Competitive Markets 2

1. Competitive Markets 2

2. 4. Related competitive markets • This section examines some situations in which two different competitive markets are related to each other.

3. 4a. Example 1. Trade between countries • Widgets are produced both inside and outside Freedonia and consumed both inside and outside Freedonia. • The market is competitive within Freedonia and outside Freedonia. • The demand in Freedonia is DF ( pF) = 200 -2 pF • and the demand outside Freedonia is D0(p) = 100 – p0 • where pF is the price in Freedonia and p0 is the price outside Freedonia (also measured in Freedonian dollars for convenience). • The supply from producers in Freedonia is SF(pF) = 2 pF -40 • and the supply from producers outside Freedonia is S0(p0) = 3p0 - 20.

4. a. First suppose that it is impossible to transport widgets into or out of Freedonia. What are the equilibrium prices and quantities in Freedonia and outside Freedonia? What are the corresponding consumers’ and producers’ surplus in Freedonia and outside Freedonia? • With no trade possible, there are two independent markets. In Freedonia, supply equals demand at price pF=60, with corresponding quantity • QF= 80, • CSF = (0.5)(100- 60)(80) =1600 and PSF = (0.5)(60- 20)(80) = 1600. • Outside Freedonia, supply equals demand at price p0 = 30, with corresponding quantity Q0 = 70, • CS0 = (0.5)(100- 30)(70) = 2450 and • PS0 = (0.5)(30 - 20/3)(70) = 2450/3 = 816.7.

5. b. Now suppose it becomes possible to transport the good into and out of Freedonia, and the cost to transport the good between Freedonia and non-Freedonia markets is zero. What are the competitive equilibrium prices, quantities and consumers’ and producers’ surpluses in Freedonia and outside Freedonia? How much is imported/exported by Freedonia? • With transportation cost zero, there is now a single world market with a single price, p, for the good. • World demand (the sum of Freedonian and non-Freedonian demands) equals world supply (the sum of Freedonian and non-Freedonian supplies) at price p = 45, and world quantity Q = 165.

6. In Freedonia, DF(45) = 110 units are demanded and SF (45) = 50 units are supplied, with corresponding surpluses • CSF= (0.5)(l00- 45)(110) = 3025 and • PSF= (0.5)(45 - 20)(50) = 625. • Outside Freedonia, D0(45) = 55 units are demanded and SF (45) = 115 units are supplied, with corresponding surpluses • CS0 = (0.5)(l00- 45)(55) = 1512.5 and • PS0 = (0.5)(45- 20/3)(l 15) 13225/6 = 2204.2. • Freedonia imports DF(45)- SF(45) = 110 - 50 = 60 units.

7. c. Suppose the cost to transport the good between Freedonia and non-Freedonian markets is $4 per unit rather than zero. What are the competitive equilibrium prices, quantities and consumers’ and producers’ surpluses in Freedonia and outside Freedonia? How much is imported/exported by Freedonia? • If the answer in part a had included a price difference no larger than $4 between the two markets, then the introduction of transportation at a cost of $4 per unit would not introduce any arbitrage opportunities, and the availability of transportation would have no effect. However, in the answer to part a, the price difference is • pF –p0= 60 - 30 = 30, • so the introduction of transportation does introduce arbitrage opportunities. Since the price is higher in Freedonia, the arbitrage opportunity is to purchase the good outside Freedonia, transport it to Freedonia, and sell it there. • Thus we know Freedonia will import widgets in the new competitive equilibrium.

8. Since some of the units sold in Freedonia come from outside Freedonia, the equilibrium prices inside and outside Freedonia must differ by the transportation cost, • c = p0 + 4. • To find the equilibrium prices, we can compute net demand in Freedonia, • DF(pF) - SF (pF) = 240 - 4 pF, • which is the demand for imports by Freedonia, and net supply from outside Freedonia, • S0(p0) - D0(p0) = 4 p0- 120, • which is the supply of imports to Freedonia. • Substituting the relationship between prices, pF = p0 + 4, into the “supply of imports equals demand for imports” equation, • 4p0 - 120 = 240 - 4 pF • and solving, we obtain p0 = 43 and pF = 47 as the equilibrium prices.

