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UNIVERSITY OF CALGARY DEPARTMENT OF ECONOMICS ECONOMICS 301 Assignment 2 Wojciech (Victor) Fulmyk Winter 2023 This assignment is due on D2L on Wednesday, April 12, 2023, at 11:59pm. Your submission has to be a single .pdf file Note: Throughout the assignment assume prices and quantities are perfectly divisible. If necessary, round your answers to 2 decimal points. 1. Suppose that you have some information about a firm’s variable and fixed costs in the short-run. Suppose a firm’s variable costs are Cv(y) = 20 √ y + 4wy2, where w is the price of the firm’s variable factor. Moreover, suppose the firm’s fixed costs are F = 20. (a) Solve for the firm’s marginal cost function. (b) Solve for the firm’s average cost, average variable cost, and average fixed cost. (c) For the rest of the question, assume w = 1. Solve for the firm’s shutdown price. (d) Suppose the price of the output good is $10. At this price, does the firm operate? If so, what are the firm’s profits? (e) Suppose the price of the output good is $14. At this price, does the firm operate? If so, what are the firm’s profits? (f) Suppose the price of the output good is $18. At this price, does the firm operate? If so, what are the firm’s profits? 2. Suppose you are analyzing a particular market. All consumers in this market have the utility function U(y1, y2) = y2 + 10y1 − y21/2. Suppose that there are many firms producing good 1, and that each of these firms has the production function y1 = 2L0.5 + 4K0.5. Assume good 2 is a composite good with price p2 = 1. (a) Derive a consumer’s Marshallian demand for good 1. Assume all consumers can always afford good 1. (b) For the rest of this question, suppose there are 5 consumers in total in the market. What is the aggregate demand for good 1? Page 1 of 5 ECON 301 Assignment 2 Winter 2023 (c) What is the price elasticity of demand for prices p1 = 1, p1 = 5, and p1 = 10? (d) Find the conditional factor demands for L and K for a typical firm producing good 1. (e) Suppose the government forces all firms operating in the market to purchase a $20 operating license. What is the cost function of a firm? (f) For the rest of the question, assume that all input prices are equal to 1. What is the supply curve of a firm in this market? (Note: Assume the long-run case, that is, that the supply is zero if profits are negative.) (g) If there was only one firm supplying the market, what would the equilibrium price and quantity be? (Note: do not assume any complicated strategic behaviour. For instance, do not assume that this is a monopoly and that this firm would act like a monopolist; rather, view this as a short-run case where the firm behaves according to the supply curve you identified in the previous part.) (h) Calculate the profits for this individual firm supplying the market. (i) Assume firms have no barriers to entry or exit. Moreover, assume there are no restrictions on the number of operating licenses issued. In the long-run, how many firms will exist in this market? What will the equilibrium price and quantity be? (j) Compare the short-run, single firm equilibrium to the long-run equilibrium. What are the prices, quantities and average costs of production under each equilibrium? What is the producer and consumer surplus under each equilibrium? 3. Suppose a consumer’s utility function for goods 1 and 2 is u(y1, y2) = min(10y1, 2y2). Suppose a firm producing good 1 has the production function f(x1, x2) = x2+x1+x2x1. Suppose the price of factor 2 is w2 = 1, and let w1 denote price of input 1. (Assume input 1 is labour and input 2 is capital). (a) Derive the consumer’s Marshallian demand for goods 1 and 2. (b) Derive the consumer’s Hicksian demand for goods 1 and 2. (c) Suppose the firm is required to produce 8 units of output. For which values of w1 would a cost-minimizing firm use i. only labour? ii. only capital? iii. both capital and labour? Page 2 of 5 ECON 301 Assignment 2 Winter 2023 4. Suppose a consumer’s utility function for goods 1 and 2 is u(y1, y2) = 4y1 + 10y2. Suppose a firm producing good 1 has the production function f(x) = 9/x−2/3, where x is the amount of some input used by the firm to produce output y1. (a) Derive the consumer’s Marshallian demand for goods 1 and 2. (b) Derive the consumer’s Hicksian demand for goods 1 and 2. (c) Find the marginal product of input x. (d) Suppose that the cost of input x is w = 9 and that the output price is p = 3. Draw the production function for this firm. Also, draw the isoprofit lines whose vertical intercepts are the points (0,12), (0,8), and (0,4). What is the slope of these isoprofit lines? (e) Suppose that the cost of input x is w = 9 and that the price of output is p = 3. How many units of input x will the firm choose to hire? How many units of output will the firm choose to produce? If there are no other costs, what would the firm’s total profit be? (f) Now, suppose that the cost of input x drops to w = 6 and that the output price remains at p = 3. Will the firm choose to increase it’s output at the new price? What will the firm’s new output level be? How many units of the input will the firm choose to hire now? If there are no other costs, what is the firm’s total profit? 