varian chapter21 cost curves.ppt - lancaster university

C t CCost Curves

Types of Cost CurvesTypes of Cost Curves

A total cost curve is the graph of a firm’s total cost functionfirm s total cost function.

A variable cost curve is the graph of fi ’ i bl f ia firm’s variable cost function.

An average total cost curve is the An average total cost curve is the graph of a firm’s average total cost functionfunction.

An average variable cost curve is the graph of a firm’s average variablegraph of a firm s average variable cost function.A fi d i h An average fixed cost curve is the graph of a firm’s average fixed cost g gfunction.

A marginal cost curve is the graph of A marginal cost curve is the graph of a firm’s marginal cost function.

How are these cost curves related to each other?each other?

How are a firm’s long-run and short-l d?run cost curves related?

Fixed Variable & Total Cost FunctionsFixed, 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’scost, does not vary with the firm s output level.

( ) i th t t l t t fi f it 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 cost function.

cv(y) depends upon the levels of the fixed inputs

fixed inputs.

Fixed Variable & Total Cost FunctionsFixed, Variable & Total Cost Functions

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

c y F c yv( ) ( ).

cv(y)c y F c yv( ) ( )

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

The firm’s total cost function isc y F c yv( ) ( ).

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

ACy F c yv( ) ( ) y

y yAFCy AVCy

( ) ( ) AFCy AVCy( ) ( ).

What does an average fixed cost curve look like? Fcurve look like?

AFCy Fy

AFC(y) is a rectangular hyperbola so

(y) g yits graph looks like ...

$/output unitp

AFC(y) 0 as y (y) y

AFC(y)

In a short-run with a fixed amount of at least one input the Law ofat least one input, the Law of Diminishing (Marginal) Returns must apply causing the firm’s averageapply, causing the firm’s average variable cost of production to increase eventually.

$/output unitp

AVC(y)

$/output unitp

AVC(y)

AFC(y)

And ATC(y) = AFC(y) + AVC(y)

$/output unitp

ATC(y) = AFC(y) + AVC(y)

ATC(y)

AVC(y)

AFC(y)

$/output unitp

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

ATC(y)

AVC(y)AFC

AFC(y)

$/output unit Since AFC(y) 0 as y ,pATC(y) AVC(y) as y

ATC(y)

AVC(y)AFC

AFC(y)

$/output unit Since AFC(y) 0 as y ,pATC(y) AVC(y) as y

And since short run AVC(y) mustAnd since short-run AVC(y) musteventually increase, ATC(y) must eventually increase in a short-run.

ATC(y)

AVC(y)

AFC(y)

Marginal Cost FunctionMarginal Cost Function

Marginal cost is the rate-of-change of variable production cost as thevariable production cost as the output level changes. That is,

MCy c yv( ) ( ).y

Marginal Cost FunctionMarginal Cost Function

The firm’s total cost function isc y F c yv( ) ( )

and the fixed cost F does not change with the output level y sowith the output level y, so

MCy c yy

v( ) ( ) ( ).

MC is the slope of both the variable

MC is the slope of both the variable cost and the total cost functions.

Marginal and Variable Cost FunctionsMarginal and Variable Cost Functions

Since MC(y) is the derivative of cv(y), c (y) must be the integral of MC(y)cv(y) must be the integral of MC(y). That is, MCy c y

yv( ) ( )

c y MCz dzy

( ) ( ) c y MCz dzv( ) ( ) .0

Marginal and Variable Cost FunctionsMarginal and Variable Cost Functionsy$/output unit

c y MCz dzv( ) ( ) 0

MC(y)0

Area is the variablecost of making y’ units

Marginal & Average Cost FunctionsMarginal & Average Cost Functions

How is marginal cost related to average variable cost?average variable cost?