9. In Freedonia, DF(47) = 106 units are demanded and SF(47) = 54 units are supplied, with corresponding surpluses • CSF = (0.5)(100 - 47)(106) = 2809 • and PSF = (0.5)(47 - 20)(54) = 729. • Outside Freedonia, D0(43) = 57 units are demanded and S0(43) = 109 units are supplied, with corresponding surpluses • CS0 = (0.5)(100- 43)(57) = 1624.5 • and PS0 = (0.5)(43 - 20/3)(109) = 11881/6 = 1980.2. • Freedonia imports DF (47) - SF(47) = 106 - 54 = 52 units.

10. 4b. Example 2. Related products with discrete demand • In the previous example, when trade became possible in part b, there were arbitrage possibilities because the original equilibrium prices from part a were different in different countries. All arbitrage opportunities will be squeezed out when we return to a new equilibrium. Since the transportation cost was 0 in part b, in order to eliminate all arbitrage opportunities the equilibrium in part b must have equal prices inside and outside the US. • A somewhat similar idea underlies the competitive equilibrium in a discrete choice problem with multiple goods. The problem involves discrete choice in the sense that the buyers decide which one of several goods to purchase. For example, a commuter in Manhattan must decide whether to use the bus, the subway, a taxi, or none (i.e., walk) for each trip to or from work. In these problems the different goods are not identical (e.g., a taxi is likely to be faster than a bus), so the prices do not need to be equal in equilibrium. The movement between markets cannot be in terms of the goods (a bus cannot become a taxi) but instead is in terms of buyers, who decide to switch between buses and taxis.

11. To keep things easy, we will use a very simplified setup. Suppose there are only two types of cars available in Freedonia, Hudsons and DeSotos. Each individual wants either one car or no car, depending on the prices. A second car is not of use to anyone, and a car is not essential. • All Hudsons are viewed as identical and all DeSotos are viewed as identical. However, everyone in Freedonia thinks a Hudson is a better car than a DeSoto, so the two brands are not viewed as identical. • Each person has two reservation prices, one for each brand of car. If a person’s reservation price for a Hudson were $10,000, then if that person owned a Hudson she would be willing to sell it at any price above $10,000, she would not sell at any price below $10,000, and she would be indifferent about selling at price $ 10,000. • With the same reservation price, if she did not own a car, and was offered a Hudson but no other option, she would buy it at any price less than $10,000, she would not buy at any price above $10,000, and she would be indifferent about buying at price $10,000.

12. Given prices pH for Hudsons and pD for DeSotos, each person has three options. • If they do not own a car, they can purchase one Hudson at price pH, or purchase one DeSoto at price pD, or not purchase any car. • If they currently own a car, they can keep it, or sell it and not buy a car, or sell it and buy the other brand of car. • Given the prices, each person chooses an action (from the three available) that maximizes her surplus. • The surplus from selling is the selling price minus the reservation value, while the surplus from buying is the reservation price minus the buying price. • If an individual decides to neither buy nor sell, her surplus is zero. • If she sells one car and buys another, her surplus is the sum of the surpluses from selling and from buying.

13. The similarity to arbitrage arises because a person without a car (either because she never had a car or because she sold it) may go to either the market for Hudsons or the market for DeSotos. • Since the goods are not identical, prices do not need to be identical in equilibrium. Rather, the “arbitrage” occurs in buyers. • If two individuals started out in identical circumstances in terms of reservation prices and initial ownership, then they must end up with identical surpluses in equilibrium. • They don’t need to do the same thing (e.g., one could buy a Hudson while the other buys a DeSoto), but they must end up with the same surplus. [If not, the one with lower surplus would prefer to make the choices made by the one with higher surplus, and the initial situation could not have been an equilibrium.]

