Econ211Chapter 08 SolutionsStLouis0405[1]

Econ211Chapter 08 SolutionsStLouis0405[1] - x h a p t e r 8...

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Unformatted text preview: x h a p t e r 8 : Profit Maximization and Competitive Supply EXERCISES 1. !e d ata in the following table give information about the price (in dollars) for which a h firm can eel1 a u nit of output and the total cost of production. a. Fill in the blanks in the table. b. Show what happens to the firm's output choice and profit if the price of the product falls from $60 t o $60. T he table below shows the h ' s revenue and cost for t he two prices. m At a price of $60, t he f should produce ten units of output to maximize profit because this is the point closest to where price equals marginal cost without having marginal 106 I Chapter 8 : Profit Maximization and Competitive Supply Q b . TC MC TVC TFC AVC I f 100 identical firms are in the market, what is the industry supply curve? For 100 firms with identical cost structures, the market supply curve is t he horizontal summation of each f r ' o utput a t each price. ims P c napter t: r rortt MUXZmtZUttOn a n& competcttue s upply i Suppose you are the manager of a watchmaking firm operating in a competitive market. Your cost of production is given by C = 200 +2 q2,where q is the level of output and C i s total cost. (The marginal cost of production is 4q. The fixed cost of production is 4. $200.) a. If the price of watches is $100, how many watches should you produce to maximize profit? Profits are maximized where marginal cost is e qual to m arginal revenue. Here, marginal revenue is e qual to $100; recall that price equals marginal revenue in a competitive market: b. What will the profit level be? Profit is equal to t otal revenue minus total cost: c. At what minimum price will the firm produce a positive output? Af irm will produce i n t he short run if t he revenues it receives are greater than its variable costs. Remember that the firm's short-run supply curve is its marginal cost curve above the minimum of average variable cost. Here, average variable cost is vc - 2 - --- 4* - 24 . Also, MC is equal to 4q. So, MC is greater than AVC for any quantity 4 4 g reater than 0. T his means that the firm produces in the short run a s long a s price is positive. 5. Suppose that a competitive firm's marginal cost of producing output q is given by M C(q) = 3 + 2q. Assume that the market price of the firm's product is $9. a. What level of output will the firm produce? To maximize profits, the firm should set marginal revenue equal to m arginal cost. Given the fact that this Grm i s operating in a competitive market, the market price it faces is equal to m arginal revenue. Thus, the firm should set the market price equal to marginal cost to maximize its profits: b. What is the firm's producer surplus? Producer surplus is equal to t he area below the market price, i.e., $9.00, nd above the a marginal cost curve, i.e., 3 + 2q. Because MC i s linear, producer surplus is a t riangle with a base equal to $6 (9 3 = 6 . The height of the triangle is 3, where P = MC. ) Therefore, producer surplus is ( . ) 6 ( )= $9. 05()3 - Chapter 8: Profit Maximization a nd Competitive Supply Price MC(q) = 3 + 10 - P = $9.00 I I I 1. c. I 2 I 3 I 4 Quantity Suppose that the average variable cost of the firm is given by AVC(q) = 3 + q. Suppose o that the firm's fixed costs are known t be $3. Will the firm be earning a positive, negative, or zero profit in the short run? Profit is equal to total revenue minus total cost. Total cost is equal to total variable cost plus fixed cost. Total variable cost is equal to CAVC)(q). Therefore, at q = 3, TVC = ( 3 + 3)(3)= $18. Fixed cost is equal to $3. Therefore, total cost equals TVC plus TFC, or Total revenue is price times quantity: TR = ($9X3) = $27. Profit is total revenue minus total cost: Therefore, the firm is earning positive economic profits. More easily, you might recall that profit equals producer surplus minus fixed cost. Since we found that producer surplus was $9 in part b, profit equals 9-3 or $6. 6. A firm produces a product in a competitive industry and has a total cost function TC = 50 4 q 2 q2 and a marginal cost function MC = 4 4 9. At the given market price of $20, t he firm is producing 5 units of output. Is the firm maximizing profit? What quantity of output should the firm produce in the long run? ++ + If the firm is maximizing profit, then price will be equal to marginal cost. P=MC results in P=20=4+4q=MC, or q=4. The firm is not maximizing profit, since it is producing too much output. The current level of profit is = profit = 20*5-(50+4*5+2*5*5) -20, and the profit maximizing level is = profit = 20*4-(50+4*4+2*4*4) -18. Given no change in the price of the product or the cost structure of the firm, the firm should produce q=O units of output in the long run since at the quantity where price is equal to marginal cost, economic profit is negative. The firm should exit the industry. Chapter 8 : Profit Maximization and Competitive Supply 7. Suppose the cost function is C(q)=4qz+16. a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost. Hint: Marginal cost is MC=8q. V ariable cost is that part of total cost that depends on q ( 4q2) a nd fixed cost is that part of total cost that does not depend on q (16). 