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Problem 1. Cost function derivation [1 point] A firm has a production function q = 4K0.75 L 0.25, where q is the amount of output produced, K is the amount of capital while L is the amount oif labor invested in production. Total cost are: TC(K, L) =rK + WL, where r = 3 is the price of a unit of capital while w = 16 is the price of a unit of labor. Minimize the total cost with respect to K and L using Lagrangian auxiliary function and express the minimized cost as a function of q. Problem 2. Multi-product firm [1 point] Consider a bakery that rents an oven for $100 to produce two types of bread - black and white. Let qo be the number of black breads the bakery produces, and qw - the number of white breads. The bakery's cost functions is: C(96,9w) = 100 +46 +24w. Both types of bread have equal share in the total output, do = 0.5 and lw = 0.5 . Find: (a) marginal costs (b) average in crement al costs (c) ray average cost (d) scale economies measure (e) scope economies measure Problem 3. Short-run versus long-run competitive equilibrium [2 points) Suppose there is a perfectly competitive industry where all the firms are identical with identical cost curves. Furthermore, suppose that a representative firm's total cost is given by the equation TC = 100+q2 + where q is the quantity of output produced by the firm. You also know that the market demand for this product is given by the equation P = 1000 - 20 where Q is the market quantity. In addition you are told that the market supply curve is given by the equation P= 100+ Q. (a) What is the equilibrium quantity and price in this market given this information? (b) The firm's MC equation based upon its TC equation is MC = 2q +1. Given this information and your answer in part (a), what is the firm's profit maximizing level of production, total revenue, total cost and profit at this market equilibrium? Is this a short-run or long-run equilibrium? Explain your answer. (c) Given your answer in part (b), what do you anticipate will happen in this market in the long-run? (d) In this market, what is the long-run equilibrium price and what is the long-run equilibrium quantity for a representative firm to produce? Explain your answer. (e) Given the long-run equilibrium price you calculated in part (d), how many units of this good are produced in this market? 1 = 100 Problem 4. Social Welfare [1 point] Suppose that in Prague the daily demand for taxi rides is Q = 2100100P where P is the price in $. Suppose also that the daily cost of operating each cab is a fixed $100 dollar rental cost per vehicle, plus a variable cost of VC(q) = 166, where q is the number of cab rides per cab per day. (a) What is the long run total cost function of each cab? (b) If the market is a constant cost competitive one, what is the long-run price of a cab ride? What is the number of rides each cab supplies and the number of cabs operating? (c) What is consumer surplus and producer surplus in the taxi cab market? (d) Does the market achieve the condition for efficiency that p= MC. Explain. (e) Suppose that a tax of $3 per ride is imposed. What is the new market price? What is number of rides per day, and number of cabs? (f) What is the deadweight cost of the tax per day

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