lecture 5. cost functions

7/30/2019 Lecture 5. Cost Functions

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Lecture 5

Chapter 10

Costs

Cost Functions

Nicholson and Snyder, Copyright 2008 by Thomson South-Western. All rights reserved.


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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 competitivemarkets

firms are price takers in input markets


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Economic Profits Total costs for the firm are given bytotal 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


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Economic Profits Economic profits are a function of theamount ofkand l employed

we could examine how a firm would choosekand l to maximize profit

derived demand theory of labor and capital

inputs

for now, we will assume that the firm hasalready chosen its output level (q0) and

wants to minimize its costs


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Cost-Minimizing Input Choices Minimum cost occurs where the RTSis

equal to w/v

the rate at which kcan be traded forl inthe production process = the rate at which

they can be traded in the marketplace


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Cost-Minimizing Input Choices Dividing the first two conditions we get

)for(/

/

kRTSkf

f

v

wl

l

The cost-minimizing firm should equate

the RTSfor the two inputs to the ratio of

their prices


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Cost-Minimizing Input Choices Cross-multiplying, we get

w

f

v

fk l

For costs to be minimized, the marginal

productivity per dollar spent should be

the same for all inputs


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

interest

kfv

fw

l

The Lagrangian multiplier shows how

the extra costs that would be incurredby increasing the output constraint

slightly


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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

kper period

C1 < C2 < C3


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C1

C2

C3

q0

The minimum cost of producing q0 is C2

Cost-Minimizing Input Choices

l per period

kper period

k*

l*

The optimal choice

is l*, k*

This occurs at the

tangency between theisoquant and the total cost

curve


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Contingent Demand for Inputs In Chapter 4, we considered an

individuals expenditure-minimization

problem to develop the compensated demand for a

good

Can we develop a firms demand for aninput in the same way?


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Contingent Demand for Inputs

In the present case, cost minimization

leads to a demand for capital and labor

that is contingent on the level of outputbeing produced

The demand for an input is a derived

demand it is based on the level of the firms output


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Cost Minimization Suppose that the production function is

Cobb-Douglas:

q= kl

The Lagrangian expression for cost

minimization of producing q0 is

= vk+ wl + (q0 - kl)


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Cost Minimization The FOCs for a minimum are

/k= v- k-1l= 0

/l = w- kl-1 = 0

/ = q0 - kl = 0


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Cost Minimization Dividing the first equation by the second

gives us

RTSkkk

vw

ll

l1

1

This production function is homothetic

the RTSdepends only on the ratio of the two

inputs


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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 thefirm is

C= C(v,w,q)

As output (q) increases, total costsincrease


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Average Cost Function The average cost function (AC) is found

by computing total costs per unit of

output

q

qwvCqwvAC

),,(),,(costaverage


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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

),,(),,(costmarginal


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Graphical Analysis of Total Costs

Suppose that k1 units of capital and l1

units of labor input are required to

produce one unit of outputC(q=1) = vk1 + wl1

To produce munits of output (assuming

constant returns to scale)C(q=m) = vmk1 + wml1 = m(vk1 + wl1)

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


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Output

Total

costs

C

With constant returns to scale, total costs

are proportional to output

AC= MC

Both ACand

MCwill beconstant


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Output

Total

costs

C

Total costs risedramatically as

output increases

after diminishing

returns set in


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Output

Average

and

marginal

costsMC

MCis the slope of the Ccurve

AC

IfAC> MC,

ACmust befalling

IfAC< MC,

ACmust berising

min AC


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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


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Properties of Cost Functions

Homogeneity

cost functions are all homogeneous of

degree one in the input prices a doubling of all input prices will not change the

levels of inputs purchased

inflation will shift the cost curves up


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Properties of Cost Functions

Nondecreasing in q, v, and w

cost functions are derived from a cost-

minimization process any decline in costs from an increase in one of

the functions arguments would lead to a

contradiction


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Contingent Demand for Inputs Contingent demand functions for all of

the firms inputs can be derived from the

cost function Shephards lemma

the contingent demand function for any input is

given by the partial derivative of the total-costfunction with respect to that inputs price


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Contingent Demand for Inputs

wCqwv

v

Cqwvk

c

c

),,(

),,(

l


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Short-Run, Long-Run Distinction

In the short run, economic actors have

only limited flexibility in their actions

Assume that the capital input is heldconstant at k1 and the firm is free to

vary only its labor input

The production function becomesq= f(k1,l)


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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 costsassociated with variable inputs (wl)


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Short-Run Total Costs Short-run costs are not minimal costs

for producing the various output levels

(except at the combination k1,l2) the firm does not have the flexibility of input

choice

to vary its output in the short run, the firmmust use non optimal input combinations

the RTSwill not be equal to the ratio of

input prices


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Short-Run Total Costs

l per period

kper period

q0

q1

q2

k1

l1 l2 l3

Because capital is fixed at k1,

the firm cannot equate RTS

with the ratio of input prices


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Short-Run Marginal and Average Costs

The short-run average total cost (SAC)

function is

SAC= total costs/total output = SC/q

The short-run marginal cost (SMC) function

is

SMC= change in SC/change in output = SC/q


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Short-Run and Long-Run Costs

Output

Total

costs

SC(k0)

SC(k1)

SC(k2)

The long-run

Ccurve can

be derived by

varying thelevel ofk

q0 q1 q2

C


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Short-Run and Long-Run Costs

Output

Costs

The geometric

relationshipbetween short-

run and long-run

ACand MCcan

also be shown

q0

q1

AC

MCSAC(k0)SMC(k

0)

SAC(k1)SMC(k1)


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Short-Run and Long-Run Costs At the minimum point of the ACcurve:

the MCcurve crosses the ACcurve

MC= ACat this point

the SACcurve is tangent to the ACcurve

SAC(for this level ofk) is minimized at the same

level of output as AC

SMCintersects SACalso at this point

AC= MC= SAC= SMC


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Important Points to Note:

A firm that wishes to minimize the

economic costs of producing a

particular level of output shouldchoose that input combination for

which the rate of technical substitution

(RTS) is equal to the ratio of the

inputs rental prices


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Repeated application of this

minimization procedure yields the

firms expansion path the expansion path shows how input

usage expands with the level of output

it also shows the relationship between output

level and total cost

this relationship is summarized by the total

cost function, C(v,w,q)


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The firms average cost (AC= C/q)and marginal cost (MC= C/q) can

be derived directly from the total-costfunction

if the total cost curve has a general cubic

shape, the ACand MCcurves will be u-

shaped


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Input demand functions can be derived

from the firms total-cost function

through partial differentiation these input demands will depend on the

quantity of output the firm chooses to

produce

are called contingent demand functions


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In the short run, the firm may not be

able to vary some inputs

it can then alter its level of productiononly by changing the employment of its

variable inputs

it may have to use nonoptimal, higher-

cost input combinations than it wouldchoose if it were possible to vary all

inputs

lecture 5. cost functions

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