Optimization, constrained optimization and applications of integrals.

Transcription

1 ams 11b Study Guide econ 11b Optimization, constrained optimization and applications of integrals. (*) In all the constrained optimization problems below, you may assume that the critical values you find are in fact the optimal values you seek. 1. Compute the producers and consumers surplus at equilibrium for the market with the supply and demand equations: supply: p =.1q + ; demand: p = 4 q 1 q2 1. Solution: First, find the equilibrium price and quantity, by setting demand equal to supply: p =.1q + = 4 q 1 q2 1 =.1q2 +.2q 3 = = q =.2 ± (.1) ( 3).2 ± 1.2 = = q = or q = Since quantity must be positive, the equilibrium quantity is q =, and the equilibrium price is p =.1 q + = 1. Now compute the Consumers Surplus and Producers Surplus. Consumers Surplus = = Producers Surplus = q (demand p) dq 4 q 1 q2 1 = 3q q2 2 q3 3 q ( p supply) dq = 1 (.1q + ) dq = q q2 2 = dq = A monopolistic firm sells one product in two markets, A and B. The daily demand equations for the firm s product in these markets are given by Q A = 1.4P A and Q B = 12.P B, where Q A and Q B are the demands and P A and P B are the prices for the firm s product in markets A and B, respectively. The firm s constant marginal cost is $4 and the its daily fixed cost is $2. 1

2 a. Find the prices that the firm should charge in each market to maximize its daily profit. Use the second derivative test to verify that the prices you found yield the absolute maximum profit. Solution: The firm s total daily output is Q A + Q B, so the firm s daily cost is C = 4(Q A + Q B ) + 2. The firm s revenue from markets A and B is R A = P A Q A and R B = P B Q B. The firm s profit function is Π = P A Q A + P B Q B C = P A (1.4P A ) + P B (12.P B ) [4(1.4P A + 12.P B ) + 2] =.4P 2 A.P 2 B + 116P A + 14P B 113. The first order conditions for an optimum value give Π PA = =.8P A = = P A = 14 Π PB = = P B + 14 = = P B = 14 The second order conditions for a maximum are Π PA P A (P A, P B) Π PB P B (P A, P B) (Π PA P B (P A, P B)) 2 > and Π PA P A (P A, P B) <. In this case we have Π PA P A =.8 < and Π PA P A (P A, P B) Π PB P B (P A, P B) (Π PA P B (P A, P B)) 2 =.8 > for all (P A, P B ), so that the second order conditions are satisfied, and because the conditions hold everywhere, the critical value of profit Π = Π(P A, P B) = 691 is the absolute maximum daily profit. To summarize: the firm s profit is maximized when PA = 14 and P B the daily demands are = 14, at which point Q A = 1.4P A = 42 and Q B = 12.P B = and the maximum profit is Π = 691. b. Use the envelope theorem (and linear approximation) to estimate the change in the Firm s max profit if the marginal cost of their product increases to $4.7. Solution: If we denote the marginal cost by µ, then the profit function can be written as Π = P A Q A + P B Q B µ 2 total daily ouptut {}}{ (Q A + Q B ) 2.

3 According to the envelope theorem dπ dµ = dπ = (Q A + Q B ) dµ PA =P A P B =P B PA =P A P B =P B = (Q A + Q B) = 92. Now, using linear approximation, we find that Π dπ dµ µ = 92 (.7) = 69. I.e., if the firm s marginal cost increases by $.7, then the max daily profit will decrease by about $ Jack s (gustatory) utility function is U(x, y, z) = ln x + 7 ln y + 18 ln z, where x is the number of fast-food meals Jack consumes in a month; y is the number of diner meals he consumes in a month; and z is the number of fancy restaurant meals he consumes in a month. The average price of a fast-food meal is p x = $4.; the average price of a diner meal is p y = $8.; and the average price of a fancy restaurant meal is p z = $3.. a. How many meals of each type should Jack consumer per month to maximize his utility, if his monthly budget for these meals is β = $12.? Solution: The objective function is the utility U(x, y, z) = ln x + 7 ln y + 18 ln z, and the constraint is the budget (or income) constraint we obtain from the prices and the budget: xp x + yp y + zp z = β = 4x + 8y + 3z = 12. Lagrangian: F (x, y, z, λ) = ln x + 7 ln y + 18 ln z λ(4x + 8y + 3z 12). Structural equations: F x = x 4λ = F y = 7 y 8λ = F z = 18 3λ =. z Solving these equations for λ gives the triple equation λ = 4x = 7 8y = 3 z. 3

