Chap. 22 COST CURVES. Cost... · 2019. 9. 5. · Cost curves (Geometry of costs) §Total cost function can be derived from the cost min. problem: c(w 1,…,w n, y) §Assume that the

Engineering Economic Analysis2019 SPRING

Prof. D. J. LEE, SNU

Chap. 22

COST CURVES

Average cost & Marginal cost

§ Total cost function can be derived from the cost min. problem: c(w1,…,wn, y)

§ Various costs can be defined based on the total cost function• variable cost, fixed cost, • average cost, marginal cost, • long-run cost, short-run cost

1


§ Let

2

( ): vector of fixed inputs, : vector of variable factors

,

f v

v f

x x

w w w=

§ Short-run cost function ( ) ( ), , , ,f v v f f fc w y x w x w y x w x= × + ×

• Short-run average cost (SAC): ( ), , fc w y x

y

• Short-run average variable cost (SAVC): ( ), ,v v fw x w y x

y

×

• Short-run average fixed cost (SAFC): f fw xy×

• Short-run marginal cost (SMC): ( ), , fc w y w

y

¶

¶


3

§ Long-run cost function ( ) ( ) ( ), , ,v v f fc w y w x w y w x w y= × + ×

• Long-run average cost (LAC): ( ),c w y

y

• Long-run marginal cost (LMC): ( ),c w y

y

¶

¶

• Note that ‘long-run average cost’ equals ‘long-run average variable cost’ and ‘long-run fixed costs’ are zero’

There is no fixed input


4

§ Example: Short-run Cobb-Douglas cost function • In a short-run, 2x k=

• Cost-min. problem: 1 1 21

1

min

. . a a

w x w k

s t y x k -

+

=

1 1

1

aa ax k y-

= ×

• SR Cost function: ( )1 1

1 2, ,aa ac w y k w k y w k-

= × +

1

21

1

1

2

1

11

aa

aa

aa

w kySAC wk y

ySAVC wk

w kSAFCy

ySMC wa k

-

-

-

æ ö= +ç ÷è ø

æ ö= ç ÷è ø

=

æ ö= ç ÷è ø

Cost curves (Geometry of costs)

§ Total cost function can be derived from the cost min. problem: c(w1,…,wn, y)

§ Assume that the factor prices to be fixed, then c(y).

§ Total cost, c(y), is assumed to be monotonic in y.§ Various cost curves: Average cost curve, Marginal

cost curve, LR cost curve, SR cost curve etc.§ How are these cost curves related to each other?

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§ SR Average cost• SR total cost = VC + FC:• SAC = SAVC + SAFC

6

( ) ( )vc y c y F= +

( ) ( ) ( ), ,v v f f fvw x w y xc y c y F w x

y y y y y

× ×= + = +

U-shapedSAC


§ Marginal cost vs. Average cost

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( ) ( ) ( )2

AC( ) c y c y y c yd y ddy dy y y

¢ -æ ö= =ç ÷

è ø

( ) ( ) ( )1 1 MC( ) AC( )c y

c y y yy y yæ ö¢= - = -ç ÷

è ø

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

AC is increasing (AC ( ) 0)

AC is decreasing (AC ( ) 0)

AC is minimum (AC ( ) 0)

y MC y AC y

y MC y AC y

y MC y AC y

¢ > Û >

¢ < Û <

¢ = Û =

• Thus,

• MC for the first small unit of amount equals AVC for a single unit of output

( ) ( ) ( ) ( ) ( )1 0 1(1) (0)1 1

1 1 1v v vc F c F cTC TCMC AVC

+ - --= = = =


§ Marginal cost vs. Average cost

8

AC decreasing AC increasing


§ Marginal cost vs. Variable cost• Since

9

( ) ( ) vdc y

MC ydy

=0

( ) ( )y

vc y MC z dz= ò

• The area beneath the MC curve up to y gives us the variable cost of producing y units of output

MC(y)

y0

Area is the variablecost of making y� units

Cost

y¢


§ Example• SR total cost function:

