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Profit Maximization
The theory of the firm is first presented in terms of general functional forms (Lectures 1-4)
and then in Lecture 6 we consider the Cobb-Douglas production function. For Lectures
1-4 the homework is to redo the previous lecture under the assumption that the production
function Q = log(K + 1) + 2 log(L + 1). Also one should be able to reproduce the lecture
without looking at your notes. Note that log always means natural log.
Q(K, L) =output. Q is a function of inputs, K = capital and L = labor.
∂Q(K, L
= QL (K, L) = QL
∂L
∂Q
= QK
∂K
That is, the partial derivatives are denoted by subscripts.
Assume that the Hessian of 2nd partials is negative definite (this implies concavity of the
production function).
PERFECTLY COMPETITIVE FIRM
P , w, and i are exogenous. P = price of output, w = wage, i = interest rate
Objective Function: MAX Π = P Q(K, L) − Lw − Ki
Profit = revenue minus cost
KT Conditions:
ΠL = P QL − w ≤ 0
ΠK = P Q K − i ≤ 0
L≥0
K≥0
ΠL · L = 0
ΠK · K = 0
Verbal Interpretation:
Wage ≥ Marginal Revenue Product of Labor
Interest Rate ≥ Marginal Revenue Product of Capital
For an Interior Solution, ΠL = ΠK = 0
QL
w
Implied Relations (assuming L, K > 0):
=
QK
i
Ratio of Marginal Products = Ratio of Payments to Factors of production
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ISOQUANT
Isocurve: Q(L, K) = Q̄
isoquant
dQ̄ = QL dL + QK dK = 0
QL
dK
=−
dL
QK
slope of the isoquant
As shown above, (assuming L, K > 0) a profit maximizing firm sets QL /QK = w/i.. So
for a profit maximizing firm, the slope of the isoquant at the profit maximizing point equals
dK
w
= − . This relationship can be understood via the following diagram:
dL
i
The straight line is the budget line. A profit maximizing firm will choose the lowest
budget line for any isoquant. At that point, the slope of the budget line and the isoquant
will be identical.
HOMOGENEITY
From the first order (KT) conditions, we can establish homogeneity:
w
i
QK ≤
P
P
Therefore, the First Order Conditions (FOC) are homogeneous of degree zero in w, P , and
i. If double w, P , and i, then QK , QL , K, and L remain the same (RTS), and so does Q.
QL ≤
8
Therefore, the maximum of Π = P Q(K, L) − Lw − Ki is homogeneous of degree 1 in w,
P , and i since Q, K, L RTS (remain the same) and thus doubling w, P , and i doubles Π.
I have said that profits are homogeneous of degree 1 in w, P , and i. This should not be
confused with constant returns to scale of the production function. Q may or may not be
homogeneous of degree 1 in K and L. We will discuss returns to scale later.
SECOND ORDER CONDITIONS
�
� P QLL P QLK
�
� P QKL P QKK
�
�
�
�
2
� = P � QLL QLK
�
� QKL QKK
�
�
� = |H|
�
H is negative definite since Q is negative definite by assumption: H1 < 0; H2 > 0. H1 is the
1x1 determinant — the upper left term; H2 is the 2x2 determinant.
COMPARATIVE STATICS
We want to find the effect of a change of an exogenous variable (w, P , or i) on an
endogenous variable (K or L) assuming that the firm is maximizing profits. That is, we
want to find the effect of a change in w, P , or i on the profit maximizing K or L, and
not on just any possible K or L (we could denote the profit maximizing K and L by K̂
and L̂, but this would clutter up the notation further). We will make use of the implicit
function theorem. For our first example, we will find the effect of an exogenous change in w
on L. To make things simpler, we will assume that both K and L are greater than 0. Both
before and after the exogenous change, the first order conditions hold. That is, the marginal
profitability of increased K or L is zero. More formally:
w changes
dΠL = ΠLL dL + ΠLK dK + ΠLw dw = 0
dΠK = ΠKL dL + ΠKK dK + ΠKw dw = 0
dΠK = P QLL dL + P QLK dK − 1 · dw = 0
dΠL = P QKL dL + P QKK dK + 0 · dw = 0
�
P QLL P QLK
P QKL P QKK
or
��
dL
dK
�
Since there are 2 linear equations (linear in terms
Cramer’s rule. First we find the effect on L
�
� dw P QLK
�
� 0 P QKK
dL =
|H|
9
=
�
dw
0
�
of dL and dK), we can solve using
�
�
�
�
dL
P QKK
=
dw
|H|
QKK < 0 by assumption and |H| = H2 > 0 by assumption. Therefore
dL
< 0.
dw
We have a downward sloping factor demand curve. Locally the derived demand curve
for the factor is always downward sloping. Next we find the effect of w on K.
