Economics 2 Spring 2018 rofessor Christina Romer rofessor David Romer SUGGESTED ANSWERS TO ROBLEM SET 2 1.a. In this problem we are dividing everything the household buys into two categories child care (C) and everything else (E). The condition for the household to be doing the best that it can is: MU C C = MU E E, where MU is the marginal utility of another unit of a good (such as another hour of child care) and is the price of a good. The intuition behind this condition is that MU X/ X is the extra utility derived by spending another dollar on good x. This is true because 1/ X is the fraction of a unit of good x a dollar will buy, and MU X is the marginal utility of another unit of the good at the current level of consumption. When MU/ is the same for all goods, there is no way for the household to rearrange its spending that will increase total utility. Anything that is gained by spending one fewer dollar on one good will just equal what is gained by spending another dollar on the other good. b. As the hint in the problem suggests, the denominator of each element in the rational spending rule is what the household must pay for another unit of the good. Normally, that is just the market price. But, when the government imposes a that is physically collected from the consumer, that amount is the market price plus the ( C + ). If the household was maximizing its utility (so the rational spending rule held) before the imposition of the and the price were to remain the same, after the imposition of the the household would find that: MU C C + ttt < MU E. E The household needs to consume less child care and more of everything else. This will drive up the MU of another unit of child care and drive down the MU of another unit of everything else, and so help to restore the optimization condition to equality. In addition, if the household had been on its budget constraint before the imposition of the, after the (and supposing that C did not change), the initial levels of consumption of child care and everything else would cost more than the household s income. To get back on its budget constraint, the household needs to reduce its consumption of both goods. Thus when the is imposed, there are two factors that reduce the amount of child care the household would want to consume if the price of child care did not change: the decrease in consumption of child care and the increase in the quantity of everything else to restore the optimization condition to equality (this is the substitution effect of the ), and the reduction in consumption of both goods to get back on the budget constraint (this is the income effect of the ). c. In part (b), we used the condition for utility maximization to show that a per unit on child care collected from the household will reduce the quantity of child care that the household demands at a given price. This corresponds to a shift down or to the left in the individual household s demand curve for child care (from d 1 to d 2). We can go a step further and say how much the will shift down the demand curve. If we think of the individual d 2 d 1 q
demand curve as showing how much an optimizing household would be willing to pay for one more unit at different levels of consumption (the vertical interpretation of the demand curve), then the will shift the individual demand curve down by the amount of the. The price that the household would be willing to pay for one more unit is lower by the amount of the, because the household knows that on top of the price it will need to send the government a check for the. 2 d. The market demand curve is the horizontal sum of the individual household demand curves. The imposition of the (collected from consumers) will shift down each household s demand curve by the amount of the. Thus, it will also shift down the market demand curve by the amount of the. The equilibrium quantity of child care bought and sold will fall (from 1 to 2). The equilibrium price received by sellers of child care will fall (from 1 to 2). The amount paid by households for an hour of child care will rise from 1 to 2+. This problem illustrates the important fact that regardless of which side physically pays the to the government, both sides feel the impact of the. 2 + 1 2 0 2 1 e. There are two ways to visualize the revenue the government receives from the. One (shown in panel (a)) is the rectangle between 2+ and 2, from 0 to 2. By construction, this rectangle is equal to the times the quantity bought and sold when there is a. The second (shown in panel (b)) makes use of the fact that the distance between and is the. Therefore, the area between the two demand curves between 0 and 2 is also government revenue. Importantly, the two ways of visualizing the revenue yield exactly the same sized area. anel (a) anel (b) 2 + 1 2 1 2 0 2 1 0 2 1 f. As with the revenue, there are two ways to do the welfare analysis and find the deadweight loss associated with the. 2 + 1 2 a b c d e f h g i 0 2 1
