Chapter 21 COST CURVES

21.1 Average Costs

Average cost AC(y): cost per unit of output. Average variable cost AVC(y): variable cost p

er unit of output. Average fixed cost AFC(y): fixed costs per un

it of output.

AC(y)=c(y)/y =cv(y)/y +F/y=AVC(y)+AFC(y)

21.1 Average Costs

y

AC

y

AC

y

AC

AFC ACAVC

AVC may decrease initially: fixed input was matched by variable inputs.

AVC is eventually increasing: output is constrained by the fixed input.

21.2 Marginal Costs

Marginal cost: the change in cost for a given change in output.

0

( ) ( )( ) lim ( )

y

c y y c yMC y c y

y

( ) ( ) ( )vMC y c y c y

c(y) =cv(y)+F

F does not change as y changes.

21.2 Marginal Costs AVC(0)=MC(0): applying L’Hospital’s rule

0 0

( ) ( )(0) lim lim (0) (0)

1v v

vy y

c y c yAVC c MC

y

MC curve crosses AVC curve from below

2

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

( ) ( )

v v v v vc y c y y c y c y c y yAVC y

y y y

MC y AVC y

y

21.2 Marginal CostsAVC’(y)<0 if MC(y)<AVC(y)AVC’(y)>0 if MC(y)>AVC(y)

By the same logic, MC curve crosses AC curve from below

21.2 Marginal Costs

y

MC

AC

AVC

AC

AVC

MC

21.2 Marginal Costs

To review the important points: The average variable cost curve may initially slope

down but need not. However, it will eventually rise, as long as there are fixed factors that constrain production.

The average cost curve will initially fall due to declining fixed costs but then rise due to the increasing average variable costs.

21.2 Marginal Costs

The marginal cost and average variable cost are the same at the first unit of output.

The marginal cost curve passed through the minimum point of both the average variable cost and the average cost curves.

21.3 Marginal Costs and Variable Costs The area under the marginal cost curve is the variable

cost.

cv(y)= [cv(y)- cv(y-1)]+ [cv(y-1)- cv(y-2)]+

[cv(y-2)- cv(y-3)]+…+ [cv(1)- cv(0)]=MC(y-1)+ MC(y-2)+ MC(y-3)…+ MC(0)

0 0

( ) (0) ( ) ( )y y

v v vc y c c y dy MC y dy

21.3 Marginal Costs and Variable Costs The area under the

marginal cost curve gives the variable costs.

y

MCMC

Variable costs

EXAMPLE: Specific Cost Curves

c(y)=y2+1

cv(y)=y2

cf(y)=1

AVC(y) = y2/y=y AFC(y) = 1/y AC(y) =(y2+1)/y= y+1/y MC(y) = 2y

y

MC

AC AVC

AC

MC

AVC

EXAMPLE: Marginal Cost Curves for Two Plants

The overall marginal cost curve is the horizontal sum of the marginal cost curves for two plants.

y1

MC

y2

MC

y1+y2

MC

MC1

MC2 MC

y1* y2

* y1*+y2*

c

21.4 Long-Run Costs

The short-run cost to produce output y must always be at least as large as the long-run cost to produce y.

c(y)≤cs(y, k*)

In fact, at one particular level of y, namely y*, we know that

c(y*)=cs(y*, k*)

21.4 Long-Run Costs

The short-run average cost curve must be tangent to the long-run average cost curve.

y

AC SAC=c(y,k*)/y

LAC=c(y)/y

y*

21.4 Long-Run Costs

The long-run average cost curve is the envelope of the short-run average cost.

21.5 Discrete Levels of Plant Size

Only a few different levels of plant size to choose from.

For any level of output y, we just choose the plant size that gives us the minimum cost of producing that output level.

21.5 Discrete Levels of Plant Size

The long-run average cost curve will be the lower envelope of the short-run average costs.

y

AC Short-run average cost curves

Long-run average cost curves

21.6 Long-Run Marginal Costs The long-run marg

inal cost curve will consist of various segments of the short-run marginal cost curves.

21.6 Long-Run Marginal Costs

The long-run average cost curve is tangent to the short-run average cost curve.

The long-run marginal cost curve crosses the short-run marginal cost curve.

yy*

AC

MC

SMCSAC

LMC

LAC