Consumer’s Surplus

1. Consumer’s Surplus

2. Discrete demand • First consider a discrete version of the problem, in which the first good must be consumed in integer amounts. • Suppose the consumer could use only one unit of the good, so the choice is between zero units of the first good (along with m/q units of the second good) and one unit of the first good (along with (m - p)/q units of the second good). • In this context we express the consumer’s attitudes toward the choice in terms of a reservation value (or valuation) for buying the first good — the highest price she is willing to pay to obtain the one unit of the first good. • Let V be this reservation value. Then she is indifferent between the bundles (0, m/q) and (1, (m - V)/q). Her demand for the first good is easily expressed in terms of its price and V: if p <V she buys one unit of the good, if p> V she does not buy the good, and if p = V she is indifferent between buying one unit and not buying.

3. What if she is able to purchase the good at a price less than V? • Then she is better off than she would be if the good were not available, and S = V - p is a dollar measure of how much better off she is. • This difference, S. is called her consumer’s surplus. • If she is offered the good at price p > V, then she does not purchase it, and she is no better off than if the good were not available, so her consumer’s surplus is 0.

4. Summarizing, for this discrete version of the problem, if a consumer has valuation V for one unit of the first good and is able to obtain it at cost p, then this opportunity yields a (consumer’s) surplus • S = maximum{0, V- p), • where maximum{b,c} means take the maximum of the numbers b and c. • The surplus is the maximum of 0 and V- p because she has the option not to purchase (0) or to purchase (V- p) the good.

5. Continuous demand • We now return to our maintained assumption that both goods may be consumed in any amount. Is it possible, to obtain an analog of the consumer’s surplus in the discrete problem, and if so, how could it be measured? • CLAIM: • If (in the relevant region) the income elasticity of demand for the first good is zero, then the area below the demand curve for the first good can be used as a dollar measure of the value to the consumer of the first good.

6. In the first figure, the shaded area represents the value to the consumer of 50 units of the first good. [Recall that areas in the figure have units of “dollars per unit of the first good” times “units of the first good per period” or “dollars per period.”] If the consumer must pay for the good, then the consumer’s surplus is the amount by which the value exceeds the required payment. • For example, if the consumer in the first figure must pay p = 30 per unit to obtain the 50 units, then her surplus is the value from the first figure minus (30)(50). This is the area below the demand curve and above the price, out to the quantity, and is illustrated in the second figure.

7. To understand why these areas represent value and surplus respectively, first observe that an income elasticity of demand for the first good of zero corresponds to a consumer with quasi-linear utility and with enough income to be at an interior solution. Thus we can use a utility function of the form u(x, y) = f(x) +y and the optimal x satisfies f ’(x) = p/q (i.e., MRS= price ratio).

8. Since q is fixed when we consider the demand curve, the set of points making up the curve can be thought of in two ways. • Thought of as the demand curve, each value for p determines a corresponding value for x. • An alternative way to think of the same set of points is as what is called the inverse demand curve. From this perspective, each value for x determines a corresponding value for p. • It is important to understand that we are dealing with the same set of (x, p) points, but we are interpreting them differently. [We will use this approach again when we deal with monopoly and oligopoly problems.] • The advantage of this alternative perspective for our current problem is that it is easy to solve for p as a function of x: • p = qf ’(x).

9. Using the inverse demand curve, call it p = P(x) = qf ‘(x), the shaded area from the first figure is given by the integral • The corresponding shaded area from the second figure is • qf(50)- qf(0) - 1500. • If this is the surplus value to the consumer, then if we added it to the consumer’s income but prevented her from purchasing any of the first good, she would be exactly as well off as with her original income and facing p = 30. • When p = 30, her optimal bundle is (50, [m 1500]/q), with corresponding utility • f(50) + [m - 1500]/q. • With the additional income but the inability to purchase the first good, her bundle is (0, (m/q) + f(50) - f(0)-1500/q), with corresponding utility • f(0) + (m/q) + f(50)- f(0) - 1500/q = f(50) + [m- 1500]/q. • Since she is exactly as well off in the two situations, the calculated value, • qf(50) - qf(0)- 1500, is the surplus.

10. NOTE: The follwoing areas are incorrect except in special cases such as when the income elasticity of demand is zero. Correct measures exist for other cases, but won’t be discussed at this point.

11. Surplus measures are commonly used to assess the impact of price changes on individuals. • For example, how much worse off would an individual be if the price rose from $1 per unit to $4 per unit. • The answer depends on the change in surplus, which is illustrated in the first figure. The shaded area is part of the consumer’s surplus at price $1 per unit but is not part of consumer’s surplus at $4 per unit. The area provides a dollar measure of how much worse off the consumer is when facing price $4 per unit rather than $1 per unit. • The area is sometimes most easily computed by thinking of flipping the axes on the diagram, and taking the integral of x as a function of p, as shown in the second figure.

12. Example 1: For u(x, y) = y + 100 ln (x), with q = 6, and m = 900, the demand for the first good is x*(p,6,900) = 600/p. • If the price of the first good increases from $1 to $4, then a dollar measure of how much worse off the consumer is, is given by

13. Example 2: Suppose in the relevant region the income elasticity of demand for the first good is zero and, with the price of the second good fixed, the quantity demanded of the first good is • x*(p) = 100 - 4p. • Also suppose the market price for the first good is $20 per unit. • What is the maximum amount this person would be willing to pay to join a “discount club” that allows her to purchase the good for $15 per unit (and has no other benefits)?

14. Because the demand curve is a straight line, the areas can be computed directly from the graph. If joining the club were free, her gain in surplus from joining the club (and facing the lower price) would be the shaded area in the figure. The area is the sum of a rectangle with area (20-- 15)(20-- 0) = 100 and a triangle with area (1/2)(20 - 15)(40 - 20) = 50, for total area 150. • If the fee to join the club were $F, then her net gain in surplus from joining the club would be $150 - $F. As long as F is less than 150 she would have a net gain in surplus. Thus she would be willing to pay up to $150 to join the club. • [At F = 150 she is indifferent between joining and not joining the club. For convenience, we ordinarily do not bother to mention this indifference, and just say she would be willing to pay up to $150.]

15. Thesecond example is related to an important class of problems called two-part tariff problems, which we will discuss in the context of monopolies