15.Perfect Competition 5 - Columbia Universitymd3405/IM_PC_5_16.pdfIntermediate Microeconomics W3211 Lecture 15: Perfect Competition 5 The Short Run and the Long Run Columbia University,

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Intermediate Microeconomics W3211

Lecture 15: Perfect Competition 5The Short Run and the Long Run

Columbia University, Spring 2016Mark Dean: [email protected]

Introduction

The Story So Far….• We have now modeled the perfectly competitive firm in

some detail

• Set up the firm’s problem

• Discussed how to split the problem into two • Cost minimization• Profit maximization

• Solved both parts

• Thought a bit about how firm behavior will change as prices change

Today• Think more about the behavior of the firm in the short and

the long run

The Short and the Long RunTechnology in the Short and the Long Run

The Long-Run and the Short-Runs

We now introduce the distinction between long run and short run

The long-run is the circumstance in which a firm is unrestricted in its choice of all input levels. In the long run a firm can choose how many workers to hire and how

many machines to use

The short-run is a circumstance in which a firm is restricted in some way in its choice of at least one input level. They have already purchased machines, and can now only decide

how many workers to hire Notice that there are many possible short runs

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Notice, there are other possible causes of the firm being in a ‘short run’ situation

i.e. being unable to change one of its inputs: temporarily being unable to install, or remove, machinery being required by law to meet affirmative action quotas having to meet domestic content regulations.


What do short-run restrictions imply for a firm’s technology?

Suppose the short-run restriction is fixing the level of input 2.

Input 2 is thus a fixed input in the short-run. Input 1 remains variable.

The Long-Run and the Short-Runsy x x 1

1/321/3

is the long-run productionfunction (both x1 and x2 are variable).The short-run production function whenx2 1 is

.x1xy 3/11

3/13/11

The short-run production function when x2 10 is

y x11/3101/3.


x1

y

Four short-run production functions

y x11/ 3101/ 3

y x11/ 351/ 3

y x11/ 321/ 3

y x11/ 311/ 3

The Short and the Long RunCost Minimization in the Short and the Long Run

Short-Run & Long-Run Total Costs

In the long-run a firm can vary all of its input levels.

Consider a firm that cannot change its input 2 level from x2’units.

How does the short-run total cost of producing y output units compare to the long-run total cost of producing y units of output?

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The long-run cost-minimization problem is

The short-run cost-minimization problem is

22110, 21

min xpxpxx

subject to f (x1,x2) y.

221101

min xpxpx

subject to f (x1, x 2) y.

13Short-Run & Long-Run Total Costs The short-run cost-min. problem is the long-run problem

subject to the extra constraint that x2 = x2’.

How does this affect costs?

If the long-run choice for x2 was x2’ then the extra constraint x2= x2’ is not really a constraint at all Long-run and short-run total costs of producing y output units are

the same.

But, if the long-run choice for x2 x2’ then the extra constraint x2 = x2’ prevents the firm from achieving its long-run production cost Short-run total cost exceed the long-run total cost of producing y

output units.


x1

x2

y

y

y

Consider three output levels.


x1

x2

y

y

y

In the long-run when the firmis free to choose both x1 andx2, the least-costly inputbundles are ...


x1

x2

y

y

y

x1 x1 x1

x2

x2

x2

Long-runoutputexpansionpath


x1

x2

y

y

y

Long-runoutputexpansionpath

x1 x1 x1

x2

x2

x2

Long-run costs are:c y w x w xc y w x w xc y w x w x

( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2

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Now suppose the firm becomes subject to the short-run constraint that x2 = x2“

Denote by cs(y) the corresponding short run cost function


x1

x2

y

y

y

x1 x1 x1

x2

x2

x2

Short-runoutputexpansionpath


( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2


x1

x2

y

y

y

x1 x1 x1

x2

x2

x2



( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2


x1

x2

y

y

y

x1 x1 x1

x2

x2

x2



( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2

Short-run costs are:c y c ys ( ) ( )


x1

x2

y

y

y

x1 x1 x1

x2

x2

x2



( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2

Short-run costs are:c y c yc y c y

s

s

( ) ( )( ) ( )


x1

x2

y

y

y

x1 x1 x1

x2

x2

x2



( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2

Short-run costs are:c y c yc y c y

s

s

( ) ( )( ) ( )

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Short-run costs are:

x1

x2

y

y

y

x1 x1 x1

x2

x2

x2



( )( )( )

1 1 2 2

1 1 2 2

1 1 2 2

c y c yc y c yc y c y

s

s

s

( ) ( )( ) ( )( ) ( )


Short-run total cost exceeds long-run total cost except for the output level where the short-run input level restriction is the long-run input level choice.

