Supply and Demand, Study notes of Economics

Download Supply and Demand and more Study notes Economics in PDF only on Docsity! Chapter 2 10 Supply and Demand Talk is cheap because supply exceeds demand. When asked “What is the most important thing you know about economics?” many people reply, “Supply equals demand.” This statement is a shorthand description of one of the simplest yet most powerful models of economics. The supply-and-demand model describes how consumers and suppliers interact to determine the quantity of a good or service sold in a market and the price at which it is sold. To use the model, you need to determine three things: buyers’ behavior, sellers’ behavior, and how buyers’ and sellers’ actions affect price and quantity. After reading this chapter, you should be able to use the supply-and-demand model to analyze some of the most important policy questions facing your country today, such as those con- cerning international trade, minimum wages, and price controls on health care. After reading that grandiose claim, you might ask, “Is that all there is to economics? Can I become an expert economist that fast?” The answer to both questions, of course, is no. In addition, you need to learn the limits of this model and which other models to use when this one does not apply. (You must also learn the economists’ secret hand- shake.) Even with its limitations, the supply-and-demand model is the most widely used eco- nomic model. It provides a good description of how markets function, and it works par- ticularly well in markets that have many buyers and many sellers, such as most agriculture and labor markets. Like all good theories, the supply-and-demand model can be tested—and possibly shown to be false. But in markets where it is applicable, it allows us to make accurate predictions easily. 1. Demand: The quantity of a good or service that consumers demand depends on price and other factors such as consumers’ incomes and the price of related goods. 2. Supply: The quantity of a good or service that firms supply depends on price and other factors such as the cost of inputs that firms use to produce the good or service. 3. Market Equilibrium: The interaction between consumers’ demand curve and firms’ supply curve determines the market price and quantity of a good or service that is bought and sold. 4. Shocking the Equilibrium: Comparative Statics: Changes in a factor that affect demand (such as consumers’ incomes), supply (such as a rise in the price of inputs), or a new government policy (such as a new tax) alter the market price and quantity of a good. 5. Elasticities: Given estimates of summary statistics called elasticities, economists can forecast the effects of changes in taxes and other factors on market price and quantity. In this chapter, we examine eight main topics: M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 10 2.1 Demand The amount of a good that consumers are willing to buy at a given price during a spec- ified time period (such as a day or a year), holding constant the other factors that influ- ence purchases, is the quantity demanded. The quantity demanded of a good or service can exceed the quantity actually sold. For example, as a promotion, a local store might sell DVDs for $1 each today only. At that low price, you might want to buy 25 DVDs, but because the store has run out of stock, you can buy only 10 DVDs. The quantity you demand is 25—it’s the amount you want—even though the amount you actually buy is only 10. Potential consumers decide how much of a good or service to buy on the basis of its price, which is expressed as an amount of money per unit of the good (for example, dollars per pound), and many other factors, including consumers’ own tastes, infor- mation, and income; prices of other goods; and government actions. Before concen- trating on the role of price in determining demand, let’s look briefly at some of the other factors. Consumers make purchases based on their tastes. Consumers do not purchase foods they dislike, works of art they hate, or clothes they view as unfashionable or uncom- fortable. However, advertising may influence people’s tastes. Similarly, information (or misinformation) about the uses of a good affects con- sumers’ decisions. A few years ago when many consumers were convinced that oatmeal could lower their cholesterol level, they rushed to grocery stores and bought large quantities of oatmeal. (They even ate some of it until they remembered that they couldn’t stand how it tastes.) The prices of other goods also affect consumers’ purchase decisions. Before deciding to buy Levi’s jeans, you might check the prices of other brands. If the price of a close substitute—a product that you view as similar or identical to the one you are consid- ering purchasing—is much lower than the price of Levi’s jeans, you may buy that other brand instead. Similarly, the price of a complement—a good that you like to consume at the same time as the product you are considering buying—may affect your decision. If you eat pie only with ice cream, the higher the price of ice cream, the less likely you are to buy pie. Income plays a major role in determining what and how much to purchase. People who suddenly inherit great wealth may purchase a Mercedes or other luxury items and would probably no longer buy do-it-yourself repair kits. Demand 11 6. Effects of a Sales Tax: How a sales tax increase affects the equilibrium price and the quantity of a good and whether the tax falls more heavily on consumers or on suppli- ers depend on the supply and demand curves. 7. Quantity Supplied Need Not Equal Quantity Demanded: If the government regulates the prices in a market, the quantity supplied might not equal the quantity demanded. 8. When to Use the Supply-and-Demand Model: The supply-and-demand model applies only to competitive markets. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 11 14 CHAPTER 2 Supply and Demand 2Economists typically do not state the relevant physical and time period measures unless these measures are particularly useful in context. I’ll generally follow this convention and refer to the price as, say, $3.30 (with the “per kg” understood) and the quantity as 220 (with the “million kg per year” understood). 3In Chapter 4, we show that the Law of Demand need not hold theoretically; however, available empirical evidence strongly supports the Law of Demand. 4We can show the same result using the more general demand function in Equation 2.2, where the demand function has several arguments: price, prices of two substitutes, and income. With several arguments, we need to use a partial derivative with respect to price because we are inter- ested in determining how the quantity demanded changes as the price changes, holding other rele- vant factors constant. The partial derivative with respect to price is 0Q/0p
-20 < 0. Thus using either approach, we find that the quantity demanded falls by 20 times as much as the price rises. zero in Equation 2.3, we find that the quantity demanded is Q
286  (20  0)
286.2 By plugging the particular values for p in the figure into the demand equation, we can determine the corresponding quantities. For example, if p
$3.30, then Q
286  (20  3.30)
220. Effect of a Change in Price on Demand. The demand curve in Figure 2.1 shows that if the price increases from $3.30 to $4.30, the quantity consumers demand decreases by 20 units, from 220 to 200. These changes in the quantity demanded in response to changes in price are movements along the demand curve. The demand curve is a con- cise summary of the answers to the question “What happens to the quantity demanded as the price changes, when all other factors are held constant?” One of the most important empirical findings in economics is the Law of Demand: Consumers demand more of a good the lower its price, holding constant tastes, the prices of other goods, and other factors that influence the amount they consume.3 One way to state the Law of Demand is that the demand curve slopes downward, as in Figure 2.1. Because the derivative of the demand function with respect to price shows the movement along the demand curve as we vary price, another way to state the Law of Demand is that this derivative is negative: A higher price results in a lower quantity demanded. If the demand function is Q
D(p), then the Law of Demand says that dQ/dp < 0, where dQ/dp is the derivative of the D function with respect to p. (Unless we state otherwise, we assume that all demand (and other) functions are continuous and differentiable everywhere.) The derivative of the quantity of pork demanded with respect to its price in Equation 2.3 is which is negative, so the Law of Demand holds.4 Given dQ/dp
-20, a small change in price (measured in dollars per kg) causes a 20-times-larger fall in quantity (mea- sured in million kg per year). This derivative gives the change in the quantity demanded for an infinitesimal change in price. In general, if we look at a discrete, relatively large increase in price, the change in quantity may not be proportional to the change for a small increase in price. However, here the derivative is a constant that does not vary with price, so the same derivative holds for large as well as for small changes in price. dQ dp = -20, M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 14 Demand 15 5Substituting pb
$4.60 into Equation 2.2 and using the same values as before for pc and Y, we find that + (2 * 12.5) = 298 - 20p. Q = 171 - 20p + 20pb + 3pc + 2Y = 171 - 20p + (20 * 4.60) + a3 * 3 1 3 b For example, let the price increase from p1
$3.30 to p2
$4.30. That is, the change in the price ∆p
p2  p1
$4.30  $3.30
$1. (The ∆ symbol, the Greek letter capital delta, means “change in” the following variable, so ∆p means “change in price.”) As Figure 2.1 shows, the corresponding quantities are Q1
220 and Q2
200. Thus if ∆p
$1, the change in the quantity demanded is ∆Q
Q2  Q1
200  220
-20, or 20 times the change in price. Because we put price on the vertical axis and quantity on the horizontal axis, the slope of the demand curve is the reciprocal of the derivative of the demand function: slope
dp/dQ
1/(dQ/dp). In our example, the slope of demand curve D1 in Figure 2.1 is dp/dQ
1/(dQ/dp)
1/(-20)
