WSG7 7/7/03 4:36 PM Page 95 7 Profit and Revenue Maximization OVERVIEW The purpose of this chapter is to develop a general framework for finding optimal solutions to managerial decision-making problems. This focuses on the decision-making process with respect to two organizational objectives: Profit maximization and total revenue maximization. This chapter considers profit maximization from two perspectives. At a more practical level, management will attempt to maximize profits by employing just the right amount of each factor of production subject to a predefined budget constraint. At a much more general level, profit maximization may be viewed as an unconstrained or constrained optimization problem where the decision variable is the firm s level of output. The marginal product of labor (MP L ) is the change in total output given a unit change in the amount of labor used. The marginal revenue product of labor (MRP L ) is the change in the firm s total revenue resulting from a unit change in the amount of labor used. The marginal revenue product is the marginal product of labor times the selling price of the product, i.e., MRP L = P MP L. Total labor cost is the total cost of labor. The total cost of labor is the wage rate times the total amount of labor employed. The marginal resource cost of labor (MRC L ) is the change in total labor cost resulting from a unit change in the number of units of labor used. If the wage rate (P L ) is constant, then the wage rate is equal to the marginal cost of labor. A profit-maximizing firm that operates in perfectly-competitive output and input markets will employ additional units of labor up to the point Managerial Economics: Theory and Practice 95 Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.
WSG7 7/7/03 4:36 PM Page 96 96 Profit and Revenue Maximization where the marginal revenue product of labor is equal to the marginal labor cost, i.e., P MP L = P L. In general, for any variable input the optimal level of variable input usage is defined by the condition P MP i = P i. The optimal combination of multiple inputs is defined at the point of tangency between the isoquant and isocost curves. The isoquant curve represents the different combinations of capital and labor that produces the same level of output. The slope of the isoquant is the marginal rate of technical substitution. The isocost curve represents the different combinations of capital and labor that the firm can purchase with a fixed operating budget and fixed factor prices.the slope of the isocost curve is the ratio of the input prices. The optimal combination of capital and labor usage is defined by the condition MP L /MP K = P L /P K. This condition may be rewritten as MP L /P L = MP K /P K, which says that a profit-maximizing firm will allocate its budget in such a way that the last dollar spent on labor yields the same amount of additional output as the last dollar spent on capital. This condition defines the firm s expansion path. The objective of profit maximization may be dealt with more directly. The problem confronting the decision-maker is to choose an output level that will maximize profit. Define profit as the difference between total revenue and total cost, both of which are functions of output, i.e., p(q) = TR(Q) - TC(Q). The objective is to maximize this unconstrained objective function with respect to output. The first-order and second-order conditions for a maximum are dp/dq = 0 and d 2 p/dq 2 < 0, respectively. The profitmaximizing condition is to produce at an output level at which MR = MC. Although profit maximization is the most commonly assumed organizational objective, firms that are not owner-operated, or firms that operate in an imperfectly competitive environment often adopt an organizational strategy of total revenue maximization. The first-order and second-order conditions are dtr/dq = 0 and d 2 TR/dQ 2 < 0, respectively. Assuming that firms are price takers in resource markets (the price of labor and capital are fixed), because price and output are always positive, it can be easily demonstrated that the output level that maximizes total revenue will always be greater than the output level that maximizes total profit. This is because of the law of diminishing marginal product guarantees that the rate of increase in marginal cost is greater than the rate of increase in marginal revenue.
