in
constrained optimization: the Kuhn-Tucker algorithm (Question 1), the Envelope Theorem (Question 2), and the Implicit Function Theorem (Question 3).
They are written in two “languages”: math and economics. If you find that you
do not understand some of the economics yet, that is fine; the problems should
still be solvable purely as math problems.
The Kuhn-Tucker Algorithm
In economics: A consumer is choosing between two goods x and y, and has
utility function u(x, y) = x + log(y). She has w dollars to spend, and the prices
of the two goods are p = (px , py ). Solve for the consumer’s demand as a function
of her wealth and prices.
In math: Solve the following optimization problem:
max x + log(y)
subject to
px x + py y  w and x, y
That is, find the solution functions x⇤ (p, w) and y ⇤ (p, w) (be careful of corner
The Envelope Theorem
(This is question 5.C.13 from MWG.)
In economics: A price-taking firm produces output q from inputs z1 and z2
according to a differentiable, concave production function f (z1 , z2 ). The price
of the output is p > 0 and the price of the inputs are (w1 , w2 ) > 0. However,
there are two unusual things about this firm. First, rather than maximizing
profits, the firm maximizes revenue. Second, the firm is cash constrained. In
particular, it only has C dollars on hand before production and, as a result, its
total expenditures cannot exceed C.
Suppose one of your econometrician friends tells you she has used repeated
observations of the firm’s revenues under various output prices, input prices,
and levels of financial constraint and has determined that the firm’s revenue
level R can be expressed as the following function of the variables (p, w1 , w2 , C):
R(p, w1 , w2 , C) = p( + log C
↵ log w1
↵) log w2 ).
and ↵ are scalars that she tells you. What is the firm’s use of output z1 when
prices are (p, w1 , w2 ) and it has C dollars of cash on hand?
In math: The following is an alternate expression for R:
R(p, w1 , w2 , C) = max pf (z1 , z2 ) s.t. w1 z1 + w2 z2  C
z1 ,z2 0
where f is unknown. Apply the envelope theorem to find z1⇤ (p, w1 , w2 , C),
the revenue-maximizing choice of input z1 .
The Implicit Function Theorem
In economics: Let f (x, y) be a concave production function with two inputs,
x and y, with prices px and py . Let the price of the output good be p. Suppose
that @f
@x , @y > 0 everywhere. Use the first order conditions and implicit function
theorem to show:1
1. An increase in the output price always increases the profit-maximizing
level of output.
2. An increase in the output price increases the demand for some input.
3. An increase in the price of an input leads to a reduction in the demand
for that input.
In math: Consider the following optimization problem:
max pf (x, y)
x,y 0
px x
py y
Use the first order conditions and implicit function theorem to show:
@f (x⇤ (p,px ,py ),y ⇤ (p,px ,py ))
@x⇤ (p,px ,py )
> 0 or
@x⇤ (p,px ,py )
< 0 and
1 You
@y ⇤ (p,px ,py )
> 0 (or both)
@y ⇤ (p,px ,py )
may assume the solutions are interior and ignore the nonnegativity constraints.