KULIAH 6

Cost Functions

Dr. Amalia A. Widyasanti

Program Pasca Sarjana Ilmu Akuntansi

FE-UI, 2010

Two Simplifying Assumptions• There are only two inputs

– homogeneous labor (l), measured in labor-hours

– homogeneous capital (k), measured in machine-hours

• entrepreneurial costs are included in capital costs

• Inputs are hired in perfectly competitive markets– firms are price takers in input markets

Economic Profits• Total costs for the firm are given by

total costs = C = wl + vk

• Total revenue for the firm is given bytotal revenue = pq = pf(k,l)

• Economic profits () are equal to = total revenue - total cost

= pq - wl - vk

= pf(k,l) - wl - vk

Economic Profits• Economic profits are a function of the

amount of k and l employed– we could examine how a firm would choose

k and l to maximize profit• “derived demand” theory of labor and capital

inputs

– for now, we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs

Cost-Minimizing Input Choices• Minimum cost occurs where the RTS is

equal to w/v– the rate at which k can be traded for l in

the production process = the rate at which they can be traded in the marketplace

Cost Minimization • Total costs = C = wl +vk• Production function: q = f(k,l) = qo• Lagrangian expression:

0

( ( , ))

:

0

0

( , ) 0

oL wl vk q f k l

FOC

L fw

l lL f

vk kL

q f k l

Cost Minimization• Suppose that the production function is

Cobb-Douglas:

q = k l

• The Lagrangian expression for cost minimization of producing q0 is

ℒ = vk + wl + (q0 - k l )

Cost Minimization

• The FOCs for a minimum are

ℒ /k = v - k -1l = 0

ℒ /l = w - k l -1 = 0

ℒ/ = q0 - k l = 0

Cost Minimization• Dividing the first equation by the second

gives us

RTSk

k

k

v

w

ll

l1

1

• This production function is homothetic– the RTS depends only on the ratio of the two

inputs– the expansion path is a straight line

Cost-Minimizing Input Choices• The inverse of this equation is also of

interest

kf

v

f

w

l

• The Lagrangian multiplier shows how the extra costs that would be incurred by increasing the output constraint slightly

q0

Given output q0, we wish to find the least costly point on the isoquant

C1

C2

C3

Costs are represented by parallel lines with a slope of -w/v

Cost-Minimizing Input Choices

l per period

k per period

C1 < C2 < C3

C1

C2

C3

q0

The minimum cost of producing q0 is C2

Cost-Minimizing Input Choices

l per period

k per period

k*

l*

The optimal choice is l*, k*

This occurs at the tangency between the isoquant and the total cost curve

The Firm’s Expansion Path• The firm can determine the cost-

minimizing combinations of k and l for every level of output

• If input costs remain constant for all amounts of k and l, we can trace the locus of cost-minimizing choices– called the firm’s expansion path

The Firm’s Expansion Path

l per period

k per period

q00

The expansion path is the locus of cost-minimizing tangencies

q0

q1

E

The curve shows how inputs increase as output increases

The Firm’s Expansion Path• The expansion path does not have to be

a straight line– the use of some inputs may increase faster

than others as output expands• depends on the shape of the isoquants

• The expansion path does not have to be upward sloping– if the use of an input falls as output expands,

that input is an inferior input

Cost Minimization• Suppose that the production function is

CES:

q = (k + l )/

• The Lagrangian expression for cost minimization of producing q0 is

ℒ = vk + wl + [q0 - (k + l )/]

Cost Minimization

• The FOCs for a minimum are

ℒ /k = v - (/)(k + l)(-)/()k-1 = 0

ℒ /l = w - (/)(k + l)(-)/()l-1 = 0

ℒ / = q0 - (k + l )/ = 0

Cost Minimization• Dividing the first equation by the second

gives us

/1111

ll

kk

kv

w

• This production function is also homothetic

Total Cost Function

• The total cost function shows that for any set of input costs and for any output level, the minimum cost incurred by the firm is

C = C(v,w,q)

• As output (q) increases, total costs increase

Average Cost Function

• The average cost function (AC) is found by computing total costs per unit of output

q

qwvCqwvAC

),,(),,( cost average

Marginal Cost Function

• The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

q

qwvCqwvMC

),,(

),,( cost marginal

Graphical Analysis of Total Costs• Suppose that k1 units of capital and l1 units of

labor input are required to produce one unit of output

C(q=1) = vk1 + wl1

• To produce m units of output (assuming constant returns to scale)

