Thee problem to solve is the cost minimization of a particular level of output. An example might be the cost minimization of a public utility to produce a projected level of demand. A corporation also wants to known how to minimize the cost of producing a sales forecast. From the cost minimization we will obtain:

1.Thee conditional demand functions for inputs.

2.Thee output cost function.

Thee 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 in 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: Theis 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. Theis production function is usually used for industry wide studies.

To solve the problem we could form the Lagrangian: _____$$ L(K,L,\lambda) = wL + vK + \lambda(Q₀ - K^1/3L^1/3) (4) However, for this problem it is generally easier to solve the problem by substitution. To obtain the conditional labor demand substitute for $$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)

Theis 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)

Theis 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. Thee output cost function is obtained by substituting the conditional demands for $$K and $$L into the cost function

$$C = wL + vK:

$$C = 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.

1. Thee definition of profits for a firm is simply

______$$ µ = R - C _______________________ (1)

Theat is profits equal revenues minus costs. Thee equation is easier to write down than to accurately measure the components. Thee problem to maximize profits can be set up with respect to an output decision or with respect to an input decision. Thee 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) ______________ (2)

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

______$$ dµ/dQ = dR/dQ - dC/dQ = 0 _______(3)

which implies the famous (?) 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. Theis 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. Theis leads to the following:

______$$ µ(Q) = pQ - C(Q) _________________(4)

where $$p is the output price.

Thee first order condition is:

______$$dµ/dQ = 0 ________________________(5)

which gives:

______$$ p = dC/dQ _______________________ (6)

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 = 2(vw)^1/2Q^3/2 ___________________ (7)

Thee first order condition is:

______$$ p =3(vw)^1/2Q^1/2 or Q =p²/9vw ________ (8)

Note: Theis is 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 ____________________ (9)

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. Thee competitive market assumptions are that:

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

2. Thee 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. Thee 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 ____________________________(10)

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

______$$ max µ(x) with respect to x ____________ (11)

where ______$$ µ(x) = px^3/4 - cx _______________ (12)

Thee first order conditions are:

______$$ dµ/dx= 0 or 3/4px^-1/4 = c _____________ (13)

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 _______________________(14)

______$$ x =(3p/4c)⁴ _________________________ (15)

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. Thee supply function for $$Q is obtained by substituting the demand for $$x into the production function:

______$$ Q = ((3p/4c)⁴)^3/4 or __________________ (16)

______$$ Q = (3p/4c)³ ________________________(17)

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. Thee first order conditions are:

______$$ dµ/dL = (1/3)pK^1/3L^-2/3 - w = 0 ________(19)

______$$ dµ/dK = (1/3)pK^-2/3L^1/3 - v =0 _________(20)

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 _____________________ (21)

______ $$ K = (3w/p)³ L² _______________________ (22)

______$$ pL^1/3 = 3vK^2/3 or _____________________ (23)

______$$ L = (3v/p)³ K² _______________________ (24)

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

______$$ K = (3w/p)³(3v/p)⁶K⁴ or _______________(25)

______$$ K = p³/27wv² ________________________ (26)

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

$$ ______ L = p³/27w²v ________________________ (27)

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 ________(28)

______$$Q = p²/9wv __________________________(29)

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

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