We shall present the first elements of duality theory. We shall show that cost minimization, output profit maximization, and input profit maximization all lead to the same decision. That is we obtain the same results from the three alternative approaches.

One input and one output: $$ q = f(x)
where $$q is output and $$x is the input. For example, the input might be labor, steel, or any other material. One way of looking at a production function with only one input is that we are holding all the other inputs fixed.

Two inputs and one output: $$ Q = f(K, L)
where $$q is output, $$K is the amount of capital and $$L is the amount of labor

Obviously, we could have a production function with n inputs and m outputs, but such a production function would be beyond the scope of this course. Economists studying policy problems general have limited data from within the firms. Economists usually obtain empirical production functions from statistical analysis of industry data. Thus the production firm represents the representative firm in the industry. The simpliest form of these econometric production functions is $$ q = x^ß . If the exponent is less than or equal to 1, the production function is concave.

If a production function has two or more inputs and there is some substitution between inputs (that is a specified amount of output can be produced by more than one combination of inputs) then the cost of producing a specified amount of output depends on which combination of inputs is chosed. The cost function is the minimum cost of producing the output over its range. The cost function is obtained by solving a cost minimization problem which also provides the conditional demands for inputs to produce each level of output.

Cost minimization is an important decision problem in industry. A corporation also wants to known how to minimize the cost of producing a sales forecast. Over the year a public utility manage wants to know how to minimize the cost of producing power for all the various levels of demand.

First let us consider the one input variable case. The problem statements is:

min$$ C with respect to $$x _____ (1)

where: $$ C = cx ____________ (2)

subject to: $$ q₀ = x^(3/4) ________ (3)

This problem is very simple to solve because for each amount of output there is only one amount of input to produce it. Therefore we simply substitute $$x = q^(4/3) to obtain:

$$ C(q) = cq^(4/3)______________ (4)

Now let us consider the two input variable case. The problem statements is:

_____ min$$ C with respect to $$K and $$L_________ $$(1)

_________ where: $$ C = wL + vK _____________ (2)

_________ subject to: $$ Q₀ = f(K,L) ___________ (3)

where

$$C is the cost defined in terms of inputs

$$w is the wage rate

$$L is the amount of labor employed

$$v is the capital rental rate

$$K is the amount of capital and

$$f(K,L) is the production function.

Note: This output might be obtained from an economic forecast. For this example we will use a variation of the Cobb-Douglas production function: $$Q = K^1/3L^1/3. This production function is usually used for industry wide studies.

We shall solve the problem by substitution. To obtain the conditional labor demand substitute for $$K:

__________$$ K = Q₀³/L____________________ (5)

__________$$ C = wL + vQ₀³/L______________ (6)

Taking the first derivative:

__________$$ dC/dL = 0 = w - vQ₀³L^-2_______ (7)

Solving for $$L:

__________ $$L²= (v/w)Q₀³ _________________ (8)

__________$$ L =(v/w)^1/2Q₀^3/2 _______________ (9)

This is the conditional demand for labor. Note the demand depends on the price ratio. To obtain the conditional demand for capital, we substitute for $$L:

__________$$ L = Q₀³/K ___________________ (10)

__________$$ C = wQ₀³/K + vK _____________ (11)

Taking the first derivative:

__________$$ dC/dK = 0 = -wQ₀³K^-2 + v _____ (12)

__________$$ K =(w/v)^1/2Q₀^3/2 _______________ (13)

This is the conditional demand for capital. Note that the demand depends on the price ratio. Because the problem has symmetry, the conditional demand for $$K can be written by inspection once the conditional demand for $$L has been obtained. The output cost function is obtained by substituting the conditional demands for $$K and $$L into the cost function

__________ $$C = wL + vK:

_______ $$C(Q) = w(v/w)^1/2Q₀^3/2 + v(w/v)^1/2Q₀^3/2

____________ $$ = 2(vw)^1/2Q^3/2 ________________ (14)

Note that for this case the cost function is an increasing function for fixed $$v and $$w.

__________$$ µ = R - C ______________________ (15)

That is profits, µ, equal revenues, R, minus costs, C. The equation is easier to write down than to accurately measure the components. The problem to maximize profits can be set up with respect to an output decision or with respect to an input decision. The form of the equations for revenue and cost are dependent on the market structure of the input and output markets. First we will consider a firm which operates in competitive input and output markets.

2. Output decision:

__________$$ µ(Q) = R(Q) - C(Q) _____________ (16)

Note that the cost function $$C(q) could be obtained from the constrained cost minimization problem. The first order condition is:

__________$$ dµ/dQ = dR/dQ - dC/dQ = 0______ (17)

which implies the well known rule that marginal revenue must equal marginal cost, $$MR = MC, to maximize profits. For this rule to be operational we must be able to measure both marginal revenue and marginal cost. This is not an easy task because accounting data provides average revenue and average cost data, but not the corresponding marginal data. In small businesses this rule is approximately satisfied by rules of thumb under the stimulus of competition.