 0.05. We can also calculate the slope in Figure 2.1 using the rise-over-run formula and the numbers we just calculated (because the slope is the same for small and for large changes): This slope tells us that to sell one more unit (million kg per year) of pork, the price (per kg) must fall by 5¢. Effects of Changes in Other Factors on Demand Curves. If a demand curve measures the effects of price changes when all other factors that affect demand are held constant, how can we use demand curves to show the effects of a change in one of these other factors, such as the price of beef? One solution is to draw the demand curve in a three- dimensional diagram with the price of pork on one axis, the price of beef on a second axis, and the quantity of pork on the third axis. But just thinking about drawing such a diagram probably makes your head hurt. Economists use a simpler approach to show the effect on demand of a change in a factor other than the price of the good that affects demand. A change in any factor other than the price of the good itself causes a shift of the demand curve rather than a movement along the demand curve. If the price of beef rises while the price of pork remains constant, some people will switch from beef to pork. Suppose that the price of beef rises by 60¢ from $4.00 per kg to $4.60 per kg but that the price of chicken and income remain at their average levels. Using the demand function 2.2, we can calculate the new demand function relating the quantity demanded to only the price:5 (2.4) The higher price of beef causes the entire pork demand curve to shift 12 units to the right from D1, corresponding to demand function 2.3, to D2, demand function 2.4, in Figure 2.2. (In the figure, the quantity axis starts at 176 instead of 0 to emphasize the relevant portion of the demand curve.) Why does the demand function shift by 12 units? Using the demand function 2.2, we find that the partial derivative of the quantity of pork demanded with respect to the Q = 298 - 20p. slope = rise run = ¢p ¢Q = $1 per kg -20 million kg per year = -$0.05 per million kg per year . M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 15 16 CHAPTER 2 Supply and Demand p, $ p er k g 220176 Effect of a 60¢ increase in the price of beef D1 D 2 232 Q, Million kg of pork per year 0 3.30 Figure 2.2 A Shift of the Demand Curve. The demand curve for processed pork shifts to the right from D1 to D2 as the price of beef rises from $4 to $4.60. As a result of the increase in beef prices, more pork is demanded at any given price. Sideways Wine In the Academy Award–winning movie Sideways, the lead character, a wine snob, wildly praises pinot noir wine, saying that its flavors are “haunting and brilliant and thrilling and subtle.” Bizarrely, the exuberant views of this fictional character apparently caused wine buyers to flock to pinot noir wines, dramatically shifting the U.S. and British demand curves for pinot noir to the right (similar to the shift shown in Figure 2.2). Between October 2004, when Sideways was released in the United States, and January 2005, U.S. sales of pinot noir jumped 16% to record levels (and 34% in California, where the film takes place), while the price remained relatively constant. In contrast, sales of all U.S. table wines rose only 2% in this period. British con- sumers seemed similarly affected. In the five weeks after the film opened in the United Kingdom, pinot sales increased 20% at Sainsbury’s and 10% at Tesco and Oddbins, which ran a Sideways promotion.6 APPLICATION 6Sources for the applications appear at the back of the book. price of beef is 0Q/0pb
20. Thus if the price of beef increases by 60¢, the quantity of pork demanded rises by 20  0.6
12 units, holding all other factors constant. To analyze the effects of a change in some variable on the quantity demanded prop- erly, we must distinguish between a movement along a demand curve and a shift of a demand curve. A change in the price of a good causes a movement along its demand curve. A change in any other factor besides the price of the good causes a shift of the demand curve. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 16 Supply 19 p, $ p er k g 220176 Supply curve, S1 300 Q, Million kg of pork per year 0 3.30 5.30 Figure 2.3 A Supply Curve. The estimated supply curve, S1, for processed pork in Canada (Moschini and Meilke, 1992) shows the relationship between the quantity supplied per year and the price per kg, holding cost and other factors that influence supply constant. The upward slope of this supply curve indicates that firms supply more of this good when its price is high and less when the price is low. An increase in the price of pork causes a movement along the supply curve, resulting in a larger quantity of pork supplied. Corresponding to this supply function is a supply curve, which shows the quantity supplied at each possible price, holding constant the other factors that influence firms’ supply decisions. Figure 2.3 shows the estimated supply curve, S1, for processed pork. Because we hold fixed other variables that may affect the quantity supplied, such as costs and government rules, the supply curve concisely answers the question “What happens to the quantity supplied as the price changes, holding all other factors con- stant?” As the price of processed pork increases from $3.30 to $5.30, holding other fac- tors (the price of hogs) constant, the quantity of pork supplied increases from 220 to 300 million kg per year, which is a movement along the supply curve. How much does an increase in the price affect the quantity supplied? By differenti- ating the supply function 2.7 with respect to price, we find that dQ/dp
40. This derivative holds for all values of price, so it holds for both small and large changes in price. That is, the quantity supplied increases by 40 units for each $1 increase in price. Because this derivative is positive, the supply curve S1 slopes upward in Figure 2.3. Although the Law of Demand requires that the demand curve slope downward, there is no “Law of Supply” that requires the market supply curve to have a particular slope. The market supply curve can be upward sloping, or vertical, horizontal, or downward sloping. A change in a factor other than price causes a shift of the supply curve. If the price of hogs increases by 25¢, the supply function becomes (2.8) By comparing this supply function to the original one in Equation 2.7, Q
88 + 40p, we see that the supply curve, S1, shifts 15 units to the left, to S2 in Figure 2.4. Q = 73 + 40p. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 19 20 CHAPTER 2 Supply and Demand p, $ p er k g 205176 Effect of a 25¢ increase in the price of hogs S1 S 2 220 Q, Million kg of pork per year 0 3.30 Figure 2.4 A Shift of a Supply Curve. An increase in the price of hogs from $1.50 to $1.75 per kg causes a shift of the supply curve from S1 to S2. At the price of processed pork of $3.30, the quantity supplied falls from 220 on S1 to 205 on S2. Alternatively, we can determine how far the supply curve shifts by partially differ- entiating the supply function 2.6 with respect to the price of hogs: 0Q/0ph
-60. This partial derivative holds for all values of ph and hence for both small and large changes in ph. Thus a 25¢ increase in the price of hogs causes a -60  0.25
-15 units change in the quantity supplied of pork at any given constant price of pork. Again, it is important to distinguish between a movement along a supply curve and a shift of the supply curve. When the price of pork changes, the change in the quantity supplied reflects a movement along the supply curve. When costs, government rules, or other variables that affect supply change, the entire supply curve shifts. SUMMING SUPPLY FUNCTIONS The total supply curve shows the total quantity produced by all suppliers at each pos- sible price. For example, the total supply of rice in Japan is the sum of the domestic and the foreign supply curves of rice. Suppose that the domestic supply curve (panel a) and foreign supply curve (panel b) of rice in Japan are as Figure 2.5 shows. The total supply curve, S in panel c, is the hor- izontal sum of the Japanese domestic supply curve, Sd, and the foreign supply curve, Sf. In the figure, the Japanese and foreign supplies are zero at any price equal to or less than p, so the total supply is zero. At prices above p, the Japanese and foreign supplies are positive, so the total supply is positive. For example, when the price is p*, the quantity supplied by Japanese firms is (panel a), the quantity supplied by foreign firms is (panel b), and the total quantity supplied is (panel c). Because the total supply curve is the horizontal sum of the domestic and foreign supply curves, the total supply curve is flatter than either of the other two supply curves. Q * = Q*d + Q*f Q*fQ*d M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 20 Market Equilibrium 21 p, P ric e pe r to n p, P ric e pe r to n p, P ric e pe r to n Qd * S d S f (ban) Qf * Q = Qd * Q* = Qd * + Qf * Qd, Tons per year Qf , Tons per year Q, Tons per year (a) Japanese Domestic Supply (b) Foreign Supply (c) Total Supply p* p* p* – S (ban) – S (no ban)S f (no ban) p – p – p – Figure 2.5 Total Supply: The Sum of Domestic and Foreign Supply. If foreigners may sell their rice in Japan, the total Japanese supply of rice, S, is the hori- zontal sum of the domestic Japanese supply, Sd, and the imported foreign supply, Sƒ. With a ban on foreign imports, the foreign supply curve, is zero at every price, so the total supply curve, is the same as the domestic supply curve, Sd. S, Sƒ, EFFECTS OF GOVERNMENT IMPORT POLICIES ON SUPPLY CURVES We can use this approach for deriving the total supply curve to analyze the effect of government policies on the total supply curve. Traditionally, the Japanese government has banned the importation of foreign rice. We want to determine how much less rice is supplied at any given price to the Japanese market because of this ban. Without a ban, the foreign supply curve is Sƒ in panel b of Figure 2.5. A ban on imports eliminates the foreign supply, so the foreign supply curve after the ban is imposed, is a vertical line at Qƒ
0. The import ban has no effect on the domestic supply curve, Sd, so the supply curve is the same as in panel a. Because the foreign supply with a ban, is zero at every price, the total supply with a ban, in panel c is the same as the Japanese domestic supply, Sd, at any given price. The total supply curve under the ban lies to the left of the total supply curve without a ban, S. Thus the effect of the import ban is to rotate the total supply curve toward the vertical axis. The limit that a government sets on the quantity of a foreign-produced good that may be imported is called a quota. By absolutely banning the importation of rice, the Japanese government sets a quota of zero on rice imports. Sometimes governments set positive quotas, > 0. The foreign firms may supply as much as they want, Qƒ, as long as they supply no more than the quota: \ 2.3 Market Equilibrium The supply and demand curves determine the price and quantity at which goods and services are bought and sold. The demand curve shows the quantities that consumers Qƒ … Q. Q S, Sƒ, Sƒ, M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 21 24 CHAPTER 2 Supply and Demand Some consumers are lucky enough to be able to buy the pork at $2.65. Other con- sumers cannot find anyone who is willing to sell them pork at that price. What can they do? Some frustrated consumers may offer to pay suppliers more than $2.65. Alternatively, suppliers, noticing these disappointed consumers, may raise their prices. Such actions by consumers and producers cause the market price to rise. As the price rises, the quantity that firms want to supply increases and the quantity that consumers want to buy decreases. This upward pressure on price continues until it reaches the equilibrium price, $3.30, where there is no excess demand. If, instead, price is initially above the equilibrium level, suppliers want to sell more than consumers want to buy. For example, at a price of pork of $3.95, suppliers want to sell 246 million kg per year but consumers want to buy only 207 million, as the figure shows. At $3.95, the market is in disequilibrium. There is an excess supply—the amount by which the quantity supplied is greater than the quantity demanded at a specified price—of 39 (= 246  207) at a price of $3.95. Not all firms can sell as much as they want. Rather than incur storage costs (and possibly have their unsold pork spoil), firms lower the price to attract additional customers. As long as price remains above the equilibrium price, some firms have unsold pork and want to lower the price further. The price falls until it reaches the equilibrium level, $3.30, where there is no excess supply and hence no more pressure to lower the price further. In summary, at any price other than the equilibrium price, either consumers or sup- pliers are unable to trade as much as they want. These disappointed people act to change the price, driving the price to the equilibrium level. The equilibrium price is called the market clearing price because it removes from the market all frustrated buy- ers and sellers: There is no excess demand or excess supply at the equilibrium price. 