Advanced Microeconomics

Topic 2: The Theory of the Firm

(Not concerned with the emergence of the firm and why firms exist, but with the
behaviour of a profit-maximizing firm. On the former, see X. Yang & Y-K Ng,
Specialization and Economic Organization, 1993, Ch.9 and references therein.)

In this lecture, we will analyze the behavioral side of the firm, namely, how would a
firm behave. We start with the profit maximization, then discuss the profit function,
and end up with the duality issues.

2.1 Profit Maximization

A basic assumption of most economic analysis of the firm behavior is that a firm acts
so as to maximize its profits, the difference of the revenue and the cost. This leads to
the fundamental condition (Production Law):
 Choose the level of output such that marginal revenue = marginal cost.

A firm must also face the decisions on how much of a specific input to use/hire.

The second fundamental condition of profit maximization is the condition of equal

long-run profits.

2.1.1 The Profit Function

Let us return to the general framework where a firm is described by a production

possibility set Y  Rm. Let y Y be a netput vector and p the associated price
vector. Here, p contains component prices for all netputs, inputs, outputs, and
quantities that can be either input or output.

Profit Function. Let Y be a production possibility set. Then the corresponding profit
function is

A graphical illustration of the profit function is as follows.

y2

(p)=p.y
Isoprofit

Y
y1

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If the firm has a single output, the profit function becomes:
(p, w) = max pf(z) – w.z
where q = f(z) is the production function of the firm. Then the first-order conditions
for this special case are (interior solutions only):

That is, the value of the marginal product of each factor must be equal to the factor's
price.

The diagram below illustrates the above FOC for single input case.

output
q = /p + (w/p) z
slope = w/p
q = f(z)

/p

input

 The second order condition is as usual: the Hessian matrix of f is negative

semidefinite at the optimal point.

Properties of Profit Functions: The above defined profit function (p) is

1. non-decreasing in output prices, non-increasing in input prices;
2. homogeneous of degree 1 in p;
3. convex in p;
4. continuous in p.

 Properties of the profit function have several uses. In particular, these

properties offer some observable implications of profit-maximizing
behavior:
o Whenever some property is not true, we can claim that the firm is
not a profit-maximizer.

Example
 (Cobb-Douglas Production, decreasing returns to scale)

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 Let w1 and w2 be the prices of the two inputs and p the price of the output.
Then

which leads to

(A derivation is placed in the Technical Appendix at the end of the note.)

Note:
 Assume that the production technology shows the constant returns to scale,
then the profit function is degenerate, namely, taking a value of either 0 or
infinite.
o What will be the profit function for CES technologies? For
example, Leontief production function?

2.1.2 Net Supply Functions and Hotelling's Lemma

Net Supply Functions - Input Demand & Output Supply Functions

The solution of the profit maximization problem:

is denoted by y = y(p), which is commonly called net supply function of the firm.
Clearly,
(p) = py(p).

In particular,
 if yi is an input, then the corresponding function yi(p) is called the input
demand function, also known as factor demand function.
 similarly, if yi is an output, then the corresponding function yi(p) is called
the output supply function, or simply supply function.

Hotelling's Lemma

If you know the profit function, then according to the following well-known lemma,
Hotelling's Lemma, it is easy to find the net supply function: just differentiate the
profit function.

Hotelling's Lemma. Let yi(p) be the firm's net supply function for good i (i = 1,…,
m). Then,

assuming that the derivative exists and that pi > 0.

Proof: Suppose y* is a profit maximizing output vector at prices p*. Then define the
function:
g(p) = (p) - py*.

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Clearly, the profit-maximizing production plan at prices p will always be at least as
profitable as the production plan y*. But, the plan y* will be a profit-maximizing plan
at prices p*, so the function g obtains a minimum value of 0 at p*, which is an interior
solution according to the (positivity) assumptions on the prices. We can then use the
first-order condition on g:

Since this is true for all choices of p*, the proof is completed.

