Economics 2 Spring 2016 rofessor Christina Romer rofessor David Romer SUGGESTED ANSWERS TO ROBLEM SET 2 1.a. Recall that the price elasticity of supply is the percentage change in quantity supplied divided by the percentage change in price. This implies a relationship between the elasticity and the slope of the supply curve: ε S =(1/slope) (/). Thus, in the range where and are similar (and so / is not far from 1), a flat supply curve is relatively elastic and a steep supply curve is relatively inelastic. Recall also that the supply curve is the marginal cost curve. Thus, supply is quite elastic if marginal cost rises slowly with quantity, and quite inelastic if marginal cost rises rapidly with quantity. Inelastic Supply Elastic Supply,MC S (also, MC),MC S (also, MC) The high load factors and long construction times suggest airlines would find it quite difficult to supply much more airplane travel in the near term. The scope to just fill empty seats is very limited. Instead, they would soon have to do things like schedule more overnight flights (requiring costly overtime) or squeeze in even more seats (requiring costly remodeling of existing planes). Thus, providing more airplane travel is likely to cause marginal cost to rise rapidly in the current situation. This suggests a steep marginal cost curve which corresponds to quite inelastic supply. b. art (a) showed that the supply of air travel is probably quite inelastic in the short run. To think about what this implies about the effects of the fall in the price of jet fuel, the key insight is that the supply curve is the marginal cost curve. The fall in jet fuel prices lowers marginal cost by some amount. Thus, there is a downward shift of the supply curve, and the size of the vertical shift of the supply curve doesn t depend on whether supply is highly elastic or highly inelastic. The diagram on the left below shows the effects of a downward shift of a highly inelastic supply curve. You can see that the changes in both price and quantity are small. For comparison, the right-hand diagram shows the effects of a downward shift of the same size of a quite elastic supply curve. The two diagrams are drawn with the same demand curve and with the same initial levels of price and quantity. You can see that the changes in price and quantity are much larger when supply is highly elastic. Inelastic Supply Elastic Supply,MC,MC S 2 1 2 1 Decrease in MC 2 Decrease in MC S 2 1 2 1 2
Intuitively: The fall in marginal cost means that Typical Airline at the old price and quantity, the typical airline (which we are assuming is a competitive firm) is no longer maximizing profits marginal revenue MC 2 exceeds marginal cost. It therefore raises the quantity it supplies at a given price. But when the marginal cost curve is very steep, marginal cost rises rapidly as it increases the quantity, and so it takes only a small increase in quantity to bring marginal cost back to marginal revenue. Thus, the increase in 1 the quantity supplied at a given price is small. That is, the fact that the supply curve is very steep means that the rightward shift caused by the fall in q 1 q 1 ' q marginal cost is very small. (In the diagram to the right, q 1' is the quantity a typical airline would supply at the old price, 1, after the fall in the price of jet fuel.) Because the rightward shift of the supply curve is small, consumers don t move down their demand curve very much, so the fall in the market price of airline travel is small. 2.a. A subsidy that is paid to consumers shifts the demand curve up by the amount of the subsidy: consumers are willing to pay more at a given quantity because they know they will be getting a check from the government. Because the subsidy is paid to consumers, the supply curve is not affected. The upward shift of the demand curve raises the equilibrium quantity (from 1 to 2). In addition, the equilibrium price, which is what firms receive, also rises (from 1 to 2). The price paid by consumers was 1, but after the subsidy the net price they pay is 2 minus the subsidy. Since the distance between Subsidy 1 2 and is the amount of the subsidy, we find this price paid by consumers by looking at the point on corresponding to the new equilibrium quantity ( 2). With a normal, downward-sloping demand curve and a normal, upward-sloping supply curve, the subsidy increases the amount suppliers receive for the good and decreases the amount that consumers pay. b. The statement is reasonable: Although the subsidy is physically paid to consumers, the introduction of the subsidy raises the price received by firms ( 2 is greater than 1), and lowers the net price paid by consumers ( 2 minus the subsidy is less than 1). Hence, just as both sides of the market bear some of the burden of a tax even if only one side physically pays the tax, the two sides share the subsidy even though only one side physically receives it. c. The amount of the subsidy per panel is the difference between the price received by sellers and the price paid by consumers. The number of units that consumers buy when the subsidy is in effect is 2. Thus, we can show the amount the government spends on the subsidy as the rectangle with height from 2 subsidy to 2 and length from 0 to 2. This is shown in panel (a) of the diagram on the next page. Another way to show the spending on the subsidy is to remember that the vertical distance between the two demand curves is the amount of the subsidy per unit. Thus, spending on the subsidy equals the area between and between 0 and 2. Since the distance between the two demand curves is the amount of the subsidy, this area is the subsidy times the new equilibrium quantity ( 2). This is shown in panel (b) of the diagram on the next page. 2 1 2 subs. 2
