Chapter 12 COSTS Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS.

Chapter 12

COSTS

Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

Definitions of Costs

• It is important to differentiate between accounting cost and economic cost– the accountant’s view of cost stresses out-

of-pocket expenses, historical costs, depreciation, and other bookkeeping entries

– economists focus more on opportunity cost

Definitions of Costs

• Labor Costs– to accountants, expenditures on labor are

current expenses and hence costs of production

– to economists, labor is an explicit cost• labor services are contracted at some hourly

wage (w) and it is assumed that this is also what the labor could earn in alternative employment

Definitions of Costs• Capital Costs

– accountants use the historical price of the capital and apply some depreciation rule to determine current costs

– economists refer to the capital’s original price as a “sunk cost” and instead regard the implicit cost of the capital to be what someone else would be willing to pay for its use

• we will use v to denote the rental rate for capital

Definitions of Costs• Costs of Entrepreneurial Services

– To an accountant, the owner of a firm is entitled to all profits, which are the revenues or losses left over after paying all input costs

– Economists consider the opportunity costs of time and funds that owners devote to the operation of their firms

• these services are inputs and some cost should be imputed to them

• part of accounting profits would be considered as entrepreneurial costs by economists

Economic Cost

• The economic cost of any input is the payment required to keep that input in its present employment– the remuneration the input would receive in

its best alternative employment

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 = TC = 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 capital and labor employed– we could examine how a firm would choose

K and L to maximize profit• “derived demand” theory of labor and capital

inputs (see Chapter 21)

• But 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

• To minimize the cost of producing a given level of output, a firm should choose a point on the isoquant at which the RTS is equal to the ratio w/v– it should equate the rate at which K can be

traded for L in the productive process to the rate at which they can be traded in the marketplace

Cost-Minimizing Input Choices

• Mathematically, we seek to minimize total costs given q = f(K,L) = q0

• Setting up the Lagrangian

L = wL + vK + [q0 - f(K,L)]

• First order conditions are

L/L = w - (f/L) = 0

L/K = v - (f/K) = 0

L/ = q0 - f(K,L) = 0

Cost-Minimizing Input Choices

• Dividing the first two conditions we get

) for ( /

/KLRTS

Kf

Lf

v

w

• The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices

q0

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

TC1

TC2

TC3

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

Cost-Minimizing Input Choices

L per period

K per period

TC1 < TC2 < TC3

TC1

TC2

TC3

q0

The minimum cost of producing q0 is TC2

Cost-Minimizing Input Choices

L per period

K per period This occurs at the tangency between the isoquant and the total cost curve

K*

L*

The optimal choice is L*, K*

Output Maximization

• The dual formulation of the firm’s cost minimization problem is to maximize output for a given level of cost

• The Lagrangian is

L = f(K,L) + D(TC1 - wL - vK)

• The first-order conditions are identical to those for the primal problem

q00

The maximum output attainable with total cost TC2 is q0

q0

q1

TC2 = wL + vK

Output Maximization

L per period

K per periodThis occurs at the tangency between the total cost curve and isoquant q0

K*

L*

The optimal choice is L*, K*

Derived Demand

• In Chapter 5, we considered how the utility-maximizing choice is affected by the change in the price of a good– we used this technique to develop the

demand curve for a good

• Can we develop a firm’s demand for an input in the same way?

Derived Demand

• To analyze what happens to K* if v changes, we must know what happens to the output level chosen by the firm

• The demand for K is a derived demand– it is based on the level of the firm’s output

• We cannot answer questions about K* without looking at the interaction of supply and demand in the output market

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 the firm may demand, 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– 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

Minimizing Costs for a Cobb-Douglas Function

• Suppose that the production function for hamburgers is

q = 10K 0.5 L 0.5

• Total costs are given by

TC = vK + wL

• Suppose that the firm wishes to minimize the cost of producing 40 hamburgers

Minimizing Costs for a Cobb-Douglas Function

• The Lagrangian expression is

L = vK + wL + (40 - 10K 0.5 L 0.5)

• The first-order conditions are

L/K = v - 5(L/K)0.5 = 0

L/L = w - 5(K/L)0.5 = 0

L/ = 40 - 10K 0.5 L 0.5 = 0

Minimizing Costs for a Cobb-Douglas Function

• Dividing the first equation by the second gives us

RTSL

K

v

w

• This production function exhibits constant returns to scale so the expansion path is a straight line

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

TC = TC(v,w,q)

• As output increases, total costs increase

Average Cost Function

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

q

qwvTCqwvAC

),,(),,( 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

qwvTCqwvMC

),,(

),,( cost marginal

Graphical Analysis ofTotal Costs

• Suppose that K1 units of capital and L1 units of labor input are required to produce one unit of output

TC(q=1) = vK1 + wL1

• To produce m units of outputTC(q=m) = vmK1 + wmL1 = m(vK1 + wL1)

TC(q=m) = m TC(q=1)

• TC is proportional to q

Graphical Analysis ofTotal Costs

Output

Totalcosts

TC

Total costs are proportional to output

AC = MC

Both AC andMC will beconstant

Graphical Analysis ofTotal Costs

• Suppose instead that total costs start out as concave and then becomes convex as output increases– one possible explanation for this is that

