Lecture 1: Introduction

1

Optimization Problems

We start with the utility maximization problem commonly seen in intermediate microeconomics. Example 1.1. Denote I a consumer’ monetary income, p1 and p2 the prices of two goods, s and x1 and x2 for their quantities. The budget set of the consumer is

B = f(x1 ; x2 ) : p1 x1 + p2 x2

Ig:

1

where D

X is the constraint set. The minimization problem can be de…ned similarly. f (x) have the same set of solutions,

Theorem 1.1. The problems maxx2D f (x) and minx2D so do minx2D f (x) and maxx2D Proof. Note that f (x) f (x).

f (y) for all y 2 D is equivalent to

f (x)

f (y) for all y 2 D.

Given this result, we can focus on how to solve maximization problems.

2

More examples

Economics studies how to make the best use of allocation of scarece resources, translated in mathematical terms, it studies various optimization problems under constraints.

2.1

Expenditure minimization

This is the ‡ side of the utility maximization problem. It studies how to sustain a certain ip utility level with the lowest expense. With the usual notation, the problem can be written as min p x; s.t. u(x) u:

2.2

Pro…t maximization

Let x be the inputs of a production function f (x); let the price of the inputs be w, and the price of good be p(f (x)); then the pro…t maximization problem can be formulated as

max p(f (x))f (x)

w x, s.t. x 2 Rn .

2

2.3

Cost minimization

Similarly, the cost minimization problem is de…ned as

min w x; s.t. f (x)

y:

2.4

Consumption-leisure choice

max u(x; l); s.t. p x

w(H

l):

3

Solution to the utility maximization problem

Here we give the intuition on solving the utility maximization problem. Let’ …rst assume s that the constraint is binding, p1 x1 + p2 x2 = I; i.e., the consumer is not satiated. The problem is now