Chapter 14 Consumer’s Surplus How to estimate preferences or utility from demand A discrete good with quasilinear utility u(x,y)=v(x)+y where y is money to be spent on other goods and its price is normalized to be 1. Have seen before that the consumer will start to buy the first unit at the r 1 so that v(1)+m- r 1 =v(0)+m or r 1 =v(1)-v(0)

Following this logic, r 1 =v(1)-v(0), r 2 =v(2)-v(1), r 3 =v(3)-v(2) and so on. Therefore, if we sum the area under the demand curve say up to the third unit, then we get r 1 + r 2 + r 3 =v(3)-v(0). This is the change in utility due to the increase in consumption of x. If the consumer can get these three units for free, then his utility change (from consuming zero unit to consuming three units for free) will be v(3)+m-(v(0)+m)=v(3)-v(0), exactly the area under the demand.

Suppose the price of x is p, then the utility change when the consumer buys three units of x is v(3)+m-3p- (v(0)+m)=v(3)-v(0)-3p. This is the area that lies between the demand and the price (up to three units) and is called the consumer’s surplus. Alternatively, can think in the following way. For the first unit, the consumer is willing to pay up to r 1 but he only pays p, so there is this surplus r 1 –p for the first unit.

Similarly, there is a surplus of r 2 –p for the second unit and r 3 –p for the third unit. So when he consumes 3 units, his consumer’s surplus is r 1 -p+r 2 -p+r 3 - p=v(3)-v(0)-3p. Can think of a continuous demand using the discrete approximation or solve max v(x)+y s.t. px+y=m.

Fig. 14.1

MRS xy =v’(x)/1=p/1 so plotting the demand (the price) is the same as plotting v’(x). If the consumer consumes up to 3 units, then his change in utility is v(3)+m- 3p-(v(0)+m)=v(3)-v(0)-3p= ∫ 0 3 v’(x)dx-3p. This is the area that lies between the demand curve and the price (up to three units) and is the consumer’s surplus. Interpret the change in consumer’s surplus the same way.

Fig. 14.2

Suppose the price changes from p to p’ and accordingly the consumer changes consumption from x to x’. The CS (up to x) is the change of utility from not buying to buying x units. Similarly, the CS (up to x’) is the change of utility from not buying to buying x’ units. Hence the difference CS(x’)-CS(x) is the change of utility from buying x units to buying x’ units. It consists of a rectangular area and a roughly triangular area.

Fig. 14.3

Suppose we have estimated a utility function from a demand, there are two useful utility function worth mentioning because they lead to measurement of the welfare change expressed in dollar units. It is convenient to have some monetary measures of utility. Consider a price change from p to p’. Then the change in income necessary to restore the consumer to his original indifference curve is called the compensating variation (CV).

Fig. 14.4

This is measuring the two indifference curves by p’. Alternatively, we can ask how much money would have to be taken away from the consumer before the price change to leave him as well off as he would be after the price change. This is called the equivalent variation (EV). This is measuring the two indifference curves by p. (mention three lines)

One example. u(x,y)=x 0.5 y 0.5 and price of y is 1. Suppose price of x changes from 1 to 2 and income is 100. CV: =(0.5m/2) 0.5 (0.5m/1) 0.5, so m=141. You have to give the consumer additional 41 dollars to make him as well off as before. EV: =(0.5m/1) 0.5 (0.5m/1) 0.5, so m=70. You have to take 30 dollars away from the consumer to make him as well off as after the price change.

Note further that for quasilinear preferences, since it does not matter which price vector we use to measure the distance between two indifference curves, so CV and EV will be the same.

Using the same example, suppose the consumer’s consumption changes from x to x’ (with some abuse of notation). CV: v(x)+m-px=v(x’)+m-p’x’+C so C=v(x)-v(x’)+p’x’-px EV: v(x’)+m-p’x’=v(x)+m-px-E so E=v(x)-v(x’)+p’x’-px=C=-ΔCS

Fig. 14.5

Producer’s surplus can be approached the same way. Can think of selling endowment. Suppose a consumer has n units of good x. When will he start selling the first unit of x? Must be v(n)+m=v(n- 1)+m+r 1. So r 1 =v(n)-v(n-1). Similarly, he will start selling the second unit at v(n-1)+m+r 2 =v(n-2)+m+2r 2, or r 2 =v(n-1)-v(n-2).

Hence if for the first unit, he is selling at r 1, then he is indifferent between selling or not selling. If the market price is p, then he gets a surplus of p-r 1 for the first unit. Thus the area between the price and the supply curve measures the producer’s surplus from selling (compared to not selling). When moving from consumer’s surplus to consumers’ surplus, we can aggregate the consumer’s surplus across consumers.

Fig. 14.6

Yet we may care about the distribution of the welfare change, i.e., which consumer gets better off and which gets worse off (a tax change making the poor better off may be preferred by the society to that making the rich better off, for instance). In this case, we may use the convenient measures of CV and EV. Say a policy change makes A worse off by 3 dollars and B better off by 4 dollars (using the CV measure), then we may want to do it depending on who A and B are (mention redistribution).

Some comments on CS. Budget constraint may kick in. Look at the case where u(x,y)=x 1/2 +y and m=1. r 4 should be 4 1/2 -3 1/2 =0.2679, but 4r 4 >1. Look at some other utility function say u(x,y)=x 0.5 y 0.5, the price of y is 1 and income 100. The demand curve is x=50/p x. If p x changes from 2 to 1, the increase of utility is = Yet the change of CS is ∫ (50/x)dx=