3 (a) (b) Subsidy Subsidy 2 2 1 2 subs. 1 2 subs. 1 2 1 2 A naïve government planner might have projected total spending on the subsidy as the subsidy per unit times the number of units before the subsidy, 1. In terms of the diagrams, this is the area between 2 and 2 subsidy from 0 to 1; alternatively, it is the area between and from 0 to 1. Since the equilibrium quantity with the subsidy is bigger than 1, actual spending is greater than this. Only if either supply or demand is perfectly inelastic is actual spending equal to the pre-subsidy quantity ( 1) times the per-unit subsidy. And actual spending can never be less than 1 times the per-unit subsidy. d. To answer this question, we need to consider the profit-maximization problem of a typical installation firm. This is shown in the two-panel diagram below. If the installation industry is competitive and begins in long-run equilibrium, then the typical installation firm initially earns zero economic profits. In our diagram, the initial market price of installations ( 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 its 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 be tangent to at q 1. The shift out in the market demand curve (from to ) caused by the subsidy raises the equilibrium price to 2, and hence shifts the marginal revenue curve for the typical firm up (to MR 2). Marginal revenue now intersects marginal cost at a higher level of output (q 2). Thus, the quantity of installations of a typical firm rises. At q 2, price is greater than ATC, so the typical firm is earning positive economic profits in the short run. 2 ATC 1 MR 2 1 2 subs. 1 2 Notice that the fact that the typical firm supplies more in the short run is what allows the market quantity to increase. There is no entry in the short run, so the rise from 1 to 2 reflects the fact that the typical firm is supplying more. q 1 q 2 q
e. The positive profits will encourage entry and hence will shift the market supply curve out in the long run. In the long run, enough firms will enter so that profits will return to zero. For this to be the case, the supply curve will have to shift out to S 3. The movement of resources into the installation industry will eventually return the equilibrium price received by firms to its initial level ( 3 is the same as 1). We know that the price will have to fall all the way back to 1 because only at 1 is price equal to both marginal cost and average total cost for a typical firm. The equilibrium quantity will rise more in the long run than in the short run (to 3). Even though each typical firm returns to producing quantity q 1, there will be more firms than before, so the market equilibrium quantity will be larger. 4 2 S 3 ATC 1 MR 2 1, 3 2 subs. 3 subs.,3 1 2 3 q 1,3 q 2 q Notice that, in contrast to what happens in the short run, in the long run the benefits of the subsidy are not shared by customers and firms. Instead, they go entirely to consumers: in the long run, the price received by firms is the same as it was before the subsidy was introduced, and the price paid by consumers is lower by the full amount of the subsidy 3.a. In this problem we are dividing everything the household buys into two categories restaurant meals (R) and everything else (E). The budget constraint is defined by the equation: q R R + q E E = Income, where the q s are the quantity of each good that the household buys and the s are the market prices. The graph of the budget constraint (BC 1) shows the various combinations of restaurant meals and everything else that the household can just afford using all of its income. If the household spends all its q R Income/ R Slope = E / R income on meals, then q R R = Income, which implies q R = Income/ R. Thus, the vertical intercept of the budget constraint is Income/ R. By similar reasoning, the horizontal intercept is Income/ E. The slope of the budget constraint is (minus) the BC 1 Income/ E q E ratio of the two prices. Since restaurant meals are on the vertical axis and everything else is on the horizontal axis, the slope is E/ R. To see the economics of this, note that if the household buys 1 less unit of everything else a movement of (minus) 1 horizontally it frees up E dollars to spend on restaurant meals. Thus, the amount of additional restaurant meals it can buy is the extra dollars it has to spend divided by the price of each restaurant meal, or E/ R. b. Recall the rational spending rule: the marginal utility derived from spending one more dollar on a good should be the same for all goods. The marginal utility from spending one more dollar on a good is the marginal utility of the good (MU) times the amount of the good the household buys with the dollar, which is 1 over its price (1/). Before the fall in the price of restaurant meals, the
household was maximizing its utility, and so it was satisfying the rational spending rule. In symbols, we can write this as: R1 = MU E E1, where R1 and E1 are the prices before the influx of new chefs. The fall in the price of restaurant meals causes the left-hand side of this equation to be larger than the right-hand side. If the household did not change its consumption choices, it would no longer be maximizing its utility. What it needs to do is consume more restaurant meals and less of everything else. As the household consumes more restaurant meals, this will drive down the marginal utility of another meal. The household will move down the marginal utility curve for restaurant meals from a point such as A to a point such as B. And as the household consumes less of everything else, it will drive up the marginal utility of another unit of everything else. The household will move up the marginal utility curve for everything else from a point such as C to a point such as D. This process will continue until the marginal utility of another dollar spent is once again the same for both types of goods. The fact that the household needs to substitute toward a good whose price has fallen in order to restore the optimization condition to balance is what we call the substitution effect of a price change. 