Now consider a perfectly competitive output market. In a perfectly competitive market the firm can sell any amount of output without influencing the price. The firm's output is assumed to be very small in relationship to the total market demand. This leads to the following:

__________$$ µ(Q) = pQ - C(Q)________________ (18)

where $$p is the output price. The first order condition is:

__________$$dµ/dQ = 0 _______________________ (19)

which gives:

__________ $$ p = dC/dQ ______________________ (20)

thus marginal revenue which in this case is the price equals the marginal cost. Now consider the following example for the cost function case:

__________$$ C = cQ^4/3 _______________________ (21)

The first order condition is:

__________$$ p =(4/3)cQ^1/3 or Q =(3p/4c)³ ________ (22)

Now consider the following example for the cost function case:

__________$$ C = 2(vw)^1/2Q^3/2 __________________ (23)

The first order condition is:

__________$$ p =3(vw)^1/2Q^1/2 or Q =p²/9vw _______ (24)

Note: (22) and (23) are the supply function for the firm.

3. Input decision under perfectly competitive markets for input and output. Consider first the one factor case.

__________$$ µ(x)= pf(x) - cx __________________ (25)

where $$x is a factor of production such as labor; $$f(x) is the production function and $$pf(x) is the revenue and $$cx is the cost of using $$x units of input. Repeating for emphasis, the competitive market assumptions are that:

1. The amount of $$x purchased does not influence the price $$c.

2. The amount of output $$Q = f(x) sold does not influence the price of output $$p.

Essentially the firm is small in the input and output markets. The no effect is a limiting assumption. Our goal is to show how profit maximization leads to the demand function for $$x and the supply function for $$Q. To achieve this goal for intermediate microeconomics we shall use a specific production function:

__________$$ Q = x^3/4_________________________ (26)

Note the form of the derived demand and supply functions depend on the form of the production function chosen. The profit maximization problem becomes:

__________$$ max µ(x) with respect to $$x __________ (27)

where

__________$$ µ(x) = px^3/4 - cx __________________ (28)

The first order conditions are:

__________$$ dµ/dx= 0 or 3/4px^-1/4 = c___________ (29)

Note that the first term is the marginal revenue of adding another unit of $$x and $$c is the marginal cost (in this market the marginal and average costs are the same.) To obtain the demand function for $$x we solve the first order conditions for $$x and obtain:

__________$$ x^1/4= 3p/4c or ____________________ (30)

__________ $$ x =(3p/4c)⁴ _______________________ (31)

Note that the amount of $$x needed to maximize profits for particular values of $$p and $$c is found be substituting the numbers in the demand equation for $$x. The supply function for $$Q is obtained by substituting the demand for $$x into the production function:

__________$$ Q = ((3p/4c)⁴)^3/4 or ________________ (32)

__________$$ Q = (3p/4c)³ ______________________ (33)

Students you are not responsible for this derivation. It is inserted for completeness.

4. Let us now consider the two input case for capital and labor using a Cobb-Douglas production function, $$f(K.L) = K^1/3L^1/3: $$ µ(K,L) = p(KL)^1/3 - wL - vK (18) where $$K is capital, $$L is labor, $$v is the capital rental rate and $$w is the wage rate. The first order conditions are:

__________$$ dµ/dL = (1/3)pK^1/3L^-2/3 - w = 0 ______ (34)

__________$$ dµ/dK = (1/3)pK^-2/3L^1/3 - v = 0 ______ (35)

Note that this is a system of two equations in two unknowns. To solve the system clear the negative powers:

__________$$ pK^1/3 = 3wL^2/3 or __________________ (36)

__________ $$ K = (3w/p)³ L² ____________________ (37)

__________$$ pL^1/3 = 3vK^2/3 or __________________ (38)

__________$$ L = (3v/p)³ K² ____________________ (39)

Top solve for $$k substitute the second equation into the first:

__________$$ K = (3w/p)³(3v/p)⁶K⁴ or ____________ (40)

__________$$ K = p³/27wv² _____________________ (41)

Similarly the demand for $$l may be derived as:

__________$$ L = p³/27w²v _____________________ (42)

To obtain the supply function for $$Q we substitute the demand for $$L and $$K into the production function:

__________$$Q = (p³/27wv²)^1/3(p³/27w²v)^1/3 or _____ (43)

__________$$Q = p²/9wv ________________________ (44)

Note: This is the supply function and it is the same one we derived from cost minimization.

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