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14 Consunmer's Surplus

14 Consunmer's Surplus - CHAPTER 1 4 CONSUMER’S SURPLUS...

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Unformatted text preview: CHAPTER 1 4 CONSUMER’S SURPLUS In the preceding chapters we have seen how to derive a consumer’s demand function from the underlying preferences or utility function. But in prac- tice we are usually concerned with the reverse problem~how to estimate preferences or utility from observed demand behavior. We have already examined this problem in two other contexts. In Chap— ter 5 we showed how one could estimate the parameters of a utility function from observing demand behavior. In the Cobvaouglas example used in that chapter, we were able to estimate a utility function that described the observed choice behavior simply by calculating the average expendi- ture share of each good. The resulting utility function could then be used to evaluate changes in consumption. In Chapter 7 we described how to use revealed preference analysis to recover estimates of the underlying preferences that may have generated some observed choices. These estimated indifference curves can also be used to evaluate changes in consumption. In this chapter we will consider some more approaches to the problem of estimating utility from observing demand behavior. Although some of the methods we will examine are less general than the two methods we 248 CONSUMER’S SURPLUS (Ch. 14) examined previously, they will turn out to be useful in several applications that we will discuss later in the book. We will start by reviewing a special case of demand behavior for which it is very easy to recover an estimate of utility. Later we will consider more general cases of preferences and demand behavior. 14.1 Demand for a Discrete Good Let us start by reviewing demand for a discrete good with quasilinear utility, as described in Chapter 6. Suppose that the utility function takes the form 12(30) + y and that the x—good is only available in integer amounts. Let us think of the y-good as money to be spent on other goods and set its price to 1. Let p be the price of the x—good. We saw in Chapter 6 that in this case consumer behavior can be described in terms of the reservation prices, 11 2 11(1) — 11(0), r2 : 11(2) — "0(1), and so on. The relationship between reservation prices and demand was very simple: if 11 units of the discrete good are demanded, then 1“,, 2 p 2 Tn+1. To verify this, let’s look at an example. Suppose that the censumer chooses to consume 6 units of the x—good when its price is p. Then the utility of consuming (6, m — 6p) must be at least as large as the utility of consuming any other bundle (:19, m — p110): 11(6)+m—6p21)(:v)+m-—p$. (14.1) In particular this inequality must hold for a: = 5, which gives us 11(6) +m—6p 2 12(5) +m—5p. Rearranging, we have v(6) — 11(5) = r6 2 p. Equation (14.1) must also hold for a: = 7. This gives us 1;(6)+m—6p212(7)+m—7p, which can be rearranged to yield p > 11(7) — 12(6) = r7. This argument shows that if 6 units of the X-good is demanded, then the price of the x-good must lie between 116 and 1‘7. In general, if n units of the x—good are demanded at price p, then Tn 2 p 2 Tn+1, as we wanted to show. The list of reservation prices contains all the information necessary to describe the demand behavior. The graph of the reservation prices forms a “staircase” as shown in Figure 14.1. This staircase is precisely the demand curve for the discrete good. CONSTRUCTING UTILlTY FROM DEMAND 249 14.2 Constructing Utility from Demand We have just seen how to construct the demand curve given the reservation prices or the utility function. But we can also do the same Operation in reverse. If we are given the demand curve, we can construct the utility function—at least in the special case of quasilinear utility. At one level, this is just a trivial operation of arithmetic. The reservation prices are defined to be the difference in utility: r1 = v(1) ~— 21(0) r2 : 11(2) — 11(1) 73 = 11(3) — 12(2) If we want to calculate 71(3), for example, we simply add up both sides of this list of equations to find m + m + T3 = 11(3) — 71(0). It is convenient to set the utility from consuming zero units of the good equal to zero, so that v(0) : 0, and therefore v(n) is just the sum of the first n reservation prices. This construction has a nice geometrical interpretation that is illustrated in Figure 14.1A. The utility from consuming 73 units of the discrete good is just the area of the first n bars which make up the demand function. This is true because the height of each bar is the reservation price associated with that level of demand and the width of each bar is 1. This area is sometimes called the gross benefit or the gross consumer’s surplus associated with the consumption of the good. Note that this is only the utility associated with the consumption of good 1. The final utility of consumption depends on the how much the consumer consumes of good 1 and good 2. If the consumer chooses 