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Econ 604 Advanced Microeconomics
Davis Spring 2006, April 20
Lecture 11
Reading.
Problems:
Chapter 12
Next time: Finish Ch. 12 (if
necessary) Ch. 13
12.3; 12.5; 12.7, 12.9
REVIEW
X. Chapter 11. Production Functions
A. Production Functions, and Marginal Productivity.
Relationship between Marginals, Averages and Totals (Problem 11.1)
B. Isoquant Maps and the Rate of Technical Substitution
RTS
(L for K)
=
-dK/dL| q=qo
RTS and Marginal Productivities
Different types of isoquant maps
Reasons for a Diminishing RTS. (quasi-concavity conditions)
(Totally differentiate RTS=fL/ fK
dRTS =
(fLK (fK) - fLfKK) /fK2 dK + (fLLfK - fLfKL)/fK2 dL
Then divide through by dL, using the fact that dK/dL = -fL/fK along an isoquant,
and Young’s theorem
dRTS/dL
=
(f2LfKK + f2KfLL-2 fKfLfLK ) /(fK)3
Observe that the above condition is negative if the products are complements
fLK>0, or if substitutes, if the own effects dominate the cross effects.)
C. Returns to Scale
Constant Returns to Scale and the RTS
q = f(X1, X2, .. , Xn)
Consider
mkq = f(mX1, mX2, .. , mXn)
k=1 we have constant returns to scale
k>1, increasing returns
k<1, diminishing returns
1
D. The Elasticity of Substitution

=
%(K/L)/ %(RTS) =
d(K/L) RTS =
dRTS (K/L)
ln(K/L)
ln(RTS)
Notice that in many instances, we can most easily calculate  by differentiating
the denominator w.r.t the top, and (given concavity conditions necessary of the implicit
function theorem) taking the inverse
E. Some Common Production Functions
1. Linear
2. Fixed Proportions
3. Cobb Douglas
4. Constant Elasticity
F. Technical Progress
Measuring Technical Progress (Problem: Distinguishing increased capital
accumulation from Technical progress
Starting with
q
we can develop
=
A(t)f(K,L)
Gq
+
q,K GK
=
GA
+ q,L GL
Where Gx = (dx/dt)/x
Thus, we can separate total grown into portions attributable to changes in
capital, and the capital stock and technology. Such an estimation process is
extremely important for identifying the relative importance of, say R&D efforts in
an economy.
PREVIEW_______________________________________________________
XI. Costs
A. Definitions of Costs
B. Cost-Minimizing Input Choices
C. Cost Functions
D. Shifts in Cost Curves
E. Short Run, Long Run Distinction
Lecture______________________________________________________
The purpose of production theory is to identify the way inputs are related to each
other in the production process. In this chapter we shift our attention to characterizing
the cost conditions for the firm, in light of the prices of input as well as outputs. In
general we will talk about 3 topics: The least cost combination of inputs, short run cost
2
curves, and long run cost curves. The material in this chapter is a preliminary for the
profit maximizing decisions of the firm, to be covered in chapter 13.
A. Definitions of Costs: Prior to discussing costs for the firm, it is important to
distinguish between costs for purposes of planning (economic costs) and out-of-pocket
costs maintained for the purposes of accounting for taxes (accounting costs).
Economic Cost: The cost of maintaining a resource in its present use. For non-marketed
goods, such as owned resources, these are measured as opportunity costs.
Accounting Cost: Explicit, out of pocket expenses.
In the development that follows, we will focus on two acquired inputs, labor, and capital,
as well as entrepreneurial services. We conventionally treat labor as an explicit, out of
pocket expense, w, the wage rate (This comports with considerable experience, but is not
always true – as the recent troubles of the domestic automoblile industry illustrate.)
Capital purchases, on the other hand, are usually fixed, and the value of owned
equipment is defined implicitly, as the rental rate v, on the machine’s best alternative use.
