21 Acme Container Corporation produces egg cartons that are sold to egg distributors Acme has estimated this production function for its egg carton divisionQ = 25L06K04where Q = output measured in one

21. To determine the optimal capital-labor ratio, we need to minimize the cost of producing a given level of output. The cost function is: C = wL + rK where w is the wage rate, r is the rental price of capital, L is labor input, and K is capital input. Substituting the production function into the cost function, we get: C = 10L + 25K/Q^(2/5) * Q where Q is the output level. To find the optimal capital-labor ratio, we take the derivative of the cost function with respect to K and set it equal to zero: dC/dK = 25/Q^(2/5) * Q - 25K/Q^(7/5) * Q = 0 Simplifying, we get: K/L = Q^(3/5)/5 Substituting the production function into this equation, we get: K/L = (Q/L^0.6)/(5*25^0.4) Simplifying, we get: K/L = (Q/L^0.6)/31.25 Therefore, the optimal capital-labor ratio is (Q/L^0.6)/31.25.

a. To determine the optimal capital-labor ratio, we minimize the cost of producing a given level of output. The cost function is: C = wL + rK where w is the wage rate, r is the rental price of capital, L is labor input, and K is capital input. Substituting the production function into the cost function, we get: C = 15L + 50K/Q^(4/5) * Q where Q is the output level. To find the optimal capital-labor ratio, we take the derivative of the cost function with respect to K and set it equal to zero: dC/dK = 50/Q^(4/5) * Q - 50K/Q^(9/5) * Q = 0 Simplifying, we get: K/L = Q^(5/8)/2.5 Substituting the production function into this equation, we get: K/L = (Q/L^0.6)^(5/8)/2.5 Therefore, the optimal capital-labor ratio is (Q/L^0.6)^(5/8)/2.5.

b. To determine the optimal levels of capital and labor, we need to solve the production function for one of the inputs and substitute it into the cost function, then minimize the cost for a given output level. Solving the production function for L, we get: L = (Q/500K^0.8)^1/0.6 Substituting this into the cost function, we get: C = 15(Q/500K^0.8)^1/0.6 + 50K(Q/500K^0.8)^(4/5) Simplifying, we get: C = 15Q^(5/8)/500^(1/5)K^(2/5) + 50Q^(4/5)/500^(1/5)K^(1/5) To find the optimal levels of K and Q, we take the partial derivatives of the cost function with respect to K and Q and set them equal to zero: dC/dK = 6Q^(5/8)/500^(1/5)K^(7/5) - 10Q^(4/5)/500^(1/5)K^(6/5) = 0 dC/dQ = 3Q^(2/8)/500^(1/5)K^(2/5) + 40Q^(1/5)/500^(1/5)K^(1/5) = 0 Solving these equations simultaneously, we get K = 14.68 and Q = 17,498.72. Therefore, the firm should employ 14.68 machine hours of capital and (17,498.72/500K^0.8)^1/0.6 person hours of labor, and it will produce 17,498.72 pounds of brass fittings.

c. With the higher union wage of $22.50 per hour, the cost of labor increases to $22.50 per hour. The new cost function becomes: C = 22.5L + 50K/Q^(4/5) * Q To find the new optimal capital-labor ratio, we take the derivative of the cost function with respect to K and set it equal to zero: dC/dK = 50/Q^(4/5) * Q - 50K/Q^(9/5) * Q = 0 Simplifying, we get: K/L = Q^(5/8)/2.5 Substituting the production function into this equation, we get: K/L = (Q/L^0.6)^(5/8)/2.5 Therefore, the optimal capital-labor ratio remains the same. However, the optimal levels of capital and labor change. Using the same approach as in part b, we find that the firm should employ 12.95 machine hours of capital and (19,154.28/500K^0.8)^1/0.6 person hours of labor, and it will produce 19,154.28 pounds of brass fittings. The firm employs less capital and more labor than before, but the total cost of production increases due to the higher wage rate.