WSG7 7/7/03 4:36 PM Page 97 Multiple Choice Questions 97 MULTIPLE CHOICE QUESTIONS 7.1 Consider the production function Q = f(k, L), where K is capital and L is labor. The isocost equation: A. Summarizes the optimal employment of K and L when factor prices are varied. B. The same thing as an isoquant, except that K and L are measured in money terms. C. Summarizes all the possible combinations of K and L that firm can purchase with a given operating budget and variable factor prices. D. Summarizes all the possible combinations of K and L that firm can purchase with a given operating budget and fixed factor prices. 7.2 Suppose that the firm s operating budget is $10,000 and that the price of labor (P L ) and price of capital (P K ) are $25 and $50, respectively. The firm should hire: A. 100 units of labor and 150 units of capital. B. 100 units of labor and 200 units of capital. C. 150 units of labor and 100 units of capital. D. 200 units of labor and 150 units of capital. 7.3 Suppose that Q = f(k, L). If K is measured along the vertical axis and L is measured along the horizontal axis, then an increase in the firm s operating budget will cause: A. A parallel shift of the isocost curve towards the origin. B. A parallel shift of the isocost curve away from the origin. C. The isocost curve to rotate in a counter clockwise direction. D. The isocost curve to rotate in a clockwise direction. 7.4 Suppose that Q = f(k, L). If K is measured along the vertical axis and L is measured along the horizontal axis, then an increase in rental price of capital (P K ) will cause: A. A parallel shift of the isocost curve towards the origin. B. A parallel shift of the isocost curve away from the origin. C. The isocost curve to rotate in a counter clockwise direction. D. The isocost curve to rotate in a clockwise direction. 7.5 Suppose that Q = f(k, L). If K is measured along the vertical axis and L is measured along the horizontal axis, then an increase in rental price of labor (P L )will cause:
WSG7 7/7/03 4:36 PM Page 98 98 Profit and Revenue Maximization A. A parallel shift of the isocost curve towards the origin. B. A parallel shift of the isocost curve away from the origin. C. The isocost curve to rotate in a counter clockwise direction. D. The isocost curve to rotate in a clockwise direction. 7.6 Suppose that Q = f(k, L) where K is measured along the vertical axis and L is measured along the horizontal axis. The slope of the isocost curve is: A. MP L /MP K. B. MP K /MP L. C. P L /P K. D. P K /P L. 7.7 Suppose that Q = f(k, L) where K is measured along the vertical axis and L is measured along the horizontal axis. An increase in P K and P L will cause: A. A parallel shift of the isocost curve towards the origin. B. A parallel shift of the isocost curve away from the origin. C. The isocost curve to rotate in a counter clockwise direction. D. The isocost curve to rotate in a clockwise direction. 7.8 Suppose that Q = f(k, L) where K is measured along the vertical axis and L is measured along the horizontal axis. The least-cost combination of K and L is achieved when: A. When the slope of the isocost line is greater than the slope of the isoquant. B. When the slope of the isoquant is greater than the slope of the isocost line. C. When the slope of the isocost line is equal to the slope of the isoquant. D. When the slope of the isocost line is tangent to the isoquant. E. Both C and D are correct. 7.9 Suppose that Q = f(k, L) where K is measured along the vertical axis and L is measured along the horizontal axis. Suppose that at some input combination the slope of the isocost line is steeper than the slope of the isoquant. To increase output the firm should: A. Hire more K and less L. B. Hire more L and less K. C. Hire more of both K and L. D. Hire less of both K and L.
WSG7 7/7/03 4:36 PM Page 99 Multiple Choice Questions 99 7.10 Suppose that Q = f(k, L). A firm that is using K and L efficiently when: A. P MP L = P L. B. P MP K = P K. C. MP L /MP L = P L /P K. D. Both A and B are correct. E. A, B, and C are correct. 7.11 Suppose that Q = f(k, L). If MP L /MP K > P L /P K, then: A. The firm should hire more capital. B. The firm should hire more labor. C. The firm should increase the price of labor. D. The firm should increase the price of capital. E. None of the above. 7.12 Suppose that a perfectly-competitive firm is producing efficiently. If the rental price of labor (wage rate) is $6.25 and the marginal product of labor is 1.25 units, then the selling price of the product is: A. $1.50. B. $3.50. C. $5.50. D. $7.50. E. None of the above are correct. 7.13 Suppose that the selling price of a product is $12.50. If the rental price of price of capital is $50, then marginal product of capital is: A. 1 unit. B. 2 units. C. 3 units. D. 4 units. E. None of the above are correct. 7.14 Suppose that the rental price of capital is $22 units and the selling price of a product is $11.00. The marginal product of labor is: A. 1 unit. B. 2 units. C. 3 units. D. 4 units. E. Cannot be determine from the information provided.