First, consider the situation before the imposition of the. Consumer surplus is the area below the original demand curve (, which is also the marginal benefit curve) and above the price ( 1), up to the quantity bought and sold ( 1). Thus it is a+b+c+f+h. roducer surplus is the area above the supply curve () and below the price ( 1), up to the quantity bought and sold ( 1). Thus it is d+e+g+i. Now consider the situation after the imposition of the. roducer surplus (again, the area above the supply curve () and below the price ( 2), up to the quantity bought and sold ( 2), is e. One approach to measuring consumer surplus is to use the difference between households willingness to pay () and the price including the ( 2+). With this approach, consumer surplus is a+b. But if we instead measure consumer surplus using the difference between the benefit they get after paying the (the demand curve ) and the before- price ( 2), consumer surplus is b+c+d, which has the same area as a+b. From part (c), we know that we can write revenue as either c+d+f+g (which is the natural way to visualize it using the first approach to consumer surplus) or a+f+g (which is the natural way to visualize it using the second approach). This information is summarized in the table below. Total surplus is the sum of consumer surplus, producer surplus, and revenue. Deadweight loss is the shortfall of total surplus from its maximum possible value. Thus, as the table shows, the causes a deadweight loss of area h+i. The causes a deadweight loss because some units of the good (the ones from 2 to 1) are not produced even though their marginal benefit is greater than the marginal cost of producing them. No Tax ( 1) Tax ( 2) Approach 1 Tax ( 2) Approach 2 Consumer Surplus a+b+c+f+h a+b b+c+d roducer Surplus d+e+g+i e e Tax Revenue c+d+f+g a+f+g Total Surplus a+b+c+d+e+f+g+h+i a+b+c+d+e+f+g a+b+c+d+e+f+g Deadweight Loss h+i h+i 3 2. If marginal cost for a typical supplier rises only mc slowly as more is produced, this means that the marginal cost curve of the typical supplier is fairly flat, as shown in the diagram. But recall that a firm s mc,s supply curve is the same as its marginal cost curve. (One way to see this is to note that the price that it would take to get the firm to supply some quantity, which we will call q 1, is the firm s marginal cost at q 1. If the price is equal to the marginal cost at q 1, the firm wants to produce all the units up to q 1, since q price exceeds marginal cost for those units, but no units beyond q 1, since price is less than marginal cost for those units.) Thus if marginal cost rises only slowly as more is produced, which means that the marginal cost curve is fairly flat, the firm s supply curve is necessarily also fairly flat. The price elasticity of supply is the percentage change in the quantity demanded divided by the percentage change in price as we move along the supply curve. As we have discussed, there is a rough relationship between slope and elasticity: in normal cases, a fairly flat supply curve (which means that the quantity supplied changes a lot in response to a given change in price) corresponds to a high price elasticity of supply, and a fairly steep supply curve (which means that the quantity supplied changes only a little in response to a given change in price) corresponds to a low price elasticity of supply. Thus if the marginal cost for a typical supplier rises only slowly as more is produced, the price elasticity of supply for the typical supplier is likely to be high. And since the market supply curve is the sum of the supply curves of the individual suppliers, if the ratio of the
percent change in the quantity supplied to the percent change in price is high for the typical firm, it is also high for the industry as a whole. Thus, the price elasticity of supply for the product is likely to be high. The diagram to the right shows the market supply curve and the effects of a shift in the demand curve (from to ): as we move along the supply curve in response to the shift in demand, price rises only slightly (from 1 to 2) while the quantity rises substantially (from 1 to 2). This corresponds to elastic supply. 2 1 4 1 2 3. The problem suggests that the market for barbershop haircuts is perfectly competitive. Therefore, to figure out what any development will do to the amount that a typical barbershop wants to supply, we must look at both the supply and demand diagram for the market and the marginal revenue and marginal cost diagram for a typical shop. a. The change in tastes will lead to an outward shift of the demand curve: because consumers prefer shorter hairstyles, which require more frequent haircuts to look good, at a given price they demand more haircuts. Thus, as the supply and demand diagram for the market shows, the demand curve shifts to the right (from to ), and both the equilibrium price and equilibrium quantity of haircuts rise (from 1 to 2 and from 1 to 2). Typical Barbershop 2 1 2 1 mr 2 1 2 q 1 q 2 q To see how the increase in the quantity of haircuts supplied comes about, look at the behavior of a typical barbershop. Because the market for haircuts is perfectly competitive, the marginal revenue curve for a typical shop is horizontal at the going market price. The initial profitmaximizing level of output occurs where the initial marginal revenue curve () intersects the initial marginal cost curve (). This occurs at quantity q 1. The change in consumers tastes does not affect the marginal cost curve of the barbershop. But the rise in the prevailing market price causes the marginal revenue curve of a typical barbershop to rise to mr 2. It intersects the marginal cost curve at q 2, which is greater than q 1. Thus, the shift up in the marginal revenue curve causes the typical barbershop to supply a greater quantity of haircuts in the short run.