4 Comparing the x-term and the y-term and clearing denominators gives 4x = 7 8y = 4y = 28x = y = 7x 1. Comparing the x-term and the z-term and clearing denominators gives 4x = 3 z = 2z = 12x = z = 12x 2. Substituting the expressions for y and z that we found into the budget constraint (F λ = ) gives ( ) ( ) 7x 12x 4x = 12 = 12x = 6 = x =, y = 3, z = Thus, Jack maximizes his utility by consuming fast food meals, 3 diner meals and 24 fancy restaurant meals in a month, resulting in a max utility of U = U(x, y, z ) = U(, 3, 24) b. By approximately how much will Jack s utility increase if his budget increases by $.? Explain your answer. Solution: Since the utility function and the prices of meals are not changing, the maximum utility, U, is a function of the budget, β. I.e., increasing the budget increases U and decreasing the budget decreases U. Now, observe that at the critical point (x, y, z ) = because F λ = {}}{ F = F (x, y, z, λ; β) = U(x, y, z ) λ (4x + 8y + 3z 12) = U(x, y, z ) = U, which means that du dβ = df dβ. Next, the envelope theorem applied to the Lagrangian function F (x, y, z, λ; β) tells us that du dβ = df dβ = df = λ, dβ x=x y=y z=z λ=λ where λ is the critical value of the multiplier λ. In this case, Finally, we use linear approximation: U du λ = 4x = 2 =.2. dβ β = λ β =.2 = 1.2. In other words, if Jack s food budget increases by $., then his max utility will increase by approximately

5 4. A firm s productions function is given by Q = 1K.4 L.7, where Q is the firm s annual output, K is the annual capital input, and L is the annual labor input. The cost per unit of capital is $1, and the cost per unit of labor is $4. a. Find the levels of labor and capital inputs that minimize the cost of producing an output of Q = 2, units. What is the minimum cost? Solution: The firm s cost is the cost of using K units of capital and L units of labor, i.e., the objective function here is C(K, L) = 1K + 4L. The constraint in this case is the output target so the Lagrangian is The first-order equations are Q = 2, = 1K.4 L.7 = 2, F (K, L, λ) = 1K + 4L λ(1k.4 L.7 2). F K = = 1 4λK.6 L.7 = F L = = 4 7λK.4 L.3 = F λ = = (1K.4 L.7 2) = Solving the first two equations for λ gives λ = 1 4K.6 L = 4 4 = 2K.6 =.7 7K.4 L.3 L.7 7 L.3 K.4 Next, clear denominators in the equation on the right and solve for K in terms of L: 17K = 4L = K = 16 7 L. Finally, substitute for K in the equation F λ = (the constraint), and solve for L: 1K.4 L.7 = 2 = ( ) { ( ) }}{ L L.7 = 2 = L 1.1 = 2 (16/7).4 ( ) 1/1.1 2 = L = = L (16/7).4 Conclusion: Cost is minimized when L and K = 16 7 L The minimum cost is C = $1K + $4L $1, 69, $2, 967, 86 = $4, 663, 774

6 b. Find the levels of labor and capital inputs that minimize the cost of producing an output of Q = q units and find the minimum cost. Express your answer in terms of q. Solution: There is no need to start over from the beginning. The only difference between this and a. is that the target output changes from 2 to q. This means that we can skip directly to the equation where 2 makes its first appearance, namely the equation marked with a (*) above, and replace the 2 that appears there by q: ( ) L L.7 = q. Now we continue as before to solve for L, then K and finally, C. First L: ( ) L L.7 = q = L 1.1 = q 1(16/7).4 = L (q) = q 1/11 1 1/11 (16/7) 4/11 = α q1/11, where α = ( ) 4/ / ( and 1 11 = and 4 11 = ) 11 Next, K (q) = 16 7 L (q) = β q 1/11, where β = 16 7 α.286. Finally, the (minimum) cost of producing q units is C (q) = 1K (q) + 4L (q) = (1β + 4α)q 1/ q 1/11.. The production function for ACME Widgets is Q = 2k 2 + kl + l 2, where k and l are the numbers of units of capital and labor input, respectively, and Q is their output, measured in 1s of widgets. The price per unit of capital input is p k = $1 and the price per unit of labor input is p l = $2. a. How many units of capital and labor input should ACME use to minimize the cost of producing 6 widgets? What is the average cost per widget? Solution: Since Q is measured in 1s of widgets, the constraint here is and the Lagrangian for this problem is 2k 2 + kl + l 2 = 6 F (k, l, λ) = 1k + 2l λ(2k 2 + kl + l 2 6). 6