11. units of the discrete good, then he will have m — pn dollars left over to purchase other things. This leaves him with a total utility of v(n) + m — pm. This utility also has an interpretation as an area: we just take the area depicted in Figure 14.1A, subtract off the expenditure on the discrete good, and add m. The term v(n) ~ pn is called consumer’s surplus or the net con- sumer’s surplus. It measures the net benefits from consuming 71. units of the discrete good: the utility v(n) minus the reduction in the expenditure on consumption of the other good. The consumer’s surplus is depicted in Figure 14.1B. 250 CONSUMER’S SURPLUS (Ch. 14) 12 3 4 5 6 QUANTiTY 12 3 4 5 6 QUANTITY A Cross surplus 3 Net surplus Reservation prices and consumer’s surplus. The gross benefit in panel A is the area under the demand curve. This measures the utility from consuming the x—good. The con- sumer’s surplus is depicted in panel B. It measures the utility from consuming both goods when the first good has to be pur- chased at a constant'price p. ' 14.3 Other Interpretations of Consumer’s Surplus There are some other ways to think about consumer’s surplus. Suppose that the price of the discrete good is p. Then the value that the consumer places on the first unit of consumption of that good is m, but he only has to pay p for it. This gives him a “surplus” of T1 ~ p on the first unit of consumption. He values the second unit of consumption at 7‘2, but again he only has to pay 19 for it. This gives him a surplus of 7"; — p on that unit. If we add this up over all 72 units the consumer chooses, we get his total consumer’s surplus: CS=r1,—p+r2—p+~-+rn—p=r1+---+’rn—np- Since the sum of the reservation prices just gives us the utility of consump- tion of good 1, we can also write this as CS 2 v(n) — pm. We can interpret consumer’s surplus in yet another way. Suppose that a consumer is consuming n units of the discrete good and paying pn dollars QUASlLlNEAR UTILITY 251 to do so. How much money would he need to induce him to give up his entire consumption of this good? Let R be the required amount of money. Then R must satisfy the equation v(0)+m+R=v(n)+m—pn. Since 21(0) : 0 by definition, this equation reduces to R = v(n) -— pn, which is just consumer’s surplus. Hence the consumer’s surplus measures how much a consumer would need to be paid to give up his entire con— sumption of some good. 14.4 From Consumer’s Surplus to Consumers' Surplus Up until now we have been considering the case of a single consumer. If sev- eral consumers are involved we can add up each consumer’s surplus across all the consumers to create an aggregate measure of the consumers’ sur- plus. Note carefully the distinction between the two concepts: consumer’s surplus refers to the surplus of a single consumer; consumers’ surplus refers to the sum of the surpluses across a number of consumers. Consumers’ surplus serves as a convenient measure of the aggregate gains from trade, just as consumer’s surplus serves as a measure of the individual gains from trade. 14.5 Approximating a Continuous Demand We have seen that the area underneath the demand curve for a discrete good measures the utility of consumption of that good. We can extend this to the case of a good available in continuous quantities by approximating the continuous demand curve by a staircase demand curve. The area under the continuous demand curve is then approximately equal to the area under the staircase demand. See Figure 14.2 for an example. In the Appendix to this chapter we show how to use calculus to calculate the exact area under a demand curve. 14.6 Quasilinear Utility It is worth thinking about the role that quasilinear utility plays in this analysis. In general the price at which a consumer is willing to purchase 252 CONSUMER’S SURPLUS (Ch. 14) PRICE x QUANTITY X QUANTITY A Approximation to gross surplus B Approximation to net surplus Approximating a continuous demand. The consumer’s surplus associated with a continuous demand curve can be ap— proximated by the consumer’s surplus associated with a discrete approximation to it. some amount of good 1 will depend on how much money he has for con— suming other goods. This means that in general the reservation prices for good 1 will depend on how much good 2 is being consumed. But in the special case of quasilinear utility the reservation prices are independent of the amount of money the consumer has to spend on other goods. Economists say that with quasilinear utility there is “no income effect” since changes in income don’t affect demand. This is what allows us to calculate utility in such a simple way. Using the area under the demand curve to measure utility will only be exactly correct when the utility function is quasilinear. But it may often be a good approximation. If the demand for a. good doesn’t change very much when income changes, then the income effects won’t matter very much, and the change in consumer’s surplus will be a reasonable approximation to the change in the consumer’s utility.1 14.7 