7.1 The concept of consumer s surplus

Transcription

1 Microeconomics I. Antonio Zabalza. University of Valencia 1 Lesson 7: Consumer s surplus 7.1 The concept of consumer s surplus The demand curve can also be interpreted as giving information on the maimum price that the consumer would pay for the marginal (last) unit of the good bought. In the following graph, p n is the maimum price that the consumer would pay for the th n unit of the good. p A B p n D O n If the market price is the valuation of the marginal unit, the sum of the valuation of all units (1, 2, 3,, n-1, n ), will be the area OAB n under the demand curve. We discussed all this last year. Now, can we give a theoretical justification of this based on consumer theory? The answer is, yes. And to show

2 Microeconomics I. Antonio Zabalza. University of Valencia 2 that we must go back from the price, good space (the demand curve) to the goods space (the indifference curve). All other goods m m 1 m 2 m mn 1 m n U(0,m) n Here on the horizontal ais we represent good and in the vertical ais all other goods. Then, the point at which the indifference curve crosses the vertical ais must be the income the consumer has, m. Suppose initially you are not consuming good (you are at point m). What would be the maimum amount of money you would be willing to give up in order to consume the first unit of? Clearly m1. What about to consume the second unit? Answer:. And so successively until the last unit, where m 2

3 Microeconomics I. Antonio Zabalza. University of Valencia 3 the maimum amount of money you would be willing to give up is mn. So, the maimum amount of money the consumer would be willing to give up to consume all n units is m. How can we represent these changes in income in the p/ space? Obviously, each change in income is the maimum price the consumer would be willing to pay for each of the corresponding units. So m1 is the first column in the following graph; m2 the second column; up to the last column mn for the last n th unit. p A m 1 m 2 m n B Demand curve O 1 2 n

4 Microeconomics I. Antonio Zabalza. University of Valencia 4 Make units infinitesimally small and then you will conclude that the maimum amount of money the consumer would be willing to give up in order to consume n units of (that is, the value that the consumer puts on these n units of ) is m, which is the area OAB n under the demand curve. Observe (this is important) that the demand curve we need to measure this value is not any demand curve, but the compensated demand curve (compensated a la Hicks). If our measures of the changes in income in the p/ space are to be consistent with those in the m/ space, along this demand curve we must necessarily be maintaining constant the level of utility. As we saw last year, the concept of consumer surplus arises from the fact that the consumer is given the possibility to buy all units of the good at the last (marginal) price. Suppose then that all units of can be bought at the same price p. How can we measure the consumer surplus of a consumer that buys n units of? The answer to this question is given by the following two graphs.

5 Microeconomics I. Antonio Zabalza. University of Valencia 5 All other goods Total value of n = Area OAB n Consumer surplus Ependiture = area ABp = area Op B n m-p n m U((m-p n ), ) p n A p B Demand curve O n

6 Microeconomics I. Antonio Zabalza. University of Valencia Interpretation of changes in the consumer s surplus Normally we are not interested in the total consumer surplus, but in the change of this consumer surplus (CS) as a result of some change in the price, the quantity consumed or any other variation in the conditions under which the consumer makes his choices. Suppose the variation we are analysing is an increase in the price of good from p 0 to p 1. What is the change in the CS as a result of this price change? p p 1 p 0 R T Demand curve 1 0 When the price increases, the CS is reduced by the two areas R and T (the total reduction of CS is R+T).

7 Microeconomics I. Antonio Zabalza. University of Valencia R: rectangle = ( p p ). The consumer still consumes 1, but he pays more for these units of. So he looses R T: triangle ( )( ) 2 p p. The consumer looses this surplus because he consumes less units of than before the price rise. So, R measures the loss from having to pay more for the units he still consumes, and R measures the loss from the reduced in consumption. There is an important difference between R and T. R is a loss to the consumer but a gain to someone else (the seller of the good). T, on the other hand, is a net loss to society: the consumer looses this value and no one gains it. 7.3 Compensating and Equivalent Variations The compensating variation (CV) and the equivalent variation (EV) are the eact measures of welfare change as a result of a price change. See first what it is that we want to measure. There is a given change in the price of good ; suppose that there is a fall in the price of. As a