2.4 Shocking the Equilibrium: Comparative Statics If the variables we hold constant in the demand and supply functions do not change, an equilibrium can persist indefinitely because none of the participants applies pres- sure to change the price. However, the equilibrium changes if a shock occurs such that one of the variables we were holding constant changes, causing a shift in either the demand curve or the supply curve. Comparative statics is the method that economists use to analyze how variables controlled by consumers and firms—here, price and quantity—react to a change in environmental variables (also called exogenous variables), such as prices of substitutes and complements, income, and prices of inputs. The term comparative statics literally refers to comparing a static equilibrium—an equilibrium at a point in time—from before the change to a static equilibrium after the change. (In contrast, economists may examine a dynamic model, in which the dynamic equilibrium adjusts over time.) COMPARATIVE STATICS WITH DISCRETE (RELATIVELY LARGE) CHANGES We can determine the comparative statics properties of an equilibrium by examining the effects of a discrete (relatively large) change in one environmental variable. We can do so by solving for the before- and after-equilibria and comparing them using math- ematics or a graph. We illustrate this approach using our beloved pork example. Suppose all the environmental variables remain constant except the price of hogs, M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 24 Shocking the Equilibrium: Comparative Statics 25 S1 S2 Q, Million kg of pork per year 3.30 3.55 e1 e2 D p, $ p er k g 1760 220205 215 Excess demand = 15 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Figure 2.7 The Equilibrium Effect of a Shift of the Supply Curve. A 25¢ increase in the price of hogs causes the supply curve for processed pork to shift to the left from S1 to S2, driving the market equilibrium from e1 to e2, and the market equilibrium price from $3.30 to $3.55. which increases by 25¢. It is now more expensive to produce pork because the price of a major input, hogs, has increased. Because the price of hogs is not an argument to the demand function—a change in the price of an input does not affect consumers’ desires—the demand curve does not shift. As we have already seen, the increase in the price of hogs causes the supply curve for pork to shift 15 units to the left from S1 to S2 in Figure 2.7. At the original equilibrium price of pork, $3.30, consumers still want 220 units, but suppliers are now willing to supply only 205, so there is excess demand of 15, as panel a shows. Market pressure forces the price of pork upward until it reaches a new equi- librium at e2, where the new equilibrium price is $3.55 and the new equilibrium quan- tity is 215. Thus the increase in the price of hogs causes the equilibrium price to rise by 25¢ a pound but the equilibrium quantity to fall by 15 units. Here the increase in the price of a factor causes a shift of the supply curve and a movement along the demand curve. We can derive the same result by using equations to solve for the equilibrium before the change and after the discrete change in the price of hogs and by comparing the two equations. We have already solved for the original equilibrium, e1, by setting quantity in the demand function 2.3 equal to the quantity in the supply function 2.7. We obtain the new equilibrium, e2, by equating the quantity in the demand function 2.3 to that of the new supply function 2.8: 286  20p
73 + 40p. Simplifying this expression, we find that the new equilibrium price is p2
$3.55. Substituting that price into either the demand or the supply function, we learn that the new equilibrium quantity is Q2
215, M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 25 26 CHAPTER 2 Supply and Demand 9The chain rule is a formula for the derivative of the composite of two functions, such as f(g(x)). According to this rule, df/dx
(df/dg)(dg/dx). See the Calculus Appendix. as panel a shows. Thus both methods show that an increase in the price of hogs causes the equilibrium price to rise and the equilibrium quantity to fall. COMPARATIVE STATICS WITH SMALL CHANGES Alternatively, we can use calculus to determine the effect of a small change (as opposed to the discrete change we just used) in one environmental variable, holding the other such variables constant. Until now, we have used calculus to examine how an argument of a demand function affects the quantity demanded or how an argument of a supply function affects the quantity supplied. Now, however, we want to know how an envi- ronmental variable affects the equilibrium price and quantity that are determined by the intersection of the supply and demand curves. Our first step is to characterize the equilibrium values as functions of the relevant environmental variables. Suppose that we hold constant all the environmental vari- ables that affect demand so that the demand function is (2.9) One environmental variable, a, in the supply function changes, causing the supply curve to shift. We write the supply function as (2.10) As before, we determine the equilibrium price by equating the quantities, Q, in Equations 2.9 and 2.10: (2.11) The equilibrium equation 2.11 is an example of an identity. As a changes, p changes so that this equation continues to hold—the market remains in equilibrium. Thus based on this equation, we can write the equilibrium price as an implicit function of the envi- ronmental variable: p
p(a). That is, we can write the equilibrium condition 2.11 as (2.12) We can characterize how the equilibrium price changes with a by differentiating the equilibrium condition 2.12 with respect to a using the chain rule at the original equilibrium,9 (2.13) Using algebra, we can rearrange Equation 2.13 as (2.14) where we suppress the arguments of the functions for notational simplicity. Equation 2.14 shows the derivative of p(a) with respect to a. dp da = 0S 0a dD dp - 0S 0p , dD(p(a)) dp dp da = 0S(p(a), a) 0p dp da + 0S(p(a), a) 0a . D(p(a)) = S(p(a), a). D(p) = S(p, a). Q = S(p, a). Q = D(p). M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:02 PM Page 26 Elasticities 29 10Economists use the elasticity rather than the slope, 0z/0x, as a summary statistic because the elasticity is a pure number, whereas the slope depends on the units of measurement. For exam- ple, if x is a price measured in pennies and we switch to measuring price using dollars, the slope changes, but the elasticity remains unchanged. so the same amount is demanded no matter what the price is, as in vertical demand curve D2 in panel b. A 25¢ increase in the price of hogs again shifts the supply curve from S1 to S2. Equilibrium quantity does not change, but the price consumers pay rises by 37.5¢ to $3.675. Thus the amount consumers spend rises by more when the demand curve is vertical instead of downward sloping. Now suppose that consumers are very sensitive to price, as in the horizontal demand curve, D3, in panel c. Consumers will buy virtually unlimited quantities of pork at $3.30 per kg (or less), but if the price rises even slightly, they will stop buying pork. Here an increase in the price of hogs has no effect on the price consumers pay; however, the equi- librium quantity drops substantially to 205 million kg per year. Thus how much the equilibrium quantity falls and how much the equilibrium price of processed pork rises when the price of hogs increases depend on the shape of the demand curve. 2.5 Elasticities It is convenient to be able to summarize the responsiveness of one variable to a change in another variable using a summary statistic. In our last example, we wanted to know whether an increase in the price causes a large or a small change in the quantity demanded (that is, whether the demand curve is relatively vertical or relatively hori- zontal at the current price). We can use summary statistics of the responsiveness of the quantity demanded and the quantity supplied to determine comparative statics prop- erties of the equilibrium. Often, we have reasonable estimates of these summary statis- tics and can use them to predict what will happen to the equilibrium in a market—that is, to make comparative statistics predictions. Later in this chapter, we will examine how the government can use these summary measures for demand and supply to pre- dict, before it institutes the tax, the effect of a new sales tax on the equilibrium price, firms’ revenues, and tax receipts. Suppose that a variable z (for example, the quantity demanded or the quantity sup- plied) is a function of a variable x (say, the price of z) and possibly other variables such as y: z
f(x, y). For example, f could be the demand function, where z is the quantity demanded, x is the price, and y is income. We want a summary statistic that describes how much z changes as x changes, holding y constant. An elasticity is the percentage change in one variable (here, z) in response to a given percentage change in another variable (here, y), holding other relevant variables (here, y) constant. The elasticity, E, of z with respect to x is (2.20) where ∆z is the change in z, so ∆z/z is the percentage change in z. If z changes by 3% when x changes by 1%, then the elasticity E is 3. Thus the elasticity is a pure number (it has no units of measure).10 As ∆x goes to zero, ∆z/∆x goes to the partial derivative 0z/0x. Economists usually calculate elasticities only at this limit—that is, for infinitesi- mal changes in x. E = percentage change in z percentage change in x = ¢z/z ¢x/x = 0z 0x x z , M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 29 30 CHAPTER 2 Supply and Demand Willingness to Surf Do you surf the Net for hours and download billions of bits of music? Would you cut back if the price of your Internet service increased? At what price would you give up using the Internet? Varian (2002) estimated demand curves for connection time by people at a uni- versity who paid for access by the minute. He found that the price elasticity of demand was -2.0 for those who used a 128 kilobits per second (Kbps) service and -2.9 for people who connected at 64 Kbps. That is, a 1% increase in the price per minute reduced the connection time used by those with high-speed access by 2% but decreased the connection time by nearly 3% for those with slow phone-line access. Thus high-speed users are less sensitive to connection prices than slow-speed users. APPLICATION DEMAND ELASTICITY The price elasticity of demand (or simply the demand elasticity or elasticity of demand) is the percentage change in the quantity demanded, Q, in response to a given percent- age change in the price, p, at a particular point on the demand curve. The price elas- ticity of demand (represented by e, the Greek letter epsilon) is (2.21) where 0Q/0p is the partial derivative of the demand function with respect to p (that is, holding constant other variables that affect the quantity demanded). For example, if e