The entrepreneur as, the risk taker and organizer of the firm is the residual claimant of
firm earnings. Notice that we distinguish entrepreneurial returns from the returns to an
organizer who also has specialized labor skills. Earnings for a skilled computer
programmer in a dot.com start up, for example would be divided into a salary and
economic profits.
Some simplifying assumptions.
In what follows we treat only two inputs, labor and capital (with profits as a residual).
Also, we will assume that labor and capital can be acquired in competitive markets at
rates w and v, respectively. (We will generalize this subsequently)
Thus, the costs for a firm are
Total costs
=
TC
=
wL
+
-
TC
wL
vK
and economic profits,

=
TR
Pf(K,L)
Note also that for the time being, P is a constant.
3
vK
B. Cost-Minimization/Profit Maximization
1. Cost minimizing input choices. The above problem can be viewed as a profit
maximization problem (where we select quantity). However, given the discussion on
production in the preceding chapter it is useful to start with the analysis as a cost
minimization problem, where K and L are selected, for a given output level, qo.
L
=
wL
+
vK
+[qo – f(K,L)]
Taking first order conditions,
L /L
=
w
-
 fL
=
0
L /K
=
v
-
 fK
=
0
L /
=
qo – f(K,L)
=
0
Dividing the first two terms
w/v
=
fL/fK
=
RTS (L for K)
That is, a firm should hire until the RTS equals the ratio of input prices. Graphically, this
is represented as a comparison of the isoquants developed in Ch. 11, with the budget
constraint, sometimes labeled isocost curves.
In the figure to the right, for
example, the firm minimizes the
costs of producing qo with costs
TC1.
K
Notice that at the point of
tangency,
K*
w/v
qo
TC1
L*
TC2
=
fL/fK
This is an important and quite
practical rule for cost
minimization. It suggests that
firms should hire resources that
TC3
L
yield the most “bang for the buck”
2. Dual Problem: Output Maximization. This same problem can be viewed as a
problem of maximizing output, subject to a total expenditure level TC1. This is the dual
problem to cost minimization.
LD
=
f(K,L) +
D (TC1
-wL
4
-
vK)
First order conditions yield the same optimal condition as in the case of cost
minimization. However, the way that the firm gets to this point is slightly different, as
indicated by the graph, show to the
K
left. In the graph it is seen that in
this case, the firm selects the
maximum quantity that can be
produced with the budget constraint
TC1. That occurs at point qo, when
again the cost minimizing condition
K*
is met.
q1
The problem illustrated in the left
parallels exactly the utility
qoo
maximization problem for the firm.
TC1
In a manner closely analogous to
L*
L
that done in the case of the
consumer, we can develop input
demand curves. In this case, however, one important difference arises. In the case of the
firm, the demand for input is derived not from any intrinsic well-being that increments of
K or L yield to the firm, but from the profits that can be realized by selling output on the
market. Changes in market conditions, such as the price of the output, will affect input
demand. Thus, we say that input demand is a derived demand.
qo
3. The Firm’s Expansion Path. One added insight that can be made from the
above analysis, is the effect of a change in total costs on the firm’s relative use of inputs.
This is called the Expansion Path. (This is analogous to the consumption path for the
consumer case). As shown below in the left panel, as a general matter, holding input
prices constant, more of both inputs will be used as output expands. However, as the
right panel illustrates, this need not be true. An inferior input is one that is used less as
the level of output expands. The notion of an inferior input is slightly less esoteric than a
Giffen good (but not much). Firms for example may shift away from certain inputs as the
scale of operation increases (the use of shovels may decrease as output increases as
output increases and the firm uses more backhoes).
K
K
Inferior Input
Expansion Path
K*
K*
q1
q1
qo
qo
qoo
TC1
L*
qoo
TC1
L*
L
5
L
Example 12.1 Minimizing Costs for a Cobb-Douglas Production Function
Suppose the production of hamburgers at a fast food restaurant is a function of grills, K
and Labor L, each hired on an hourly basis. The production function is Cobb-Douglas
Q
=
10K.5L.5
Capital and labor can be rented in competitive markets at rates v and w, respectively.