4 b. 4 Show the average cost, marginal cost, and average variable cost curves on a graph. Average cost is u-shaped. Average cost is relatively large a t first because the firm is not able to s pread the fixed cost over very many units of output. As output increases, average fixed costs will fall relatively rapidly. Average cost will increase a t some point because the average fixed cost will become very small and average variable cost is increasing as q increases. Average variable cost wlll i ncrease because of d iminishmg r eturns to the variable factor labor. MC and AVC are linear, and both pass through the origin. Average variable cost is everywhere below average cost. Marginal cost is everywhere above average variable cost. If the average is rising, then the marginal must be above the average. Marginal cost will hit average cost a t its minimum point. c. Find the output that minimizes average cost. T he minimum average cost quantity is w here MC is equal to AC: d. At what range of prices will the firm produce a positive output? T he firm w dl s upply positive levels of output a s long a s P=MC>AVC, o r a s long a s the firm is covering its variable costs of production. In this case, marginal cost is everywhere above average variable cost so the firm will supply positive output a t any positive price. e. At what range of prices will the firm earn a negative profit? T he firm will earn negative profit when P=MC<AC, o r a t any price below minimum average cost. In part c above we found that the minimum average cost quantity was q=2. P lug q=2 i nto the average cost function to find AC=16. T he firm will therefore earn negative profit if price is below 16. '\ fl '1I Chapter 8: Profit M aximiation a nd Competitive Supply f. At w h a t r a n g e of prices will t h e f i r m e a r n a p ositive profit? In part e we found that the firm would earn negative profit a t any price below 16. The firm therefore earns positive profit at3 long a s price it3 above 16. 8. A c ompetitive f i r m h a s t h e following s h o r t r u n cost function: C(q)= q 3 - 8q2 + 30q + 5 . a. F ind MC, AC, a n d AVC a n d sketch t h e m o n a g raph. The functions can be calculated as follows: Graphically, all three cost functions are u-shaped in that cost declines initially as q i ncrea~es,a nd then cost increases as q increases. Average variable cost is below average cost. Marginal cost will be initially below AVC and will then increase to h it AVC a t its minimum point. MC wdl be initially below AC and will also hit AC a t its minimum point. b. A t w h a t range of prices will t h e firm supply zero output? The f r will find it profitable to produce in the short run as long as price is greater im than or equal to average variable cost. If price is less than average variable cost then the firm will be better off shutting down in t he short run, as it will only lose its bed cost and not fixed plus some of variable cost. Here we need to find the minimum average variable cost, which can be done in two different ways. You can either set marginal cost equal to average variable cost, or you can differentiate average variable cost with respect to q and set this equal to zero. In both cases, you can solve for q and then plug into AVC to find the minimum AVC. Here we will set AVC equal to MC: Hence, the firm eupplies zero output if P 4 4 . c. I dentify t h e firm's supply curve o n your graph. The firm supply curve is the MC curve above the point where MC=AVC. The firm will produce a t the point where price equals MC as long a s MC is greater than or equal to AVC. d. At w h a t price would t h e firm supply exactly 6 u n i t s of output? The f r maximizes profit by choosing the level of output such that P=MC. To find im the price where the firm would supply 6 units of output, set q equal to 6 and solve for MC: Chapter 8 : Profit Maximization and Competitive Supply I 9. a. Suppose that a firm's production function is q = 9 x 2in the short run, where there are fixed costs of $1,000 and x is the variable input, whose cost is $4,000 per unit. What is the total cost of producing a level of output q. In other words, identify the total cost function C(q). T he total cost function C(x) = fixed cost + v ariable cost = 1 000 + 4000x. Since the variable input costs $4,000 per unit, the variable cost is 4000 times the number of units, or 4000x. Now rewrite the production function to express x in terms of q so ! I2 We can then substitute this into the above cost function to find C(q): 81 t hat x = -. b. Write down the equation for the supply