Problem Set Chapter 9 Solutions

Transcription

1 Problem Set Chapter 9 Solutions 1. Ch 9, Problem 1 Last year, the account ledger for an owner of a small drugstore showed the following information about her annual receipts and expenditures; Revenues: $1, Wages for hired labor: $300,000 Utilities: $0,000 Purchases of drugs and other supplies for the store: $500,000 Wages paid to herself: $100,000 She pays a competitive wage to her workers, and the utilities and drugs and other supplies are all obtained at market prices. She already owns the building, so she pays no money for its use. If she were to close the business she could avoid all of her expenses and, of course, would have no revenue. However, she could rent out her building for $00,000. She could also work elsewhere herself. Her two employment alternatives include working as a lawyer, earning wages of $100,000, or working at a local restaurant, earning $0,000. Determine her accounting profit and her economic profit if she stays working in the drugstore business. If the two are different explain the difference. Accounting Profits: Total Revenues: 1,000,000 Total Accounting Cost: 300, , , , 000 Profits: 80,000 Economic Profits: Total Revenues: 1,000,000 Total Economic Costs: 300, , , , ,000 Profits -10,000 So, if she continues to work at the grocery store, she earns an accounting profit of 80,000 plus the salary she pays herself 100,000. However, she earn negative economic profits if she continues to work at the grocery store (-10,000). But if she exits the business, her salary as a lawyer would be $100,000, and she would receive $00,000 rent for the building. She would therefore be better off by $10,000 if she worked as a lawyer.. Based on Ch 9, Problem 9.9 Earl s Car Wash is a small business that operates in the perfectly competitive car washing industry in Louisville, Colorado. The short-run total cost of production is

2 ( ) STC = +, Economics 3070 where is the number of cars washed per day. The prevailing market price is $ 0 per car. a. How many cars should Earl wash to maximize profit? Earl needs to maximize the profit function: MR Max π = P + +. ( ) π = P 10. = 0 P = SMC If price equals 0 then: 0 = = 50 Therefore, Earl should wash 50 cars to maximize profit. b. What is Earl s maximum daily profit? Earl s profit is given by π = TR TC. = ( ) [ ] π = P ( 0)( 50) ( )( ) ( )( ) = 10 Therefore, Earl s maximum daily profit is $10. c. Graph SMC, SAC, and the profit-maximizing quantity. On this graph, indicate the maximum daily profit. The equations for short-run marginal cost and short-run average cost are ( ) SMC ( ) SAC STC = = STC 40 = =

3 The two curves and profit-maximizing quantity ( = 50) diagram below: are depicted in the Economics 3070 $ SMC SAC The shaded area represents the Earl s profit. It is a rectangle whose height is the difference between the market price ( P = 0) and the average cost of the 50 th unit SAC 50 = 15.8 and whose width is the 50 units being produced. ( ( ) ) d. Assume that all of the $ 40 per day fixed costs are sunk. Derive an expression for Earl s short-run supply curve. Then graph the curve. The first step is to identify the minimum price needed to induce Earl to wash cars; that is, we need to identify P min. In the short-run, P min is the minimum average ANSC. If all fixed costs are sunk, then nonsunk cost ( ) ANSC ( ) = AVC ( ) = = Since ANSC is increasing in, minimum ANSC occurs at = 0 and equals 10. Therefore, P = min 10. For P Pmin Earl does not wash any cars, but for P Pmin Earl operates at a point where P = SMC : P = = 5P 50 Therefore, Earl s short-run supply curve is

4 s ( P) 0 = 5P 50 The graph below depicts this supply curve: if P 10 if P 10 P s ( P ) e. Assume, instead, that if Earl produces zero output, he can rent or sell his fixed assets thereby avoiding all his fixed costs. Derive an expression for Earl s short-run supply curve. Then graph the curve. If all fixed costs are nonsunk, then ANSC ( ) = SAC ( ) 40 = Minimum ANSC occurs at the point where ( ) SMC( ) = ANSC = : 40 = 0.1 = 400 Since ( 0 ) = 14 ANSC, P 14. min = = 0 For P Pmin Earl does not wash any cars, but for P Pmin Earl operates at a point where P = SMC. Therefore, Earl s short-run supply curve is

5 ( P) The graph below depicts this supply curve: s 0 = 5P 50 if P 14 if P 14 P s ( P )

6 3. Ch 9, Problem 9.1 The coal industry consists of 60 producers, all of whom have an identical shortrun total cost curve, ( ) 64 STC = +, where is the monthly output of a firm and $ 64 is the monthly fixed cost. Assume that $ 3 of the firm s monthly $ 64 fixed cost can be avoided if the firm produces zero output in a month. The market demand curve for coal production is where ( P ) ( P) 400 P D = 5, D is monthly demand at price P. Derive an expression for the market supply curve in this market and determine the short-run equilibrium price. We will begin by deriving the short-run supply curve for a single firm. The firm s average nonsunk cost is given by the following equation: To find the shut-down price ( P ) ANSC ( ) 3 + = 3 = + min, we find the minimum level of ANSC. This occurs at the quantity at which ANSC = SMC : Since ( 4 ) = 3/4 + ( )( 4) = = 4 3 = = 4 ANSC, P 16. min = For P Pmin the firm does not produce any output, but for P Pmin the firm operates at a point where P = SMC : P = 4 = Therefore, each firm s short-run supply curve is 1 4 P