C(q=m) = vmk1 + wml1 = m(vk1 + wl1)

C(q=m) = m C(q=1)

Graphical Analysis of Total Costs

Output

Totalcosts

C

With constant returns to scale, total costsare proportional to output

AC = MC

Both AC andMC will beconstant

Graphical Analysis of Total Costs• Suppose that total costs start out as concave

and then becomes convex as output increases– one possible explanation for this is that there is a

third factor of production that is fixed as capital and labor usage expands

– total costs begin rising rapidly after diminishing returns set in

Graphical Analysis of Total Costs

Output

Totalcosts

C

Total costs risedramatically asoutput increasesafter diminishingreturns set in

Graphical Analysis of Total Costs

Output

Average and

marginalcosts MC

MC is the slope of the C curve

AC

If AC > MC, AC must befalling

If AC < MC, AC must berising

min AC

Shifts in Cost Curves

• Cost curves are drawn under the assumption that input prices and the level of technology are held constant– any change in these factors will cause

the cost curves to shift

Some Illustrative Cost Functions

• Suppose we have a Cobb-Douglas technology such that

q = f(k,l) = k l

• Cost minimization requires that

l

k

v

w

l

v

wk

Some Illustrative Cost Functions

• If we substitute into the production function and solve for l, we will get

///

/1 vwql

• A similar method will yield

//

/

/1 vwqk

Some Illustrative Cost Functions

• Now we can derive total costs as

///1),,( wBvqwvkqwvC l

where //)(B

which is a constant that involves only the parameters and

Some Illustrative Cost Functions

• Suppose we have a CES technology such that

q = f(k,l) = (k + l )/

• To derive the total cost, we would use the same method and eventually get

/)1(1/1//1 )(),,( wvqwvkqwvC l

1/111/1 )(),,( wvqqwvC

Input Substitution• A change in the price of an input will

cause the firm to alter its input mix

• The change in k/l in response to a change in w/v, while holding q constant is

vw

kl

Input Substitution• Putting this in proportional terms as

)/ln(

)/ln(

/

/

)/(

)/(

vw

k

k

vw

vw

ks

l

l

l

gives an alternative definition of the elasticity of substitution– in the two-input case, s must be nonnegative– large values of s indicate that firms change their

input mix significantly if input prices change

Partial Elasticity of Substitution• The partial elasticity of substitution between

two inputs (xi and xj) with prices wi and wj is given by

)/ln(

)/ln(

/

/

)/(

)/(

ij

ji

ji

ij

ij

jiij ww

xx

xx

ww

ww

xxs

• Sij is a more flexible concept than – it allows the firm to alter the usage of inputs other

than xi and xj when input prices change

Size of Shifts in Costs Curves

• The increase in costs will be largely influenced by

– the relative significance of the input in the production process

– the ability of firms to substitute another input for the one that has risen in price

Technical Progress• Improvements in technology also lower

cost curves

• Suppose that total costs (with constant returns to scale) are

C0 = C0(q,v,w) = qC0(v,w,1)

Technical Progress• Because the same inputs that produced

one unit of output in period zero will produce A(t) units in period t

Ct(v,w,A(t)) = A(t)Ct(v,w,1)= C0(v,w,1)

• Total costs are given by

Ct(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t)

= C0(v,w,q)/A(t)

Shifting the Cobb-Douglas Cost Function

• The Cobb-Douglas cost function is

///1),,( wBvqwvkqwvC l

where //)(B

• If we assume = = 0.5, the total cost curve is greatly simplified:

5.05.02),,( wqvwvkqwvC l

Shifting the Cobb-Douglas Cost Function

• If v = 3 and w = 12, the relationship is

qqqC 12362),12,3(

– C = 480 to produce q =40– AC = C/q = 12– MC = C/q = 12

Shifting the Cobb-Douglas Cost Function

• If v = 3 and w = 27, the relationship is

qqqC 18812),27,3(

– C = 720 to produce q =40– AC = C/q = 18– MC = C/q = 18

Shifting the Cobb-Douglas Cost Function

• Suppose the production function is5.05.003.05.05.0

)( ll kektAqt

– we are assuming that technical change takes an exponential form and the rate of technical change is 3 percent per year