�
�
� P QLL dw �
�
�
� P QKL 0 �
dK =
= −P QKL dw
|H|
dK
−P QKL
=
dw
|H|
dK
< 0 ⇔ QKL > 0
dw
This result is surprising to many economics students. While it is true (in the absence
of crowding), more labor will increase the productivity of capital, it may not increase the
marginal productivity of capital. That is, QKL , the effect of an increase in L on the marginal
product of capital, may be less than 0. Let us look at the following graphs where the output
function is drawn in, the higher curve is due to more labor, and the marginal product of the
capital is the slope of the curve.
In the first diagram, more labor has no effect on the marginal productivity of capital; in
the second marginal productivity increase; and in the third it decreases.
When wage goes up, L goes down. If more labor decreases the marginal productivity of
capital (as in the third diagram), then less labor increases the marginal product of capital,
and more K is employed. If more labor increases the marginal productivity of capital (as in
second diagram), then when L goes down, marginal productivity of capital will also decrease,
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and K will go down as well. This would be the case for a Cobb-Douglas production function,
Q = ALB K C .
Comparative statics when i changes
dΠL = ΠLL dL + ΠLK dK + ΠLi di = 0
dΠK = ΠKL dL + ΠKK dK + ΠKi di = 0
dΠL = P QLL dL + P QLK dK + 0 · di = 0
dΠK = P QKL dL + P QKK dK − di = 0
�
� 0 P QLK
�
� di P QKK
The effect of a change in i on L=dL =
|H|
Cross demands are always equal:
�
�
�
�
= −P QLK di
dL
−P QLK
dK
=
=
di
|H|
dw
By a similar process one can also show that:
dK
P QLL
=
di
|H|
IMPLICIT AND EXPLICIT DEMAND FUNCTIONS
Note that the first order conditions make the profit maximizing L and K implicit functions of w, P , and i. Since QLL and QKK are negative and QLL QKK > QLK QLK by
assumption, in principle, L and K can be solved as explicit functions of w, P , and i:
L = L∗ (w, P, i)
K = K ∗ (w, P, i)
These derived factor demand functions are the profit maximizing amounts of L and K given
w, P , and i. An asterisk, ∗, will be used in this course to denote an explicit optimizing
function of the exogenous variables.
Characteristics of the implicit function (such as homogeneity) hold true for the explicit
function as well:
L∗ (T w, T P, T i) = L∗ (w, P, i) and K ∗ (T w, T P, T i) = K ∗ (w, P, i)
because the first order conditions are homogeneous of degree 0 in w, P , and i.
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dK
Note that we use the implicit function theorem to find
. Suppose that we had solved
di
the first order conditions explicitly for K. Then the partial of the explicit function K ∗ with
respect to the partial of i would yield the same answer as the total derivative of the implicit
dK
function with respect to the derivative of i,
. All of this can be illustrated via a simple
di
example.
Suppose that Q = log(L + 1) + 2 log(K + 1). Then
Π = P log(L + 1) + 2P log(K + 1) − Lw − Ki
KT conditions (FOC)
P
−w ≤0
L+1
2P
ΠK =
−i≤0
K +1
ΠL =
L≥0
ΠL · L = 0
K≥0
ΠK · K = 0
P
− 1.
w
P
The partial of this explicit function, L∗ , with respect to the partial of w = L∗w = − 2 .
w
We can solve for L as an explicit function of P , w, and i. For L > 0, L∗ (P, w, i) =
Alternatively, we can make use of the implicit function theorem
dΠL = ΠLL dL + ΠLK dK + ΠLw dw = 0
dΠK = ΠKL dL + ΠLL dK + ΠKw dw = 0
−P
dL + 0 · dK − dw = 0
(L + 1)2
2P
dΠK = 0 · dL −
dK + 0 · dw = 0
(K + 1)2
dΠL =
Hence,
dL
(L + 1)2
=−
dw
P
But by the first order conditions, L + 1 =
P
dL
P
. So,
=− 2
w
dw
w
Thus the total derivative of the implicit function L with respect to the derivative w is
equivalent to the partial derivative of the explicit function L∗ with respect to w.
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THE EFFECT OF A CHANGE IN P ON L
dL =
Since QL =
�
�
�
�
dΠL = P QLL dL + P QKL dK + QL dP = 0
dΠK = P QLK dK + P QKK dK + QK dP = 0
�
−QL dP P QKL ��
−QK dP P QKK �
(−QL QKK P + QK QKL P )dP
=
|H|
|H|
w
i
−wQKK + iQKL
and QK = , dL =
dP
P
P
|H|
QKK < 0, therefore −wQKK > 0.
