14.Perfect Competition 4 - Columbia University

3/31/2016

1

Intermediate Microeconomics W3211 Lecture 14: Cost Functions and Optimal Output

Columbia University, Spring 2016 Mark Dean: mark.dean@columbia.edu

Introduction

2

The Story So Far....

3

? We have set up the firm's problem

? Converts inputs to outputs in order to maximize profit

?

Thought about how to solve the firm's problem when there is only one input

? E.g. economists who only need labor to produce paper

? Thought about how to solve the problem when the firm needs to use more than one input to produce output

? Suggested that we can solve this problem by splitting it into two 1. Construct the cost function for the firm, by finding the lowest cost way of producing each output (the cost minimization problem) 2. Choose the output level that maximizes profit given these costs (the profit maximization problem)

? Figured out how to solve the firm's cost minimization problem

Today

4

? Describe the relationship between returns to scale and cost functions

? Solve the second part of the firm's problem:

? Choose the output level that maximizes profit given costs

? i.e. the profit maximization problem

? Think about comparative statics

? i.e. what happens when we change the prices of inputs and outputs

Returns to Scale and Cost Functions

5

Returns to Scale and Cost Functions6

?

In the last lecture we defined returns to scale for production functions

?

Decreasing returns to scale: doubling the inputs less than doubles the output

, , ,...

, , ,...

?

Constant returns to scale: doubling the inputs exactly doubles the output

, , ,...

, , ,...

?

Increasing returns to scale: doubling the inputs more than doubles the output

, , ,...

, , ,...

1

3/31/2016

Returns-to-Scale

7

One input, one output

Output Level

2f(x')

y = f(x)

f(2x') f(x')

Decreasing returns-to-scale

x'

2x'

x

Input Level

Returns-to-Scale

8

One input, one output

Output Level

y = f(x) 2y'

Constant

y'

returns-to-scale

x'

2x'

x

Input Level

Returns-to-Scale

9

One input, one output

Output Level

Increasing returns-to-scale

y = f(x)

f(2x')

2f(x') f(x')

x'

2x'

x

Input Level

Returns to Scale and Cost Functions10

? We showed that, a Cobb Douglas production function

, , ,...

...

Will exhibit ? Decreasing returns to scale if ? Constant returns to scale if ? Increasing returns to scale if

...1 ...1

...1

Returns to Scale and Cost Functions11

? Ultimately, what we are interested in is the firm's costs ? oSppeticmifaicl aolulyt,ptuhteir marginal costs, which will determine ? Wacoerestslsahtoiowneshdipthbaet,twinetehen rceatusernos ftoonsceainlepautn,dthmatatrhgeinraelwas ? Because of the inverse function theorem

,

1

? Ec.ogs.t,sdecreasing returns to scale means increasing marginal

Returns to Scale and Cost Functions12

? What are the marginal costs for a firm with Cobb Douglas production function? ? In the worked example, we showed that for technology

, ? Costs were given by

,, It turns out (and you should check) that for technology

Costs are given by

,, Where A is a constant which depends on

and

2

3/31/2016

Returns to Scale and Cost Functions13

,, What does this mean for marginal costs? Take the first derivative with respect to y

,, = Now take the second derivative

,, =

1

Is this positive or negative? i.e. are marginal costs increasing or decreasing?

Returns to Scale and Cost Functions14

,, =

1

Is this positive?

Only if

is greater than 1

i.e. only if

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download