This says that the long-run total cost curve always has one point in common with any particular short-run total cost curve.


y

$

c(y)

yyy

cs(y)

Fw x2 2

A short-run total cost curve always hasone point in common with the long-runtotal cost curve, and is elsewhere higherthan the long-run total cost curve.

The Short and the Long RunCost Curves in the Short and the Long Run

Types of Cost Curves

We are now going to think a little bit more about the cost curves of a firm

In order to do so, we are going to differentiate between two different types of cost Fixed Costs: these do not change regardless of how much the firm

produces Variable Costs: these do change depending on how much the firm

produces

Typically, in the long run, all costs are variable If the firm produces no output it uses no input

In the short run, the firm may have some fixed costs If they are ‘forced’ to use a certain amount of one input, then they

have to pay for that input regardless of how much they produce

Types of Cost Curves

We now have lots of different types of costs Total vs Fixed vs Variable Long run vs Short run Costs vs Average Costs vs Marginal Costs

How are these cost curves related to each other?

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Fixed, Variable & Total Cost Functions

F is the total cost to a firm of its short-run fixed inputs. F, the firm’s fixed cost, does not vary with the firm’s output level.

cv(y) is the total cost to a firm of its variable inputs when producing y output units. cv(y) is the firm’s variable costfunction. cv(y) depends upon the levels of the fixed inputs.

c(y) is the total cost of all inputs, fixed and variable, when producing y output units. c(y) is the firm’s total cost function;

c y F c yv( ) ( ).

Fixed, Variable & Total Cost Functions What do these various cost curves look like? Fixed Variable Total

y

$

F

y

$

cv(y)

y

$

F

cv(y)

y

$

F

cv(y)

c(y)

F

c y F c yv( ) ( )

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Av. Fixed, Av. Variable & Av. Total Cost Curves

What about average costs? (Remember )

For y > 0, the firm’s average total cost function is

AC yFy

c yy

AFC y AVC y

v( )( )

( ) ( ).

Av. Fixed, Av. Variable & Av. Total Cost Curves What does an average fixed cost curve look like?

AFC(y) is a rectangular hyperbola so its graph looks like ...

AFC yFy

( )

$/output unit

AFC(y)

y0

AFC(y) 0 as y

Av. Fixed, Av. Variable & Av. Total Cost Curves What about average variable costs?

Well, as we have seen, this will depend on whether the firm has increasing, decreasing, or constant returns to scale

In the short run, at least some of the inputs are fixed

We therefore typically assume that there will be diminishingreturns to scale (at least eventually) If we fix the number of computers, at some point we will have

decreasing returns to scale if we keep adding economists

Av. Fixed, Av. Variable & Av. Total Cost Curves Think of a Cobb Douglas Production function

If 1 then the firm will exhibit decreasing returns to scale in the short run i.e. if is fixed at )

This is true even if the firm exhibits increasing returns to scale in the long run i.e 1

$/output unit

AVC(y)

y0

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$/output unit

AFC(y)

AVC(y)

y0

$/output unit

AFC(y)

AVC(y)

ATC(y)

y0

AFC(y) = ATC(y) - AVC(y)

AFC

$/output unit

AFC(y)

AVC(y)

ATC(y)

y0

Two things to notice:

AFC

$/output unit

AFC(y)

AVC(y)

ATC(y)

y0

Two things to notice:1. Since AFC(y) 0 as y ,

ATC(y) AVC(y) as y

AFC

$/output unit

AFC(y)

AVC(y)

ATC(y)

y0

Two things to notice:1. Since AFC(y) 0 as y ,

ATC(y) AVC(y) as y 2. since short-run AVC(y) must eventually increase, ATC(y) must eventually increase in a short-run

AFC

Marginal Cost Function

What about marginal cost?

Well, marginal fixed cost is zero

So marginal costs are just equal to marginal variable costs

MC yc y

yv( )

( ).

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Relationship Between Marginal and Total Cost Since MC(y) is the derivative of cv(y), cv(y) must be the integral

of MC(y). Fundamental Theorem of Calculus

MC yc y

yv( )

( )

y

vv cdzzMCyc0

)0()()(

y

FdzzMCyc0

)()(

Marginal and Variable Cost

MC(y)

y0

c y MC z dzv

y( ) ( )

0

y

Area is the variablecost of making y’ units

$/output unit

Marginal & Average Cost Functions

How is marginal cost related to average variable cost?