Interpreting the Change in Consumer’s Surplus We are usually not terribly interested in the absolute level of consumer’s surplus. We are generally more interested in the change in consumer’s 1 Of course, the change in consumer’s surplus is only one way to represent a change in utility#the change in the square root of consumer’s surplus would be just as good. But it is standard to use consumer’s surplus as a standard measure of utility. INTERPRETING THE CHANGE lN CONSUMER’S SURPLUS 253 surplus that results from some policy change. For example, suppose the price of a good changes from p’ to p”. How does the consumer’s surplus change? In Figure 14.3 we have illustrated the change in consumer’s surplus as— sociated with a change in price. The change in consumer’s surplus is the difference between two roughly triangular regions and will therefore have a roughly trapezoidal shape. The trapezoid is further composed of two subregions, the rectangle indicated by R and the roughly triangular region indicated by T. Demand curve Change in consumer's surplus X" X! - X Change in consumer’s surplus.- The change in consumer’s- surplus will be the difference between two roughly triangular areas, and thus will have a roughly. trapeindal. shape. The rectangle measures the loss in surplus due to the fact that the con— sumer is now paying more for all the units he continues to consume. After the price increases the consumer continues to consume 93” units of the good, and each unit of the good is now more expensive by p” — p’. This means he has to spend (30” — p’):1:” more money than he did before just to consume at” units of the good. But this is not the entire welfare loss. Due to the increase in the price of the x—good, the consumer has decided to consume less of it than he was before. The triangle T measures the value of the lost consumption of the x—good. The total loss to the consumer is the sum of these two effects: R measures the loss from having to pay more for the units he continues to consume, and T measures the loss from the reduced consumption. 254 CONSUMER’S SURPLUS (Ch. 14) EXAMPLE: The Change in Consumer’s Surplus Question: Consider the linear demand curve 19(1)) 2 20 — 2p. When the price changes from 2 to 3 what is the associated change in consumer’s surplus? Answer: When p = 2, D(2) = 16, and when p = 3, D(3) = 14. Thus we want to compute the area of a trapezoid with a height of 1 and bases of 14 and 16. This is equivalent to a rectangle with height 1 and base 14 (having an area of 14), plus a triangle of height 1 and base 2 (having an area of 1). The total area will therefore be 15. 14.8 Compensating and Equivalent Variation The theory of consumer’s surplus is very tidy in the case of quasilinear utility. Even if utility is not quasilinear, consumer’s surplus may still be a reasonable measure of consumer’s welfare in many applications. Usually the errors in measuring demand curves outweigh the approximation errors from using consumer’s surplus. But it may be that for some applications an approximation may not be good enough. In this section we’ll outline a way to measure “utility changes” without using consumer’s surplus. There are really two separate issues involved. The first has to do with how to estimate utility when we can observe a number of consumer choices. The second has to do with how we can measure utility in monetary units. We’ve already investigated the estimation problem. We gave an example of how to estimate a Cobb-Douglas utility function in Chapter 6. In that example we noticed that expenditure shares were relatively constant and that we could use the average expenditure share as estimates of the Cobb- Douglas parameters. If the demand behavior didn’t exhibit this particular feature, we would have to choose a more complicated utility function, but the principle would be just the same: if we have enough observations on demand behavior and that behavior is consistent with maximizing some— thing, then we will generally be able to estimate the function that is being maximized. , Once we have an estimate of the utility function that describes some observed choice behavior we can use this function to evaluate the impact of proposed changes in prices and consumption levels. At the most funda— mental level of analysis, this is the best we can hope for. All that matters are the consumer’s preferences; any utility function that describes the con— sumer’s preferences is as good as any other. However, in some applications it may be convenient to use certain mon- etary measures of utility. For example, we could ask how much money we COMPENSATING AND EQUIVALENT VARIATION 255 would have to give a consumer to compensate him for a change in his con- sumption patterns. A measure of this type essentially measures a change in utility, but it measures it in monetary units. What are convenient ways to do this? Suppose that we consider the situation depicted in Figure 14.4. Here the consumer initially faces some prices (p’{, 1) and consumes some bundle (331*, $3). The price of good 1 then increases from p’{ to 151, and the consumer changes his consumption to (501, :32). How much does this price change hurt the consumer? 