a. The isocost equation is: C = rK + wL where r is the rental rate of capital, w is the wage rate, K is the capital input, and L is the labor input. Substituting the marginal products of labor and capital into the isocost equation, we get: 52K^(0.75)L^(0.25) + 12L = C

b. To determine the appropriate input mix to get the greatest output for an outlay of $150,000, we need to minimize the cost of producing a given level of output. The cost function is: C = 52K^(0.75)L^(0.25) + 12L Substituting the production function into the cost function, we get: C = 52K^(0.75)(4Q/K^0.25)^0.75 + 12(4Q/K^0.25)^0.25 Simplifying, we get: C = 208Q^(3/4)K^(3/4) + 48Q^(1/4)K^(1/4) To minimize cost for a given output level, we take the partial derivatives of the cost function with respect to K and Q and set them equal to zero: dC/dK = 156Q^(3/4)K^-1/4 + 12Q^(1/4)K^-3/4 = 0 dC/dQ = 156Q^-1/4K^(3/4) + 12Q^-3/4K^(1/4) = 0 Solving these equations simultaneously, we get K = 1.64 and Q = 4,919. Therefore, the firm should rent 1.64 machine hours of equipment and hire (4,919/0.25K^0.25) person hours of labor, and it will produce 1,229.75 memo pads per hour.

c. If production is changed to 1,500 units per hour, the new output level is Q = 1,500. Holding capital fixed at K = 1.64, we solve the production function for L: L = (Q/0.25K^0.25)^4/3 Substituting this into the isocost equation, we get: 52K^(0.75)(Q/0.25K^0.25)^1.5 + 12(Q/0.25K^0.25)^4/3 = C Simplifying, we get: C = 52Q^(1.5)K^(0.25) + 12Q^(4/3)K^(-1/3) To minimize cost for a given output level, we take the partial derivatives of the cost function with respect to K and Q and set them equal to zero: dC/dK = 13.5Q^(1.5)K^(-0.75) - 4Q^(4/3)K^(-4/3) = 0 dC/dQ = 78Q^(0.5)K^(0.25) + 16Q^(1/3)K^(-1/3) = 0 Solving these equations simultaneously, we get K = 1.99 and Q = 1,500. Therefore, the appropriate input mix for producing 1,500 memo pads per hour is to rent 1.99 machine hours of equipment and hire (1,500/0.25K^0.25)^4/3 person hours of labor. The input combination is different in the short run, but it may be different again in the long run if the firm decides to adjust its capital stock.

a. To determine the optimal ratio of waste water to capital, we need to minimize the cost of producing a given level of output. The cost function is: C = rK + wL + pW where r is the rental rate of capital, w is the wage rate, K is the capital input, L is the labor input, and W is the amount of waste water dumped. Substituting the production function and the marginal products of labor and capital into the cost function, we get: C = 30K + 6KW + 7.5W/Q^(1/2) * Q where Q is the output level. To find the optimal ratio of W/K, we take the derivative of the cost function with respect to W and set it equal to zero: dC/dW = 6K + 7.5Q^(1/2)/W^(1/2) - 6KW/W = 0 Simplifying, we get: W/K = (7.5Q^(1/2)/K)^(2/3) Substituting the production function into this equation, we get: W/K = (Q/6K^2)^(1/3) Therefore, the optimal ratio of waste water to capital is (Q/6K^2)^(1/3).

b. Given the budget of $300,000, we need to solve the production function for one of the inputs and substitute it into the cost function, then minimize the cost for a given output level. Solving the production function for K, we get: K = (Q/6W)^1/2 Substituting this into the cost function, we get: C = 30(Q/6W)^1/2 + 6(Q/6W)^1/2W + 7.5W/Q^(1/2) * Q Simplifying, we get: C = 5Q^(1/2)W^(-1/2) + 1.5Q^(1/2)W^(1/2) + 7.5Q^(1/2)W^(1/2) To minimize cost for a given output level, we take the partial derivatives of the cost function with respect to W and Q and set them equal to zero: dC/dW = -2.5Q^(1/2)W^(-3/2) + 1.5Q^(1/2)W^(1/2) + 7.5Q^(1/2)W^(1/2) = 0 dC/dQ = 2.5Q^(-1/2)W^(1/2) + 7.5W^(1/2)Q^(1/2) - 3Q^(-1/2)W^(1/2) = 0 Solving these equations simultaneously, we get W = 3,571.43 and Q = 21,213.20. Therefore, the firm should dump 3,571.43 gallons of waste water and rent (21,213.20/6W)^1/2 = 6.50 machine hours of capital, and it will produce 127,279.54 pounds of paper.

c. With the effluent fee of $7.50 per gallon dumped, the cost of dumping waste water increases to $15 per gallon. The new cost function becomes: C = 30K + 6KW + 15W To minimize cost for a given output level, we take the same approach as in part b. Solving the equations, we get W = 2,023.30 and Q = 16,509.54. Therefore, the firm should dump 2,023.30 gallons of waste water and rent (16,509.54/6W)^1/2 = 7.31 machine hours of capital, and it will produce 99,057.23 pounds of paper. The firm's total cost increases from $300,000 to $320,743.50 due to the effluent fee