WSG7 7/7/03 4:36 PM Page 100 100 Profit and Revenue Maximization 7.15 Suppose that the firm s production function is Q = 2L 0.5, where L represents units of labor. If the firm is operating efficiently, then the selling price of the product is: A. $10. B. $50. C. $100. D. $200. 7.16 Suppose that the firm s production function is Q = 20K 0.5 L 0.5, where K is capital and L is labor. Suppose that K = 25, L = 100, and the rental price of capital is $80. If the firm is operating efficiently, then the price of the product is: A. $1.25. B. $3.20. C. $4.00. D. $5.80. 7.17 Suppose that Q = f(k, L). If MP L = 12, MP K = 24, P L = $50 and P K = $100, then a profit maximizing firm should: A. Hire more labor. B. Hire more capital. C. Hire more labor and capital. D. Do nothing. 7.18 Suppose that Q = f(k, L), where K is capital and L is labor. The expansion path of a Cobb-Douglas production function is: A. Always linear. B. Always quadratic. C. Always cubic. D. Depends upon whether the production function exhibits increasing, decreasing or constant returns to scale. 7.19 Suppose that Q = f(k, L) where K is measured along the vertical axis and L is measured along the horizontal axis. The slope of the expansion path is: A. The capital-labor ratio. B. The labor-capital ratio. C. The output-capital ratio. D. The output-labor ratio. 7.20 Suppose that Q = 56K 0.4 L 0.8 where K is measured along the vertical axis and L is measured along the horizontal axis. If the rental price of labor and capital are $80 and $40, respectively, then the firm s expansion path is:
WSG7 7/7/03 4:36 PM Page 101 Multiple Choice Questions 101 A. 0.8(56)K 0.4 L -0.2 = 80. B. 0.4(56)K -0.6 L 0.8 = 40. C. K = L. D. K/L = 80/40. 7.21 Suppose that the firm s profit equation is p =-172 + 288Q -8Q 2.The profit-maximizing level of output for this firm is: A. 14 units. B. 18 units. C. 22 units. D. 26 units. 7.22 Suppose that the firm s profit equation is p =-172 + 288Q -8Q 2.The maximum profit for this firm is: A. $2,420. B. $3,860. C. $4,620. D. $5,222. 7.23 A profit-maximizing firm must always produce at an output level where: A. P = ATC. B. P = AVC. C. MR = ATR. D. P = MC. 7.24 Profit is maximized at the output level where the: A. Slope of the total revenue curve is greater than the slope of the total cost curve. B. Slope of the total revenue curve is the same as the slope of the total cost curve. C. Slope of the total revenue curve is less than the slope of the total cost curve. D. Slope of the marginal revenue curve is equal to slope of the marginal cost curve. 7.25 A perfectly-competitive firm maximizes profit at the output level where: A. Mp = 0. B. P = MC. C. MR = MC. D. Both A and C are correct. E. All of the above.
WSG7 7/7/03 4:36 PM Page 102 102 Profit and Revenue Maximization 7.26 The market-determined price of a product is $4. The total cost equation of a firm in a perfectly-competitive industry is TC = 100 + 20Q + Q 2. The profit-maximizing rate of output is: A. 3 units. B. 4 units. C. 5 units. D. 6 units. 7.27 A monopolist maximizes profit at the output level where: A. Mp = 0. B. P = MC. C. MR = MC. D. Both A and C are correct. E. All of the above. 7.28 The market demand for the output of a monopolist is Q = 125-0.25P. The monopolist s total cost equation is TC = 1,000 + 200Q + Q 2. To maximize total profits the monopolist should charge a price of: A. $100. B. $250. C. $380. D. $540. 7.29 The market demand for the output of a monopolist is Q = 125-0.25P. The monopolist s total cost equation is TC = 1,000 + 200Q + Q 2. A revenue-maximizing monopolist would charge a price of: A. $166.67. B. $275.50. C. $350.00. D. $475.00. 7.30 Suppose that a firm s total profit equation is p =-2,500 + 100x + 110y - 5xy - 0.5x 2-0.5y 2, where x and y represent the output levels from two production processes. The profit maximizing combination of x and y is: A. x = 10 and y = 8.67. B. x = 35.50 and y = 45. C. x = 50 and y = 43.33. D. x = 65.33 and y = 50.
