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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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Graphical Analysis of Total Costs
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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Graphical Analysis of Total Costs
Output
Total
costs
C
Total costs risedramatically as
output increases
after diminishing
returns set in
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Graphical Analysis of Total Costs
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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Important Points to Note:
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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Important Points to Note:
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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Important Points to Note:
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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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 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