Marginal & Average Cost FunctionsMarginal & Average Cost FunctionsSince AVCy c yv( ) ( ),y

AVC y y MC y c yv( ) ( ) ( ) 1VC y

yy C y c y

yv( ) ( ) ( ). 2

yy C y c y

yv( ) ( ) ( ). 2

Therefore,AVCy( )

0 y MC y c yv

( ) ( ).as

yy C y c y

yv( ) ( ) ( ). 2

Therefore,AVCy( )

0 y MC y c yv

( ) ( ).as

MC y c y AVC yv( ) ( ) ( ). asAVC y( )

y( ) ( )y

Marginal & Average Cost FunctionsMarginal & Average Cost FunctionsMCy AVCy( ) ( ).

asAVC y( )

0 MCy AVCy( ) ( ).

$/output unitp

AVC(y)

$/output unitp

MCy AVCy AVCy( ) ( ) ( )

0MCy AVCyy

( ) ( )

AVC(y)

$/output unitp

MCy AVCy AVCyy

( ) ( ) ( )

AVC(y)

$/output unitp

MCy AVCy AVC yy

( ) ( ) ( )

AVC(y)

$/output unitp

MCy AVCy AVC yy

( ) ( ) ( )

The short-run MC curve intersectsth h t AVC f

MC(y)the short-run AVC curve frombelow at the AVC curve’s

minimum.

AVC(y)

Marginal & Average Cost FunctionsMarginal & Average Cost FunctionsSimilarly, since ATCy c y( ) ( ),y, y

y( ) ,

ATC y y MC y c y( ) ( ) ( ) 1

y C y yy

( ) ( ) ( ). 2

y( ) ,

ATC y y MC y c y( ) ( ) ( ) 1

y C y yy

( ) ( ) ( ). 2

Therefore,ATCy( )

ATCyy( )

0 y MCy c y

( ) ( ).as

y( ) ,

ATC y y MC y c y( ) ( ) ( ) 1

y C y yy

( ) ( ) ( ). 2

Therefore,ATCy( )

ATCyy( )

0 y MCy c y

( ) ( ).as

MCy c y ATCy( ) ( ) ( ). as

ATCy( )0

y( ) ( )

$/output unitp

MCy ATCy( ) ( )asATCy( )

0 MCy ATCy( ) ( )

ATC(y)ATC(y)

Marginal & Average Cost FunctionsMarginal & Average Cost Functions

The short-run MC curve intersects the short-run AVC curve from belowthe short run AVC curve from below at the AVC curve’s minimum.A d i il l h h MC And, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC curve’s minimum.minimum.

$/output unitp

ATC(y)

AVC(y)

Short-Run & Long-Run Total Cost Curves

A firm has a different short-run total cost curve for each possible short-cost curve for each possible shortrun circumstance.S h fi b i f Suppose the firm can be in one of just three short-runs;j

x2 = x2or x2 = x2 x2 < x2 < x2.or x2 x2 x2 < x2 < x2 .or x2 = x2.

F = w x cs(y;x2)

F = w2x2s 2

F = w x cs(y;x2)

F = w2x2F = w2x2

cs(y;x2)

F = w x cs(y;x2)

F = w2x2F = w2x2

A larger amount of the fixedinput increases the firm’s

cs(y;x2)input increases the firm sfixed cost.

F = cs(y;x2)

F = w2x2F = w2x2

A larger amount of the fixedinput increases the firm’s

cs(y;x2)input increases the firm sfixed cost.

Why doesWhy doesa larger amount of

the fixed input reduce theF

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

Short-Run & Long-Run Total Cost

MP is the marginal physical productivityCurves

MP1 is the marginal physical productivityof the variable input 1, so one extra unit ofinput 1 gives MP1 extra output units.Therefore, the extra amount of input 1, pneeded for 1 extra output unit is

units of input 11MP/1 units of input 1.1MP/1

units of input 11MP/1 units of input 1.Each unit of input 1 costs w1, so the firm’sextra cost from producing one extra unit

extra cost from producing one extra unitof output is

units of input 11MP/1 units of input 1.Each unit of input 1 costs w1, so the firm’sextra cost from producing one extra unit

MC wMP

1 .extra cost from producing one extra unitof output is

MC wMP

is the slope of the firm’s total cost curve1 cost curve.

MC wMP

is the slope of the firm’s total cost curve1 cost curve.