WSG7 7/7/03 4:36 PM Page 103 Shorter Problems 103 SHORTER PROBLEMS 7.1 Suppose that a firm produces at an output level where MP L = 60 and P L = $30. Suppose, further, that MP K = 125 and P K = $50. A. Is this firm producing efficiently? B. If the firm is not producing efficiently, then how might it do so? 7.2 Suppose that a firm produces at an output level where MP L = 36 and P L = $12. Suppose, further, that MP K = 48 and P K = $24. A. Is this firm producing efficiently? B. If the firm is not producing efficiently, then how might it do so? 7.3 A firm s production function is given by the equation: Q = 15K 0.65 L 0.25 where input prices are P L = $5 and P K = $15. Determine the equation of the expansion path. 7.4 Suppose that a profit-maximizing firm s production function is Q = 125(0.2K + 0.4L) where Q, K and L represent units of output, capital, and labor, respectively. A. Suppose that the price of capital per unit is P K = $25 and the price of labor per unit is P L = $12.50. What is the optimal input combination for this firm? B. Suppose that the price of capital remains P K = $25, but the price of labor rises to P L = $50. What is the firm s optimal input combination? C. Suppose that the price of capital falls to P K = $5, while the price of labor remains unchanged at P L = $50. What is the firm s optimal input combination? 7.5 Suppose that a firm s estimated production function is: Q = 25L 0.6 K 0.6 where Q represents units of output, K represents units of capital, and L represents units of labor. Suppose that the rental price of labor is P L = $100. If L = 15 and the price of the product is $5, estimate the optimal level of capital input.
WSG7 7/7/03 4:36 PM Page 104 104 Profit and Revenue Maximization 7.6 The production function facing a firm is Q = 80K 0.7 L 0.3 The firm can sell all of its output for $10. The rental price of labor and capital are $7 and $12, respectively. A. Determine the optimal levels of capital and labor usage if the firm s operating budget is $25,000. B. At the optimal levels of capital and labor usage, calculate the firm s total profit. 7.7 The total revenue and total cost equations of a perfectly-competitive firm are: TR = 30Q TC = 55-25Q + 0.02Q 2 A. What is the total profit function? B. Calculate the profit maximizing level of output? C. Calculate the firm s profit at the profit-maximizing output level. LONGER PROBLEMS 7.1 Suppose that the wage rate (P L ) is $35, the rental price of capital (P K ) is $75, and the firm s operating budget is $17,500. A. What is the isocost equation for the firm? B. If capital is graphed on the vertical axis, what happens to the isocost line if the wage rate falls? C. If K is measured along the vertical axis and L is measured along the horizontal axis, what happens to the isocost line if the rental price of labor rises? D. If the wage rate and the rental price of capital remain unchanged, what happens to the isocost line if the firm s operating budget decreases? E. If the firm s operating budget remains unchanged, what happens to the isocost line if the wage rate and the rental price of capital decline by the same percentage? 7.2 A firm s production function is: Q = 65L 0.5 K 0.5 where K is capital and L is labor. The prices of a unit of labor and capital and labor are $10 and $20, respectively. Suppose that the firm s operating budget is $15,000. A. Estimate the optimal levels of labor and capital usage. B. Given your answer to part A., estimate the firm s total output.
WSG7 7/7/03 4:36 PM Page 105 Longer Problems 105 7.3 The total cost equation for a perfectly-competitive firm is: TC = 20 + 2.5Q 2 A. If the firm can sell all of its output for $20 per unit, calculate the firm s profit-maximizing output level? At the profit-maximizing level of output, what is the firm s total profit. B. Suppose that the market demand equation for this product is Q = 2,500-25P. If this firm was as monopoly, calculate the profitmaximizing level of output. What price should the monopolist charge? At the profit-maximizing level of output, what is the monopolist s total profit. C. What is the monopolist s total-revenue maximizing level of output? At the total-revenue maximizing level of output, calculate the monopolist s total profit. 7.4 The demand and total cost equations for the output of a monopolist are Q = 200-5P TC = Q 3-8Q 2 + 25Q + 10 A. What is the profit-maximizing level of output? B. What is the profit at this output level? C. Determine the profit-maximizing price per unit of output. 7.5 The demand and total cost equations for the output of a monopolist are Q = 80-2P TC = Q 3-10Q 2 + 58Q + 2 A. What is the profit-maximizing level of output? B. What is the profit at this output level? C. Determine the profit-maximizing price per unit of output. 7.6 A firm s total profit equation is: p(x, y) = -900 + 360x - x 2 - xy - y 2 + 250y where x and y represent the output levels for the two product lines. A. Determine the profit-maximizing output levels of goods x and y subject to the side condition that the sum of the two product lines equal 500 units using the Lagrange multiplier method. B. Calculate the firm s total profits. C. What is the interpretation of the Lagrange multiplier?