b. In the short run, the licensing fee has no effect on the short-run decision of the typical barbershop. To see this, notice first that the fee involves barbershops costs and has nothing to do with consumers tastes, so it does not shift the demand curve. Now consider the typical barbershop. The licensing fee does not depend on the quantity of haircuts the barbershop supplies. Thus, it is an increase in the barbershop s fixed costs, and does not change its marginal costs of supplying additional haircuts. As a result, the quantity of haircuts where marginal revenue equals marginal cost (and that therefore maximizes the shop s profits) is the same as before. Thus, the licensing fee has no effect on the quantity of haircuts that a typical barbershop wants to supply in the short run. As shown in the diagram below, none of the curves relevant to the short-run decision of the firm (,,, ) have changed. Thus, q 1 is also unchanged. 5 Typical Barbershop 1 1 atc 2 atc 1 1 q 1 q (Of course, the licensing fee does affect the level of profits in the short run. Average total cost for a typical firm has risen because of the increase in fixed cost. This can be shown in the diagram as a rise in the average total cost curve for a typical firm (from atc 1 to atc 2). Notice that the average total cost curve shifts up more at low levels of q than at high levels. This is true because at low levels of q, the rise in fixed costs is spread over fewer units. If the market for haircuts was initially in longrun equilibrium with the typical shop earning zero profits, this means that the fee causes profits to become negative in the short run.) c. A rise in the wage of barbers shifts the marginal cost curve for a typical barbershop up (from to mc 2). It now costs more to supply each additional haircut than before, because the barber who does the haircut is paid more. Since the market supply curve is just the horizontal sum of the individual firms marginal cost curves, it too shifts up by the same amount in the short run (from to S 2). The shift up in the market supply curve raises the equilibrium price (from 1 to 2) and decreases the equilibrium quantity (from 1 to 2). Notice that the equilibrium price rises by less than the increase in costs in the short run because the demand curve is downward sloping. To see how this decrease in the market equilibrium quantity comes about, look at the behavior of a typical barbershop. The initial profit-maximizing level of output occurs where the initial marginal revenue curve () intersects the initial marginal cost curve (). This occurs at quantity q 1. The rise in wages shifts the marginal cost curve for a typical shop up (to mc 2). The rise in the prevailing market price causes the marginal revenue curve of a typical barbershop to rise to mr 2. It intersects the new marginal cost curve at q 2. q 2 is lower than q 1 because the rise in the market price in the short run is less than the rise in marginal cost. That is, the shift up in the marginal revenue curve is less than the shift up in the marginal cost curve in the short run. As a result, the quantity of haircuts the typical barbershop wants to supply in the short run falls.