Thus, the budget constraint is
TC
=
vK
+
wL
Suppose that the firm wishes to produce 40 hamburgers. Then the Lagrangian is
L
=
wL
+
vK
+[40 - 10K.5L.5]
First order conditions are
L /L
=
w
-
5(K/L).5 =
0
L /K
=
v
-
 5(L/K).5fK
=
0
L /
=
40
- 10K.5L.5
=
0
Dividing the first two terms
w/v
=K/L =
RTS (L for K)
Thus, for example if w = v = $4 equal amounts of K and L would be used. Substituting,
(with Q = 40)
40
TC
=
L
10L
=
=
$4(4) + $4(4)
4, so K = 4. and
=
$32.
Notice that other input combinations can be used to produce 40 hamburgers. For
example, set L = 8 and K = 2,
Q
=
10(2).5(8) .5
=
40
However, the cost of this input set is
$4(2) +
$4(8) =
$40.
6
Solving for , note that

=
L/MPL =
K/MPK
With K = L = 4 and MPL = MPK = 5, this implies that the marginal cost of an extra
hamburger is 1/1.25 = $.80, since an extra hamburger can be produced with 1/5 of an
hour of either K or L.
Alternatively, taking the inverse,
1/
=
MPL/L =
MPK/K
Thus, again using our parameters and extra $1 spent on either K or L would increase
output by 1/.8 = 1.25 hamburgers. Intuitively, (.25 of an input unit can be purchased for
$1, and that yields .25*5 = 1.25 hamburgers).
Finally, notice the expansion path for this production function. Since the Cobb-Douglas
function is homogenous of degree 1 the firm enjoys constant returns to scale, and the
expansion path is linear. (Indeed the expansion path would be linear even if the wv)
Question: Suppose v = 12 and w = 4. What should be true about MPL and MPK in the
cost minimization output level? (Work in class)
C. Cost Functions. Given the above analysis regarding the optimal input combinations,
we can now analyze the firm’s cost functions. The total cost function shows the
minimum total cost associated with an output level, q, and input costs v and w.
TC
=
TC(v,w,q)
We often find it useful to characterize costs on a per unit basis. This is called average
costs
AC(v,w,q)
=
TC(v,w,q)/q
Finally, a characterization of costs optimization will require identification of marginal
costs
MC(v,w,q)
=
TC(v,w,q)/ q
Total, Marginal and Average Cost Functions exhibit some standard interrelationships. In
the general case marginal costs first diminish (due to gains from specialization) and then
increase (due to crowding, or the law of diminishing returns). On a totals curve, the
marginal cost is illustrated as the slope of the line tangent to the TC curve. Thus, the TC
curve increases first at an increasing rate, and then at a decreasing rate, as shown in the
top panel of the figure below.
In marginal cost, average cost space, we see the marginal cost curve “driving”
average costs, as discussed in the previous chapter. Thus, the marginal cost curve cuts the
average cost curve at it’s minimum point.
7
180
160
140
120
100
80
60
40
20
0
TC
0
5
10
15
20
35
MC
30
25
20
15
AC
10
5
0
0
5
10
15
20
Observe that some of these relationships disappear for simpler cost functions. For
example, consider these relationships for a linear TC function, say
TC
=
20+2Q
Similarly, suppose that there is no initial stage of gains from specialization, but that costs
simply increase at an increasing rate (as would be the case, for example, with a quadratic
function). Consider, for example,
TC
=
20+2Q2
D. Shifts in Cost Curves. The cost function TC (v, w, q) is graphed as a cost curve (e.g. in
a two-dimensional relationship) of costs against quantity. Changes in v or w will shift
this function. In this section we consider the effects of such changes on output.
