Producer Behavior

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Theory of Producer Behavior 2

The fundamental relation guiding producer behavior is the production function

the production function is the relationship between inputs of labor and capital and the amount of production the causal order is from inputs to output

The cost function is obtained by using the cost minimization principle under the condition of constant production levels

Cont… 3

combining the cost function and sales function, the profit function is defined as the difference between sales and cost functions.

Optimal output is determined by applying the profit maximization principle to the profit function

Therefore, cost minimization is a necessary condition for profit maximization.

The production function

where q is the amount of production and x1, x2, …, xm are factor inputs necessary for production.

Factors of production are land, labor, and

capital.

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Cont…

q = f(x1, x2, …, xm)

The definition of the cost equation is:

where C is cost and ri is the unit cost for the i-th factor input The cost minimization hypothesis stipulates that

a firm chooses a combination of factor inputs so as to minimize costs under the condition of constant production levels and fixed factor input prices

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Cont…

C = ∑irixi

To get the optimal amount of factor inputs under the condition of constant levels of production, the Lagrange multiplier method is utilized.

The evaluation function is:

first-order conditions for cost minimization are:

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Cont…

V = ∑irixi + λ(q − f(x1 x2, …, xm))

∂V/∂xi = ri − λ∂f/∂xi = 0 ∂V/∂λ = q0 − f(x , x , …, x ) = 0

From the first equation, we get:

This is called the law of equal marginal products per unit cost of input (or the least-cost rule).

Using FOC, factor demand functions are obtained as:

where q0 is a parameter

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Cont…

1/λ = (∂f/∂xi)/ri

xi = g(r1, r2, …, rm | q0 )

Substituting xi in cost equation, the cost function is obtained as the function of factor input prices of r1, r2, …, rm and constant quantity of q0:

This is the cost function satisfied by cost minimization under the condition of constant production levels.

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Cont…

C = ∑i rixi = h(r1, r2, …, rm | q0 )

When we change the levels of production while keeping the least-cost rule, the cost function changed to

This is the cost function in which the variables are factor input prices and quantity produced.

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Cont…

C = H(r1, r2, …, rm, q)

By applying the profit maximization principle, the optimal production level of the firm can be determined.

The profit function is:

where π is profit and p is the market price Differentiating both sides by q and setting dπ/dq

as zero, we get

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Cont…

π = pq − C(q)

The solution of the quantity of q satisfies the

optimal production level with maximum profit. This equation is the same as the MR (marginal

revenue) = MC (marginal cost) condition:

When dp/dq is equal to zero, the market becomes a competitive market, and the equilibrium condition of the competitive market becomes p = dC/ dq, where market price is equal to marginal cost.

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Cont…

dπ/dq = p + q(dp/dq) − dC/dq = 0

MR = p(1 + (q/p)dp/dq) = dC/dq = MC

There are three issues to bear in mind here about producer behavior:

(1) the profit maximization hypothesis;

(2) the difference between “short-run” and “long- run”

(3) the plausibility of the linear homogeneity of the production function

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Cont…

It is important to bear in mind, however, that economic analysis extends beyond the scope of individual firms and managers. It is true that firms may have different strategies and priorities, but a common goal is profit maximization.

Profit maximization is also treated as an analytical principle. When corporate behavior is explained sufficiently through the profit-maximizing principle, then the principle is relevant even if firms in the real world don’t focus solely on profit maximization.

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Cont…

We usually think that the short-run is a short period of time, say one day, one week, or at most one year, and that the long-run is a long period, say five years, ten years or one hundred years.

Short-run in microeconomics refers to the period when capital stocks are constant. That is, the capital stock in the production function is treated as being constant.

Long-run in microeconomics is defined as the period during which the amount of capital stock varies and a firm must consider the optimal investment behavior.

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Cont…

In macroeconomics, the terms short-run and long-run have different meanings.

The short-run is the period when the process of price adjustment continues, and

the long-run is the period when price adjustments are completed.

it is important to bear in mind that the two terms are defined differently in microeconomics and macroeconomics.

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Cont…

Linear Homogeneity of the production function Example

there is a factory that produces 10 units per month using 10 laborers and 10 units of capital. When building a new factory that has the same amount of laborers and capital as the existing factory, we might assume that 10 units of output will be produced in the new factory. This illustrates the concept of linear homogeneity: when all the factor inputs are doubled, the amount of production also doubles.

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Cont…

Models of Producer Behavior 17

In the case of competitive markets for labor and capital, the behavioral principle for firms is to determine the amount of labor and capital needed so as to minimize the cost of labor and capital while maintaining a constant level of production

In terms of the goods and services market, firms determine the amount of production that will maximize profits.

Cost minimization is a necessary condition for profit maximization

Cobb–Douglas production function

where Q is quantity of output, N is labor input, and K is capital input, and A, β1, and β2 are parameters to be estimated.

To estimate A, β1, and β2 there are two methods.

The first is to estimate the production function directly The second is to derive the parameters of the production

function from estimates of the cost function.

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Cont…

Q = AKβ1Nβ2

In estimating the cost function, we use the cost minimization principle to include market conditions.

The assumption of cost minimization is that the firm determines the combination of labor and capital inputs needed to minimize the cost given a constant level of quantity produced.

Therefore, a constant amount of quantity (Q0), wages (w), and interest rates (r) are treated as exogenous variables.