8 Microeconomics I. Antonio Zabalza. University of Valencia 8 result, there is an increase in the utility of the consumer. What we want to do is to measure this change in utility. The problem is that utility functions, even if known, are only ordinal concepts. Thus a change in utility does not mean anything. What we have to do is to transform this change in utility into a change in income: this is what the CV and the EV do. Suppose as above that we consider good and all other goods. This is equivalent to considering good and money. Or good and a composite good (a basket of all other goods) the price of which is 1. This allows us to consider all distances on the vertical ais as measures of income. 0 Suppose the price of good goes down from p to p, where p < p. As a result, the equilibrium of the consumer moves from q to q, and the level of utility from u to u, where u > u. By how much has welfare improved? There are two ways to answer this question, depending on the sort of conceptual eperiment that we carry out to evaluate the move of utility in terms of money. a) Suppose the consumer is already at q. That is, he has already reacted to the price change. Then

9 Microeconomics I. Antonio Zabalza. University of Valencia 9 we ask: How much money we would have to take away from the consumer after the price change to return him to the level of utility he had before the price change? To answer this, 1 put yourself at q, with p. The amount of money that you would have to take away from the consumer is given by the distance AB. This distance is the Compensating Variation (CV). b) Suppose the consumer is still at q. That is, he has not still reacted to the price change. Then we ask: How much money we would have to give the consumer before the price change to make him as well off as he would be after the price change? To answer this, put yourself at 0 q, with p. The amount of money that you would have to give the consumer is given by the distance AC. This distance is the Equivalent Variation (EV). These two concepts are the eact measures of welfare change induced by a change in prices. They are in fact two income effects, and they differ because in one of them (CV) we use the final price, and in the other (EV) we use the initial price.

10 Microeconomics I. Antonio Zabalza. University of Valencia 10 All other goods C EV A CV B q q 0 p u u 1 p

11 Microeconomics I. Antonio Zabalza. University of Valencia Calculating gains and losses Problem 6, Lesson 7 Suppose that a consumer has a utility function U( 1, 2) = 1 2. He originally faces prices (1,1) and has income 100. Then the price of good 1 increases to 2 while the price of good 2 remains constant at 1. What are the compensating and equivalent variations of this price change? The utility function is a CD utility function. We know from a previous lesson that the demand functions in this case are: m m 1 = and 2 = 2p1 2p2 Therefore, the initial equilibrium q in this case is 1 = 50 and 2 = 50 Also, the final equilibrium q is = 25 and = Let us first calculate the CV. We put the consumer after the price change has taken place, and ask: At prices (2,1), how much money would it be necessary to give to the consumer to make him as well off as he was before the price change? Utility before the price change: U(50,50)=50.

12 Microeconomics I. Antonio Zabalza. University of Valencia 12 What income m would the consumer need so that with prices (2,1) he is able to obtain a utility level of 50? m m 50 2*2 2*1 = Raise both sides of the equation to the square m m = 4 2 ( m ) 2 = m = Therefore the CV is the amount of money that, in addition of the income the consumer already has (100), we would have to give him to reach m. CV = = To calculate the EV, we put the consumer before the price change, and ask: At prices (1,1), how much money would the consumer need to make him as well off as he would be with prices (2,1)? Utility after the price change: U(25,50)= What income m would the consumer need so that with prices (1,1) he is able to obtain a utility level of ?

13 Microeconomics I. Antonio Zabalza. University of Valencia 13 ( m ) m m *1 2*1 = Raise both sides of the equation to the square m m = = 5000 m = 70.7 Therefore the EV is the amount of money that we would take away from the consumer so that he just remains with m. EV = = 29.3 Naturally, what happens here is a fall in utility. Therefore both measures have to be interpreted as negative numbers. The fall in utility from u to u is measured as a fall in income of according to the CV measure, and of 29.3 according to the EV measure.

14 Microeconomics I. Antonio Zabalza. University of Valencia 14 2 m = CV 100 EV m = q q u = u(50,50) u = u(25,50)