-2, then a 1% increase in the price results in a 2% decrease in the quantity demanded. We can use Equation 2.12 to calculate the elasticity of demand for a linear demand function (holding fixed other variables that affect demand), where a is the quantity demanded when price is zero, Q
a  (b  0)
a, and -b is the ratio of the fall in quantity to the rise in price: the derivative dQ/dp. The elasticity of demand is (2.22) For the linear demand function for pork, Q
a
bp
286  20p, at the initial equilibrium where p
$3.30 and Q
220, the elasticity of demand is The negative sign on the elasticity of demand of pork illustrates the Law of Demand: Less quantity is demanded as the price rises. The elasticity of demand con- cisely answers the question “How much does quantity demanded fall in response to a 1% increase in price?” A 1% increase in price leads to an e% change in the quantity demanded. At the equilibrium, a 1% increase in the price of pork leads to a -0.3% fall in the quantity of pork demanded: A price increase causes a less than proportionate fall in the quantity of pork demanded. e = b p Q = -20 * 3.30 220 = -0.3. e = dQ dp p Q = -b p Q . Q = a - bp, e = percentage change in quantity demanded percentage change in price = ¢Q / Q ¢p / p = 0Q 0p p Q , M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 30 Elasticities 31 p, $ p er k g a /2 = 143a/5 = 57.2 D a = 286220 Q, Million kg of pork per year 0 11.44 a/b = 14.30 3.30 a/(2b) = 7.15 Elastic: ε < –1 ε = –4 Unitary: ε = –1 ε = –0.3 Inelastic: 0 > ε > –1 Perfectly inelastic Perfectly elastic⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Figure 2.9 Elasticity Along the Linear Pork Demand Curve. With a linear demand curve such as the pork demand curve, the higher the price, the more elastic the demand curve (e is larger in absolute value—a larger negative number). The demand curve is perfectly inelastic (e
0) where the demand curve hits the horizontal axis, is perfectly elastic where the demand curve hits the vertical axis, and has unitary elasticity (e
-1) at the midpoint of the demand curve. Some recent studies have found that residential users who pay a flat rate (no per-minute charge) for service have a very inelastic demand for dial-up service. That is, few dial-up users will give up their service if the flat fee rises. Residential customers are more sensitive to the flat-fee price of broadband service, with elas- ticities ranging from -0.75 to -1.5 (Duffy-Deno, 2003). That is, if the price of broadband service increases 10%, between 7.5% and 15% fewer households will use a broadband service. Elasticities Along the Demand Curve. The elasticity of demand varies along most demand curves. The elasticity of demand is different at every point along a downward- sloping linear demand curve; however, the elasticities are constant along horizontal, vertical, and log-linear demand curves. On strictly downward-sloping linear demand curves—those that are neither verti- cal nor horizontal—the elasticity of demand is a more negative number the higher the price. Consequently, even though the slope of the linear demand curve is constant, the elasticity varies along the curve. A 1% increase in price causes a larger percentage fall in quantity near the top (left) of the demand curve than near the bottom (right). Where a linear demand curve hits the quantity axis (p
0 and Q
a), the elastic- ity of demand is e
-b(0/a)
0, according to Equation 2.22. The linear pork demand curve in Figure 2.9 illustrates this pattern. Where the price is zero, a 1% increase in price does not raise the price, so quantity does not change. At a point where the elasticity of demand is zero, the demand curve is said to be perfectly inelastic. As a phys- ical analogy, if you try to stretch an inelastic steel rod, the length does not change. The change in the price is the force pulling at demand; if the quantity demanded does not change in response to this pulling, the demand curve is perfectly inelastic. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 31 34 CHAPTER 2 Supply and Demand Other Demand Elasticities. We refer to the price elasticity of demand as the elasticity of demand. However, there are other demand elasticities that show how the quantity demanded changes in response to changes in variables other than price that affect the quantity demanded. Two such demand elasticities are the income elasticity of demand and the cross-price elasticity of demand. As income increases, the demand curve shifts. If the demand curve shifts to the right, a larger quantity is demanded at any given price. If instead the demand curve shifts to the left, a smaller quantity is demanded at any given price. We can measure how sensitive the quantity demanded at a given price is to income by using the income elasticity of demand (or income elasticity), which is the percent- age change in the quantity demanded in response to a given percentage change in income, Y. The income elasticity of demand is where j is the Greek letter xi. If the quantity demanded increases as income rises, the income elasticity of demand is positive. If the quantity demanded does not change as income rises, the income elasticity is zero. Finally, if the quantity demanded falls as income rises, the income elasticity is negative. By partially differentiating the pork demand function 2.2, Q
171  20p + 20pb + 3pc + 2Y, with respect to Y, we find that 0Q/0Y
2, so the income elasticity of demand for pork is j
2Y/Q. At our original equilibrium, quantity Q
220 and income Y
12.5, so the income elasticity is 2  (12.5/220) ≈ 0.114, or about one-ninth. The posi- tive income elasticity shows that an increase in income causes the pork demand curve to shift to the right. Income elasticities play an important role in our analysis of consumer behavior in Chapter 5. Typically, goods that consumers view as necessities, such as food, have income elasticities near zero. Goods that they consider to be luxuries generally have income elasticities greater than one. j = percentage change in quantity demanded percentage change in income = ¢Q / Q ¢Y / Y = 0Q 0Y Y Q , Substituting that expression into the elasticity definition, we learn that the elas- ticity is Because the elasticity is a constant that does not depend on the particular value of p, it is the same at every point along the demand curve. 2. Differentiate the log-linear demand curve to determine dQ/dp, and substitute that expression into the definition of the elasticity of demand: Differentiating the log-linear demand curve, ln Q
ln A + e ln p, with respect to p, we find that (dQ/dp)/Q
e/p. Multiplying both sides of this equation by p, we again discover that the elasticity is constant: dQ dp p Q = e Q p p Q = e. dQ dp p Q = eApe-1 p Q = eApe-1 p Ape = e. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 34 Elasticities 35 Substitution May Save Endangered Species One reason that many species—including tigers, rhinoceroses, pinnipeds, green turtles, geckos, sea horses, pipefish, and sea cucumbers—are endangered, threat- ened, or vulnerable to extinction is that certain of their body parts are used as aphrodisiacs in traditional Chinese medicine. Is it possible that consumers will switch from such potions to Viagra, a less expensive and almost certainly more effective alternative treatment, and thereby help save these endangered species? We cannot directly calculate the cross-price elasticity of demand between Viagra and these endangered species because their trade is illicit and not reported. However, in Asia, harp seal and hooded seal genitalia are also used as aphrodisiacs, and they may be legally traded. Before 1998, Viagra was unavailable (effectively, it had an infinite price). When it became available at about $15 to $20 Canadian per pill, the demand curve for seal sex organs shifted substantially to the left. According to von Hippel and von Hippel (2002, 2004), 30,000 to 50,000 seal organs were sold at between $70 and $100 Canadian in the years just before 1998. In 1998, the price per unit fell to between $15 and $20, and only 20,000 organs were sold. By 1999–2000 (and thereafter), virtually none were sold. This evidence suggests a strong willingness to substitute at current prices: a positive cross-price elasticity between seal organs and the price of Viagra. Thus Viagra can perhaps save more than marriages. APPLICATION 13Jargon alert: Graduate-level textbooks generally call these goods gross substitutes (and the goods in the previous example would be called gross complements). The cross-price elasticity of demand is the percentage change in the quantity demanded in response to a given percentage change in the price of another good, po. The cross-price elasticity may be calculated as When the cross-price elasticity is negative, the goods are complements. If the cross- price elasticity is negative, people buy less of one good when the price of the other, sec- ond good increases: The demand curve for the first good shifts to the left. For example, if people like cream in their coffee, as the price of cream rises, they consume less cof- fee, so the cross-price elasticity of the quantity of coffee with respect to the price of cream is negative. If the cross-price elasticity is positive, the goods are substitutes.13 As the price of the second good increases, people buy more of the first good. For example, the quantity demanded of pork increases when the price of beef, pb, rises. By partially differentiat- ing the pork demand function 2.2, Q
171  20p + 20pb + 3pc + 2Y, with respect to the price of beef, we find that 0Q/0pb
20. As a result, the cross-price elasticity between the price of beef and the quantity of pork is 20pb/Q. At the original equilib- rium where Q
220 million kg per year, and pb