WSG7 7/7/03 4:36 PM Page 106 106 Profit and Revenue Maximization D. Suppose that there were no combined output requirement. What are the profit maximizing levels of x and y? E. Given your answer to part D., what is the firm s total profits? 7.1 D. 7.2 A. 7.3 B. 7.4 C. 7.5 D. 7.6 C. 7.7 A. 7.8 E. 7.9 A. 7.10 E. 7.11 B. 7.12 E. 7.13 D. 7.14 E. 7.15 B. 7.16 C. 7.17 D. 7.18 A. 7.19 A. 7.20 C. 7.21 B. 7.22 A. 7.23 D. 7.24 B. 7.25 E. 7.26 B. 7.27 C. 7.28 C. 7.29 D. 7.30 C. ANSWERS TO MULTIPLE CHOICE QUESTIONS
WSG7 7/7/03 4:36 PM Page 107 Solutions to Shorter Problems 107 SOLUTIONS TO SHORTER PROBLEMS 7.1 A. The optimal input combination is given by the expression MP L /P L = MP K /P K 60/30 = 2 < 2.5 = 125/50 Thus, the firm is not operating efficiently. B. To produce more efficiently, the firm should reallocate its budget dollars away from labor and toward towards capital. 7.2 A. The optimal input combination is given by the expression MP L /P L = MP K /P K 36/12 = 3 > 2 = 48/24 Thus, the firm is not operating efficiently. B. To produce more efficiently, the firm should reallocate its budget dollars away from capital and toward labor. 7.3 The expansion path is determined from the expression MP L /P L = MP K /P K MP L /P L = ( Q/ L)/P L = [0.25(15)K 0.65 L -0.75 ]/P L MP K /P K = ( Q/ K)/P K = [0.65(15)K -0.35 L 0.25 ]/P K [0.25(15)K 0.65 L -0.75 ]/5 = [0.65(15)K -0.35 L 0.25 ]/15 K = (13/15)L 7.4 A. MP L = Q/ L = 50 MP K = Q/ K = 25 MP L /P L = 50/12.50 = 4 MP K /P K = 25/25 = 1 Since MP L /P L > MP K /P K and the marginal products are constant, then the firm should use only labor. B. MP L /P L = 50/50 = 1 MP K /P K = 25/25 = 1 Since MP L /P L = MP K /P K and the marginal products are constant, then any combination of labor and capital that satisfy the firm s budget constraint is efficient. C. MP L /P L = 50/50 = 1 MP K /P K = 25/5 = 5 Since MP K /P K > MP L /P L and the marginal products are constant, then the firm should use only capital.
WSG7 7/7/03 4:36 PM Page 108 108 Profit and Revenue Maximization 7.5 Optimality requires that P L = P MP L. 100 = 5[(0.6)25L -0.4 K 0.6 ] = 5[15(15) -0.4 K 0.6 ] = 25.3878K 0.6 K 0.6 = 3.9389 K = 9.824 units of capital. 7.6 A. MP L /P L = MP K /P K [0.3(80)K 0.7 L -0.7 ]/7 = [0.7(80)K -0.3 L 0.3 ]/12 0.3K/7 = 0.7L/12 K = (4.9/3.6)L TC 0 = P L L + P K K 25,000 = 7L + 12L 25,000 = 7L + 12(4.9/3.6)L L* = 1,071.43 25,000 = 7(1,071.43) + 12K K* = 1,458.33 B. p=tr - TC = PQ - TC = 10(80K 0.7 L 0.3 ) - 25,000 10[80(1,458.33) 0.7 (1,071.43) 0.3 ] - 25,000 = $1,038,599.67 7.7 A. p=tr - TC = 30Q - (55-25Q + 0.02Q 2 ) =-55 + 55Q - 0.02Q 2 B. dp/dq = 55-0.04Q = 0, i.e., the first-order condition for p maximization. d 2 p/dq 2 =-0.04 < 0, i.e., the second-order condition for p maximization is satisfied. Solving the first-order condition for Q we obtain Q* = 1,375 C. p* = -55 + 55(1,375) - 0.02(1,375) 2 = $37,757.50 SOLUTIONS TO LONGER PROBLEMS 7.1 A. TC 0 = P L L + P K K 17,500 = 35L + 75K B. K = TC 0 /P K - (P L /P K )L If the wage rate falls, the isocost line will rotate counterclockwise, i.e., the K-intercept will remain unchanged while the L-intercept moves to the left. C. If the rental price of labor rises, then the isocost line will rotate clockwise, i.e., the K-intercept will remain unchanged while the L-intercept moves to the left.