1. Homogeneity. A first observation pertains to the effects of increasing w and v
together, proportionally. Given an optimal input combination, it is easy to verify that a
cost function is homogenous of degree 1. For example, if given w, and v, suppose that L1
and K1 represent the optimal combination of inputs to produce a quantity q1. Then
TC1
=
vK1
+
wL1
8
If both inputs increase by a factor t, the optimal input combination (to produce the same
quantity q1) is unaffected. Thus
tvK1
+
twL1
=
t(vK1 +
wL1)
=
t TC1
Similarly, average and marginal costs are homogeneous of degree 1. For average costs,
observe that if TC’ =
tTC, then
AC’
Similarly,
=
tTC/q
d(TC’)/dq
=
=
tAC
d(tTC)/dq
=
tdTC/dq
=
tMC
2. Change in the Price of One Input. Now consider the more interesting problem
of changing a single input price. The issue here is more complicated, because
changing one input price changes the optimal input mix, which in turn will require
construction of a new expansion path.
a. Direction of Effect. Consider first the likely effects qualitatively.
Intuitively, increasing the price of an input raises the total costs of
production. More formally
L
=
vK
+
wL
+[qo – f(K,L)]
By the envelope theorem
TC/v
=
 L/v =
K
>
0
=
L/w =
L
>
0
and
TC/w
While input substitution effects may damp this effect, costs would certainly not
fall. Were costs to fall, the firm could not have been optimizing in the first place.
For exactly the same reason average costs should also increase.
Marginal costs will also generally increase as well. A possibility exists that an input may
be inferior, which may, perhaps surprisingly, cause marginal costs to fall. (That is, the
cost of a input increases, and you use so much less of that input that marginal costs fall.)
MC
=
TC/q
=
 L/q =

Consider the effects on MC of a change in the rental price of capital
MC/v
=
2 L/(qv)
=
9
2 L/(vq)
=
K/(q)
This latter term is positive or negative depending on whether Capital is inferior or
not.
b. Input Substitution. Consider now more formally, the effects of a change in
relative input prices on the optimal K/L. We examine the derivative
(K/L) / (w/v)
along a given isoquant. Expressed in proportional terms yields s,
s
=
[(K/L)(w/v)]/ [(w/v)(K/L)]
which, of course, is the elasticity of substitution (where the input price ratio
substitutes for the RTS).
In the two input case, s must be nonnegative, and an increase in w/v must increase
the ratio K/L, or (in the case of perfect complements) leave it unchanged.
c. Partial Elasticity of Substitution. In the more general case (with multiple
inputs) we write, for inputs Xi and Xj
sij
=
[(Xi/ Xj)(wj/ wi)]/ [(wj/ wi) (Xi/ Xj)]
where output and other input prices are held constant. We term above sij as a
partial elasticity of substitution. Non-negativity may not hold in this case, as the
increase in, say, wj may cause increased use of a third factor, which may result in
decreased used of Xj as well as Xi.
To be concrete, consider a production process consisting of capital, labor
and, say, energy. An increase in the price of energy may result in decreased use
of capital and increased use of labor. The partial elasticity of substitution is
useful for studying the derived demand for inputs, and can be calculated if the
production function is known. However, we won’t pursue it in detail here.
d. Quantitative Size of Shifts in Cost Curves. From the preceding discussion we
know that increases in an input price will increase total, average and (in most
instances) marginal costs. Consider briefly the quantitative impact of increasing
input prices. We confine comments to some intuitive observations.
1) Costs will increase a lot if the input is an important part of
production. (e.g., a labor cost increase will affect operations at a
call center very considerably, since labor is such a large percentage
of the total costs for the firm)
2) Costs will increase less if the input has good substitutes. For
example, an increase in copper prices did not affect importantly
10
electricity distribution costs, since suppliers could easily switch
from copper to aluminum. On the other hand, gold jewelry costs
move very closely with the price of gold, since no substitutes for
this critical input exist.
3. Technical Progress. Costs will also decrease with technical improvements,
since the technical improvements allow the same output to be produced with
fewer inputs. This is easy to illustrate in the case of constant returns to scale.