Marginal & Average Cost FunctionsSince AVC y

c yy

v( )( )

,

AVC yy

y MC y c y

yv( ) ( ) ( )

. 1

2

Marginal & Average Cost FunctionsSince AVC y

c yy

v( )( )

,

AVC yy

y MC y c y

yv( ) ( ) ( )

. 1

2

Therefore,

MC yc y

yAVC yv( )

( )( ).

as

AVC yy

( )

0

Av. Fixed, Av. Variable & Av. Total Cost Curves What does this look like in practice?

To make things more interesting, let’s think about a production function which has both increasing and decreasing returns to scale

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A Non-Concave Production Function

Production Function is

Convex

Production Function is

Concave

*

y*

Av. Fixed, Av. Variable & Av. Total Cost Curves What does this look like in practice?

To make things more interesting, let’s think about a production function which has both increasing and decreasing returns to scale

Implies Marginal Cost first decreases then increases

$/output unit

y

AVC(y)

MC(y)

$/output unit

y

AVC(y)

MC(y)

MC y AVC yAVC y

y( ) ( )

( )

0

$/output unit

y

AVC(y)

MC(y)

MC y AVC yAVC y

y( ) ( )

( )

0

$/output unit

y

AVC(y)

MC(y)

MC y AVC yAVC y

y( ) ( )

( )

0

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$/output unit

y

AVC(y)

MC(y)

MC y AVC yAVC y

y( ) ( )

( )

0

The short-run MC curve intersectsthe short-run AVC curve frombelow at the AVC curve’s

minimum.

Marginal & Average Cost FunctionsSimilarly, since ATC y

c yy

( )( )

,

ATC yy

y MC y c y

y

( ) ( ) ( ).

12

Marginal & Average Cost FunctionsSimilarly, since ATC y

c yy

( )( )

,

ATC yy

y MC y c y

y

( ) ( ) ( ).

12

MC yc y

yATC y( )

( )( ).

as

ATC y

y( )

0

$/output unit

y

MC(y)

ATC(y)

MC y ATC y( ) ( )

as

ATC yy

( )

0

Marginal & Average Cost Functions

The short-run MC curve intersects the short-run AVC curve from below at the AVC curve’s minimum.

And, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC curve’s minimum.

This is also intuitive If marginal cost is below average, the average must be going down If marginal cost is above average, the average must be going up

$/output unit

y

AVC(y)

MC(y)

ATC(y)

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Short-Run & Long-Run Total Cost Curves Remember, a firm has a different short-run total cost curve for

each possible short-run circumstance. By ‘circumstance’ we mean ‘level of the fixed input’

Suppose the firm can be in one of just three short-runs;x2 = x2

or x2 = x2 x2 < x2 < x2.or x2 = x2.

y0

F = w2x2

F

cs(y;x2)

$

yF

0

F = w2x2

F

F = w2x2

cs(y;x2)

cs(y;x2)

$

yF

0

F = w2x2F = w2x2

A larger amount of the fixedinput increases the firm’sfixed cost.

cs(y;x2)

cs(y;x2)

$

F

yF

0

F = w2x2F = w2x2

A larger amount of the fixedinput increases the firm’sfixed cost.

Why doesa larger amount of the fixed

input reduce the slope of the firm’s total cost curve?

cs(y;x2)

cs(y;x2)

$

F

yF

0

F = w2x2F = w2x2

F

F = w2x2

cs(y;x2)

cs(y;x2)

cs(y;x2)

$

F

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Short-Run & Long-Run Total Cost Curves The firm has three short-run total cost curves.

In the long-run the firm is free to choose amongst these three since it is free to select x2 equal to any of x2, x2, or x2.

How does the firm make this choice?

yF

0

F

y y

For 0 y y, choose x2 = ?

cs(y;x2)

cs(y;x2)

cs(y;x2)

$

F

yF

0

F

y y

For 0 y y, choose x2 = x2.

cs(y;x2)

cs(y;x2)

cs(y;x2)

$

F

yF

0

F

y y


For y y y, choose x2 = ?

cs(y;x2)

cs(y;x2)

cs(y;x2)

$

F

yF

0

F

y y


For y y y, choose x2 = x2.

cs(y;x2)

cs(y;x2)

cs(y;x2)

$

F

yF

0

F

y y



For y y, choose x2 = ?

cs(y;x2)

cs(y;x2)

cs(y;x2)

$

F

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yF

0

F

cs(y;x2)

y y



For y y, choose x2 = x2.

cs(y;x2)

cs(y;x2)

$

F

yF

0

cs(y;x2)

cs(y;x2)

F

cs(y;x2)

y y



For y y, choose x2 = x2.

c(y), thefirm’s long-run totalcost curve.

