201 ECON Shomu Banerjee CONSUMPTION EXTERNALITIES & INEFFICIENCY The First Fundamental Welfare Theorem that we studied earlier states that : Under fairly weak assumptions1 about consumer preferences, at a Walrasian equilibrium, the allocation attained is Pareto efficient and individually rational.2 As stated, this statement is not exactly true; it is only true so long as (a) all market are Walrasian (perfectly competitive) and there is no imperfect competition (b) there are no public goods (c) there are no externalities (d) there are no information asymmetries, and (e) there are no non-convexities.3 The aim of this note is to present an example where the First Fundamental Welfare Theorem fails when there is a negative consumption externality. Suppose A and B are the only two consumers in our economy, there are two goods x and y, and As endowment is a = (0, 1) while Bs is b = (1, 0). Further suppose that their utility functions are given by uA(xa, ya, xb) = xa + ya 0.5xb and uB(xb, yb) = xbyb. Bs preferences are Cobb-Douglas, while As preferences are linear in her own consumption of xa and ya. But xb also enters As utility function as a bad; consumer A has no part in choosing xb but derives a negative utility from the other persons consumption of it.4 So from As point of view, her indifference curves are linear with slope 1. Therefore in this Edgeworth box economy (shown in Figure 1 on the next page) the Walrasian equilibrium is at point E where both goods are priced at $1 and (xa, ya) = (1/2, 1/2) and (xb, yb) = (1/2, 1/2).
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These details are not important for you to know but just in case youre curious, preferences must not be satiated (i.e., have any bliss points) or have thick indifference curves. 2 Pareto-efficient means that it is not possible to make anyone better off without hurting someone. Individually rational means no person is worse at off the Walrasian allocation as compared to their initial endowment point; this ensures that people want to voluntarily participate in the competitive market mechanism. 3 This is somewhat technical and has to do with such things as the presence of increasing returns in productionIve included this for the sake of completeness. 4 This may be interpreted as the envy or jealousy that A feels when she sees B consume xb.
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Ob
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However, the allocation at E is not Pareto-efficient. To see this, look at this twoperson economy from societys (or as economists say, from a social planners) point of view. For such a planner who knows that xa + xb = 1, As utility function is actually uA = xa + ya 0.5(1 xa) = 1.5xa + ya 0.5, so As preferences are linear in xa and ya with slope 1.5 (shown by the green Ob 1
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solid green indifference curve in Figure 2, as opposed to the black dashed indifference curve which has a slope of -1). In other words, the marginal rate of substitution between the two goods for A is not really 1 because of the negative externality, it is 1.5. To find the contract curve in this Edgeworth box, we need to set the social marginal rates of substitution (i.e., the marginal rates of substitution from societys point of view) for each person equal to each other. For A, we have seen this to be 1.5; for B, it is simply -yb/xb. Setting them equal to each other and using the fact that xb = 1 - xa and yb = 1 - ya, we get the equation of the contract curve: ya = -0.5 + 1.5xa. This is shown by the blue line in Figure 2. It is clear graphically that point E is not Pareto-efficient. To check this out, note that at E, ua = 3/4 and ub = 1/4 = 24/96. At point F where (xa, ya) = (7/12, 3/8) and (xb, yb) = (5/12, 5/8) yields ua =3/4 (so consumer A is no worse) but ub is higher at 25/96.