12. Cost Curves.pdf

(1)

Engineering Economic Analysis

2019 SPRING Prof. D. J. LEE, SNU

Chap. 22 COST CURVES

(2)

Average cost & Marginal cost

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

(3)

Average cost & Marginal cost

§ Let

( )

: 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 ^f

c w y x = w x w y x× + w x×

• Short-run average cost (SAC): ^{c w y x}

(

^{, ,} ^f

)

y

• Short-run average variable cost (SAVC): ^{w x w y x}^v ^v ( ^{, ,} ^f )

y

×

• Short-run average fixed cost (SAFC): ^{w x}^f _y^× ^f

• Short-run marginal cost (SMC): ^{c w y w}( ^{, ,} ^f )

y

¶

(4)

Average cost & Marginal cost

§ Long-run cost function

( )

^, ^v ^v

( )

^, ^f ^f

( )

^,

c 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

(5)

Average cost & Marginal cost

§ Example: Short-run Cobb-Douglas cost function

• In a short-run, x₂ = k

• Cost-min. problem: ^{1 1} ²

1 1

min

. . ^a ^a

w x w k s t y x k ^-

+

=

1 1

1

a

a a

x k y

-

= ×

• SR Cost function: ^{c w y k}

(

^{, ,}

)

⁼ ^{w k}¹ ââ^-¹ ^× ^y¹â ⁺ ^{w k}²

1

2 1

1

2

1

a a

w k SAC w y

k y

SAVC w y k SAFC w k

y SMC w y

a k

-

= æ öç ÷è ø +

= æ öç ÷è ø

=

= æ öç ÷è ø

(6)

Cost curves (Geometry of costs)

§ Total cost function can be derived from the cost min. problem: c(w₁,…,w_n, 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?

(7)

Cost curves (Geometry of costs)

§ SR Average cost

• SR total cost = VC + FC:

• SAC = SAVC + SAFC

( ) v ( )

c y = c y + F

( ) v ( ) w x w y x^v ^v

(

^{, ,} ^f

)

f f

c y c y F w x

y y y y y

× ×

= + = +

U-shaped SAC

(8)

§ Marginal cost vs. Average cost

( ) ( ) ( )

2

AC( ) c y c y y c y

d y d

dy dy y y

¢ -

æ ö

= ç ÷ =

è ø

( ) ( ) ( )

1 1

c y MC( ) AC( )

c y y y

y y y

æ ö

= ç ¢ - ÷ = -

è ø

( ) ( )

AC is increasing (AC ( ) 0) AC is decreasing (AC ( ) 0) AC is minimum (AC ( ) 0)

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 ⁽¹⁾ ⁽⁰⁾ ( )¹ ( )⁰ ( )¹ ( )¹

1 1 1

v v v

c F c F c

TC TC

MC - + - - AVC

= = = =

(9)

§ Marginal cost vs. Average cost

AC decreasing AC increasing

(10)

§ Marginal cost vs. Variable cost

• Since

( ) ( )

v MC y dc y

= dy

0

( ) ( )

y

c yv = òMC z dz

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

MC(y)

y 0

Area is the variable

cost of making y units Cost

y¢

(11)

§ Example

• SR total cost function: ^{c y}( ) ⁼ ^y² ⁺¹

• Variable cost:

• Fixed cost:

( ) ²

c yv = y

( ) ¹

cf y =

( ) ( )

AVC 2 / AFC 1/

y y y y

y y

= =

=

( )

AC y = +y 1/ y

( )

MC y = 2y

(12)

§ Example: C-D Technology

• Recall that

( ) ( )

1

1 a

a a

c y Ky F AC y Ky F

y

-

= +

( ₁^, ₂^, ) ¹ ₁ ₂ ¹

b a

a b

a b a b

a b a a a b a b a b

c w w y A w w y

b b

- + -+

+ + + +

é ù

æ ö æ ö

ê ú

= êç ÷è ø +ç ÷è ø ú

ë û

• For a fixed w₁, w₂, ^{c y}_{( )}= ^Ky^{a b}¹⁺ ^,^{a b}+ £¹

( ) ( )

1

1 a b a b

a b a b

AC y Ky MC y K y

a b

- - +

=

= +

• In the short-run, recall that ^{c w y k}

(

^{, ,}

)

⁼ ^{w k}¹ ââ^-¹ ^× ^y¹â ⁺ ^{w k}²

(13)

§ Example: MC curves for two plant

• How much should you produce in each plant?

c₁(y) Plant 1

y₁ c₂(y) Plant 2

y₂

( )^* ( )^*

1 1 2 2

MC y = MC y

{ } ( ) ( )

1, 2 1 1 2 2

1 2

min . .

y y c y c y

s t y y y + + =

2 1

y = y y-

( )

1 1 1 2 1

miny c y + c y y-

( ) ( )

1 1 2 2 2 2

1 1 2 1 1

0 and 1

dc y dc y dy dy

c

y dy dy dy dy

¶ = + = = -

¶

• Therefore, the optimality condition is

(14)

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: c y ks

( )

^,

• 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 ( ) ^s

(

^, ( )

)

c y = c y k y How this looks graphically?

(15)

Long-run vs. Short-run Cost Curves

• 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

£

= =

(16)

Long-run vs. Short-run Cost Curves

• 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

£

= =

y_min

( ) ( )

*

min min

*

min min

Note that LAC SAC , ,

that is, LAC SAC ,

y y k

¹

£

LAC(ymin)

(

^*

)

^min

argmin SAC ,y k = y

SAC(ymin)

(17)

Long-run vs. Short-run Cost Curves

(18)

Long-run vs. Short-run Cost Curves

• Discrete levels of plant size: k k k k₁, , ,₂ ₃ ₄

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

(19)

Long-run vs. Short-run 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.

(20)

Long-run vs. Short-run Cost Curves

§ LR marginal cost

• This has to hold no matter how many different plant sizes there are !

LAC(y) Cost

SACs

(21)

Short-Run & Long-Run Marginal Cost Curves

LAC(y) LMC(y)

Cost

y

SRMCs

(22)

Long-run vs. Short-run Cost Curves

§ LR marginal cost

• 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*)