Note:
 Of course, we can directly apply the Envelope Theorem.
 Try to gain some intuition by looking into the simple case:
(p, w) = maxz (p f(z) - w z)
A geometrical intuition is as follows:

Profits
(p)

(p) = p y* - w z*

(p*)

p* Output Prices (p)

2.2 Comparative Statics Analysis

 Economists refer to the analysis of how an economic variable responds to changes

in its environment as comparative statics.
 The term “comparative” refers to comparing a “before” and a “after”
solution.
 The term “statics” refers to the idea that the comparison is made after all
adjustments have been “worked out;” that is, we must compare one
equilibrium situation to another.
 In optimization, it is known as sensitivity analysis.

2.2.1 Comparative Statics of Input Demand Functions

Case 1: Single input: maxz p f(z) - w z

Assume that f is differentiable. Let z = z(p, w) be the input demand function. Then the
first-order and the second-order conditions are:

 As the FOC holds for all p, we differentiate it w.r.t. w and get

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  Key implication of this result:
 Input demand curve slopes downward in its own price

 Similarly, we can also differentiate the FOC w.r.t. p:

 Key implications:
 Input demand curve shifts upward when output price increases
 (Subject to no-shutting down), a one percentage increase/decrease in
output price has the same (positive/negative) effect on output as a
one percentage decrease/increase in input price

Now, if look at the issue from the profit function:

(p, w) = p f(z(p, w)) - wz(p, w),
then

This just verifies the Hotelling's lemma.

Case 2: Two inputs: max pf(z1, z2) - (w1 z1 + w2 z2)

Denote the input demand functions as z1(w1, w2) and z2(w1, w2) (we deliberately drop
off the output price argument for ease of discussion.) The FOCs are as follow:

Differentiating w.r.t. w1, we have

Differentiating w.r.t. w2, we have

Therefore, we get

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The matrix on left-hand side of the last equation is known as a substitution matrix as
it specifies how the firm substitutes one input for another as the input prices change.

The second-order condition for (strict) profit maximization is the Hessian matrix H is
a symmetric negative definite matrix. From Linear Algebra, we know that H-1 must
also be a symmetric, negative definite matrix. This result leads to the following
important properties of the input demand functions:
1. zi/wi < 0 for i = 1, 2, since the diagonal entries of a negative definite
matrix must be negative;
2. zi/wj = zj/wi, by the symmetry of the matrix.

Case 3: General case - multiple inputs

We can normalize p = 1. The FOC is

f(z(w)) = w
Differentiate it w.r.t. w leads to
2f(z(w))z(w) = I  Hz(w) = I
Solving this for the substitution matrix, we have
z(w) = [H(f(z))-1|z = z(w)
From this identity, we will have similar results as for the case of two inputs.

2.2.2 Comparative Statics Using the Profit Function

Implications of Properties of the Profit Function

We now get back to the key properties of the profit function:

 monotonicity, homogeneous of degree 1 and convexity

 Monotonicity implies that partial derivative of (p) with respect to the price of
good i is negative if the good i is an input and positive if the good i is an output.

 Homogenous of degree 1 implies that

 Scaling all prices by a common positive factor will not change the optimal
choice of the firm, i.e., the net supply functions are homogenous of degree 0.

2.2.3 The LeChatelier Principle

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 We are motivated to look into the short-run response of s firm's supply behavior
as compared to the long-run response.
 Since there is no fixed input in the long run, we would expect that the firm
will respond more (net supply) to a price change in the long run than in the
short run. (Trying to place the issue in the context of labor and capital.)
 This is the so-called LeChatelier Principle.

Let us consider the case of single output and two inputs with a production function
q = f(z1, z2)
Assume that z2 = z20 is a fixed input. The profit function then becomes

Then the FOC is simply

p f1(z1, z20) = w1
and the (sufficient) second-order condition is p f11 < 0. The corresponding input
demand function (solving z1 from the FOC) is
z1 = z1S(w1, p, z20)
(Note that w2 does not enter this demand curve). Now differentiating the following
equality (by plugging z1S into the FOC):
p f1(z1S, z20) = w1
w.r.t. w1, we have

Recall that for the long-run case (both inputs are variable inputs), we have the
following result (without assuming p=1):

From linear algebra, we will have the following:

Remember that the second-order conditions are: f11 < 0, f22 < 0 and f11 f22 - f122 > 0.
Therefore,

Since both are negative, the above inequality says that the
change in z1, due to a change in its price is larger, in absolute value, when z2 is
variable (the long-run) than z2 is fixed (short-run).