'I Optimal bundle at ' price pf x1 Slope=afi1 I _ B The compensating and the equivalent variations. Panel A shows the compensating variation (CV), and panel B shows the equivalent variation (EV). One way to answer this question is to ask how much money we would have to give the consumer after the price change to make him just as well off as he was before the price change. In terms of the diagram, we ask how far up we would have to shift the new budget line to make it tan- gent to the indifference curve that passes through the original consumption point (ILJJE). The change in income necessary to restore the consumer to his original indifference curve is called the compensating variation in income, since it is the change in income that will just compensate the con— sumer for the price change. The compensating variation measures how much extra money the government would have to give the consumer if it wanted to exactly compensate the consumer for the price change. Another way to measure the impact of a price change in monetary terms is to ask how much money would have to be taken away from the consumer 256 CONSUMER’S SURPLUS (Ch. 14) before the price change to leave him as well off as he would be after the price change. This is called the equivalent variation in income since it is the income change that is equivalent to the price change in terms of the change in utility. In Figure 14.4 we ask how far down we must shift the original budget line to just touch the indifference curve that passes through the new consumption bundle. The equivalent variation measures the maximum amount of income that the consumer would be willing to pay to avoid the price change. In general the amount of money that the consumer would be willing to pay to avoid a price change would be different from the amount of money that the consumer would have to be paid to compensate him for a price change. After all, at different sets of prices a dollar is worth a different amount to a consumer since it will purchase different amounts of consumption. In geometric terms, the compensating and equivalent variations are just two different ways to measure “how far apart” two indifference curves are. In each case we are measuring the distance between two indifference curves by seeing how far apart their tangent lines are. In general this measure of distance will depend on the slope of the tangent linesithat is, on the prices that we choose to determine the budget lines. However, the compensating and equivalent variation are the same in one important caseithe case of quasilinear utility. In this case the indifference curves are parallel, so the distance between any two indifference curves is the same no matter where it is measured, as depicted in Figure 14.5. In the case of quasilinear utility the compensating variation, the equivalent variation, and the change in consumer’s surplus all give the same measure of the monetary value of a price change. EXAMPLE: Compensating and Equivalent Variations r 1 Suppose that a consumer has a utility function u(:i:1,;c2) = $12 3322. He originally faces prices (1, 1) and has income 100. Then the price of good 1 increases to 2. What are the compensating and equivalent variations? We know that the demand functions for this Cobb—Douglas utility func- tion are given by m (171 = --— 2P1 m 352 =1 -—. 2192 Using this formula, we see that the consumer’s demands change from ($1333) 2 (50,50) to (531,332) = (25, 50). To calculate the compensating variation we ask how much money would be necessary at prices (2,1) to make the consumer as well off as he was consuming the bundle (50,50)? If the prices were (2,1) and the consumer COMPENSATING AND EQUIVALENT VARIATION 257 Indifference indifference"- curves Utility differ- ence Quasilinear preferences. With quasilinear preferences, the distance between two indifference curves is independent of the slope of the budget lines. had income m, we can substitute into the demand functions to find that the consumer would optimally choose the bundle (m / 4, m / 2). Setting the utility of this bundle equal to the utility of the bundle (50, 50) we have 1 1 m E m 5 l _1_ (r) (a) =502502< m = rum/i m 141. Solving for m gives us Hence the consumer would need about 141 ~ 100 = $41 of additional money after the price change to make him as well off as he was before the price change. In order to calculate the equivalent variation we ask how much money would be necessary at the prices (1,1) to make the consumer as well off as he would be consuming the bundle (25,50). Letting m stand for this amount of money and following the same logic as before, m: Flux/fie 70. Solving for m gives us Thus if the consumer had an income of $70 at the original prices, he would be just as well off as he would be facing the new prices and having an income of $100. The equivalent variation in incom...
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