Partial derivatives, linear approximation and optimization

Transcription

1 ams 11b Study Guide 4 econ 11b Partial derivatives, linear approximation and optimization 1. Find the indicated partial derivatives of the functions below. a. z = 3x 2 + 4xy 5y 2 4x + 7y 2, z x = 6x + 4y 4 z yx = z xy = 4 b. F (u, v, w) = 60u 2/3 v 1/6 w 1/2 F u = 40u 1/3 v 1/6 w 1/2 2 F w u = 2 F u w = 20u 1/3 v 1/6 w 1/2 c. w = x 2 z ln(y 2 + z 3 ) w x = 2xz ln(y 2 + z 3 ) w y = x 2 2y z y 2 + z = 2x2 yz 3 y 2 + z 3 w xx = 2z ln(y 2 + z 3 ) w yz = 2x2 y(y 2 + z 3 ) 3z 2 (2x 2 yz) = 2x2 y 3 4x 2 yz 3 (y 2 + z 3 ) 2 (y 2 + z 3 ) 2 w xyz = w yzx = 4xy(y2 2z 3 ) (y 2 + z 3 ) 2 ( ) = 2x2 y(y 2 2z 3 ) (y 2 + z 3 ) 2 d. q(u, v) = u2 v 3uv 3 2u + 3v q u = (2uv 3v3 )(2u + 3v) 2(u 2 v 3uv 3 ) = 2u2 v + 6uv 2 9v 4 (2u + 3v) 2 (2u + 3v) 2 q v = (u2 9uv 2 )(2u + 3v) 3(u 2 v 3uv 3 ) = 2u3 18u 2 v 2 18uv 3 (2u + 3v) 2 (2u + 3v) 2 2. The monthly cost function for ACME Widgets is C = 0.02Q 2 A Q A Q B Q 2 B + 35Q A + 28Q B , where Q A and Q B are the monthly outputs of type A widgets and type B widgets, respectively, measured in 100s of widgets. The cost is measured in dollars. a. Compute the marginal cost of type A widgets and the marginal cost of type B widgets, if the monthly outputs are type A widgets and type B widgets. C/ Q A = 0.04Q A Q B + 35 and C/ = 0.01Q A Q B Next, remembering the units, Q A = 25000/100 = 250 and Q B = 36000/100 = 360, so C C = 48.6 and = Q A 1

2 b. Suppose that production of type A widgets is held fixed at 25000, and production of type B widgets is increased from to Use your answer to part a. to estimate the change in cost to the firm. Approximation formula: C C Q B, since we are assuming in this case that Q A = 0. Now, Q B = 50/100 = 0.5, so C (52.1)(0.5) = c. Suppose that production of type A widgets is increased from to 25060, and production of type B widgets is increased from to Use your answer to part a. to estimate the change in cost to the firm. General Approximation formula: C C Q A Q A + C Q B. In this case we have Q A = 60/100 = 0.6 and Q B = 40/100 = 0.4, so C C Q A + C Q B = (48.6)(0.6) + (52.1)(0.4) = 50. Q A 3. The demand function for a firm s product is given by Q = 30 6Y + 5p s, where 3p + 5 Q is the monthly demand for the firm s product, measured in 1000 s of units, Y is the average monthly disposable income in the market for the firm s product, measured in 1000s of dollars, p s is the average price of a substitute for the firm s product, measured in dollars, p is the price of the firm s product, also measured in dollars. a. Find Q, Q Y, Q ps and Q p when the monthly income is $2500 and the prices are p s = 17 and p = 15. Round your (final) answers to two decimal places. First, writing Q = Q(p, p s, Y ), and remembering that Y is measured in $1000s, we have Next, the partial derivatives are Q(15, 17, 2.5) = Q Y = 30 3p (6Y + 5p s ) = 90 1/2 (3p + 5)(6Y + 5p s ) = Q 1/2 Y (15, 17, 2.5) = 9 50 = 0.18, Q ps = 30 3p (6Y + 5p s ) = 75 1/2 (3p + 5)(6Y + 5p s ) = Q 1/2 p s (15, 17, 2.5) = 3 20 = 0.15 and Q p = Y + 6Y + 5p s ( 1) (3p + 5) = 90 5ps = Q 2 (3p + 5) 2 p (15, 17, 2.5) = 9 25 = = 6.

3 b. Compute the income-elasticity of demand for the firm s product at the point in part a. η Q/Y = Q Y Y Q = η Q/Y p=15 ps=17 Y =2.5 = = c. Use linear approximation and your answer to a. to estimate the change in demand for the firm s product if the price of the firm s product increases to $16 and the price of substitutes increases to $18, but income remains fixed. Q Q p p + Q ps p s + Q Y Y = = I.e., demand will decrease by about 210 units. d. Use your answer to part b. to estimate the percentage change in demand for the firm s product if the average income increases to $2600 while the prices stay the same as they were in part a. The percentage change in income is % Y = % = 4%, so Find the critical points of the functions below. % Q η Q/Y % Y = % = 0.3%. a. f(x, y) = 3x 2 12xy + 19y 2 2x 4y + 5. First order conditions: f x = 6x 12y 2 = } 0 f y = 12x + 38y 4 = 0 Now, f x = 0 = 6x = 12y + 2, so 12x = 24y + 4. Plugging this into the second equation gives = (24y + 4) + 38y 4 = 0 = 14y 8 = 0 = y 0 = 4 7 = x 0 = ( 31 So there is one critical point, 21, 4 ). 7 b. g(s, t) = s 3 + 3t st + 2. First order conditions: g s = 3s t = 0 g t = 6t + 12s = 0 Now, g t = 0 = t = 2s, and plugging this into the first equation gives 3s 2 24s = 0 = 3s(s 8) = 0 = two solutions: s 1 = 0 and s 2 = 8. So there are two critical points in this case, (s 1, t 1 ) = (0, 0) and (s 2, t 2 ) = (8, 16). } 3