curve. T he firm supplies output where P=MC so the marginal cost curve is the supply curve, or c. 800% 81 P=-. If price is $1000, how many units will the firm produce? What is the level of profit? Illustrate on a cost curve graph. To figure this out, set price equal to marginal cost to find: Profit is 1000*10.125-(1000+(4000* 10.125*10.125)/81) = 4062.5. Graphically, the firm produces where the price line hits the MC curve. Since profit is positive, this will occur a t a quantity where price is greater than average cost. To find profit on the graph, take the difference of the revenue box (price times quantity) and the cost box (average cost times quantity). This rectangle is the profit area. 10. Suppose you are given the following information about a particular industry: @ = 6500 - lOOP Market demand @ = 1200P Market supply Firm total cost function Firm marginal cost function. Assume that all firms a re identical, and that the market is characterized by pure competition. a. Find the equilibrium price, the equilibrium quantity, the output supplied by the firm, and the profit of the f irm E q d b r i u m price and quantity are found by setting market supply equal to market demand, so that 6500-100P=1200P. Solve to find P =5 a nd substitute into either equation t o find Q=6000. To find the output for the firm s et price equal to marginal cost so t h a t 5 = or 29 - a nd q=500. 200 Profit of the firm is total revenue minus total cost 500 n = pq - C (q)= 5 (500)- 7 22 - -= 5 28. 200 Notice that since the total output in t he market is 6000, and the firm output is 500, there must be 6000/500=12 f irms in the industry. 1 13 " , Chapter 8: Profit Maximization and Competitive Supply b. Would you expect to see entry into or exit from the industry in the long-run? Explain. What effect will entry or exit have on market equilibrium? E ntry because the firms in the industry are making positive profit. As firms enter, the supply curve for the industry will shift down and to t he right and the equilibrium price will fall, all else the same. This will reduce each firm's profit down to zero until there is no incentive for further entry. c. What is t he lowest price a t which each firm would sell its output in the long run? Is profit positive, negative, or zero a t this price? Explain. I n the long run the firm will not sell for a price that is below minimum average cost. At any price below minimum average cost, profit is negative and the firm is better off selling its fixed resources and producing zero output. To find the minimum average cost, set marginal cost equal to average cost and solve for q: Therefore, the firm will not sell for any price less than 3.8 in the long run. d. What is the lowest price a t which each firm would sell its output in the short run? Is profit positive, negative, or zero a t this price? Explain. The firm will sell for any positive price, because a t any positive price marginal cost will be above average variable cost (AVC=q/2000). Profit ie negative as long as price is below minimum average cost, or as long as price is below 3.8. + + 11. Suppose that a competitive firm has a total cost function C (q)= 450 15q 2q2 a nd a marginal cost function M C(q) = 15 4 q . If the market price is P=$116 p er unit, find the level of output produced by the firm Find the level of profit and the level of producer surplus. + The firm should produce where price is equal to marginal cost so that P=115=15+4q=MC a nd q=25. Profit is n = 1 1 5 (25)- 450 - 15(25)- 2 (25') = 8 00. Producer surplus is profit plus fixed cost, which is 1250. Note that producer surplus can also be found graphically by calculating the area below the price line and above the marginal cost (supply) curve, so that PS=0.5*(115-15)*25=1250. 12. A number of stores offer film developing a s a service to their customers. Suppose that each store that offers this service has a cost function C (q)= 50 + 0 .5q + 0 .08qz a nd a marginal cost MC = 0.5 + 0 .16q. a. If the going rate for developing a roll of film is $8.60, is the industry in long run equilibrium? If not find the price associated with long run e quilibrium F irst find the profit maximizing quantity associated with a price of $8.50 by setting price equal to marginal cost so that MC=0.5+0.16q=8.5=P, or q=50. Profit is then 8.5*50-(50+0.5*50+0.08*50*50)=$150.he industry is not in long run e quhbrium T because profit is greater than zero. In a long run equilibrium, firms produce where price is equal to minimum average cost and there is no incentive for entry or exit. To Chapter 8 : Profit Maximization and Competitive Supply f ind the minimum average cost point, set marginal cost equal to average cost and solve for q: To find the long run price in the market, substitute q=25 i nto either m argnal cost or average cost to get P=$4.50. b. S u p p o s e n o w t h a t a n e w technology is d eveloped w h i c h will r e d u c e t h e c o s t of f i l m developing by 