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.
1
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)
x,y
subject to
px x + py y  w and x, y
0.
That is, find the solution functions x⇤ (p, w) and y ⇤ (p, w) (be careful of corner
solutions).
2
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,
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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
(1
↵) 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 .
3
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
@f
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:
1.
@f (x⇤ (p,px ,py ),y ⇤ (p,px ,py ))
@p
2.
@x⇤ (p,px ,py )
@p
> 0 or
3.
@x⇤ (p,px ,py )
@px
< 0 and
1 You
>0
@y ⇤ (p,px ,py )
@p
> 0 (or both)
@y ⇤ (p,px ,py )
@py
<0
may assume the solutions are interior and ignore the nonnegativity constraints.
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