7 The first order conditions are Solving the first two equations for λ gives F k = = 1 λ(4k + l) = F l = = 2 λ(k + 1l) = F λ = = 2k 2 + kl + l 2 = 6 λ = 1 4k + l = 2 k + 1l. Dividing by and clearing denominators on the right we find that 2k + 2l = 2k + l = 1l = 18k = l = 6k. Substituting for l in the constraint we have 2k 2 + k ( ) ( ) 2 6k 6k + = 6 = 2k2 = 6 = k 2 = 6.2. Since capital input must be positive, the cost-minimizing levels of capital and labor input for producing 6 widgets are k = 2. and l = 3. It follows that the (minimum) cost of producing 6 widgets is c = = $1, and the average cost per widget is c = 1 6 $.14. b. By approximately how much will ACME s cost rise if they raise their output from 6 widgets to 6 widgets? Justify your answer. Solution: Linear approximation tells us that c dc Q, and by the envelope dq theorem, we have dc /dq = λ, the critical value of the multiplier. In this problem we have λ = 1 4k + l = Also, if output increases by widgets, then Q =.. Thus we have c dc dq Q = λ Q = c. By approximately how much will ACME s minimum cost increase (from part a.) if the cost per unit of capital increases to $11? Use the envelope theorem and linear approximation. Solution: First observe that similarly to problem 3b., if we denote the price of capital by p k, then dc = df, dp k dp k 7

8 where F = F (k, l, λ ; p k ) and F = F (k, l, λ; p k ) = p k k + 2l λ(2k 2 + kl + l 2 6). Now use the envelope theorem to find df /dp k : dc = df = df = k dp k dp k dp k k=k l=l λ=λ Finally, use this and linear approximation to find that k=k l=l λ=λ = k = 2.. c dc dp k p k = 2. 1 = 2. I.e., if the price per unit of capital increases by $1, then the cost will increase by about $2. 6. The annual output for a luxury hotel chain is given by Q = 3K 2/ L 1/2 R 1/4, where K, L and R are the capital, labor and real estate inputs, all measured in $1,, s, and Q is the average number of rooms rented per day. The hotel chain s annual budget is B = $69 million. a. How should they allocate this budget to the three inputs in order to maximize their annual output? What is the maximum output? Solution: We want to maximize the output, Q = 3K 2/ L 1/2 R 1/4, subject to the budget constraint K + L + R = 69, since the inputs are all being measured in millions of dollars. The Lagrangian for this problem is and the first order conditions are The first two equations imply that F (K, L, R, λ) = 3K 2/ L 1/2 R 1/4 λ(k + L + R 69), F K = = 12K 3/ L 1/2 R 1/4 = λ, F L = = 1K 2/ L 1/2 R 1/4 = λ, F R = = 7.K 2/ L 1/2 R 3/4 = λ, F λ = = K + L + R = K 3/ L 1/2 R 1/4 = 1K 2/ L 1/2 R 1/4 = 12L1/2 K 3/ = 1K2/ L 1/2, after canceling the common factor of R 1/4. Clearing denominators gives 12L = 1K = L = 1.2K. 8

9 Likewise, comparing the first and third equations implies that 12K 3/ L 1/2 R 1/4 = 7.K 2/ L 1/2 R 3/4 = 12R1/4 K 3/ and clearing denominators gives = 7.K2/ R 3/4, 12R = 7.K = R =.62K. Substituting for R and L in the fourth equation (the constraint) gives Thus, the critical values for the inputs are K + 1.2K +.62K = 69 = 2.87K = 69. K = = 24, L = 1.2K = 3 and R =.62K = 1, and the hotel chain s maximum output is Q = Q(24, 3, 1) b. What is the critical value of the multiplier when output is maximized? Solution: When output is maximized, the critical value of λ is λ = 12(K ) 3/ (L ) 1/2 (R ) 1/ c. Use your answer to b. to compute the approximate change in the firm s maximum output if their annual budget increases by $,? Explain your answer. Solution: As in problem 3b., it follows from the envelope theorem that where B is the budget. It follows that dq db = λ, Q λ B , since we measure the budget in the same units (millions of dollars) as the inputs, so an increase of $,, means that B =.. 9