With CRS
TCo
=
C0(q,v,w)
=
qCo(v,w)
Now consider a production function that allows for technical progress, as we did
at the end of chapter 11.
q
=
A(t)f(K,L)
Where at time t0, A(t0) = 1. Thus, unit costs at time t (in terms of output qo)
become
Ct(v, w)
=
[C0(v, w)qo]/ [A(t)qo] =
Co(v, w)/ A(t)
Total Costs are
TCt (q, v, w) =
Ct(v, w)qo
=
TCo/A(t)
(Notice, in the above example, I’m abusing the notation a bit by defining q0 =
f(K,L). We do not consider inter-temporal output effects, so q need not be
subscripted. However, even holding output fixed, I need some way to indicate
that fewer inputs were required to produce that output. Thus, we define the
production function in terms of initial period outputs.
Example 12.2 Cobb-Douglas Cost Function. Consider again the Cobb-Douglas
production function qo= 10K ½L ½. Although generating costs from a production function
can be tedious, the process is rather straightforward here.
Given v and w, sellers minimize the cost of producing q0 when
w/v
=
K/L
Given
q= 10K ½ L ½
q/K= 10(L/K) ½
Substituting
11
q/K= 10(w/v) ½
Solving for vK renders
vK = (q/10)(wv) ½
Reasoning similarly
wL = (q/10)(wv) ½
Thus
TC
=
=
=
vK
+
(q/10)(wv) ½ +
(2q/10)(wv) ½
wL
(q/10)(wv) ½
which is the cost function for our Cobb-Douglas production relation. Thus, for example,
suppose that w= v = $4.
Average costs are
.2(wv) ½
TC/q =
so, with w= v = $4,
AC
=
.2(4) = .8
Marginal costs are
TC/q
=
.2(wv) ½
Identical to average costs.
Input Price Changes. Now consider the effects of an input price change. Suppose v = $9
and w = $4. Then
TC’
=
=
(2q/10)(4*9) ½
1.2q
Hence AC and MC increase from $0.80 to $1.20. Observe in this case that it was
unnecessary to go back and recalculate the cost minimizing input choices here. That is
because we have an expression for the cost function. Input changes are accounted for in
this expression automatically.
Technical Progress Suppose we can write the production function in terms of a temporal
element, as we did in Example 11.4.
qt
=
A(t)f(K,L)
=
e.05tf(K,L)
12
= e.05tqo
Then total costs at any time t are
TCt
=
TC0/A(t)
e.-05t[.2qo(wv) ½]
=
Thus, after, say, 10 years of progress, costs are
TC10
=
.607TC0.
.121qo(wv) ½
=
With w = v = 4,
TC10
=
.48q0
So average and marginal costs have fallen by 40% (from 80 to 48. 48/80 = 6/10)
E. Short Run, Long Run Distinction. It is conventional in economics to distinguish
between a long run, when all inputs are variable, from a short run, where at least one
input is fixed. The former is termed the “planning horizon,” where all inputs are variable.
The latter, the “operating horizon” assesses the optimal level of short run output. Here
we assess the effects of having some factors in fixed supply by holding capital constant,
at a level K1.
Thus
q
=
f(K1, L)
1. Short Run Total Costs Short run total costs become
STC
=
vK1
+
wL
2. Fixed and Variable Costs. Costs for the fixed capital stock K1 are
termed Short Run Fixed Costs (SFC). Labor costs, wL are short run variable costs
(SVC).
3. Nonoptimality of Short-Run Costs. Importantly, short run costs are not
the minimal costs for producing various output levels, because the firm cannot adjust
optimally the capital stock
The figure to the left illustrates.
K
STC2
Although the firm optimally uses
labor and capital for production
level q1, either decreasing or
increasing output from L1, K1
requires deviation from the efficient
set of input combinations, because
K is fixed at K1.
K1
q2
STCo
q1
L0
L1
L2
STC1
qo
L
13
4. Short-Run Marginal and Average Costs. We will often refer to short run costs on a per
unit basis. Thus, we develop definitions that parallel the long run case
SATC
=
STC/q
SMC
=
STC/q
5. Short-Run Average Fixed and Variable Costs. Similarly, it is instructive to break total
costs into fixed costs and variable cost components.