$4 per kg, the cross-price elasticity is 20  (4/220) ≈ 0.364. As the price of beef rises by 1%, the quantity of pork demanded rises by a little more than one-third of 1%. percentage change in quantity demanded percentage change in price of another good = ¢Q / Q ¢po / po = 0Q 0po po Q . M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 35 36 CHAPTER 2 Supply and Demand SUPPLY ELASTICITY Just as we can use the elasticity of demand to summarize information about the responsiveness of the quantity demanded to price or other variables, we can use the elasticity of supply to summarize information about the responsiveness of the quan- tity demanded. The price elasticity of supply (or supply elasticity) is the percentage change in the quantity supplied in response to a given percentage change in the price. The price elasticity of supply (h, the Greek letter eta) is (2.25) where Q is the quantity supplied. If h
2, a 1% increase in price leads to a 2% increase in the quantity supplied. The definition of the elasticity of supply, Equation 2.25, is very similar to the defi- nition of the elasticity of demand, Equation 2.21. The key distinction is that the elas- ticity of supply describes the movement along the supply curve as price changes, whereas the elasticity of demand describes the movement along the demand curve as price changes. That is, in the numerator, supply elasticity depends on the percentage change in the quantity supplied, whereas demand elasticity depends on the percentage change in the quantity demanded. If the supply curve is upward sloping, 0p/0Q > 0, the supply elasticity is positive: h > 0. If the supply curve slopes downward, the supply elasticity is negative: h < 0. For the pork supply function 2.7, Q
88 + 40p, the elasticity of supply of pork at the orig- inal equilibrium, where p
$3.30 and Q
220, is As the price of pork increases by 1%, the quantity supplied rises by slightly less than two-thirds of a percent. The elasticity of supply varies along an upward-sloping supply curve. For example, because the elasticity of supply for the pork is h
40p/Q, as the ratio p/Q rises, the supply elasticity rises. At a point on a supply curve where the elasticity of supply is h
0, we say that the supply curve is perfectly inelastic: The supply does not change as the price rises. If 0 < h < 1, the supply curve is inelastic (but not perfectly inelastic): A 1% increase in price causes a less than 1% rise in the quantity supplied. If h > 1, the supply curve is elastic. If h is infinite, the supply curve is perfectly elastic. The supply elasticity does not vary along constant-elasticity supply functions, which are exponential or (equivalently) log-linear: Q
Bph or ln Q
ln B + h ln p. If h is a positive, finite number, the constant-elasticity supply curve starts at the origin, as Figure 2.11 shows. Two extreme examples of both constant-elasticity of supply curves and linear supply curves are the vertical supply curve and the horizontal supply curve. A supply curve that is vertical at a quantity, Q*, is perfectly inelastic. No matter what the price is, firms supply Q*. An example of inelastic supply is a perishable item such as already picked fresh fruit. If the perishable good is not sold, it quickly becomes worthless. Thus the seller will accept any market price for the good. A supply curve that is horizontal at a price, p*, is perfectly elastic. Firms supply as much as the market wants—a potentially unlimited amount—if the price is p* or h = dQ dp p Q = 40 * 3.30 220 = 0.6. h = percentage change in quantity supplied percentage change in price = ¢Q / Q ¢p / p = 0Q 0p p Q , M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 36 Elasticities 39 What would be the effect of ANWR production on the world equilibrium price of oil given that e
0.4, h
0.3, the pre-ANWR daily world production of oil is Q1
82 million barrels per day, the pre-ANWR world price is p1
$50 per barrel, and daily ANWR pro- duction would be 0.8 million barrels per day? For specificity, assume that the supply and demand curves are linear and that the introduction of ANWR oil would cause a parallel shift in the world supply curve to the right by 0.8 million barrels per day. Answer 1. Determine the long-run linear demand function that is consistent with pre- ANWR world output and price: At the original equilibrium, e1 in the figure, p1
$50 and Q1
82. There the elasticity of demand is e
(dQ/dp)(p1/Q1)
(dQ/dp)(50/82)
-0.4. Using algebra, we find that dQ/dp equals -0.4(82/50)
-0.656, which is the inverse of the slope of the demand curve, D, in the figure. Knowing this slope and that demand equals 82 at $50 per barrel, we can solve for the intercept, because the quantity demanded rises by 0.656 for each dollar by which the price falls. The demand when the price is zero is 82 + (0.656  50)
114.8. Thus the equation for the demand curve is Q
114.8  0.656p. 2. Determine the long-run linear supply function that is consistent with pre-ANWR world output and price: Where S1 intercepts D at the original equilibrium, e1, the elasticity of supply is h
(dQ/dp)(p1/Q1)
(dQ/dp)(50/82)
0.3. Solving, we find that dQ/dp
0.3(82/50)
0.492. Because the quantity supplied falls by SOLVED PROBLEM 2.3 A number of studies estimate that the long-run elasticity of demand, e, for oil is about 0.4 and the long-run supply elas- ticity,h, is about 0.3. Analysts agree less about how much ANWR oil will be produced. The Department of Energy’s Energy Information Service (EIS) predicts that production from the ANWR would average about 800,000 barrels per day (the EIS estimates that the ANWR’s oil would increase the volume of pro- duction by about 0.7% in 2020). That production would be about 1% of the worldwide oil production, which averaged about 82 million barrels per day in 2004 (and was only slightly higher in 2005 and 2006). A report of the U.S. Department of Energy predicted that ANWR drilling could lower the price of oil by about 50¢ a barrel or 1%, given that the price of a barrel of oil was slightly above $50 at the beginning of 2007. Severin Borenstein, an economist who is the director of the U.C. Energy Institute, concluded that the ANWR might reduce oil prices by up to a few percentage points but that “drilling in ANWR will never noticeably affect gasoline prices.” In the following solved problem, we can make our own calculations of the price effect of drilling in the ANWR. Here and in many of the solved problems in this book, you are asked to determine how a change in a variable or policy affects one or more variables. In this problem, the policy changes from not allowing to permitting drilling in the ANWR, which affects the world’s equilibrium price of oil. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 39 40 CHAPTER 2 Supply and Demand 0.492 for each dollar by which the price drops, the quantity supplied when the price is zero is 82  (0.492  50)
57.4. Thus the equation for the pre-ANWR supply curve, S1 in the figure, is Q
57.4 + 0.492p. 3. Determine the post-ANWR long-run linear supply function: The oil pumped from the ANWR would cause a parallel shift in the supply curve, moving S1 to the right by 0.8 to S2. That is, the slope remains the same, but the intercept on the quantity axis increases by 0.8. Thus the supply function for S2 is Q
58.2 + 0.492p. 4. Use the demand curve and the post-ANWR supply function to calculate the new equilibrium price and quantity: The new equilibrium, e2, occurs where S2 inter- sects D. Setting the right-hand sides of the demand function and the post- ANWR supply function equal, we obtain an expression in the new price, p2: We can solve this expression for the new equilibrium price: p2 ≈ $49.30. That is, the price drops about 70¢, or approximately 1.4%. If we substitute this new price into either the demand curve or the post-ANWR supply curve, we find that the new equilibrium quantity is 82.46 million barrels per day. That is, equilibrium output rises by 0.46 million barrels per day (0.56%), which is only a little more than half of the predicted daily ANWR supply, because other suppliers will decrease their output slightly in response to the lower price. Comment: Our estimate of a small drop in the world oil price if ANWR oil is sold would not change substantially if our estimates of the elasticities of supply and demand were moderately larger or smaller. The main reason for this result is that the ANWR output would be a very small portion of worldwide supply—the new supply curve is only slightly to the right of the initial supply curve. Thus drilling in the ANWR cannot insulate the American market from international events that roil the oil market. A new war in the Persian Gulf could shift the worldwide supply curve to the left by 3 million barrels a day or more (nearly four times the ANWR production). Such a shock would cause the price of oil to soar whether or not we drill in the ANWR. 58.2 + 0.492p2 = 114.8 - 0.656p2. p, $ p er b ar re l Q, Millions of barrels of oil per day 50 49.30 S1 S 2 e1 e2 D 57.4 8258.2 82.46 114.8 M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 40 Effects of a Sales Tax 41 15For specificity, we assume that the price firms receive is p
(1  a)p*, where p* is the price consumers pay and a is the ad valorem tax rate on the price consumers pay. However, many governments (including U.S. and Japanese governments) set the ad valorem sales tax, b, as an amount added to the price sellers charge, so consumers pay p*
(1  b)p. By setting a and b appropriately, the taxes are equivalent. Here p
p*/(1  b), so (1  a)
1/(1  b). For example, if b
1/3, then a
1/4. 2.6 Effects of a Sales Tax How much a tax affects the equilibrium price and quantity and how much of the tax falls on consumers depends on the elasticities of demand and supply. Knowing only the elasticities of demand and supply, we can make accurate predictions about the effects of a new tax and determine how much of the tax falls on consumers. In this section, we examine three questions about the effects of a sales tax: 1. What effect does a sales tax have on equilibrium prices and quantity? 2. Is it true, as many people claim, that taxes assessed on producers are passed along to consumers? That is, do consumers pay for the entire tax, or do producers pay part of it? 3. Do the equilibrium price and quantity depend on whether the tax is assessed on consumers or on producers? TWO TYPES OF SALES TAXES Governments use two types of sales taxes. The most common sales tax is called an ad valorem tax by economists and the sales tax by real people. For every dollar the con- sumer spends, the government keeps a fraction, a, which is the ad valorem tax rate. Japan’s national sales tax is a
5%. If a consumer in Japan buys a Nintendo Wii for $500, the government collects a  $500
5%  $500
$25 in taxes, and the seller receives (1  a)  $500
$475.15 The other type of sales tax is a specific or unit tax, where a specified dollar amount, t, is collected per unit of output. The federal government collects t
18.4¢ on each gallon of gas sold in the United States. EQUILIBRIUM EFFECTS OF A SPECIFIC TAX To answer our three questions, we must extend the standard supply-and-demand anal- ysis to take taxes into account. Let’s start by assuming that the specific tax is assessed on firms at the time of sale. If the consumer pays p for a good, the government takes t and the seller receives p  t. Suppose that the government collects a specific tax of t