5 MU E A B D C MU E q R q E In addition, the fall in the price of restaurant meals makes the budget constraint steeper. (It changes from BC 1 to BC 2.) This is true because R is q R now smaller, so the slope is a larger negative BC number. One can also see this by simply thinking 2 about the end points. Since the price of everything else hasn t changed, the amount of everything else the household can buy if it buys no restaurant meals which is the intercept on the everything BC 1 else axis doesn t change. On the other hand, if the household buys only restaurant meals, the fall in q E their price means that it can buy more than before. The intercept on the restaurant meals axis is higher than before. This shift in the budget constraint shows that the fall in the price of restaurant meals, in effect, makes the household richer: it can afford better combinations of restaurant meals and everything else than before. This makes the household want to consume more of both types of goods. The fact that the household consumes more restaurant meals because the fall in their price changes the budget constraint for the better is what we refer to as the income effect of a price change. c. If the marginal utility of restaurant meals declines rapidly as more are consumed, this implies that the household has a strong preference about the quantity of meals: it gets a lot of utility
from the first meals, but little from later ones. As a result, the household is likely to change its quantity demanded relatively little in response to a change in the price. To see this, recall that the household had been choosing the quantities of restaurant meals and everything else so that 6 R1 = MU E E1. Now consider the fall in the price of restaurant meals. If declines quickly as the quantity rises, this implies that the marginal utility curve for restaurant meals is relatively steep. As a result, it takes a relatively small increase in the quantity of meals to restore the rational spending rule. Therefore, if the marginal utility of restaurant meals declines rapidly, the quantity demanded is likely to be relatively insensitive to the fall in price. With rapidly declining marginal utility, a small rise in quantity leads to a large fall in marginal utility 1 2 q R1 q R2 q R 4. The problem states that the firm is competitive. Therefore, to figure out what any development does to the amount it produces, we must look at both the supply and demand diagram for the market and the marginal revenue and marginal cost diagram for the typical firm. Note that it s not enough to look just at the marginal revenue-marginal cost diagram. We can use that diagram to analyze how various developments affect the amount the firm wants to produce at a given price. But to know how much the firm actually produces, we need to know what happens to the price of its product, and so we need to look at the supply and demand diagram for the market. a. A rise in the price of an input shifts the marginal cost curve for the firm up (from to MC 2). It now costs more to supply each additional unit of output, because one of the inputs used to produce it is more expensive. Since the market supply curve is just the horizontal sum of the individual firms marginal cost curves, it 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 so shifts the firm s marginal revenue curve up. Notice that the equilibrium price rises by less than the increase in costs in the short run because the demand curve is downward sloping. Because the firm is competitive, its marginal revenue curve is horizontal at the going market price. 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 the prevailing market price causes the marginal revenue curve of the firm to rise to MR 2. It intersects the new marginal cost curve (MC 2) at q 2. q 2 is lower than q 1 because the rise in the market price (and hence the upward shift in the marginal revenue curve) in the short run is less than the rise in costs. As a result, the amount produced by the firm falls. S 2 MC 2 2 1 MR 2 2 1 q 2 q 1 q
The result that the typical firm produces less makes sense. Since there is no exit or entry in the short run, if the market equilibrium quantity falls, it must be the case that the typical firm is producing less. b. Before the shift in demand, the going price is 1, and so the initial marginal revenue curve is. To maximize profits, the firm produced where marginal revenue equaled marginal cost, which occurred at quantity q 1. The shift out of the market demand curve for the product (from to ) leads to an increase in both the equilibrium price of the good (from 1 to 2) and the equilibrium quantity (from 1 to 2). The increase in price causes the marginal revenue curve for the typical firm to shift up (from to MR 2). rofits are now maximized at the level of output where the original marginal cost curve () intersects MR 2. This occurs at quantity q 2. Note that this change in the quantity supplied by a typical firm because of the change in the price of the product corresponds to the movement along the original supply curve (). 7 2 1 MR 2 1 2 q 1 q 2 q c. Because the firm has to pay the tax regardless of how much it produces, the tax is a change in the firm s fixed costs. The rise in fixed costs has no effect on the marginal cost curve of the typical firm. As a result, it has no effect on the market supply curve. Therefore, nothing happens to the equilibrium price (and hence nothing happens to the marginal revenue curve facing the firm) in the short run. That is, fixed costs are irrelevant to the short-run decision of the firm, and so the tax has no impact on the amount the firm produces in the short run. 1 1 q 1 q Of course, although the tax doesn t affect the profit-maximizing quantity of output (which occurs where marginal revenue and marginal costs are equal), it reduces the level of profits. As a result, if the industry was initially in long-run equilibrium with firms earning zero profits, profits