10

( ) 2 1c y y= +

• Variable cost:• Fixed cost:

( ) 2vc y y=

( ) 1fc y =

( )( )

2AVC /

AFC 1/

y y y y

y y

= =

=

( )AC 1/y y y= +

( )MC 2y y=


§ Example: C-D Technology• Recall that

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( )

( )

1

1

a

aa

c y Ky F

FAC y Kyy

-

= +

= +

( )1 1

1 2 1 2, ,b a

a ba b a ba b a b a b a ba ac w w y A w w y

b b

-- + ++ + + +é ùæ ö æ öê ú= +ç ÷ ç ÷ê úè ø è øë û

• For a fixed w1, w2, ( )1

, 1a bc y Ky a b+= + £

( )

( )

1

1

a ba b

a ba b

AC y Ky

KMC y ya b

- -+

- -+

=

=+

• In the short-run, recall that ( )1 1

1 2, ,aa ac w y k w k y w k-

= × +


§ Example: MC curves for two plant

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• How much should you produce in each plant?

c1(y)

Plant 1

y1 c2(y)

Plant 2

y2

( ) ( )* *1 1 2 2MC y MC y=

{ }( ) ( )

1 21 1 2 2,

1 2

min

. . y y

c y c y

s t y y y

+

+ =

2 1y y y= -( ) ( )

11 1 2 1min

yc y c y y+ -

( ) ( )1 1 2 2 2 2

1 1 2 1 1

0 and 1dc y dc y dy dyc

y dy dy dy dy¶

= + = = -¶

• Therefore, the optimality condition is

Long-run vs. Short-run Cost Curves

§ In the long run, all inputs are variable• LR problem: Planning the type and scale investment

• SR problem: Optimal operation

§ Given a fixed factor: Plant size k• SR cost function: ( ),sc y k• In LR, let the optimal plant size to produce y be k(y)

Øfirm’s conditional factor demand for plant size!

§ Then LR cost function is ( ) ( )( ),sc y c y k y= How this looks graphically?

13


• For some given level of output y*

• Since SR cost min problem is just a constrained version of the LR cost min problem, SR cost curve must be at least as large as the LR cost curve for all y

( )( )

( )( )

* * *

* *

: optimal plant size for

, : SR cost function for a given

, : LR cost function

s

s

k k y y

c y k k

c y k y

Þ =

Þ

Þ

( ) ( )( ) ( ) ( )

*

* * * * *

, for all level of

, when

s

s

c y c y k y

c y c y k k k y

£

= =

14


• Also

• Hence SR and LR cost curves must be tangent at y*

( ) ( )( ) ( ) ( )

*

* * * * *

LAC SAC , for all level of

LAC SAC , when

y y k y

y y k k k y

£

= =

15

ymin

( ) ( )( ) ( )

*min min

*min min

Note that LAC SAC , ,

that is, LAC SAC ,

y y k

y y k

¹

£LAC(ymin)

( )*minargmin SAC , = y k y

SAC(ymin)


16


17

• Discrete levels of plant size: 1 2 3 4, , ,k k k k

• LR average cost curve is the lower-envelop of SR average cost curves


§ LR marginal cost• When there are discrete levels of the fixed factor, the firm will

choose the amount of the fixed factor to minimize costs.

• Thus the LRMC curve will consist of the various segments of the SRMC curves associated with each different level of the fixed factor.

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§ LR marginal cost• This has to hold no matter how many different plant sizes

there are !

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LAC(y)

Cost

y

SACs

Short-Run & Long-Run Marginal Cost Curves

LAC(y)

LMC(y)Cost

y

SRMCs

20


§ LR marginal cost

21

• LR cost function

• Differentiating LR cost function w.r.t. y

• Since k* is the optimal at y=y*,

• Thus LRMC at y* equals to SR marginal cost at (k*, y*)

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Chap. 22 COST CURVES. Cost... · 2019. 9. 5. · Cost curves (Geometry of costs) §Total cost function can be derived from the cost min. problem: c(w 1,…,w n, y) §Assume that the