If input 2 is a complement to input 1 thenIf input 2 is a complement to input 1 thenMP1 is higher for higher x2.Hence MC is lower for higher xHence, MC is lower for higher x2.

That is, a short-run total cost curve starts,higher and has a lower slope if x2 is larger.

F = cs(y;x2)

F = w2x2F = w2x2

F = w x

F = w2x2 cs(y;x2)

cs(y;x2)F

The firm has three short-run total cost curvescost curves.

In the long-run the firm is free to h h h i ichoose amongst these three since it

is free to select x2 equal to any of x2, 2 y 2x2, or x2.

How does the firm make this choice? How does the firm make this choice?

For 0 y y, choose x2 = ? cs(y;x2)

cs(y;x2)

cs(y;x2)F

For 0 y y, choose x2 = x2. cs(y;x2)

cs(y;x2)

cs(y;x2)F

For y y y, choose x2 = ?s 2

cs(y;x2)

cs(y;x2)F

For y y y, choose x2 = x2.s 2

cs(y;x2)

cs(y;x2)F

For y y y, choose x2 = x2.For y y choose x = ?

For y y, choose x2 = ?cs(y;x2)

cs(y;x2)F

For y y y, choose x2 = x2.For y y choose x = x

For y y, choose x2 = x2 .cs(y;x2)

cs(y;x2)F

cs(y;x2)For 0 y y, choose x2 = x2.

s 2For y y y, choose x2 = x2.For y y choose x = x

cs(y;x2)For y y, choose x2 = x2 .

cs(y;x2) c(y) theF

c(y), thefirm’s long-

t t lF

run totalcost curve.

The firm’s long-run total cost curve consists of the lowest parts of theconsists of the lowest parts of the short-run total cost curves. The long run total cost curve is the lowerlong-run total cost curve is the lower envelope of the short-run total cost curves.

If input 2 is available in continuous amounts then there is an infinity ofamounts then there is an infinity of short-run total cost curves but the long run total cost curve is still thelong-run total cost curve is still the lower envelope of all of the short-run total cost curves.

cs(y;x2)s 2

cs(y;x2)

cs(y;x2) c(y)F

Short-Run & Long-Run Average Total Cost Curves

For an o tp t le el the long r n 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 Therefore, the long run av. total cost curve must always give the lowest possible av total production costpossible 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.

firm s short run av. total cost curves.

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

x2 = x2or x2 = x2 (x2 < x2 < x2)2 2 ( 2 2 2 )or x2 = x2then the firm’s three short-runthen the firm s three short run average total cost curves are ...

$/output unit

ACs(y;x2)

The firm’s long-run average total cost curve is the lower envelope ofcost curve is the lower envelope of the short-run average total cost curvescurves ...

$/output unit

ACs(y;x2)s(y; 2 )

ACs(y;x2)

AC(y)The long-run av. total costcurve is the lower envelopecurve 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 thecurve the lower envelope of the firm’s short-run marginal cost curves?curves?

A: No.

The firm’s three short-run average total cost curves aretotal cost curves are ...

$/output unit

ACs(y;x2)

AC (y;x )ACs(y;x2 )

$/output unit MCs(y;x2) MCs(y;x2)ACs(y;x2)

s(y; 2 ) s(y; 2 )

AC (y;x )ACs(y;x2)ACs(y;x2)

MC ( )MCs(y;x2)

s(y; 2 ) s(y; 2 )

MC ( )AC(y)

MCs(y;x2)

s(y; 2 ) s(y; 2 )

MC ( )AC(y)

MCs(y;x2)

s(y; 2 ) s(y; 2 )

MC ( )MCs(y;x2)

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

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

s(y; 2 ) s(y; 2 )

MC ( )MCs(y;x2)

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

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

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

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

So for the continuous case, where x22can be fixed at any value of zero or more, the relationship between themore, the relationship between the long-run marginal cost and all of the short run marginal costs is

short-run marginal costs is ...

$/output unitSRACs

$/output unitSRMCs

MC(y)$/output unit

MC(y)SRMCs

yFor each y > 0 the long run MC equals the

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

varian chapter21 cost curves.ppt - lancaster university

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