If QKL > 0, then dL > 0. Note that dL might be greater than zero even if QKL < 0.
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LE CHATELIER PRINCIPLE (SKIP)
Le Chatelier Principle
Long-Run Changes (in absolute value) > Short-Run Changes
We will determine the effect of a change in wage on the amount of labor employed, first in
the short run and then in the long run. Recall the following:
w changes
dΠL = ΠLL dL + ΠLK dK + ΠLw dw = 0
dΠK = ΠKL dL + ΠKK dK + ΠKw dw = 0
dΠK = P QLL dL + P QLK dK − 1 · dw = 0
dΠL = P QKL dL + P QKK dK + 0 · dw = 0
First assume K is fixed. What is the effect of a change in w?
dΠL = P QLL dL − dw = 0
dL
1
=
<0
dw
P QLL
Now suppose K is not fixed, then again
dΠL = P QLL dL + P QLK dK − dw = 0
dΠK = P QKL dK + P QKK dK + 0 · dw = 0
or
�
��
� �
�
P QLL P QLK
dL
dw
=
P QKL P QKK
dK
0
�
�
� dw P QLK �
�
�
� 0 P QKK �
dL =
|H|
dL
P QKK
P QKK
<0
=
= 2
dw
|H|
P QLL QKK − P 2 Q2LK
The inequality holds since |H| > 0 and QKK < 0.
P 2Q
P QKK
1
=
2
2
LL QKK − P QLK
P QLL −
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P Q2LK
QKK
≤
1
<0
P QLL
CONSTANT RETURNS TO SCALE
Constant returns to scale: Q is homogeneous of degree 1 in K and L if
Q(T K, T L) = T Q(K, L) = T 1 Q(K, L)
(1)
For example, if all inputs are doubled, then output is doubled. The following CobbDouglas production function is an example of constant returns to scale.
1
2
Q = AK 3 L 3
If we multiply both K and L by T we get
1
2
1
2
Q = A(T K) 3 (T L) 3 = T AK 3 L 3 = T Q = T 1 Q.
P roposition : The first derivative of a function which is homogeneous of degree 1 in K
and L is itself homogeneous of degree 0 in K and L.
P roof : We take the derivative of Q(T K, T L) = T Q(K, L) with respect to K, and get:
T Q1 (T K, T L) = T QK (K, L)
or Q1 (T K, T L) = QK (K, L) = T 0 QK (K, L).///
(2)
This is just a special case of Euler’s Theorem. Note the convention regarding derivatives. When the first (or later) argument is just a single variable, we often denote the partial
derivative by the variable. Hence, QK (K, L). When the first (or later) argument is a more
complex expression, we use a number subscript. Hence, T Q1 (T K, T L).
P roposition : When there are constant returns to scale, marginal product of capital times
capital plus the marginal product of labor times labor is equal to the total product and a
perfectly competitive firm has 0 profits.
P roof : Take derivative of both sides of (1) with respect to T :
KQ1 (T K, T L) + LQ2 (T K, T L) = KQK (K, L) + LQL (K, L) = Q(K, L)
(3)
The first equality holds by (2). The last equality is the derivative of the right hand side of
(1) with respect to T . Equivalently, P KQK + P LQL = P Q(K, L).
But by the first order conditions from profit maximization, we have the following:
P KQK = iK
and P LQL = wL
Therefore, Cost = iK + wL = P KQL + P QL = P Q(K, L) = Revenue
That is, there are zero profits when there are constant returns to scale.///
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Note that when there are constant returns to scale, the first order conditions will give us
ratios but not the amounts of K and L.
Please note that Euler’s theorem says that if the function is homogeneous of degree 1
with respect to certain variables, then the derivatives of the function are homogeneous of
degree 0 with respect to the same variables and vice versa. Do not conflate this with the
homogeneity discussed earlier. Earlier we showed that the first order conditions with respect
to K and L were homogeneous of degree 0 with respect to P , w, and i (not with respect to
K and L). Inspection of the profit equation then showed that the maximum profit equation
was homogeneous of degree 1 with respect to P , w, and i. However, this homogeneity would
not in general be true if the firm were not maximizing profits.
If there are constant returns to scale, then the hessian of second derivatives is negative
semi-definite. For example, the hessian of 2nd derivatives of the Cobb-Douglas production
1
2
function (Q = AK 3 L 3 ) looks like
�
�
� 2 −5 2 2 −2 −1 �
� − K 3 L3
K 3 L 3 ��
� 9
9
� 2
�
� K − 23 L− 13 − 2 K 13 L− 43 �
�
�
9
9
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