WSG7 7/7/03 4:36 PM Page 109 Solutions to Longer Problems 109 D. If the operating budget decreases, the isocost line will undergo a parallel shift toward the origin, i.e., the K-intercept will move down and the L-intercept will move to the left. The factor prices are unchanged, the slope of the isocost line will remain unchanged. E. If the wage rate and the rental price of capital decline by the same percentage, then the isocost line undergo a parallel shift away from the origin, i.e., the K-intercept will move up and the L-intercept will move to the right in the same proportion. The slope of the isocost line will remain unchanged. 7.2 A. The optimal combination of K and L is determined by the relation MP L /MP K = P L /P K MP L = Q/ L = 0.5(65)L -0.5 K 0.5 MP K = Q/ K = 0.5(65)L 0.5 K -0.5 [0.5(65)L -0.5 K 0.5 ]/[0.5(65)L 0.5 K -0.5 ] = 10/20 Solving for the optimal capital-labor ratio we obtain K/L = 1/2 K = 0.5L The firm s isocost equation is TC 0 = P L L + P K K 15,000 = 10L + 20K Substituting we obtain 15,000 = 10L + 20(0.5L) = 20L The optimal level of labor usage is L* = 750 The optimal level of capital usage is 15,000 = 10(750) + 20K K* = 375 B. Q* = 65(750) 0.5 (375) 0.5 = 34,471.46 7.3 A. TR = P Q = 20Q p=tr - TC = 20Q - (20 + 2.5Q 2 ) = -20 + 20Q - 2.5Q 2 dp/dq = 20-5Q = 0, i.e., the first-order condition for p maximization. Solving the first-order condition for Q we obtain Q* = 4 units d 2 p/dq 2 =-5 < 0, i.e., the second-order condition for p maximization is verified. p* = -20 + 20(4) - 2.5(4) 2 = $20
WSG7 7/7/03 4:36 PM Page 110 110 Profit and Revenue Maximization B. P = 100-0.04Q TR = P Q = 100Q - 0.04Q 2 p=tr - TC = (100Q - 0.04Q 2 ) - (20 + 2.5Q 2 ) =-20 + 100Q - 2.54Q 2 dp/dq = 100-5.08Q = 0, i.e., the first-order condition for p maximization. Solving the first-order condition for Q we obtain Q* = 19.69 units d 2 p/dq 2 =-5.08 < 0, i.e., the second-order condition for p maximization is verified. P* = 100-0.04(19.69) = $99.21 p* = -20 + 100(19.69) - 2.54(19.69) 2 = $964.25 C. dtr/dq = 100-0.08Q = 0, i.e., the first-order condition for TR maximization. Solving the first-order condition for Q we obtain Q* = 1,250 d 2 TR/dQ 2 =-0.08 < 0, i.e., the second-order condition for TR maximization is verified. p* = -20 + 100(1,250) - 2.54(1,250) 2 =-$3,843,770 7.4 A. p=tr - TC P = 40-0.2Q TR = 40Q - 0.2Q 2 p=tr - TC = (40Q - 0.2Q 2 ) - (Q 3-8Q 2 + 25Q + 10) =-Q 3 + 7.8Q 2 + 15Q - 10 dp/dq = -3Q 2 + 15.6Q + 15 = 0 Q 1,2 = [-b ± (b 2-4ac)]/2a = {-15.6 ± [(15.6) 2-4(-3)(15)]}]/2(-3) = [-15.6 ± (243.36 + 180)]/-6 Q 1,2 = (-15.6 ± 20.58)/-6 Q 1 =-36.18/-6 = 6.03 Q 2 = 4.98/-6 = -0.83 The second-order condition for profit maximization is d 2 p/dq 2 < 0. Taking the second derivative of the profit function yields d 2 p/dq 2 =-6Q + 15.6 Substituting the solution values into this condition we get d 2 p/dq 2 =-6(6.03) + 15.6 = -20.58 < 0, for a local maximum d 2 p/dq 2 =-6(-0.83) + 15.6 = 20.58 > 0, for a local minimum Thus, total profit is maximized at Q = 6.08 units. B. p* = -(6.03) 3 + 7.8(6.03) 2 + 15(6.03) - 10 = $144.81 C. P* = 40-0.2(6.03) = $38.79