$1.05 per kg of processed pork from pork producers. Because of the tax, suppliers keep only p  t of price p that consumers pay. Thus at every possible price paid by consumers, firms are willing to supply less than when they received the full amount consumers paid. Before the tax, firms were willing to supply 206 million kg per year at a price of $2.95 as the pretax supply curve S1 in Figure 2.12 shows. After the tax, firms receive only $1.90 if con- sumers pay $2.95, so they are not willing to supply 206. For firms to be willing to sup- ply 206, they must receive $2.95 after the tax, so consumers must pay $4. As a result, the after-tax supply curve, S2, is t
$1.05 above the original supply curve S1 at every quantity, as the figure shows. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 41 44 CHAPTER 2 Supply and Demand 17See www.aw-bc.com/perloff, Chapter 2, “Incidence of a Tax on Restaurant Meals,” for another application. of the tax on firms is the amount by which the price to firms falls: 1 - dp/dt. The sum of the incidence of the tax to consumers and firms is dp/dt + 1 - dp/dt
1. That is, the increase in price to consumers plus the drop in price to firms equals the tax. The demand elasticity for pork is e
-0.3 and the supply elasticity is h
0.6, so the incidence of a specific tax on consumers is dp/dt
h/(h e)
0.6/[0.6  (-0.3)]
0.6/0.9
2/3, and the incidence of the tax on firms is 1  2/3
1/3. Thus a discrete change in the tax of ∆t
t  0
$1.05 causes the price that con- sumers pay to rise by ∆p
p2  p1
$4.00  $3.30
[h/(h e)]∆t
2/3  $1.05
70¢ and the price to firms to fall by 1/3  $1.05
35¢, as Figure 2.2 shows. The sum of the increase to consumers plus the loss to firms is 70¢ + 35¢
$1.05
t. Equation 2.28 shows that, for a given supply elasticity, the more elastic the demand, the less the equilibrium price rises when a tax is imposed. Similarly, for a given demand elasticity, the smaller the supply elasticity, the smaller the increase in the equilibrium price that consumers pay when a tax is imposed. For example, in the pork example, if the supply elasticity were h
0 (a perfectly inelastic vertical supply curve), dp/dt
0/[0  (-0.3)]
0, so none of the incidence of the tax falls on consumers, and the entire incidence of the tax falls on firms.17 THE SAME EQUILIBRIUM NO MATTER WHO IS TAXED Our third question is, “Does the equilibrium or the incidence of the tax depend on whether the tax is collected from producers or consumers?” Surprisingly, in the supply- and-demand model, the equilibrium and the incidence of the tax are the same regard- less of whether the government collects the tax from producers or from consumers. We’ve already seen that firms are able to pass on some or all of the tax collected from them to consumers. We now show that, if the tax is collected from consumers, they can pass the producers’ share back to the firms. Suppose the specific tax t
$1.05 on pork is collected from consumers rather than from producers. Because the government takes t from each p that consumers spend, producers receive only p  t. Thus the demand curve as seen by firms shifts downward by $1.05 from D to Ds in Figure 2.13. The intersection of D2 and S determines the after-tax equilibrium, where the equi- librium quantity is Q2 and the price received by producers is p2  t. The price paid by consumers, p2 (on the original demand curve D at Q2), is t above the price received by producers. We place the after-tax equilibrium, e2, bullet on the market demand D in Figure 2.11 to show that it is the same as the e2 in Figure 2.12. Comparing Figure 2.13 to Figure 2.12, we see that the after-tax equilibrium is the same regardless of whether the tax is imposed on consumers or producers. The price to consumers rises by the same amount, ∆p
70¢, and the incidence of the tax, ∆p/∆t
2/3, is the same. A specific tax, regardless of whether the tax is collected from consumers or produc- ers, creates a wedge equal to the per-unit tax of t between the price consumers pay, p, and the price producers receive, p  t. In short, regardless of whether firms or con- sumers pay the tax to the government, you can solve tax problems by shifting the sup- ply curve, shifting the demand curve, or inserting a wedge between the supply and demand curves. All three approaches give the same answer. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 44 Effects of a Sales Tax 45 p, $ p er k g Q2 = 206 Q1 = 220176 T = $216.3 million Q, Million kg of pork per year0 p2 = 4.00 p1 = 3.30 p2 – τ = 2.95 e1 e2 Da Ds S D Figure 2.13 Effects of a Specific Tax and of an Ad Valorem Tax on Consumers. Without a tax, the demand curve is D and the supply curve is S. A specific tax of t
$1.05 per kg collected from consumers shifts the demand curve to Ds, which is parallel to D. The new equilibrium is e2 on the original demand curve D. If instead an ad valorem tax of a
26.25% is imposed, the demand curve facing firms is Da. The gap between D and Da, the per-unit tax, is larger at higher prices. The after-tax equilibrium is the same with both of these taxes. THE SIMILAR EFFECTS OF AD VALOREM AND SPECIFIC TAXES In contrast to specific sales taxes, which are applied to relatively few goods, govern- ments levy ad valorem taxes on a wide variety of goods. Most states apply an ad val- orem sales tax to most goods and services, exempting only a few staples such as food and medicine. There are 6,400 different ad valorem sales tax rates across the United States, which can go as high as 8.5% (Besley and Rosen, 1999). Suppose that the government imposes an ad valorem tax of a, instead of a specific tax, on the price that consumers pay for processed pork. We already know that the equi- librium price is $4 with a specific tax of $1.05 per kg. At that price, an ad valorem tax of a
$1.05/$4
26.25% raises the same amount of tax per unit as a $1.05 specific tax. It is usually easiest to analyze the effects of an ad valorem tax by shifting the demand curve. Figure 2.13 shows how an ad valorem tax shifts the processed pork demand curve. The ad valorem tax shifts the demand curve to Da. At any given price p, the gap between D and Da is ap, which is greater at high prices than at low prices. The gap is $1.05 (= 0.2625  $4) per unit when the price is $4, and $2.10 when the price is $8. Imposing an ad valorem tax causes the after-tax equilibrium quantity, Q2, to fall below the original quantity, Q1, and the after-tax price, p2, to rise above the original price, p1. The tax collected per unit of output is t
ap2. The incidence of the tax that falls on con- sumers is the change in price, ∆p
(p2  p1), divided by the change in the per-unit tax, ∆t
ap2  0, that is collected, ∆p/(ap2). The incidence of an ad valorem tax is generally shared between consumers and producers. Because the ad valorem tax of a
26.25% has exactly the same impact on the equilibrium pork price and raises the same amount of tax per unit as the $1.05 specific tax, the incidence is the same for both types of taxes. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 45 46 CHAPTER 2 Supply and Demand (As with specific taxes, the incidence of the ad valorem tax depends on the elasticities of supply and demand, but we’ll spare your having to go through that in detail.) 2.7 Quantity Supplied Need Not Equal Quantity Demanded In a supply-and-demand model, the quantity supplied does not necessarily equal the quan- tity demanded because of the way we defined these two concepts. We defined the quantity supplied as the amount firms want to sell at a given price, holding constant other factors that affect supply, such as the price of inputs. We defined the quantity demanded as the quantity that consumers want to buy at a given price, if other factors that affect demand are held constant. The quantity that firms want to sell and the quantity that consumers want to buy at a given price need not equal the actual quantity that is bought and sold. We could have defined the quantity supplied and the quantity demanded so that they must be equal. If we had defined the quantity supplied as the amount firms actually sell at a given price and the quantity demanded as the amount consumers actually buy, supply would have to equal demand in all markets because we defined the quantity demanded and the quantity supplied as the same quantity. It is worth emphasizing this distinction because politicians, pundits, and the press are so often confused on this point. Someone insisting that “demand must equal supply” must be defining demand and supply as the actual quantities sold. Because we define the quan- tities supplied and demanded in terms of people’s wants and not actual quantities bought and sold, the statement that “supply equals demand” is a theory, not merely a definition. This theory says that the quantity supplied equals the quantity demanded at the intersection of the supply and demand curves if the government does not intervene. Not all government interventions prevent markets from clearing by equilibrating the quantity supplied and the quantity demanded. For example, as we’ve seen, a govern- ment tax affects the equilibrium but does not cause a gap between the quantity demanded and the quantity supplied. However, some government policies do more than merely shift the supply or demand curve. For example, a government may control price directly. This policy leads to either excess supply or excess demand if the price the government sets differs from the mar- ket clearing price. We illustrate this result with two types of price control programs. The government may set a price ceiling at so that the price at which goods are sold may be no higher than . When the government sets a price floor at , the price at which goods are sold may not fall below . We can study the effects of such regulations using the supply-and-demand model. Despite the lack of equality between the quantity supplied and the quantity demanded, the supply-and-demand model is useful in analyzing this market because it predicts the excess demand or excess supply that is observed. PRICE CEILING Price ceilings have no effect if they are set above the equilibrium price that would be observed in the absence of the price controls. If the government says that firms may charge no more than per gallon of gas and firms are actually charging p