$

F

Short-Run & Long-Run Total Cost Curves The firm’s long-run total cost curve consists of the lowest parts of

the short-run total cost curves. The long-run total cost curve is the lower envelope of the short-run total cost curves.

Short-Run & Long-Run Total Cost Curves If input 2 is available in continuous amounts then there is an

infinity of short-run total cost curves but the long-run total cost curve is still the lower envelope of all of the short-run total cost curves.

$

yF

0

F

cs(y;x2)

cs(y;x2)

cs(y;x2) c(y)

F

Short-Run & Long-Run Average Total Cost Curves

For any output level y, the long-run total cost curve always gives the lowest possible total production cost.

Therefore, the long-run av. total cost curve must always give the lowest possible av. total production cost.

The long-run av. total cost curve must be the lower envelope of all of the firm’s short-run av. total cost curves.

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Short-Run & Long-Run Average Total Cost Curves

E.g. suppose again that the firm can be in one of just three short-runs;

x2 = x2or x2 = x2 (x2 < x2 < x2)or x2 = x2then the firm’s three short-run average total cost curves are ...

y

$/output unit

ACs(y;x2)

ACs(y;x2)

ACs(y;x2)

Short-Run & Long-Run Average Total Cost Curves The firm’s long-run average total cost curve is the lower

envelope of the short-run average total cost curves ...

y

$/output unit

ACs(y;x2)

ACs(y;x2)

ACs(y;x2)

AC(y)The long-run av. total costcurve is the lower envelopeof the short-run av. total cost curves.

Short-Run & Long-Run Marginal Cost Curves Q: Is the long-run marginal cost curve the lower envelope of the

firm’s short-run marginal cost curves?

Short-Run & Long-Run Marginal Cost Curves Q: Is the long-run marginal cost curve the lower envelope of the

firm’s short-run marginal cost curves?

A: No.

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Short-Run & Long-Run Marginal Cost Curves The firm’s three short-run average total cost curves are ...

y

$/output unit

ACs(y;x2)

ACs(y;x2)

ACs(y;x2)

y

$/output unit

ACs(y;x2)

ACs(y;x2)ACs(y;x2)

MCs(y;x2) MCs(y;x2)

MCs(y;x2)

y

$/output unit

ACs(y;x2)

ACs(y;x2)ACs(y;x2)

MCs(y;x2) MCs(y;x2)

MCs(y;x2)AC(y)

y y

+ Short-Run & Long-Run Marginal Cost Curves Below y’, will choose ′, despite the fact it has higher marginal

cost than either ′′ or ′′′

Because it gives lower total cost

y

$/output unit

ACs(y;x2)

ACs(y;x2)ACs(y;x2)

MCs(y;x2) MCs(y;x2)

MCs(y;x2)AC(y)

y y

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y

$/output unit

ACs(y;x2)

ACs(y;x2)ACs(y;x2)

MCs(y;x2) MCs(y;x2)

MCs(y;x2)

MC(y), the long-run marginalcost curve.

Short-Run & Long-Run Marginal Cost Curves For any output level y > 0, the long-run marginal cost of

production is the marginal cost of production for the short-run chosen by the firm.

y

$/output unit

ACs(y;x2)

ACs(y;x2)ACs(y;x2)

MCs(y;x2) MCs(y;x2)

MCs(y;x2)

MC(y), the long-run marginalcost curve.

Short-Run & Long-Run Marginal Cost

For any output level y > 0, the long-run marginal cost is the marginal cost for the short-run chosen by the firm.

This is always true, no matter how many and which short-run circumstances exist for the firm.


For any output level y > 0, the long-run marginal cost is the marginal cost for the short-run chosen by the firm.

So for the continuous case, where x2 can be fixed at any value of zero or more, the relationship between the long-run marginal cost and all of the short-run marginal costs is ...


AC(y)

$/output unit

y

SRACs

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AC(y)

$/output unit

y

SRMCs


AC(y)

MC(y)$/output unit

y

SRMCs

For each y > 0, the long-run MC equals theMC for the short-run chosen by the firm.

Summary

Summary

• Today we focused on the difference between the short and the long run • How technology changes• How costs change

Documents

15.Perfect Competition 5 - Columbia Universitymd3405/IM_PC_5_16.pdfIntermediate Microeconomics W3211 Lecture 15: Perfect Competition 5 The Short Run and the Long Run Columbia University,