4 c. h(u, v) = u 3 + v 3 3u 2 3v + 5. First order conditions: h u = 3u 2 6u = 0, h v = 3v 2 3 = 0. The first equation factors as 3u(u 2) = 0, which has two solutions u 1 = 0 and u 2 = 2. The second equation factors as well, giving 3(v 2 1) = 0, which has the two solutions v 1 = 1 and v 2 = 1. The first equation places no restrictions on the variable v, while the second equation places no restrictions on the variable u, so the critical points of the function h(u, v) are (u 1, v 1 ) = (0, 1), (u 1, v 2 ) = (0, 1), (u 2, v 1 ) = (2, 1) and (u 2, v 2 ) = (2, 1). 5. Use the second derivative test to classify the critical values of the functions in the previous problem. a. Second derivative test: f xx = 6 f yy = 38 f xy = 12 = D = = 84 > 0. Since D > 0 and f xx > 0, it follows that f ( 31, ) = 50 is a relative minimum value. (In 21 fact, since the second derivatives are all constant, this is the absolute minimum value.) b. Second derivative test: g ss = 6s g tt = 6 g st = 12 = D(s, t) = 36s 144. Since D(0, 0) = 144 < 0, the first critical point yields a saddle point on the graph of g(s, t), i.e., g(0, 0) = 2 is neither max nor min. Since D(8, 16) = 144 > 0 and g ss (8, 16) = 48 > 0, it follows that g(8, 16) = 254, is a relative minimum value. Can you show that g(8, 16) = 254 is not the absolute minimum? c. Second derivative test: h uu = 6u 6 h vv = 6v h uv = 0 = D(u, v) = 36v(u 1). Evaluating the discriminant at the four critical points we find that i. D(0, 1) = 36 < 0, so h(0, 1) = 3 is neither a local minimum value nor a local maximum value; ii. D(0, 1) = 36 > 0 and h uu (0, 1) = 6 < 0, so h(0, 1) = 7 is a local maximum value; 4

5 iii. D(2, 1) = 36 > 0 and h uu (2, 1) = 6 > 0, so h(2, 1) = 1 is a local minimum value; and iv. (D(2, 1) = 36, so h(2, 1) = 3 is neither a local minimum value nor a local maximum value. 6. ACME Widgets produces two competing products, type A widgets and type B widgets. The joint demand functions for these products are Q A = 100 3P A + 2P B and Q B = P A 2P B and ACME s cost function is C = 20Q A + 30Q B Find the prices that ACME should charge to maximize their profit, the corresponding output levels and the max profit. Justify your claim that the prices you found yield the absolute maximum profit. ACME s profit function is (i) Critical point(s): Now, Π PB Π = P A Q A + P B Q B C = 100P A 3P 2 A + 2P A P B + 60P B + 2P A P B 2P 2 B [ 20(100 3P A + 2P B ) + 30(60 + 2P A 2P B ) ] = 3P 2 A + 4P A P B 2P 2 B + 100P A + 80P B Π PA = 6P A + 4P B = 0 Π PB = 4P A 4P B + 80 = 0 = 0 = 4P B = 4P A + 80, and plugging this into the first equation gives 6P A + (4P A + 80) = 0 = 2P A = 0 = P A = 90 = P B = 110. So there is only one critical point, and the critical prices are P A = 90 and P B = 110, with corresponding output/demand levels Q A = 50 and Q B = 20. (ii) Second derivative test: Π PA P A = 6 Π PB P B = 4 Π PA P B = 4 = D = ( 6) ( 4) 16 = 8 > 0. Since D > 0 and Π PA P A = 6 < 0, and the second derivatives are all constant, it follows that Π(90, 110) = 3900 is the absolute maximum profit. 7. An electronics retailer has determined that the number N of laptops she can sell per week is N = 9x 4 + x + 20y 5 + y 5 }

6 where x is her weekly expenditure on radio advertising and y is her weekly expenditure on internet advertising, both measured in $100s. Her weekly profit is $400 per sale, less the cost of advertising. Find the amount of money that the retailer should spend on radio and internet advertising, respectively, to maximize her weekly profit. Verify that the point you found yields a relative maximum value. What is the maximum profit? The first step is to find the weekly profit function, P, which is $400 times the number of laptops sold minus the cost of advertising: ( 36x P = 400N 100(x + y) = x + 80y ) 5 + y x y. Next, first-order conditions: ( ) 144 P x = 0 = 100 ((4 + x) 1 2 ) 400 P y = 0 = 100 (5 + y) 1 2 = 0 = 144 (4 + x) 2 = 1 = (4 + x)2 = 144 = 0 = 400 (5 + y) 2 = 1 = (5 + y)2 = 400 It follows that 4 + x = ±12 and 5 + y = ±20, and since both x and y must be nonnegative, we conclude that the critical numbers are x = 12 4 = 8 and y = 20 5 = 15. The corresponding profit is P = 100 ( ) = To verify that this is the maximum profit, we use the second derivative test: so P xx = 288 (4 + x) 3, P yy = 800 (5 + y) 3 and P xy = 0, D(x, y) = = D(8, 15) = (4 + x) 3 (5 + y) > 0 3 and P xx (8, 15) = < 0 3 which shows that P is a maximum. 6