$1, the gov- ernment’s price control policy is irrelevant. However, if the equilibrium price, p, is above the price ceiling the price that is actually observed in the market is the price ceiling. The U.S. experience with gasoline illustrates the effects of price controls. In the 1970s, OPEC reduced supplies of oil—which is converted into gasoline—to Western countries. As a result, the total supply curve for gasoline in the United States—the hor- p, p = $5 p pp p M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 46 Quantity Supplied Need Not Equal Quantity Demanded 49 20The minimum wage could raise the wage enough that total wage payments, wL, rise despite the fall in demand for labor services. If workers could share the unemployment—everybody works fewer hours than he or she wants—all workers could benefit from the minimum wage. See Problem 40. PRICE FLOOR Governments also commonly use price floors. One of the most important examples of a price floor is the minimum wage in labor markets. The minimum wage law forbids employers from paying less than a minimum wage, w. Minimum wage laws date from 1894 in New Zealand, 1909 in the United Kingdom, and 1912 in Massachusetts. The Fair Labor Standards Act of 1938 set a federal U.S. minimum wage of 25¢. The U.S. federal minimum wage is currently $5.15 an hour, but Congress is debating a substantial increase. The statutory monthly minimum wage ranges from the equivalent of 19 in the Russian Federation to 375 in Portugal, 1,154 in France, and 1,466 in Luxembourg. If the minimum wage binds—exceeds the equi- librium wage, w*—the minimum wage may cause unemployment, which is a persistent excess supply of labor.19 For simplicity, suppose that there is a single labor market in which everyone is paid the same wage. Figure 2.15 shows the supply and demand curves for labor services (hours worked). Firms buy hours of labor service—they hire workers. The quantity measure on the horizontal axis is hours worked per year, and the price measure on the vertical axis is the wage per hour. With no government intervention, the market equilibrium is e, where the wage is w* and the number of hours worked is L*. The minimum wage creates a price floor, a hor- izontal line, at w. At that wage, the quantity demanded falls to Ld and the quantity sup- plied rises to Ls. As a result, there is an excess supply or unemployment of Ls  Ld. The minimum wage prevents market forces from eliminating this excess supply, so it leads to an equilibrium with unemployment. The original 1938 U.S. minimum wage law caused massive unemployment in Puerto Rico (see www.aw-bc.com/perloff, Chapter 2, “Minimum Wage Law in Puerto Rico”). It is ironic that a law designed to help workers by raising their wages may harm some of them by causing them to become unemployed. Such a minimum wage law benefits only those who remain employed.20 In 2005, the government announced new price controls on basic food com- modities. Food shortages grew worse. More than a third of the populace is mal- nourished. Only international food aid has kept millions of these people alive. In 2006, the inflation rate (largely prices of non-controlled goods) exceeded 1,000% and the government introduced a new watchdog agency to monitor prices and incomes—a combination likely to exacerbate the situation. 19The U.S. Department of Labor maintains at its Web site (www.dol.gov) an extensive history of the federal minimum wage law, labor markets, state minimum wage laws, and other information. For European minimum wages, see www.fedee.com/minwage.html. Where the minimum wage applies to only some labor markets (Chapter 10) or where only a single firm hires all the work- ers in a market (Chapter 15), a minimum wage might not cause unemployment. Card and Krueger (1997) provide evidence that recent rises in the minimum wage had negligible (at most) effects on employment in certain low-skill labor markets. M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 49 50 CHAPTER 2 Supply and Demand w , W ag e pe r ho ur Ld L* Ls Minimum wage, price floor S D L, Hours worked per year Unemployment e w * w— ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Figure 2.15 Minimum Wage. In the absence of a minimum wage, the equilibrium wage is w* and the equilibrium number of hours worked is L*. A minimum wage, w, set above w*, leads to unemployment—persistent excess supply—because the quantity demanded, Ld, is less than the quantity supplied, Ls. 2.8 When to Use the Supply-and-Demand Model As we’ve seen, supply-and-demand theory can help us to understand and predict real- world events in many markets. Through Chapter 10, we discuss competitive markets in which the supply-and-demand model is a powerful tool for predicting what will happen to market equilibrium if underlying conditions—tastes, incomes, and prices of inputs—change. The types of markets for which the supply-and-demand model is use- ful are described at length in these chapters, particularly Chapter 8. Briefly, this model is applicable in markets in which: ■ Everyone is a price taker: Because no consumer or firm is a very large part of the market, no one can affect the market price. Easy entry of firms into the market, which leads to a large number of firms, is usually necessary to ensure that firms are price takers. ■ Firms sell identical products: Consumers do not prefer one firm’s good to another. ■ Everyone has full information about the price and quality of goods: Consumers know if a firm is charging a price higher than the price others set, and they know if a firm tries to sell them inferior-quality goods. ■ Costs of trading are low: It is not time consuming, difficult, or expensive for a buyer to find a seller and make a trade or for a seller to find and trade with a buyer. Markets with these properties are called perfectly competitive markets. Where there are many firms and consumers, no single firm or consumer is a large enough part of the market to affect the price. If you stop buying bread or if one of the many thousands of wheat farmers stops selling the wheat used to make the bread, the price of bread will not change. Consumers and firms are price takers: They cannot affect the market price. In contrast, if there is only one seller of a good or service—a monopoly (Chapter 11)— that seller is a price setter and can affect the market price. Because demand curves slope downward, a monopoly can increase the price it receives by reducing the amount of a M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 50 Summary 51 good it supplies. Firms are also price setters in an oligopoly—a market with only a small number of firms—or in markets where they sell differentiated products and a con- sumer prefers one product to another (Chapter 13). In markets with price setters, the market price is usually higher than that predicted by the supply-and-demand model. That doesn’t make the model generally wrong. It means only that the supply-and- demand model does not apply to markets with a small number of sellers or buyers. In such markets, we use other models. If consumers have less information than a firm, the firm can take advantage of con- sumers by selling them inferior-quality goods or by charging a much higher price than that charged by other firms. In such a market, the observed price is usually higher than that predicted by the supply-and-demand model, the market may not exist at all (con- sumers and firms cannot reach agreements), or different firms may charge different prices for the same good (Chapter 18). The supply-and-demand model is also not entirely appropriate in markets in which it is costly to trade with others because the costs of a buyer’s finding a seller or of a seller’s finding a buyer are high. Transaction costs are the expenses of finding a trad- ing partner and making a trade for a good or service other than the price paid for that good or service. These costs include the time and money spent to find someone with whom to trade. When transaction costs are high, trades may not occur; or if they do occur, individual trades may occur at a variety of prices (Chapter 18). Thus the supply-and-demand model is not appropriate in markets in which there are only one or a few sellers (such as electricity), firms produce differentiated products (such as music CDs), consumers know less than sellers about quality or price (such as used cars), or there are high transaction costs (such as nuclear turbine engines). Markets in which the supply-and-demand model has proved useful include agricul- ture, finance, labor, construction, services, wholesale, and retail—markets with many firms and consumers and where firms sell identical products. Summary 1. Demand: The quantity of a good or service demanded by consumers depends on their tastes, the price of a good, the price of goods that are substitutes and complements, con- sumers’ income, information, government regulations, and other factors. The Law of Demand—which is based on observation—says that demand curves slope downward. The higher the price, the less quantity is demanded, hold- ing constant other factors that affect demand. A change in price causes a movement along the demand curve. A change in income, tastes, or another factor that affects demand other than price causes a shift of the demand curve. To get a total demand curve, we horizontally sum the demand curves of individuals or types of consumers or countries. That is, we add the quantities demanded by each individ- ual at a given price to get the total demanded. 2. Supply: The quantity of a good or service supplied by firms depends on the price, the firm’s costs, government regulations, and other factors. The market supply curve need not slope upward but usually does. A change in price causes a movement along the supply curve. A change in the price of an input or government regulation causes a shift of the supply curve. The total supply curve is the hor- izontal sum of the supply curves for individual firms. 3. Market Equilibrium: The intersection of the demand curve and the supply curve determines the equilibrium price and quantity in a market. Market forces—actions of consumers and firms—drive the price and quantity to the equilibrium levels if they are initially too low or too high. 4. Shocking the Equilibrium: Comparative Statics: A change in an underlying factor other than price causes a shift of the supply curve or the demand curve, which alters the equilibrium. Comparative statics is the method that economists use to analyze how variables controlled by consumers and firms—such as price and quantity— react to a change in environmental variables such as prices of substitutes and complements, income, and prices of inputs. 5. Elasticities: An elasticity is the percentage change in a variable in response to a given percentage change in another variable, holding all other relevant variables con- stant. The elasticity of demand, e, is the percentage change M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 51 54 CHAPTER 2 Supply and Demand a. the demand curve is perfectly inelastic? b. the demand curve is perfectly elastic? c. the supply curve is perfectly inelastic? d. the supply curve is perfect elastic? e. the demand curve is perfectly elastic and the supply curve is perfectly inelastic? Use graphs and math to explain your answers. 19. On July 1, 1965, the federal ad valorem taxes on many goods and services were eliminated. Comparing prices before and after this change, we can determine how much the price fell in response to the tax’s elimination. When the tax was in place, the tax per unit on a good that sold for p was ap. If the price fell by ap when the tax was elim- inated, consumers must have been bearing the full inci- dence of the tax. Consequently, consumers got the full benefit of removing the tax from those goods. The entire amount of the tax cut was passed on to consumers for all commodities and services that were studied for which the taxes were collected at the retail level (except admissions and club dues) and for most commodities for which excise taxes were imposed at the manufacturer level, including face powder, sterling silverware, wristwatches, and handbags (Brownlee and Perry, 1967). List the condi- tions (in terms of the elasticities or shapes of supply or demand curves) that are consistent with 100% pass- through of the taxes. Use graphs to illustrate your answer. 20. Essentially none of the savings from removing the federal ad valorem tax were passed on to consumers for motion picture admissions and club dues (Brownlee and Perry, 1967; see Question 19). List the conditions (in terms of the elasticities or shapes of supply or demand curves) that are consistent with 0% pass-through of the taxes. Use graphs to illustrate your answer. *21. Do you care whether a 15¢ tax per gallon of milk is col- lected from milk producers or from consumers at the store? Why or why not? *22. Usury laws place a ceiling on interest rates that lenders such as banks can charge borrowers. Low-income house- holds in states with usury laws have significantly lower levels of consumer credit (loans) than comparable house- holds in states without usury laws (Villegas, 1989). Why? (Hint: The interest rate is the price of a loan, and the amount of the loan is the quantity.) Problems *23. Using the estimated demand function for processed pork in Canada, Equation 2.2, show how the quantity demanded, Q, at a given price changes as per capita income, Y, increases slightly (that is, calculate the partial derivative of quantity demanded with respect to income). How much does Q change if income rises by $100 a year? *24. Suppose that the inverse demand function for movies is p
