ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

ADVANCED NE2000 COMPATIBLE CARD DRIVER INFORMATION
PROFESSIONAL EXCELLENCE ADVANCED SKILLS TRAINING FOR
ADVANCED TOOLING SYSTEMS UK LTD PLANT LIST

053B ADVANCED CONVERSATION QUIZ 1 MARCH 17 2006 NAME
11 Supplementary Record Form Advanced Livestock Record Rabbits 20to
111 DERIVING DYNAMIC PROGRAMMING ADVANCED MATH THIS LECTURE REALLY

EC9101 - Seminar in Microeconomics

Advanced Microeconomics

Topic 6: The Theory of the Firm

Primary Readings: DL – Chapter 3

(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, what would a firm behave. We start with the profit maximization, then discuss the profit function, and end up with the duality issues.

6.1 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.

6.1.1 The Profit Function

Let us return to the general framework where a firm is described by a production possibility set Y  R^m. 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

y₁

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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. (Do you see that this is a special case of MR = MC?)

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

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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

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

Properties of the profit function have several uses. In particular, these properties offer some observable implications of profit-maximizing behavior:
- 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)

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.
- What will be the profit function for CES technologies? For example, Leontief production function?

6.1.2 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 y_i is an input, then the corresponding function y_i(p) is called the input demand function, also known as factor demand function.
similarly, if y_i is an output, then the corresponding function y_i(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 y_i(p) be the firm's net supply function for good i (i = 1,…, m). Then,

assuming that the derivative exists and that p_i > 0.

Proof: Suppose y* is a profit maximizing netput vector at prices p*. Then define the function:

g(p) = (p) - py*.

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) = max_z (p f(z) - w z)

A geometrical intuition is as follows:

Profits

(p)

(p) = p y* - w z*

(p*)

p* Output Prices (p)

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

6.2 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.

6.2.1 Comparative Statics of Input Demand Functions

Case 1: Single input: max_z 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:

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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

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

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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:

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

Differentiating w.r.t. w₁, we have

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM 

Differentiating w.r.t. w₂, we have

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM 

Therefore, we get

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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:

z_i/w_i < 0 for i = 1, 2, since the diagonal entries of a negative definite matrix must be negative;
z_i/w_j = z_j/w_i, 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.

6.2.2 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.

6.2.3 The LeChatelier Principle

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 an 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 = z₁^S(w₁, p, z₂⁰)

(Note that w₂ does not enter this demand curve). Now differentiating the following equality (by plugging z₁^S into the FOC):

p f₁(z₁^S, 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):

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

From linear algebra, we will have the following:

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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".

6.3 Duality in Production

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 z_i/w_i < 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.

6.3.1 Duality in Production

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

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM 

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:

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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

Consider the cost function:

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

Then,

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

which leads to the following production function:

6.3.2 The Geometry of Duality

Price 1

Isocost

z' z

Price 2

Input 2

Isoquant

w w'

Input 1

The slope conditions are:

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

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

Now, if (z₁^*, z₂^*) is a cost-minimizing point at prices (w₁^*, w₂^*), we know that it must satisfy the first-order condition:

ADVANCED MICROECONOMICS TOPIC 6 THE THEORY OF THE FIRM

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.

6.3.3 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.