6 Typical Barbershop S 2 mc 2 2 1 2 1 mr 2 2 1 q 2 q 1 q 4.a. The fact that Scottish fishing boats need to travel farther from shore on each trip and to use more intensive fishing techniques will raise the marginal cost of the typical fishing company at any given level of output: because of the need to travel farther on each trip and to use more labor, the additional costs associated with producing one more unit of output at a given level of output are higher than before. This is shown by the upward shift in the marginal cost curve in the right-hand diagram below (from to mc 2). Typical Firm S 2 atc 2 mc 2 atc 1 ATC 2 2 1 2 ATC 1, 1 mr 2 2 1 q 2 q 1 q b. The rise in marginal cost for the typical fishing company shifts the industry supply curve up (from to S 2 in the left-hand diagram above). One way to realize this is to remember that the industry supply curve is the industry marginal cost curve; thus if each firm s marginal cost curve shifts up, the marginal cost curve of the industry shifts up. As the left-hand diagram shows, the upward shift of the supply curve causes the price of haddock to rise (from 1 to 2) and the equilibrium quantity of haddock to fall (from 1 to 2). Since the problem states that the industry began in long-run equilibrium, we know that the typical fishing company was initially earning zero economic profits. In our diagram, the initial market price of haddock ( 1) is determined by the intersection of the initial market supply curve () and the initial market demand curve (). The typical competitive firm can sell as much as it wants at the going market price, so the initial marginal revenue curve () for the firm is a horizontal line at 1. The typical firm produces where intersects the firm s marginal cost curve (), which occurs at q 1. For profits to be zero, the typical firm s average total cost curve must also intersect and at q 1. The upward shift of the marginal cost curve causes the atc curve of the typical fishing company to shift up (from atc 1 to atc 2). The amount of the upward shift is the same as the amount of the upward shift of the mc curve. The reason for this is that if the marginal cost of producing each
unit increases by a certain amount, the average cost per unit rises by that same amount. (A good way to make sure your drawing is accurate is to check that at q 1, both atc and mc have shifted up by the same amount.) As the left-hand diagram shows, because the demand curve slopes down, the amount that the price increases is less than the amount that marginal cost increases. (In terms of the diagram, the vertical distance between 1 and 2 is less than the vertical distance between and S 2.) This means that the upward shift in the marginal revenue curve of the typical firm (from to mr 2) is less than the upward shift of the atc curve (from atc 1 to atc 2). The right-hand diagram shows that when mr shifts up by less than mc does, the amount the typical firm wants to produce falls (from q 1 to q 2). It also shows that at the new profitmaximizing level of output for the typical firm, marginal revenue (which is equal to price) is less than average total cost (ATC 2). Thus, the typical company is now earning negative profits. c. The fact that the typical firm is earning negative profits will cause exit. As the number of firms falls, the industry supply curve shifts to the left: the fact that the industry supply curve is the sum of the supply curves of the individual firms means that when there are fewer firms, the quantity supplied at a given price is lower. As the supply curve shifts to the left, the market price of haddock (and hence the marginal revenue curve of the typical fishing company) rises. The process continues until the typical company is again earning zero profits. As the diagrams below show, this occurs when the price has risen by the same amount as the increase in marginal cost. In this situation, with the industry supply curve given by S 3 in the left-hand diagram and the price given by 3, marginal revenue in the right-hand diagram has risen by the same amount as average total costs, and so the typical firm is back to earning zero profits. As the right-hand diagram shows, although there are fewer firms than before and hence less total output, the output of each of the firms that remains in the industry is the same as it was before the rise in marginal cost. According to Wikipedia, the implication of our analysis that overfishing reduces profits and so leads to exit matches what we observe in the data: The Scottish demersal fleet [that is, boats fishing for haddock and related fish] has been facing economically difficult times for several years due to the decline of cod and haddock in the North Sea. The fleet has declined from around 800 vessels in 1992 to just over 400 in 2004. 7 Typical Firm S 3 S 2 atc 2 mc 2 atc 1 mr 3 mr 2 3 2 1 ATC 3, 3 2 1 3 2 1 q 2 q 1 q 3 q d. In the long run, the typical fishing company earns zero economic profits, not positive economic profits. Recall that economic profits are the difference between revenues and costs, where costs account for the opportunity costs of all inputs. If economic profits are positive, this means that the revenues from using the inputs in the industry are greater than their opportunity costs. In such a situation, other firms would enter the industry, since they could get more from the inputs than in their next best use. As they did this, the industry supply curve would shift out, the price of the output would fall, and the profits of the typical firm would decline.