Shifting the Cobb-Douglas Cost Function

• The cost function is then

t

t ewqvtAqwvC

qwvC03.05.05.00 2

)(),,(

),,(

– if input prices remain the same, costs fall at the rate of technical improvement

Short-Run, Long-Run Distinction

• In the short run, economic actors have only limited flexibility in their actions

• Assume that the capital input is held constant at k1 and the firm is free to vary only its labor input

• The production function becomes

q = f(k1,l)

Short-Run Total Costs

• Short-run total cost for the firm is

SC = vk1 + wl

• There are two types of short-run costs:– short-run fixed costs are costs associated

with fixed inputs (vk1)

– short-run variable costs are costs associated with variable inputs (wl)

Short-Run Total Costs

• Short-run costs are not minimal costs for producing the various output levels– the firm does not have the flexibility of input

choice– to vary its output in the short run, the firm

must use nonoptimal input combinations– the RTS will not be equal to the ratio of

input prices

Short-Run Total Costs

l per period

k per period

q0

q1

q2

k1

l1 l2 l3

Because capital is fixed at k1,the firm cannot equate RTSwith the ratio of input prices

Cost Functions

• Cost Function:- the value of the conditional factor demands

- the minimum cost of producing y unit of output

• Short-run cost function:

- the factors of production are fixed at predetermined levels

- the price vectors and the variable vectors are composed of : FIXED AND

VARIABLE FACTORS

( , ) ( , )c w y wx w y

( , , ) ( , , ) ( , , )f v v f f f fc w y x w x w y x w x w y x

,

,

( )

( )

v f

v f

w w w

x x x

SVC Fixed Cost

Total, Average, and Marginal Costs

• Short-Run Total Cost = STC =

• Short-Run Average Cost = SAC =

• Short-run average variable cost= SAVC =

• Short-run average fixed cost = SAFC =

• Short-run marginal cost = SMC =

( , , ) ( , , ) ( , , )f v v f f f fc w y x w x w y x w x w y x

( , , )fc w y x

y

( , , )v v fw x w y x

y

f fw x

y

( , , )fc w y x

y

Long Run Cost

Long run cost:

Long-run average cost:

Long-run marginal cost:

Note: Long-run fixed cost are zero

( , ) ( , ) ( , ) ( , , ( , ))v v f f fc w y w x w y w x w y c w y x w y

( , )c w yLAC

y

( , )c w yLMC

y

Short-Run and Long-Run Costs

Output

Total costs

SC (k0)

SC (k1)

SC (k2)

The long-runC curve canbe derived byvarying the level of k

q0 q1 q2

C

Geometry of Costs (3)

output

AFCOutput naik AFC turun

output

AVC Output naik AVC naik

output

AC

MinimumEfficient scale

Short-Run and Long-Run Costs

Output

Costs

The geometric relationshipbetween short-run and long-runAC and MC canalso be shown

q0 q1

AC

MCSAC (k0)SMC (k0)

SAC (k1)SMC (k1)

Long-run and Short-run Cost Curve

output

AC

y*

AC(y*,z*)

Short-Run and Long-Run Costs

• At the minimum point of the AC curve:– the MC curve crosses the AC curve

• MC = AC at this point

– the SAC curve is tangent to the AC curve• SAC (for this level of k) is minimized at the same

level of output as AC• SMC intersects SAC also at this point

AC = MC = SAC = SMC

Exercise (1)

• Suppose that a firm uses two inputs x1 and x2 with cobb-douglas technology

If the firm is restricted to operate at level of k. Calculate:

(a) Short-run cost

(b) SAC

(c) SAVC

(d) SAFC

(e) SMC

(f) Long-run cost

0.3 0.71 2y ax x

Exercise (2)

Production function for the book:

Where: q = the number of pages in the finished book; S = the number of working hours spent by Smith, and J = the number of hours spent working by Jones.

Smith values his labor as $ 3 per working hour. He has spent 900 hours preparing the first draft. Jones, whose labor is valued at $ 12 per working hour, will revise Smith’s draft to complete the book.

a. How many hours will Jones have to spend to produce a finished book of 150 pages?

b. What is the marginal cost of th 150th page of the finished book?

1/ 2 1/ 2q S J