14. Consider a market for used cars, with two brands of cars, Hudsons and DeSotos. There are two types of individuals: • 1000 identical types S individuals who have reservation prices $5000 for a DeSoto and $7,000 for a Hudson; • and 800 identical type B individuals who have reservation prices $6500 for a DeSoto and $9,000 for a Hudson. • [The reservation values are for a single car. For two or more cars, an individual’s overall reservation value matches the highest reservation value ‘. for a single car among those in the group. It is not the sum of reservation values.]

15. a. Suppose there are 1000 used cars, 400 of which are DeSotos and 600 of which are Hudsons. None of the individuals own a car initially. All of the cars have just been brought into the country, and it is your job to give them away (at no charge). In order to maximize the total value (this corresponds to surplus at price $0) how should you allocate the cars among the individuals? How many of each type of individual should get a car, and what brand of car should they get? This is a non-market question. There are no prices involved. You are just giving the cars away. [Hint: start from an allocation in which some type S get a Hudson, some type S get a DeSoto, some type S get nothing, some type B get a Hudson, some type B get a DeSoto, and some type B get nothing. Could you take a car away from someone and give it to someone else in such a way that total value increases? (Recall there are no prices. You can change the allocation in any way you want without compensating anyone for the change.) Could you swap cars between two individuals in such a way that total surplus increases? Keep making changes until it is impossible to increase total value.]

16. b. Now suppose the 1000 Type S individuals each own one of the cars initially. Note there are now three “types” of individuals: those of type S who own Hudsons, those of type S who own DeSotos, and those of type B. Find competitive equilibrium prices pH and pD, and the resulting equilibrium allocation of cars (i.e., which individuals end up with which cars?). [Hint: recall the competitive equilibrium allocation maximizes total surplus.]

17. Now suppose that, in addition to the individuals of types S and B, 500 individuals of a third type, type C, also exist. Type C individuals have reservation prices $12,000 for Hudsons and $10,000 for DeSotos. • c. Repeat question (a) with the additional type C individuals. • d. Repeat question (b) with the additional type C individuals.

18. 5. Intervention in competitive markets • 5a. Taxes or subsidies • There are three ways to solve general tax/subsidy problems. Suppose there is a tax of t per unit (for a subsidy, t would be negative). • i. If we let p be the price firms receive, net of the tax, then consumers pay p + t, and the equilibrium condition is S(p) = D(p + t). • This is done from the perspective of the firms, whose behavior, S(p), has not changed. From their perspective, it is as if consumer behavior has changed from D(p) to D(p + t). At every quantity, it is as if the demand curve has been shifted down by t.

19. ii. If we let P be the price consumers pay, including the tax, then firms receive P - t net of the tax, and the equilibrium condition is • S(P - t) = D(P). • This is done from the perspective of the consumers, whose behavior, D(P), has not changed. From their perspective, it is as if firm behavior has changed from S(P) to S(P - t). At every quantity, it is as if the supply curve has been shifted up by t.

20. iii. If we let P be the price consumers pay, including the tax, and let p be the price firms receive, net of the tax, then the equilibrium conditions are • S(p) = D(P) and P - p = t. • This recognizes that neither the consumers nor the firms have changed their behavior. Neither the demand curve nor the supply curve has shifted. What has happened is that demanders and suppliers face prices which differ by the tax, t.

21. We will solve the problem for the demand function D(P) = 300 - 60P, the supply function S(p) = 120p -150 and a tax of $0.75 per unit in each of the three ways. In the equilibrium without the tax, P = p, and S(p) = D(P), or 120p - 150 = 300 -60P. The solution is P*= p*= 2.5. Now consider the three approaches when t = $0.75. • S(P) = D(p + 0.75) or 120p - 150 = 300 - 60(p + 0.75). The solution is p = 2.25. This is the price firms receive, net of the tax. Consumers pay 2.25 + 0.75 = 3. • S(P - 0.75) = D(P) or 120(P - 0.75) - 150 = 300 - 60P. The solution is P = 3. This is the price paid by consumers, including the tax. Net of the tax, firms receive 3 - 0.75 = 2.25. • S(p) = D(P) (or 120p - 150 = 300 - 60P) and P - p = 0.75. The solution to this system of equations is P = 3 and p = 2.25.