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

– total costs begin rising rapidly after diminishing returns set in

Graphical Analysis ofTotal Costs

Output

Totalcosts

TC

Total costs risedramatically asoutput risesafter diminishingreturns set in

Graphical Analysis ofTotal Costs

Output

Average and

marginalcosts

AC

MC

MC is the slope of the TC curve

If AC > MC, AC must befalling

If AC < MC, AC must berising

min AC

Shifts in Cost Curves

• The 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

Homogeneity• The total cost function is homogeneous

of degree one in input prices– if all input prices were to increase in the

same proportion (t), the total costs for producing any given output level would also be multiplied by t

– a simultaneous increase in input prices does not change the ratio of input prices

• cost-minimizing combination of K and L unchanged

Homogeneity• The average and marginal cost functions

will also be homogeneous of degree one in input prices

• In a “pure” inflationary period (one where the prices of all inputs rise at the same rate), there will be no incentive for firms to alter their input choices

Change in the Priceof One Input

• If the price of only one input changes, the firm’s cost-minimizing choice of inputs will be affected– a new expansion path must be derived

Change in the Priceof One Input

• An increase in the price of one input must increase TC for any output level

• AC will also rise

• If the input is not inferior, MC will also rise

Change in the Priceof One Input

• A change in the price of an input means that the firm must alter its cost-minimizing choice of inputs– in the two-input case, an increase in w/v will

be met by an increase in K/L

• Define the elasticity of substitution as

)/ln(

)/ln(

/

/

)/(

)/(

vw

LK

LK

vw

vw

LKs

– s must be nonnegative

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 because 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

• If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs

Technical Progress

• Improvements in technology also lower cost curves

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

TC0 = TC0(q,v,w) = C0(v,w)q

where C0(v,w) is the initial cost of producing one unit of output

Technical Progress• If the production function is

q = A(t)f(K,L)

then unit costs at time t are given by

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

and total costs are given by

TCt(q,v,w) = Ct(v,w)q = TC0/A(t)

• TC falls over time at the rate of technical change– AC and MC also fall at the rate of A(t)

Cobb-Douglas Cost Function

• Suppose that the production function isq = 10K 0.5L0.5

• First-order conditions for cost minimization require that

w/v = K/L

• Dividing by K yields

q/k = 10(v/w)0.5

Cobb-Douglas Cost Function

• This means thatK = (q/10)w 0.5v -0.5

• Multiplying both sides by v, we get

vK = (q/10)w 0.5v 0.5

• A similar chain of substitutions yields

wL = (q/10)w 0.5v 0.5

Cobb-Douglas Cost Function

• BecauseTC = vK + wL

we have

TC = (2/10)qw 0.5v 0.5

• If w = v = $4, then

TC = 0.8q

Cobb-Douglas Cost Function• Because the production function exhibits constant

returns to scale, AC and MC will be constantAC = TC/q = 0.8

MC = TC/q = 0.8

• If v rises to $9, TC, AC, and MC riseTC = (2/10)qw 0.5v 0.5 = 1.2q

AC = TC/q = 1.2

MC = TC/q = 1.2

Cobb-Douglas Cost Function• If we experience technical progress then

q = A(t)f(K,L) = e0.05tf(K,L)

• Total costs at any time are given by

TCt = TC0/A(t) = e-0.05t[0.2qw 0.5v 0.5]

• After 10 years,

TC10 = 0.48q

– costs fall by 40 percent from their previous level of 0.8q

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

STC = vK1 + wL

• There are two types of short-run costs:– short-run fixed costs (SFC) are costs

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

associated with variable inputs

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

K1

L1 L2 L3

q0

q1

q2

Because capital is fixed at K1,the firm cannot equate RTSwith the ratio of input prices

Short-Run Marginal and Average Costs

• The short-run average total cost (SATC) function is

SATC = total costs/total output = STC/q

• The short-run marginal cost (SMC) function is

SMC = change in STC/change in output = STC/q

Short-Run Average Fixed and Variable Costs

• Short-run average fixed costs (SAFC) are

SAFC = total fixed costs/total output = SFC/q

• Short-run average variable costs areSAVC = total variable costs/total output = SVC/q

Relationship between Short-Run and Long-Run Costs

Output

Total costs

STC (K0)

STC (K1)

STC (K2)

The long-runTC curve canbe derived byvarying the level of K

q0 q1 q2

TC

Relationship between Short-Run and Long-Run Costs

Output

Costs

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

q0 q1

AC

MCSATC (K0)SMC (K0)

SATC (K1)SMC (K1)

Relationship between 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 SATC curve is tangent to the AC curve• SATC (for this level of K) is minimized at the same

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

– the following are all equal:

AC = MC = SATC = SMC

Important Points to Note:

• A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs’ rental prices

Important Points to Note:• Repeated application of this minimization

procedure yields the firm’s expansion path– the expansion path shows how input usage

expands with the level of output

• The relationship between output level and total cost is summarized by the total cost function [TC(q,v,w)]– the firm’s average cost (AC = TC/q) and

marginal cost (MC = TC/q) can be derived directly from the total cost function

Important Points to Note:• All cost curves are drawn on the

assumption that the input prices and technology are held constant– when an input price changes, cost curves

shift to new positions• the size of the shifts will be determined by the

overall importance of the input and by the ease with which the firm may substitute one input for another

– technical progress will also shift cost curves

Important Points to Note:

• In the short run, the firm may not be able to vary some inputs– it can then alter its level of production only by

changing the employment of its variable inputs

– it may have to use nonoptimal, higher-cost input combinations than it would choose if it were possible to vary all inputs