are negative after the tax is introduced. Because of this, in the long run the tax causes exit, and so affects the position of the industry supply curve, the market price of the good, and hence each firm s marginal revenue curve and profit-maximizing level of output. But none of this happens in the short run. 5.a. True. Bad weather that destroys much of the French wine grape crop will lead to an inward shift of the supply curve for French wine: because many grapes have been destroyed, the quantity supplied at a given price will be lower than before. The shift in the supply curve will lead to a movement along the demand curve for French wine, from a point such as A to a point such as B. French wine and California wine are presumably substitutes. As a result, as households reduce their consumption of French wine, the marginal utility of California wine increases. Thus, to maximize their utility, households will purchase more California wine at a given price. This corresponds to a shift out of the demand curve for California wine. 8 French Wine California Wine S 2 2 1 B A 2 1 D2 2 1 1 2 b. False. It s true that the total utility one gets from food exceeds the total utility one gets from vacations, since someone who consumed no food would die. But if a household allocated its spending so that it spent more on food than on vacations, it might not be obtaining the highest possible level of total utility. Suppose the household considered spending $1 less on food and $1 more on vacations. The utility it would lose from consuming less food would be the marginal utility of food (MU F) times the amount of food it could buy with $1 (1/ F). Likewise, the utility it would gain from consuming more vacations would be the marginal utility of vacations (MU V) times the amount of vacation it could buy with $1 (1/ V). So if MU V/ V exceeded MU F/ F, this change would raise the household s total utility. (And if MU V/ V was less than MU F/ F, spending $1 less on vacations and $1 more on food would raise total utility.) Thus, total utility the sum of utility from food and utility from vacations is maximized when the marginal utility from spending one more dollar is the same for food and vacations: MU F F = MU V V. To think about why a household might actually spend more on vacations than on food, think about the marginal utility from each good. Because of the necessity of food for survival, the marginal utility of the first few units of food is surely very high and surely higher than the marginal utility of the first vacation. But for many households, the marginal utility of food falls off quickly, and it may even become negative at some point. (An extreme example is to think of salt instead of food: salt is necessary for survival, so the marginal utility of the first gram or so of daily salt consumption is extremely high. But the marginal utility of salt consumption falls off
very fast after several grams per day, the marginal health and taste benefits are probably negative. And the cost of the amount of salt needed for survival and to flavor food is just pennies per day.) And although the marginal utility of the first bit of vacation consumption may not be all that high, for many families the marginal utility falls off slowly. For such a household, if its income is high enough, it is utility maximizing to spend more on vacations than on food. 9 MU FOOD MU VAC. MU VACATION MU FOOD q FOOD q VACATION c. False. A firm s goal is to maximize its profits, which are total revenue minus total costs. For a competitive firm, total revenue is the quantity it produces times the market price of the good, and so marginal revenue (the addition to revenues from producing one more unit) is just the price. Making the difference between price and marginal cost as large as possible wouldn t maximize profits. At any point where the difference between price and marginal cost is positive, if the firm produces another unit, the additional revenue from selling the unit (which is the market price) exceeds the additional cost associated with producing the unit (marginal cost). Since the additional revenue exceeds the additional cost, producing the unit increases profits. Thus, producing at a point where price exceeds marginal cost cannot be profit maximizing. To maximize profits, the firm needs to produce at the point not where the difference between price and marginal cost is as large as possible, but at the point where the difference is zero. Only at that point is the firm not able to increase its profits by either slightly increasing or slightly decreasing the amount it produces. Finally, recall that we assume that the marginal cost curve is upward sloping. Thus, any competitive firm that followed the rule produce at the point where the difference between price and marginal cost is as large as possible would produce nothing at all! All of this is shown in the diagram below. q A (which is zero) is the level of production where the difference between price and marginal cost is as large as possible. q B is the level of production where the difference between price and marginal cost is zero. The shaded triangle shows the amount that the firm s would be lower by if it produced q A rather than q B. 1 q A q B q