22. All three approaches yield the same results. • Note that the government’s method of collecting the tax (i.e., whether it is paid by consumers on top of the price or by firms before the product is sold) does not affect the equilibrium prices, P and p. • In all cases, the price paid by consumers goes up by $0.50 from $2.50 to $3 while the price received by firms goes down by $0.25 from $2.50 to $2.25. • The economic tax incidence is not affected by the collection method.

23. If the tax had been based on the value rather than the number of units, such as the Tennessee sales tax of 9.25% or a value-added tax as used in many countries, then there is a different relationship between P and p. • If the tax is 100T% (in Tennessee , T = 0.0925), then • P = (1 + T)p. • Using this relationship instead of P = p + t, the analysis follows the same pattern as that for a per-unit tax.

24. Who gains and who losses with the tax? • consumers: Those who still buy after the tax face a higher price so they are worse off. Those who are willing to buy at the pre-tax price but not at the post-tax price are worse off. Those unwilling to buy at the pretax price are unaffected. • firms: Those who still sell after the tax face a lower price (this assumes supply is upward sloping) so they are worse off. Those who are willing to sell at the pre-tax price but not at the post-tax price are worse off. Those unwilling to sell at the competitive price are unaffected. • Government revenues have increased. The beneficiaries would depend on the use of this government revenue.

25. In the long run, what ways are devised to evade the tax? What alternative industries arise because of the tax? • Deadweight loss due to a tax or subsidy is the difference between the total surplus without and the total surplus with the tax or subsidy. When computing deadweight loss, government revenue is counted as part of surplus (and government losses with a subsidy are subtracted from the sum of producers’ and consumers’ surplus to obtain net total surplus).

26. Example 1: Consider a market with demand function D(p) = 900 - 2p and supply function S(p) =2p- 100 where p is the price. Suppose a $10 per unit subsidy is introduced. What are the equilibrium prices and quantity with the subsidy? • Let Pc be the price paid by consumers and let pf be the price received by firms. With a $10 per unit subsidy, Pc = pf - 10. • In equilibrium, D(Pc) = S(pf) and Pc = pf - 10, or • 900 - 2Pc =2 pf -100 and Pc = pf - 10. • One of the various ways to solve this is to substitute Pc = pf - 10 into the demand function and solve the “supply equals demand” equation for pt-. With that substitution, the equilibrium condition becomes • 900 - 2(pf - 10) = 2pf -100, or 900 - 2pf + 20 = 2pf -100, or 1020= 4pf, with solution pf = 255 as the equilibrium price received by firms. • The equilibrium price paid by consumers is Pc = pf -10 = 255 - 10 = 245, and the equilibrium quantity is S(255) = (2)(255) - 100 = 410 [or equivalently, the equilibrium quantity is D(245) = 900 - 490 = 410].

27. Example 2: Suppose demand and supply functions are D(p) = 400 - p and S(p) = 2p - 50 respectively, where p is the price in dollars per kilogram. If a tax of $t per kilogram is introduced into the market, what is the resulting competitive equilibrium? [I.e., solve for the equilibrium prices and quantity as functions of the parameter t.] • What fraction of the tax is passed on to consumers in the form of higher prices, and what fraction is borne by the firms in the form of lower prices?

28. Let Pc be the price paid by consumers and let pf be the price received by firms. With a $t per unit tax, Pc = pf + t. In equilibrium, • D(Pc) = S(pf) and Pc = pf + t or 400 - Pc = 2pf - 50 and Pc = pf + t. • One of the various ways to solve this is to substitute Pc = pf + t into the demand function and solve the “supply equals demand” equation for pf. • With that substitution the equilibrium, condition becomes • 400 - (pf + t) = 2pf -50 or 400 - pf - t = 2pf - 50 or 450 - t = 3pf • with solution pf = 150 - t/3 as the equilibrium price received by firms. • The equilibrium price paid by consumers is • Pc = pf + t = 150 - t/3 + t = 150 + 2t/3, • and the equilibrium quantity is • S(150 - t/3) =(2)(150- t/3) - 50 = 250 - 2t/3 • or equivalently, the equilibrium quantity is • D(150 + 2t/3) = 400- (150 + 2t/3) = 250 – 2t/3.