25%. Assuming t h a t t h e i n d u s t r y is i n l o n g r u n equilibrium, h o w m u c h would a n y o n e s t o r e be w illing to p a y t o p u r c h a s e t h i s n e w technology? The new total cost function and marginal cost function can be found by multiplying the old functions by 0.75 (or 75%) and are as follows: The firm will set marginal cost equal to price, which is $4.50 in the long r u n equilibrium. Solve for q to f md t hat the firm wlll develop approximately 34 rolls of film (rounding down). If q=34 t hen profit is $33.39. This is the most the firm would be willing to pay for the new technology. Note that if a ll firms adopt the new technology and produce more output, then price in the market will fall and profit for each firm will be reduced to zero. 13. Consider a c ity t hat h a s a n u m b e r of h o t d o g s s t a n d s o p e r a t i n g t h r o u g h o u t t he d o w n t o w n area. S u p p o s e t h a t e a c h v e n d o r h a s a m a r g i n a l c ost of $1.60 p e r h o t d o g sold, a n d n o fixed cost. S u p p o s e t h e m a x i m u m n u m b e r of h o t d o g s a n y o n e v e n d o r c a n sell i n a d a y is 100. a. I f the p r i c e of a hot d o g is $2, h o w m a y h o t d o g s d o e s e a c h v e n d o r w a n t to s ell? Since marginal cost is e qual to 1.5 and the price is e qual to 2, t he hot dog vendor will want to sell as many hot dogs a s possible, or in other words, 100 hot dogs. b. I f the i n d u s t r y is p erfectly competitive will t h e p r i c e r e m a i n a t $ 2 for a h o t dog? I f not, w h a t will t h e p r i c e be? The price should fall to $1.50 so that price is equal to marginal cost. Each hot dog vendor will have a n incentive to lower the price of a hot dog below $2 s o they can sell more hot dogs than their competitors. No hot dog vendor will sell a hot dog for a price below marginal cost, so the price wdl f all until it reaches $1.50. c. I f each v endor sells exactly 100 h o t d o g s a d a y a n d t h e d e m a n d f o r h o t d o g s f r o m v e n d o r s i n t h e city is Q=4400-1200P, h o w m a n y v e n d o r s a r e t h e r e ? If price is 1.50 then Q=4400-1200*1.5=2600 i n total. If each vendor sells 100 hot dogs then there are 26 vendors. d. S u p p o s e t h e city d e c i d e s t o r e g u l a t e h o t d o g v e n d o r s by i s s u i n g permits. If t h e city i s s u e s only 20 permits, a n d if e a c h v e n d o r c o n t i n u e s t o sell 100 h o t d o g s a d ay, w h a t p r i c e will a h o t d o g sell for? If there are 20 vendors selling 100 hot dogs each then the total number sold is 2000. If Q=2000 t hen P=$2, from the demand curve. Chapter 8: Profit Maximization a nd Competitive Supply e. Suppose the city decided to sell the permits. What is the highest price a vendor would pay for a permit? At the new price of $2 per hot dog the vendor is making a profit of $0.50 per hot dog, or a total of $50. This is the most they would pay on a per day basis. 14. A sales tax of $ 1 per unit of output is placed on one firm whose product sells for $5 i n a competitive industry. a. How will this tax affect the cost curves for the firm? With the imposition of a $1 tax on a single firm, all i ts cost curves shift up by $1. Total cost is cost becomes TC+tq, or TC+q since t=l. Average cost is now AC+l. Mar@ now MC+l. b. What will happen to the firm's price, output, and profit? Since the firm is a price-taker in a competitive market, the imposition of the t x on only a ic one firm does not change the market price. S n e t he firm's s hort-run supply curve is i ts marginal cost curve above average variable cost and that marginal cost curve has shifted up (inward), the firm supplies less to the market a t every price. Profits are lower a t every quantity. c. Will there be entry or exit in the industry? If the tax is placed on a single h, h will go out of business. In the long run, t hat price in the market will be below the minimum average cost point of this h . 15. A sales tax of 10 percent is placed on half the firms (the polluters) in a competitive industry. The revenue is paid to the remaining firms (the nonpolluters) as a 10 percent subsidy on the value of output sold. a. h suming t hat all firms have identical constant long-run average costs before the sales tax-subsidy policy, what do you expect to happen to the price of the product, the output of each of the firms, and industry output, in the short run and the long run? (Hint: How does price relate to industry input?) The price of the product depends on the quantity produced by all firms in t he industry. The immediate response to t he sales-tax=subsidy policy is a reduction in q uantity by polluters and an increase in quantity by non-polluters. If a long-run competitive equilibrium existed before the sales-tax=subsidy policy, price would have been equal to marginal cost and long-run minimum average cost. For the polluters, the price after the sales tax is