 The above conclusion is sometimes referred to as the "second law of

demand".

2.3 Duality in Production

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In our discussion of the cost function, we are more or less relying on the specification
of production. This is one part of the duality of production and cost functions. In this
section, we will discuss the derivation of the production function from the cost
function - the so-called duality in production.

For simplicity, we will focus on the case of two inputs. In this case, the first-order
conditions are:

From these, we get the conditional input demand functions:

The corresponding cost function is denoted by c(w1, w2, q).

Note that the second-order sufficient conditions implies that zi/wi < 0, i = 1, 2. But
from Shepherd's lemma, we will have

meaning that the second partials of the cost function w.r.t. input prices are negative.

2.3.1 Duality in Production

We know that the cost function is homogeneous of degree 1 in input prices. Then zi is
homogeneous of degree 0 since zi is the first partial of c. Then according to Euler's
Theorem, we have

Eliminating w1 and w2 (taking the second term of each eq. to the right hand side and
divide each side of the first eq. by that of the second) reveals that
c11c22 - c122 = 0
saying that the determinant of the cross-partials of c with respect to input prices is 0.

Derivation of Production Function from Cost Function (Duality Result)

In fact, this is quite simple. Since the conditional input demand functions are
homogenous of degree 0 in input prices, then we must have:

where w = w2/w1. This leads to two equations with four variables, z1, z2, w and q. By
eliminating variable w and solve for q will generate the production function we need.

Example

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Consider the cost function:

Then,

which leads to the following production function:

.

2.3.2 The Geometry of Duality

Price 1 Input 2

Isoquant

Isocost

w' z

z' z z
w w'
Price 2 Input 1

The slope conditions are:

 The slope at a point (w1*, w2*) on an isocost function is

 For the isoquant defined by q = f(z), the slope at a point z* is given by

Now, if (z1*, z2*) is a cost-minimizing point at prices (w1*, w2*), we know

that it must satisfy the first-order condition:

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 This result is exactly the feature of duality:
 the slope of the isoquant curve gives the ratio of the input prices
while the slope of the isocost curve gives the ratio of input level.

2.3.3 The Importance of Duality

 The duality of cost and production functions is important for reasons other than
the mathematical elegance.
 Economists will have occasions to estimate conditional input demand and cost
functions. There are two basic ways to approach this problem.
 One way is to estimate, by some procedure, the underlying production
function for some activity and to then calculate, by inverting the implied first-
order conditions, the conditional input demand curves (holding output
constant). The cost function can be calculated also.
 This, however, is a very arduous procedure. Production functions are
largely unobservable. The data points will represent a sampling of input
and output levels that will have taken place as different times, as input or
output prices changed.

 It would seem to make sense to start with estimating the cost functions or the
conditional input demand curves directly; i.e., some functional form of the
cost function could be asserted, say a logarithm linear function, and costs
could be estimated directly.
 However, this procedure would always be subject to the criticism that the
estimated cost or demand functions were beasts without parents, i.e., they
were derived from fictitious, or nonexistent, production processes. And
this would be a serious criticism indeed.

 However, the duality results rescue this simpler approach. We can be assured
that if a cost function satisfies some elementary properties, i.e., homogenous
of degree 1 and concavity in input prices, then there is in fact some real,
unique underlying production function. Therefore, the cost function will be
more plausible.

Technical Appendix for Topic 2

The Profit Function for Cobb-Douglas Technology

First, the first order conditions are:

which immediately lead to

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Substituting z2 into the second first-order condition, we get

So,

Now,

Therefore the profit function is given by:

as required.

Additional References:

Hicks, J. (1946) Value and Capital. Clarendon Press, Oxford, England.

Hotelling, H. (1932) “Edgeworth’s taxation paradox and the nature of demand and
supply function,” Journal of Political Economy, 40, 577-616.
Samuelson, P. (1947) Foundations of Economic Analysis. Harvard University Press,
Cambridge, Massachusetts.
Silberberg, E. (1990) The Structure of Economics - A Mathematical Analysis. Second
Edition. McGraw-Hill, New York. (Chapters 4, 7, 8 & 9)
Varian, H. R. (1992) Microeconomic Analysis. Third Edition. W.W. Norton &
Company, New York. (Chapters 2, 3, & 6)

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