29. The consumer price with the $t per unit tax is • 150 + 2t/3 instead of 150, • so 2/3 of the tax is passed on to consumers in the form of higher prices. • The firm price with the $t per unit tax is • 150 – t/3 instead of 150, • so 1/3 of the tax is borne by the firms in the form of lower prices.

30. Example 3: Consider a market with aggregate demand • D(p) = 240 - 2p where p is the price. • In the following questions all firms have cost function • C(0) = 0 and C(q) = q2 + 30q + 100 if q >0 • where q is the output of the firm.

31. a. If two firms act as competitive price takers, what is the equilibrium price and output for each firm? • Note C(0) = 0, so the 100 is relevant to the shut down decision. AC is minimized at q = 10 (where AVC = MC) with minimum average cost 50. Thus the firm will shut down if p < 50. The problem for each firm is to maximize pq - (q2 + 30q + 100). The firm’s supply function is • 0 0≤p<50 0 if 0≤p<50 • s(p) = {0 or 10 p = 50. Aggregate supply is (p) = {0, 10, or 20 if p = 50 • p>50 p-30 if p>50 • The equilibrium condition is QS= QD or p - 30 = 240 - 2p. • Immediately we have p*= 90 (Note 90>50, so we used the correct formula for supply), Q*= 60, and q*= 30 per firm.

32. b. If the government imposes a tax of $15 per unit of the good produced, how would the outcome in part (a) change? • Let p be the price paid by consumers. From each firm’s perspective, the problem is to maximize pq - 15q -(q2+30q+100). Thus • 0 0≤p<65 0 if 0≤p<65 • s(p) = { 0 or 10 p = 65 Aggregate supply is (p) = {0, 10, or 20 if p = 65 • p>65 p-30 if p>6 • The equilibrium condition is QS= QD or p - 45 = 240 - 2p. • Thus we have p*= 95 (note 95>65, so we used the correct formula for supply), Q*=50 and q* =25 per firm. • The introduction of tax increased the price paid by consumers to 95, decreased the price received by firms to 80,and decreased the market output to 50.

33. c. If the government imposes a tax of $15 per unit of the good produced, what is the long run competitive equilibrium price, quantity, and number of active firms? What is the deadweight loss due to the tax? • The long-run equilibrium price for consumers is 65 at which the average cost (with tax) is minimized. All firms that are active must have production 10 units and earn zero profit. • Thus equilibrium market output is D(65) = 110 and each of 110/10 = 11 firms produces 10 units. • To compute deadweight loss, note without the tax the long-run equilibrium price would be 50 (the same as firms receive with the tax) with aggregate output 140. • DWL = (1/2)(140 - 110)(15) = 225.

34. Example 4: Taxes or subsidies may also be applied to factors of production. These taxes or subsidies on inputs may also have an effect on output markets. Consider a single firm with production function q=K0.5L0.5 • where q is the amount of output, K is the amount of capital used as input, and L is the amount of labor used as input. • The price per unit of capital is $100 and the price per unit of labor is $36. • Suppose the amount of capital is fixed at K = 9 in the short run.

35. What is the firm’s short-run supply function? • We must first determine how much variable input is needed to produce output q. The short-run production function is q=90.5L0.5 = 3L0.5. • Solving for L, L = (q/3)2 =q2/9 is the amount of labor needed to produce q. • Variable cost is wL = (36)(q2/9) = 4q2. • The $900 cost of capital (rK = (100)(9) = 900) is sunk in the short run, so the relevant cost function is variable cost, VC(q) = 4q2. • Average variable cost is AVC(q) = 4q and marginal cost is MC(q) =8q, which is larger than AVC(q) for every q > 0. • The “minimum” of AVC is 0, so the firm will have nonzero supply for every p> 0. Setting p = MC(q) = 8q, and solving for q, we obtain q = p/8, which is the optimal output when price is p. • The firm’s supply function is s(p) = p/8.