120  Q1 for college students and p
120  2Q2 for other town residents. What is the town’s total demand function (Q
Q1 + Q2 as a function of p)? Use a diagram to illustrate your answer. 25. The demand function for movies is Q1
120  p for col- lege students and Q2
120  2p for other town residents. What is the total demand function? Use a diagram to illus- trate your answer. (Hint: By looking at your diagram, you’ll see that some care must be used in writing the demand function.) 26. In the application “Aggregating the Demand for Broadband Service” (based on Duffy-Deno, 2003), the demand function is Qs
15.6p  0.563 for small firms and Ql
16.0p  0.296 for larger firms, where price is in cents per kilobyte per second and quantity is in millions of kilobytes per second (Kbps). a. What is the total demand function for all firms? Suppose that the supply curve for broadband service is horizontal at 40¢ per Kbps (firms will supply as much service as desired at that price). b. What is the quantity demanded by small firms, large firms, and all firms? 27. In the application “Aggregating the Demand for Broadband Service” (based on Duffy-Deno, 2003), the demand function is Qs
15.6p-0.563 for small firms and Ql
16.0p0.296 for larger ones. As the graph in the appli- cation shows, the two demand functions cross. What are the elasticities of demand for small and large firms? Explain. *28. Green, Howitt, and Russo (2005) estimate the supply and demand curves for California processing tomatoes. The supply function is ln Q
0.2 + 0.55 ln p, where Q is the quantity of processing tomatoes in millions of tons per year and p is the price in dollars per ton. The demand function is ln Q
2.6  0.2 ln p + 0.15 ln pt, where pt is the price of tomato paste (which is what processing toma- toes are used to produce) in dollars per ton. In 2002, pt
110. What is the demand function for processing toma- toes, where the quantity is solely a function of the price of processing tomatoes? Solve for the equilibrium price and the quantity of processing tomatoes (rounded to two dig- its after the decimal point). Draw the supply and demand curves (note that they are not straight lines), and label the equilibrium and axes appropriately. 29. The U.S. Tobacco Settlement between the major tobacco companies and 46 states caused the price of cigarettes to jump 45¢ (21%) in November 1998. Levy and Meara M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 54 Problems 55 (2005) find only a 2.65% drop in prenatal smoking 15 months later. What is the elasticity of demand for prena- tal smokers? *30. Calculate the price and cross-price elasticities of demand for coconut oil. The coconut oil demand function (Buschena and Perloff, 1991) is Q
1,200  9.5p + 16.2pp + 0.2Y, where Q is the quantity of coconut oil demanded in thousands of metric tons per year, p is the price of coconut oil in cents per pound, pp is the price of palm oil in cents per pound, and Y is the income of con- sumers. Assume that p is initially 45¢ per pound, pp is 31¢ per pound, and Q is 1,275 thousand metric tons per year. 31. When the U.S. government announced that a domestic mad cow was found in December 2003, analysts estimated that domestic supplies would increase in the short run by 10.4% as many other countries barred U.S. beef. An esti- mate of the price elasticity of beef demand is -1.6 (Henderson, 2003). Assuming that only the domestic sup- ply curve shifted, how much would you expect the price to change? (Note: The U.S. price fell by about 15% in the first month, but that probably reflected shifts in both supply and demand curves.) 32. Keeler et al. (2004) estimate that the U.S. Tobacco Settlement between major tobacco companies and 46 states caused the price of cigarettes to jump by 45¢ per pack (21%) and overall per capita cigarette consumption to fall by 8.3%. What is the elasticity of demand for cigarettes? Is cigarette demand elastic or inelastic? 33. In a commentary piece on the rising cost of health insur- ance (“Healthy, Wealthy, and Wise, Wall Street Journal, May 4, 2004, A20), economists John Cogan, Glenn Hubbard, and Daniel Kessler state, “Each percentage- point rise in health-insurance costs increases the number of uninsured by 300,000 people.” Assuming that their claim is correct, demonstrate that the price elasticity of demand for health insurance depends on the number of people who are insured. What is the price elasticity if 200 million people are insured? What is the price elasticity if 220 million people are insured? W 34. Using calculus, determine the effect of an increase in the price of beef, pb, from $4 to $4.60 on the equilibrium price and quantity in the Canadian pork example. (Hint: Conduct an analysis that differs from that in Solved Problem 2.1 in that the shock is to the demand curve rather than to the supply curve.) Illustrate your compara- tive statics analysis in a figure. 35. Solved Problem 2.3 claims that a new war in the Persian Gulf could shift the world oil supply curve to the left by 3 million barrels a day or more, causing the world price of oil to soar regardless of whether we drill in the ANWR. How accurate is that claim? Use the same type of analysis as in the solved problem to calculate how much such a shock would cause the price to rise with and without the ANWR production. 36. A subsidy is a negative tax in which the government gives people money instead of taking it from them. If the gov- ernment applied a $1.05 specific subsidy instead of a spe- cific tax in Figure 2.12, what would happen to the equilibrium price and quantity? Use the demand function and the after-subsidy supply function to solve for the new equilibrium values. What is the incidence of the subsidy on consumers? 37. Besley and Rosen (1998) find that a 10¢ increase in the federal tax on a pack of cigarettes leads to an average 2.8¢ increase in state cigarette taxes. What implications does their result have for calculating the effects of an increase in the federal cigarette tax on the quantity demanded? As of 2005, the U.S. federal cigarette tax was 39¢ per pack, and the federal tax plus the average state tax was 84.5¢ per pack. Given the current federal tax and an estimated elas- ticity of demand for the U.S. population of -0.3, what is the effect of a 10¢ increase in the federal tax? How would your answer change if the state tax does not change? 38. Green et al. (2005) estimate that the demand elasticity is -0.47 and the long-run supply elasticity is 12.0 for almonds. The corresponding elasticities are -0.68 and 0.73 for cotton and -0.26 and 0.64 for processing toma- toes. If the government were to apply a specific tax to each of these commodities, what would be the consumer tax incidence for each of these commodities? *39. Use calculus to show that an increase in a specific sales tax t reduces quantity by less and tax revenue more, the less elastic the demand curve. [Hint: The quantity demanded depends on its price, which in turn depends on the spe- cific tax, Q(p(t)), and tax revenue is R
p(t)Q(p(t)).] 40. An increase in the minimum wage could raise the total wage payment, W
wL(w), where w is the minimum wage and L(w) is the demand function for labor, despite the fall in demand for labor services. Show that whether the wage payments rise or fall depends on the elasticity of demand of labor. 41. Lewit and Coate (1982) estimate that the price elasticity of demand for cigarettes is -0.42. Suppose that the daily market demand for cigarettes in New York City is Q = 20,000p0.42 and that the inverse market supply curve of cigarettes in the city is p
1.5pw, where pw is the whole- sale price of cigarettes. (That is, the inverse market supply curve is a horizontal line at a price, p, equal to 1.5pw. Retailers sell cigarettes if they receive a price that is 50% higher than what they pay for the cigarettes so as to cover their other costs.) a. Assume that the New York retail market for cigarettes is competitive. Calculate the equilibrium price and quantity of cigarettes as a function of the wholesale M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 55 56 CHAPTER 2 Supply and Demand price. Let Q* represent the equilibrium quantity. Find dQ*/dpw. b. Now suppose that New York City and State each impose a $1.50 specific tax on each pack of cigarettes, for a total of $3.00 per pack on all cigarettes possessed for sale or use in New York City. The tax is paid by the retailers. Show using both math and a graph how the introduction of the tax shifts the market supply curve. How does the introduction of the tax affect the equi- librium retail price and quantity of cigarettes? c. With the quantity tax in place, calculate the equilib- rium price and quantity of cigarettes as a function of wholesale price. How does the presence of the quan- tity tax affect dQ*/dpw? W 42. Due to a recession that lowered incomes, the 2002 market prices for last-minute rentals of U.S. beachfront proper- ties were lower than usual (June Fletcher, “Last-Minute Beach Rentals Offer Summer’s Best Deals,” Wall Street Journal, June 21, 2002, D1). Suppose that the inverse demand function for renting a beachfront property in Ocean City, New Jersey, during the first week of August is p
1,000  Q + Y/20, where Y is the median annual income of the people involved in this market, Q is quan- tity, and p is the rental price. The inverse supply function is p
Q/2 + Y/40. a. Derive the equilibrium price, p*, and quantity, Q*, in terms of Y. b. Use a supply-and-demand analysis to show the effect of decreased income on the equilibrium price of rental homes. That is, find dp*/dY. Does a decrease in median income lead to a decrease in the equilibrium rental price? W M02_PERL7945_01_SE_02V2.QXD 6/27/07 3:03 PM Page 56