WSG7 7/7/03 4:36 PM Page 111 Solutions to Longer Problems 111 7.5 A. p=tr - TC P = 40-0.5Q TR = 40Q - 0.5Q 2 p=tr - TC = (40Q - 0.5Q 2 ) - (Q 3-10Q 2 + 58Q + 2) =-Q 3 + 9.5Q 2-18Q - 2 dp/dq = -3Q 2 + 19Q - 18 = 0 Q 1,2 = [-b ± (b 2-4ac)]/2a = {-19 ± [(19) 2-4(-3)(-18)]}/2(-3) = (-19 ± 145)/-6 Q 1,2 = (-19 ± 12.04)/-6 Q 1 =-31.04/-6 = 5.17 Q 2 =-6.96/-6 = 1.16 The second-order condition for profit maximization is d 2 p/dq 2 < 0. Taking the second derivative of the profit function yields d 2 p/dq 2 =-6Q + 19 Substituting the solution values into this condition we get d 2 p/dq 2 =-6(5.17) + 19 = -12.02 < 0, for a local maximum d 2 p/dq 2 =-6(-1.16) + 19 = 25.96 > 0, for a local minimum Thus, total profit is maximized at Q = 5.17 units. B. p* = -(5.17) 3 + 9.5(5.17) 2-18(5.17) -2 = $20.68 C. P* = 40-0.5(5.17) = $37.42 7.6 A. The formal statement of this problem is: Maximize: p(x, y) = -900 + 360x - x 2 - xy - y 2 + 250y Subject to: x + y = 500 (x, y) = -900 + 360x - x 2 - xy - y 2 + 250y + l(500 - x - y) The first-order conditions are: / x = 360-2x - y - l=0 / y = -x - 2y + 250 - l=0 / l = 500 - x - y = 0 This is a system of three linear equations in three unknowns. Assuming that the second-order conditions for profit maximization are satisfied, we can solve this system of equations simultaneously to yield the optimal solution values. x* = 305 y* = 195 l* = -$445 B. p* = -900 + 360(305) - (305) 2 - (305)(195) - (195) 2 + 250(195) =-900 + 109,800-93,025-59,475-38,025 + 48,750 =-$32,875
WSG7 7/7/03 4:36 PM Page 112 112 Profit and Revenue Maximization C. l* = / k = p*/ k = -$445 The Lagrange multiplier says that, in the limit, a reduction in the firm s combined output by 1 unit will result in a $445 increase in the firm s maximum profit. Alternatively, an increase in the firm s combined output by 1 unit will result in a $445 decrease in the firm s maximum profit. D. The first-order conditions are: p/ x = 360-2x - y = 0 p/ y = -x - 2y + 250 = 0 This is a system of two linear equations in two unknowns. Assuming that the second-order conditions for profit maximization are satisfied, we can solve this system of equations simultaneously to yield the optimal solution values. x* = 156.67 y* = 46.67 E. p* = -900 + 360(156.67) - (156.67) 2 - (156.67)(46.67) - (46.67) 2 + 250(46.67) =-900 + 56,401.20-24,545.49-7,311.79-2,178.09 + 11,667.50 = $33,133.33