7 s ( P) 0 if P 16 = 1 4 P if P 16 Since there are 60 firms, the market supply curve is simply S ( P) 0 = 15P if P 16 if P 16 To find the equilibrium price, we equate market supply and market demand: ( P ) = D( P ) 15P = 400 5P 0P = 400 P = 0 Therefore, the equilibrium price is $0. S 4. Ch 9, Problem 9.17 Suppose a competitive, profit-maximizing firm operates at a point where its short-run average cost curve is upward sloping. What does this imply about the firm s economic profits? If the profit-maximizing firm operates at a point where its short-run average cost curve is downward sloping, what does this imply about the firm s economic profits? If the firm operates at a point where its SAC curve is rising, it must mean that the SMC curve is above the SAC curve. Since the firm sets P = SMC, it must be the case that P > SAC. Therefore, the firm earns positive economic profit. If the firm operates at a point where the SAC curve is falling, it must mean that SMC < SAC. Since the firm sets P = SMC, it must be the case that P < SAC. Therefore, the firm earns negative economic profit. However, the fact that the firm is still operating means that SMC ANSC ; that is, the firm s revenue is covering all of its nonsunk costs but only a portion of its sunk costs. This is why the firm is better off operating albeit at a loss than shutting down.

8 5. Ch 9, Problem 9.0 A firm s short-run supply curve is given by s ( P ) 0 = 3P 30 if P < 10 if P 10 What is the equation of the firm s marginal cost curve SMC ( )? We know that if the firm produces positive output, it produces where case, when the firm produces positive output, = 3P 30 P = SMC. In this P = This means that the equation of the firm s short-run marginal cost curve is 10 1 ( ) = + 10 SMC Based off Ch 9, Problem 9.16 The screw and bolt market contains many identical firms, each with a short-run total cost function STC() = , where is the firm s annual output (and all of the firm s $400 fixed cost is sunk). The market demand curve for this industry is D(P) = 6.5 P/ Where P is the market price. Each firm in the industry is currently earning zero economic profit. How many firms are in this industry, and what is the market equilibrium price? The two equilibrium conditions that must hold in the short run are: a. Profit Maximization, which implies that P*=MC(*) b. Market Supply = Market Demand In addition, we have been told that we must satisfy the zero profit condition. P*=AC(*) c. So, we know that the profit max and zero profit condition together mean that: P=SACmin From a: P* = -5 + *

9 From b. P* = 400/* 5 + * So a. and b. we have two equations and unknons (P* and *). First, lets slove for * So each firm will produce * = 400/ * 5 + * ( *) = 400 * = 0 We can sub the * back into the AC(*) or MC(*) to get P* So, market equilibrium price will be P* = 35. P* = MC(*) = -5 + (0) = 35 To determine the number of firms we need to figure out the market output and divide that by the amount each firm produces. Market Output: D(P) : / = = 45 So the number of firms are: 45 / 0 = Based on Ch 9, Problem 9.4 The Brussels sprouts industry is perfectly competitive, and each producer has the long-run total cost function 1 3 TC ( ) = The market demand curve for Brussels sprouts is ( P) 00 P D = 100. What is the long-run equilibrium price in this industry? At this price, how much would an individual firm produce? How many firms are in the Brussels sprouts market in a long-run competitive equilibrium? In a long-run equilibrium, the equilibrium price and equilibrium number of firms 1. = MC( ). = AC( ) P (profit maximization) P (zero economic profit) D P = n (market clearing) 3. ( ) P, equilibrium output of each firm n satisfy the following three conditions:,

10 The first step is to apply conditions 1 and by setting ( ) = AC( ) MC : = = 6 = 18 = Therefore, each individual firm produces = 9 units of Brussels sprouts. To find the equilibrium price, we simply plug = 9 MC into ( ) ( 9) = 40 1( 9) + ( 9) MC : = = 13 Therefore, the long-run equilibrium price is P = 13. To find the equilibrium number of firms, we apply condition by plugging P = 13 into the market demand curve: D D n n ( P) ( 13) = 00 ( 100)( 13) ( 9) n = P = = 900 = 100 Therefore, the equilibrium number of firms is n = Based on Ch 9, Problem 9.3 The long-run total cost for production of exam booklets is given by TC wr 3 = ( ), Where is the annual output of a firm, w is the wage rate for skilled assembly labor, and r is the price of capital services. The demand for labor for an individual firm is

11 Lwr (,, ) = r 3 ( ) w a) In a long-run competitive equilibrium, how much output will each firm produce. b) In a long-run competitive equilibrium, what will be the market price? Note that your answer will be expressed as a function of w. c) In a long-run competitive equilibrium, how much skilled labor will each firm demand? Again, your answer will be in terms of w. d) Suppose that the market demand curve is given by D(P) = 10,000/p. What is the market equilibrium quantity as a function of w? e) What is the long-run equilibrium number of firms as a function of w? f) Using your answers to (c) and (e), determine the overall demand for skilled labor in this industry as a function of w. g) Suppose the supply curve for the skilled labor used in this industry is G(w) = 50w. At what value of w does the supply of skilled labor equal the demand for skilled labor. NOTE: These answers assume the price of capita, r, is equal to 1. a) To determine long-run output we can use the profit maximizing and zero profit condition. MC(*) = AC (*) ( 10 0 * + ( *) ) = ( * + 3( *) ) wr wr b. 0 * ( *) = * * * = 10 P* = MC(*) = wr(10 40(10) + 3(10) w( ) w(0) c.

12 10*10 0* Lwr (,, ) = w = = w w 3

13 d. 10,000 10, DP ( *) = P* = 0 w = w e. 500 DP ( *) 50 n* = = w = * 10 w f. Demand for skilled labor: Demand for Lobor by each firm * number of firms * = w w w g. D L = S L 5000 = 50w w w = 100 w = 10