SAFC
=
SFC/q
SAVC
=
SVC/q
Where SFC and SVC denote short run fixed and short run variable costs, respectively.
6. Relationship Between Short-Run and Long-Run Cost Curves. Consider the
function
STC(q, K)
=
total cost.
Now increasing the allowed capital stock generates a new STC curve. The family of such
curves (generated by application the envelope theorem) illustrate the minimum cost
combinations output at each production level. Taking the minimum points for each
individual curve generates the long run total cost curve, as illustrated below in total cost
space (on the left) and in unit cost space (on the right)
i
TC
STC(Ko)
STC(K2)
STC(K1) TC
TC
STC(K0)
SMC(K0)
STC(K2)
SMC(K2)
STC(K1)
q1
q2
q3
Q
q0
q1
SMC(K1)
q2
7. Per-Unit Cost Curves. An important aspect of the unit cost curves, shown on
the right above is that the optimal point of operation for a firm in the short run typically
will not be at the point of minimum short run costs. Only if the firm is at the optimal
scale of operation (e.g., at the base of the LRATC schedule) will the firm operate at the
point of minimum costs. Otherwise the firm will use relatively too much or too little of
the inputs available in fixed supply. The Long Run Equilibrium condition for efficient
operation is as follows
14
Q
AC
=
MC
=
SATC =
SMC.
Example 12.3 Short Run Cobb-Douglas Cosst. Consider the effects of holding capital
fixed at a level K1 in terms of the Cobb Douglas production function we used in the
previous example.
q
=
10K1 ½ L ½
Thus the total productivity of Labor is
L
=
q2/[100K1]
Short run costs are
STC
=
=
vK1
vK1
+
+
wL
wq2/[100K1]
+
wq2/[400]
Thus, for example, if K1 = 4.
STC
=
4v
To calculate total costs, we need v and w. Let w = v = $4 and let K = 1, 4, and 9. This
yields the following costs
q
0
10
20
30
40
50
60
70
80
90
100
STC(K=1)
4
8
20
40
68
104
148
200
260
328
404
STC(K=4)
16
17
20
25
32
41
52
65
80
97
116
STC(K=9)
36.00
36.44
37.78
40.00
43.11
47.11
52.00
57.78
64.44
72.00
80.44
TC
0
8
16
24
32
40
48
56
64
72
80
(Recall, that the TC figures are given by TC = .2q(wv)1/2)
Notice in the table that there exists only a single point of overlap between any
short run total costs curve, and the long run cost curve. Further, except for that single
point of tangency, short run costs exceed TC.
An Envelope Derivation. In fact, the long run total cost curve can be derived from the
short run curves, by application of the envelope theorem. Consider again the STC
expression, but where v = w = $4.
15
STC
=
4K
+
4q2/[100K]
But now let K vary. Differentiating w.r.t. K
STC/K =
4
-
4q2/[100K2]
Setting this derivative to zero (since we wish to minimize STC at each output level), yields
4
=
4q2/[100K2]
=
q/10.
Solving
K
Substituting back into the STC function yields the TC function (or the envelope of
minimum STC points)
TC
=
=
=
4q/10 +
.4q
+
.8q
4q2/[100(q/10)]
.4q
Per Unit Cost Function. Recall that long run marginal and average costs were
constant at $0.80.
SATC =
vK1/q +
wq/[100K1]
SMC
2wq/[100K1]
=
Cost
3
2
SMC
=
4K1/q +
=
q/[25K1]
SAC (K=1)
2.5
Again setting v = w = 4
SATC
SMC(K=1)
SMC(K=4)
SAC(K=4)
1.5
.08q/K1
1
The chart to the left illustrates these
curves for the cases where K = 1, 4,
and 9. Notice particularly in the chart
that each short run curve is tangent to
the MC=AC curve only once.
0.5
SAC(K=9)
AC=M
C
0
qty
16