below long-run average cost; therefore, in the long run, they will exit the industry. Furthermore, after the subsidy, the non-polluters earn economic profits that will encourage the entry of non-polluters. If this is a constant cost industry and the loss of the polluters' output is compensated by an increase in the non-polluters' output, the price will remain constant. b. Can such a policy always be achieved with a balanced budget in which tax revenues are equal to subsidy payments? Why or why not? Explain. As the polluters exit and non-polluters enter the industry, revenues from polluters decrease and the subsidy to t he non-polluters increases. T i imbalance occurs when hs the &st polluter leaves the industry and persists ever after. If the taxes and subsidies are re-adjusted with every entering iirm and exiting firm, then tax revenues from polluting firms will shrink and the non-polluters get smaller and smaller subsidies. Chapter 9: The Analysis of Competitive Markets CHAPTER 9 THE ANALYSIS OF COMPETITIVE MARKETS I TEACHING NOTES With the exception of Chapter 1, Chapter 9 is the most straightforward and easily understood chapter in the text. The chapter begins with a review of consumer and producer surplus in section 9.1. If you have postponed these topics, you should carefully explain the definition of each. Section 9.2 discusses the basic concept of efficiency in competitive markets by comparing competitive outcomes with those under market failure. A more detailed discussion of efficiency is presented in Chapter 16. Sections 9.3 to 9.6 present examples of government policies that cause the market equilibrium to differ £rom the competitive, efficient equilibrium. The instructor can pick and choose among sections 9.3 to 9.6 depending on time constraints and personal preference. The presentation in each of these sections follows the same format: there is a general discussion of why market intervention leads to deadweight loss, followed by the presentation of an important policy example. Each section is discussed in one review question and applied in at least one exercise. Exercise (1) focuses on minimum wages presented in Section 9.3. Exercises (4) and (5) reinforce discussion of price supports and production quotas from Section 9.4. The use of tariffs and quotas, presented in Section 9.5, can be found in Exercises (3), (6), (7), (8), ( ll), a nd (12). Taxes and subsidies (Section 9.6) are discussed in Exercises (2), (9), and (14). Exercise (10) reviews natural gas price controls in Example 9.1, a continuation of Example 2.7. Exercise (4) may be compared to Example 9.4 and discussed as an extension of Example 2.2. REVIEW QUESTIONS 1 What is meant by deadweight loss? Why does a price ceiling usually result in a . deadweight loss? Deadweight loss refers to the benefits lost to either consumers or producers when markets do not operate efficiently. The term deadweight denotes that these are benefits unavailable to any party. A price ceiling wdl tend to result in a deadweight loss because at any price below the market equilibrium price, quantity supplied wdl be below the market equilibrium quantity supplied, resulting in a loss of surplus to producers. Consumers w dl purchase less than the market equilibrium quantity, resulting in a loss of surplus to consumers. Consumers will also purchase less than the quantity they demand at the price set by the c e h g . The surplus lost by consumers and producers is not captured by either group, and surplus not captured by market participants is deadweight loss. 2. Suppose the supply curve for a good is completely inelastic. If the government imposed a price ceiling below the markefclearing level, would a deadweight loss result? Explain. When the supply curve is completely inelastic, the imposition of an effective price ceiling transfers all loss i n producer surplus to consumers. Consumer surplus increases by the difference between the market-clearing price and the price ceiling times the market-clearing quantity. Consumers capture al decreases in total revenue. l Therefore, no deadweight loss occurs. 3. How can a price ceiling make consumers better off! Under what conditions might it make them worse off? If the supply curve is perfectly inelastic a price ceiling wlll increase consumer surplus. If the demand curve is inelastic, price controls may result in a net loss of consumer surplus because consumers willing to pay a higher price are unable to purchase the price-controlled good or service. The loss of consumer surplus is greater than the transfer of producer surplus to consumers. If demand is elastic (and supply is relatively inelastic) consumers in the aggregate will enjoy an increase in consumer surplus. I ...
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This note was uploaded on 10/11/2009 for the course ECON 191 taught by Professor Chen during the Spring '08 term at HKUST.

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