# Problem Set 8: Costs? Econ 11 Winter 2015 Due Date: March 5, 2015 Panem is divided into 12 Districts and a capital city (The Capitol). Each district specializes in the production of one type of resource. Research Paper

Problem Set 8: Costs? Econ 11 Winter 2015 Due Date: March 5, 2015 Panem is divided into 12 Districts and a capital city (The Capitol). Each district specializes in the production of one type of resource. 1. District 1s primary industry is manufacturing luxury items (Q) for the Capitol. The cost function of luxury items can be described by the following equation: T C = 120 ? 12Q + 25Q2 (a) Find the variable cost and ?xed cost. Sketch a graph of these curves. (b) Find the average variable cost, average ?xed cost and average cost functions. Sketch a graph of these curves below the total cost graphs. (c) Find the marginal cost function and sketch its curve in the graph above. (d) Show that the MC curve and SAC (short-run average cost) curve intersect at the minimum point of SAC. Does this conclusion hold in general? Provide intuition for your conclusion. (e) If the wage paid at District 1 is \$20, obtain an equation for the MPL (marginal product of labor). 2. District 2s main industry is masonry. To build their forti?cations, District 2 employs workers (L) and mortar mixers (K). Production of building structures (Q) can be represented by the following equation: Q(K, L) = L1/2 K District 2 has already payed for 3 months of rent for the mixers, hence their number cant be adjusted. (a) Write the cost minimization problem faced by District 2 and obtain the demand for workers. (b) Derive the total cost function of building structures. (c) District 2 rented 2 mixers at a price of \$2. If they pay \$4 to each worker, how many workers should District 2 hire if it wants the minimum cost of producing 2 building structures? ? For errors or corrections, please e-mail me to ccantu@ucla.edu. 1 (d) Find the equations for District 2s ST C, F C, V C, AF C, AV C, SAC and M C. Show your results in a graph. (e) President Snow, the ruler of Panem and all the Districts, imposes regulations on all building structures that increase the costs of production by 100 (independent of the quantity of structures produced), how would this a?ect your graphs? (f) District 2 decides to employ only non-rebels to produce the structures. To incentivize them to work, they raise the wage to \$8. How would this a?ect your graphs? 3. District 3s main industry is technology and they specialize in the production of televisions and computers. Total production of computers can be represented by the following function: Q(K, L) = K 1/6 L2/6 (1) (a) Write the cost minimization problem faced by District 3 and ?nd the contingent demand functions for K and L. (b) Derive the total cost function of District 3. (c) The start of the rebellion in all Panem has increased the demand for computers. District 3 wants the total cost of the total quantity of computers produced to be no more than 9000. If the price of capital and the wage for workers is 10, how many units of capital will District 3 rent? (d) Derive their ATC and MC curves. What would happen if the wage of workers was raised to 20? Would this have the same directional e?ect on the curves as an increase of 10 in the rent of capital? Show your results in a graph. 4. District 4s industry is ?shing. Total ?sh caught (Q) depends on the number of ?shermen (L) and ?shing boats (K) according to the following function: 5KL Q= K +L (a) Derive an expression for the marginal rate of technical substitution. (b) Derive expressions for the average and marginal products of labor. (c) Does this production function exhibit decreasing, constant, or increasing returns to scale? (d) Suppose that input prices are given by w = 4 and r = 1. Derive the conditional demands for K and L. (e) Obtain the long run average cost function. (f) If in the short run District 4 only rents 4 boats, obtain the short run average total cost, average variable cost, and marginal cost functions. 2 5. District 5s main industry is power and electricity. Electircty is produced in two power plants (hydroelectric and nuclear) according to the following equations: QH = K 1/4 L1/8 QN = min{K, L} The rental rate of capital is (r) and the wage paid to workers is (w). To produce any electricity at all, each plant requires one unit of capital, i.e. there is a ?xed cost of 1r, which is sunk in the short run but not sunk in the long run. Each power plant employs L workers and rents K units of capital (in addition to the one unit needed as ?xed cost). For each power plant answer the following questions. (a) In the short run each power plant is committed to hire a ?xed number of capital K(+1), and can vary its output only by employing an appropriate amount of labor. Find each power plant short-run total, average, and marginal cost functions. (b) In the long run each power plant can vary both capital and labor. Find each power plant long-run total, average, and marginal cost functions. (c) To link the short-run and the long-run curves, take the short-run average cost curve, and for a given Q, ?nd K (as a function of Q) that minimizes short-run average cost, reducing it to a function of Q, r and w. Verify that is the same as the long-run average cost function.

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