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Cont…

The definition of cost is

We can define cost minimization behavior under constant levels of production in mathematical form as:

To get optimal conditions for N, K, and λ, we derive the first-order condition for minimization through the following equations

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Cont…

C = wN + rK

V = wN + rK + λ(Q0 − AKβ1 Nβ2 )

By solving, we get

This is the law of equal marginal productivity per dollar for the Cobb–Douglas production function

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Cont…

∂V/∂N = w − λβ2 AKβ1 Nβ2 = 0 ∂V/∂K = r − λβ1 AKβ1 Nβ2 = 0 ∂V/∂λ = Q0 − AKβ1 Nβ2 = 0

wN/β2 = rK/β1

Labor and capital demand functions can be obtained as:

Estimating the equations by the regression method yields the estimates of the Cobb– Douglas production function

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Cont…

N=(1/A)1/(β1+β2)+(β1/β2)-β1/(β1+β2)+(w/r)-β1/(β1+β2)+Q01/(β1+β2)

K=(1/A)1/(β1+β2)+(β1/β2)β2/(β1+β2)+(w/r)β2/(β1+β2)+Q0

1/(β1+β2)

The causal order is from w, r, and Q0 to N and K. The causal order is different for the production

and cost functions. We will now specify the cost function and

estimate its parameters. In the case of variable levels of production, we

use the following equations:

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Cont…

N = constant1(w/r)-β1/(β1+β2)Q01/(β1+β2)

K = constant2(w/r)β2/(β1+β2)Q01/(β1+β2)

Taking the logarithms for both sides of equations

Therefore, when 1/(β1 + β2) = K0, − β1/(β1 + β2) = K1, and β2/(β1 + β2) = K2, we get β1 = −K1/K0 and β2 = K2/K0. The structural parameters β1 and β2 are estimated in

the case of variable levels of production.

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Cont…

n = log(constant1) − β1/(β1 + β2) log(w/r) + 1/(β1 + β2)q k = log(constant2) + β2/(β1 + β2) log(w/r) + 1/(β1 + β2)q

When using the Cobb–Douglas production function, the cost function is written as:

It is a highly nonlinear function including A, β1, β2, and wage and interest rates.

To estimate the equation, linear approximation is necessary, as it is difficult to estimate equation in the nonlinear form to get the estimates of A, β1, β2.

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Cont…

C = [(1/A)(β1/β2)(w/r)]-1/(β1+β2) [(β1/β2)(w/r)-β1 w + (β1/β2)(w/r)β2 r] Q01/(β1+β2)

Generally, to estimate the cost function, we specify it as a function of the wage rate, capital costs, and quantity, and then estimate it directly. First, we specify the production function in general as:

The cost function

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Cont…

Q = f(N, K)

C = Nw + Kr

Under the condition of cost minimization, capital demand and labor demand functions are derived as the function of wage, capital costs and quantity as:

Adding these to the cost definition, we get

Thus, cost is a function of wages, unit capital cost and quantity produced. This is the general cost function.

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Cont…

K = K (w, r, Q) N = N (w, r, Q

C = Nw + Kr = C(w, r, Q)

Generating Data - Monte Carlo Method 28

Cobb–Douglas type production function

Let’s consider the following estimating equation

Which is the logarithmic transformation of production function

Q = AKβ1Nβ2

qi = α + β1ki + β2ni + εi

The definition of cost is

We can define cost minimization behavior under constant levels of production in mathematical form as:

To get optimal conditions for N, K, and λ, we derive the first-order condition for minimization through the following equations

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Cont…

C = wN + rK

V = wN + rK + λ(Q0 − AKβ1 Nβ2 )

By solving, we get

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Cont…

∂V/∂N = w − λβ2 AKβ1 Nβ2 = 0 ∂V/∂K = r − λβ1 AKβ1 Nβ2 = 0 ∂V/∂λ = Q0 − AKβ1 Nβ2 = 0

wN/β2 = rK/β1

Labor and capital demand functions can be obtained as:

Taking the logarithms for both sides of equations

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Cont…

N=(1/A)1/(β1+β2)+(β1/β2)-β1/(β1+β2)+(w/r)-β1/(β1+β2)+Q01/(β1+β2)

K=(1/A)1/(β1+β2)+(β1/β2)β2/(β1+β2)+(w/r)β2/(β1+β2)+Q01/(β1+β2)

k = − 1/(β1 + β2) logA + β2/(β1 + β2) log(β1/β2) + β2/(β1 + β2) log(w/r) + 1/(β1 + β2) q + e1

n = − 1/(β1 + β2) logA – β1/(β1 + β2) log(β1/β2) - β1/(β1 + β2) log(w/r) + 1/(β1 + β2) q + e2

Cont… 32

Generate the following virtual data set n=500 β1 = 0.2, β2 = 0.8 α = 1, q ~ N(12, 32) log(r/w) ~ N(0, 22) k ~ N(12, 32) n ~ N(12, 0.62) ε1 ~ N(0, 22), ε2 ~ N(0, 12), ε3 ~ N(0, 12) r ~ Uniform Distribution (0.01, 0.2)

Exercise I 33

Estimate the labor demand function, the capital demand function and the production function

Test the hypothesis

k = B0 + B1 log(r/w) + B2q + ε

n = A0 + A1 log(r/w) + A2q + ε

qi = α + β1ki + β2ni + εi