5. Define all of the following concepts in your own words and describe the relationship between all concepts on the same line. (a) Isoquant, Technical rate of substitution, and marginal product. (b) Returns to scale and economies of scale. (c) Cost function, conditional factor demands, and factor demands. (d) Marginal cost, Average cost, average variable cost, and average fixed cost. 6. If we price all inputs at their opportunity cost, then why do we say that the payment to some inputs is an economic rent? What is the difference between an economic rent and any other opportunity cost? 7. Suppose a firm has a production function that is strictly convex and well-behaved. Suppose the firm has just two inputs into the production process. Additionally, suppose the firm is operating where the absolute value of its technical rate of subsititution is less than the input price ratio (w1/w2), where w1 is the price of input 1 and w2 is the Page 3 of 5 ECON 301 Assignment 2 Winter 2023 price of input 2. How can this firm increase its output without changing the amount it spends on its inputs? Additionally, provide an argument for why a firm operating where the absolute value of its technical rate of subsititution is less than (w1/w2) cannot be profit maximizing. 8. Suppose the market for Widgets is perfectly competitive market. Suppose all con- sumers in the marhet are identical and have a utility function betweenWidgets (W ) and and the composite good (CG) defined by U(W,CG) = W 6CG2. Suppose that the ag- gregate demand and industry supply curves for Widgets are given by QD = 2000−20P and QS = 60P . Suppose the government provides a subsidy of $20 per Widget to all sellers in the market. (a) Find the equilibrium quantity of Widgets demanded and supplied; find the equilib- rium market price paid by buyers; find the equilibrium after-subsidy price received by firms. (b) Suppose each consumer in the market has a budget m = 100. How many Widgets will each consumer consume in equilibrium with the subsidy? In order for all produced Widgets to be purchased, how many identical consumers would we need? (c) Find consumers’ surplus and producers’ surplus in the absence of the subsidy. What is the net economic benefit in the absence of a subsidy? (d) Find consumers’ surplus and producers’ surplus in the presence of the subsidy. What is the impact of the subsidy on the government budget? What is the net economic benefit under the subsidy program? (e) Does the subsidy result in a deadweight loss? If so, how much is it? 9. Suppose we have a competitive industry with many firms, all of which have the cost function c(y) = y2 + 1 when y > 0 and c(0) = 0. Suppose that. initially, aggregate demand is D(p) = 84−p. Note that the number of firms has to be an interger amount, but that prices and quantities do not have to be intergers. (a) Solve for an individual firm’s supply curve? If there are n firms in the market, what is the market supply curve? (b) In the long-run, what is the minimum price at which this product will be supplied? (c) What is the equilibrium number of firms in the market? (d) What is the equilibrium price? What is the output of every firm in the market? Page 4 of 5 ECON 301 Assignment 2 Winter 2023 (e) What is the aggregate output of all firms in the market at the equilibrium? (f) Now, suppose that the market demand curve shifts to D(p) = 84.5−p. What will be the equilibrium number of firms be now? What will be the equilibrium price? What will be the output of each firm? What will be the profit of each firm in the market? (g) Now, suppose that the market demand curve shifts to D(p) = 85− p. What will be the equilibrium number of firms? What will be the equilibrium price? What will be the output of each firm in the market? What will be the profit of each firm in the market? 10. True or False. For full marks, you MUST explain your answer! (a) Suppose you have a firm that uses a production process with two inputs. Suppose that in the short run, input 1 is variable and input 2 is fixed. If the price of the input 2 goes up, the firm’s short-run average variable cost, average cost, and marginal cost curves will all shift upward. (b) Suppose you have a firm that uses a production process with two inputs. Suppose that in the short run, input 1 is variable and input 2 is fixed. If the price of the input 1 goes up, the firm’s short-run average variable cost, average cost, and marginal cost curves will all be unaffected. Page 5 of 5