5.a. If the amount of milk that the government allows to be produced is less than the equilibrium quantity, then the quantity produced will be determined by the production limit rather than by the intersection of the supply and demand curves. Thus, the quantity produced will fall (from 1 to 2 1 UOTA). With the quantity supplied limited to UOTA (and no restriction on the price), the price will rise to the point on the demand curve corresponding to that quantity that is, it will rise to 2. (To see this, think about what would happen if the price were less than 2. At such a price, the quantity demanded would exceed the amount that producers are allowed to produce. Thus, some UOTA 1 S2 Excess Supply consumers who wanted to buy at the low price would not be able to get milk, and so they would bid the price up; and producers would notice that there was unmet demand at that price, and so it would make sense for them to charge more.) Also, notice that at price 2, the amount that producers want to supply, S2, is greater than the quantity demand. The quota, however, prevents them from supplying the amount they want. The difference between the quantity producers want to supply and the amount they are allowed to supply is shown by the distance marked Excess Supply in the diagram. b. No, the revenues of dairy farmers will not necessarily rise as a result of the policy. Revenues are equal to price times quantity. Thus, revenues before the imposition of the quota are shown by the light blue rectangle with base from the origin to 1 and height from the origin to 1. After the imposition of the quota, they are given by the light red rectangle with base from the origin to UOTA and height from the origin to 2. There is nothing that guarantees that the second rectangle is larger than the first; in fact, in the case shown, it is smaller. What determines which rectangle is larger that is, whether the revenues of dairy farmers go up or down is the price elasticity of demand. The quota causes us to move up along the demand curve. If the demand curve is inelastic, this means that the percent increase in the price will be greater than the percent fall in the quantity, and so the revenues of dairy farmers will rise. But if the demand curve is elastic, then the percent increase in the price will be smaller than the percent fall in the quantity, and so the revenues of dairy farmers will fall. 2 1 UOTA 1 S2 Excess Supply 8 c. The quota will definitely cause a deadweight loss. Before the imposition of the quota, consumer surplus is a+b+c and producer surplus is d+e+f. At first glance, it looks as though after the imposition of the quota, consumer surplus is a and producer surplus is b+d+f. We have to be careful however. Because of the quantity restriction, we need to check that there is no misallocation among consumers (that is, that the good is allocated to the consumers with the highest marginal benefit) and that there is no misallocation among producers (that is, that each 2 1 ' f a b d c e UOTA 1 S2 Excess Supply
firm is producing up to the point where its marginal cost is the same as that of all other firms, so that the good is being produced as efficiently as possible). In the case of consumer surplus, the assumption of no misallocation is appropriate: any consumer who wants to buy milk at price 2 can do so, and so the milk is only sold to the consumers who get the greatest MB specifically, the consumers with MB greater than 2. In the case of producers, however, there is no reason to think that the milk is produced as efficiently as possible. With the quota, there is excess supply: with the price of milk equal to 2, each farm would like to produce to the point where its marginal cost is equal to 2 (which is what causes the quantity supplied to be S2). But since the quantity that is actually produced is only UOTA, the most efficient way to produce is to have each farm produce only to the point where its marginal cost is equal to the point on the industry s marginal cost curve at UOTA that is, until marginal cost is equal to the price marked ' in the diagram. Some method other than price has to be used to determine how much each farm gets to supply. With the determination of who is producing not being made by price, it is very unlikely that each farm will produce exactly the amount that makes its marginal cost equal to '. For example, the government might decide how much each farm is allowed to produce by a lottery, or according to who had political connections, or using some other nonmarket system. Thus, we can only say that producer surplus will be less than b+d+f, but we cannot say how much less. The table below shows the welfare accounting. Total surplus is the sum of consumer surplus and producer surplus. Deadweight loss is the shortfall of total surplus from its maximum possible value. Thus, as the table shows, the causes a deadweight loss that is greater than area c+e. There are two sources of the deadweight loss. First, some units of the good (the ones from UOTA to 1) are not produced even though their marginal benefit is greater than the marginal cost of producing them. Second, the production that does occur is not done as efficiently as possible: when there is an excess supply and no mechanisms to make sure that the marginal cost of the last unit produced is the same across firms, there is some misallocation among producers. No uota ( 1) uota ( UOTA) Consumer Surplus a+b+c a roducer Surplus d+e+f <b+d+f Total Surplus a+b+c+d+e+f <a+b+d+f Deadweight Loss >c+e 9