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Unconstrained Optimization: The Profit Function 287 Equation (7.21) expresses profit not directly as a function of output, but as a function of the inputs employed in the production process, in this case capital and labor. Equation (7.21) allows us to examine the profit-maximizing conditions from the perspective of input usage rather than output levels. Taking partial derivatives of Equation (7.21) with respect to capital and labor, the first-order conditions for profit maximiza-tion are ∂π = P Ê ∂Qˆ −P = 0 ∂π = P Ê ∂Qˆ −P = 0 The second-order condition for a profit maximum is ∂2 π < 0; ∂2 π < 0;Ë ∂2 π¯Ë ∂2 π¯ −Ë ∂∂2 π ¯ > 0 Equations (7.22) may be rewritten as P ¥ MP = P P ¥ MP = P (7.22a) (7.22b) (7.23) (7.24a) (7.24b) The term on the left-hand side of Equations (7.24) is called the marginal revenue product of the input while the term on the right, which is the rental price of the input, is called the marginal resource cost of the input. Equa-tions (7.24) may be expressed as MRP = MRCK MRP = MRCL (7.25a) (7.25b) Equations (7.24) are easily interpreted. Equation (7.24a), for example, says that a firm will hire additional incremental units of capital to the point at which the additional revenues brought into the firm are precisely equal to the cost of hiring an incremental unit of capital. Since the marginal product of capital (and labor) falls as additional units of capital are hired because of the law of diminishing marginal product, and since MRPK < MRCK, hiring one more unit of capital will result in the firm losing money on the last unit of capital hired. Hiring one unit less than the amount of capital required to satisfy Equation (7.24a) means that the firm is for going profit that could have been earned by hiring additional units of capital,since MRPK > MRCK. Problem 7.9. The production function facing a firm is Q = K.5 .5 288 Profit and Revenue Maximization The firm can sell all of its output for $4. The price of labor and capital are $5 and $10, respectively. a. Determine the optimal levels of capital and labor usage if the firm’s operating budget is $1,000. b. At the optimal levels of capital and labor usage,calculate the firm’s total profit. Solution a. The optimal input combination is given by the expression MP = MP L K Substituting into this expression we get (0.5K0.5L−0.5 ) (0.5K−0.5 0.5 ) 5 10 K = 0.5L Substituting this value into the budget constraint we get 1,000 = 5L + 10K 1,000 = 5L + 10(0.5L) L* = 80 1,000 = 5(80)+ 10K K*= 40 b. π = TR − TC = P(K0.5L0.5) − TC = 4[(40)0.6(80)0.4] − 1,000 = −$821.11 MONOPOLY We continue to assume that total cost is an increasing function of output [i.e.,TC =TC(Q)].Now,however,we assume that the selling price is a func-tion of Q, that is, P = P(Q) (7.26) where dP/dQ < 0. This is simply the demand function after applying the inverse-function rule (see Chapter 2). Substituting Equation (7.26) into Equation (7.10) yields π(Q) = P(Q)Q−TC(Q) (7.27) For a profit maximum, the first- and second-order conditions for Equa-tion (7.16) are, respectively, dπ dQ = P +QË dP¯ − dTC = 0 (7.28) Unconstrained Optimization: The Profit Function 289 or P +QÊ dPˆ = MC (7.29) The term on the left-hand side of Equation (7.29) is the expression for marginal revenue.The second-order condition for a profit maximum is d2π =QË d2P ¯ + 2 dP − d2TC < 0 (7.30) If we assume that the demand equation is linear, then Equation (7.26) may be written as P = a+bQ (7.31) where b < 0. Substituting Equation (7.31) into (7.27) yields π(Q) = (a+bQ)Q−TC(Q) = aQ+bQ2 −TC(Q) (7.32) The first- and second-order conditions become dπ = a+ 2bQ− dTC = 0 where a+ 2bQ = MR (7.33) Note that the marginal revenue equation is similar to the demand equation in that it has the same vertical intercept but twice the (negative) slope.Note also that, by definition, Equation (7.31) is the average revenue equation, that is, AR = TR = aQ+bQ2 = a+bQ = P The second-order condition for a profit maximum is d2π = 2b− d2TC < 0 (7.34) The conditions for profit maximization assuming a linear demand curve are shown in Figure 7.9. The cost functions displayed in Figure 7.9a are essentially the same as those depicted in Figure 7.9. All of the cost func-tions represent the short run in production in which 7.8, the prices of the factors of production are assumed to be fixed.The fundamental difference between the two sets of figures is that in the case of perfect competition the firm is assumed to be a “price taker,” in the sense that the firm owner can sell as much product as required to maximize profit without affecting the market price of the product.The conditions under which this occurs will be 290 Profit and Revenue Maximization b D 0 Q1 Q2 Q* C Q4 Q c MR, MC C MC P* P5 D AR 0 Q1 Q3 Q* Q5 MR FIGURE 7.9 Profit maximization: monopoly. Unconstrained Optimization: The Profit Function 291 discussed in Chapter 8.In the case of monopoly,on the other hand,the firm is a “price maker,” since increasing or decreasing output will raise or lower the market price.The simple explanation of this is that because the monop-olist is the only firm in the industry, increasing or decreasing output will result in a right- or a left-shift in the market supply curve. As always, the firm maximizes profit by producing at an output level where MR = MC, which in Figure 7.9 occurs at an output level of Q*. As before, total profit is optimized at output levels Q1 and Q*, where dπ/dQ = 0.At both Q1 and Q* the first-order conditions for profit maximization are satisfied, however only at Q* is the second-order condition for profit max-imization satisfied. In the neighborhood around point D in Figure 7.9b, while the slope of the profit function is positive, it is falling (i.e., d2π/dQ2 < 0).At point C d2π/dQ2 > 0, which is the second-order condition for a local minimum.Again,note that at D¢ the profit-maximizing condition MC = MR, with MC intersecting the MR curve from below, is satisfied. At C¢, MC = MR but, MC intersects MR from above, indicating that this point corre-sponds to a minimum profit level. Note that the marginal cost curve in Figure 7.9c reaches its minimum value at output level Q3, which corresponds to the inflection point on the total cost function in Figure 7.9a. Unlike the case of perfect competition, however, while the selling price is by definition equal to average revenue, in the case of monopoly the price is greater than marginal revenue. Once output has been determined by the firm,the selling price of the product will be defined along the demand curve. In fact, any market structure in which the firm faces a downward-sloping demand curve for its product will exhibit this characteristic. Only in the case of perfect competition, where the demand curve for the product is perfectly elastic,will the condition P = MR be satisfied for a profit-maximizing firm. Problem 7.10. The demand and total cost equations for the output of a monopolist are Q = 90− 2P TC =Q3 −8Q2 +57Q+ 2 a. Find the firm’s profit-maximizing output level. b. What is the profit at this output level? c. Determine the price per unit output at which the profit-maximizing output is sold. Solution a. Define total profit as π =TR−TC ... - tailieumienphi.vn
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