36. b. In order to boost employment, suppose the government introduces a subsidy scheme in which it pays the firm $9 for each unit of labor the firm uses. What is its new short-run supply function? • The effect of the $9 subsidy is to lower the effective price of labor for the firm to $27 = $36 - $9 per unit. The corresponding net variable cost is NVC(q) = (27)L = (27)(q2/9) = 3q2. • The remaining steps and reasoning are exactly as in part (a), but using the new net variable cost function. Average net variable cost is • ANVC(q) = 3q and marginal net cost is MNC(q) =6q, • which is larger than ANVC(q) for every q > 0. • The “minimum” of ANVC is 0, so the firm will have nonzero supply for every p > 0. Setting p = MNC(q) = 6q, and solving for q, we obtain q = p/6, which is the optimal output when price is p. • The firm’s supply function is s*(p) =p/6.

37. Suppose in the short run, there are 96 identical firms, and aggregate demand is D(p) = 1920-48p • c. What are the competitive equilibrium price and quantity without the subsidy? • The aggregate supply function is the sum of the supply functions for the individual firms. Since the firms are identical, • S(p) = 96s(p) = 12p. • At the equilibrium price, S(p) = D(p), or 12p =1920 – 48p. • The equilibrium price is p*= 32, with corresponding aggregate quantity • Q* = D(p*) = 1920 - 48(32) = 384. • Each firm produces s(32) = 4 units of output.

38. d. What are the competitive equilibrium price and quantity with the subsidy? • The aggregate supply function is the sum of the supply functions for the individual firms. Since the firms are identical, • S*(p)=96s*(p)= 16p. • At the equilibrium price, S*(p)= D(p), or • 16p = 1920 - 48p. • The equilibrium price is p**= 30, with corresponding aggregate quantity • Q** = D(p**) = 1920 - 48(30) = 480. • Each firm produces s*(30) = 5 units of output.

39. e. Comparing the equilibrium with the subsidy to the equilibrium without the subsidy, what are the changes in consumers’ surplus, producers’ surplus, total surplus, and labor use due to the subsidy? Without the subsidy, consumers’ surplus is (0.5)(40 - 32)(384) = 1536, producers’ surplus is (0.5)(32 - 0)(384) = 6144, and total surplus is 1536 + 6144 = 7680. To determine labor use, note each firm is producing s(32) = 4 units of output, which requires (4)2/9=16/9 units of labor. Total labor use is (96)(16/9) = 512/3 = 170.7.

40. With the subsidy, consumers’ surplus is (0.5)(40 - 30)(480) = 2400 and producers’ surplus is (0.5)(30 - 0)(480) = 7200. To determine labor use, note each firm is producing s*(30)=5 units of output, which requires (5)2/9 =25/9 units of labor. Total labor use is (96)(25/9) =800/3 = 266.7. To compute total surplus, we must determine the total government expenditure to provide the subsidy. Each of the 96 firms uses 25/9 units of labor, at a subsidy of $9 per unit, for a total subsidy of 2400. Thus total surplus is CS + PS - total subsidy = 7200.

41. The introduction of the subsidy increases consumers’ surplus by 2400 – 1536= 864, increases producers’ surplus by 7200 - 6144 = 1056, increases labor use by (800/3) - (512/3) = 96 units, and decreases total surplus by 7680 - 7200 = 480.

42. NOTE: The decrease in total surplus could also have been computed with the help of the diagram. The surplus maximizing output is Q = 384, attained without the subsidy. With the subsidy, output is too large by 480 - 384 = 96 units. In order to supply Q = 480 without the subsidy, price would need to be 40. Thus the triangle above the demand curve, below the original supply curve, and between outputs 384 and 480 represents the inefficiency in the subsidized equilibrium. Its area is (0.5)(40 - 30)(480 - 384) = 480, the same as computed above.