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Chapter 8 PUBLIC GOODS

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Chapter 8 PUBLIC GOODS Chapter 8 PUBLIC GOODS 1. Introduction In the last chapter we analyzed some examples of external effects, and discussed the calculation of appropriate taxes and subsidies to correct ex- ternality problems. In this chapter we carry the externality phenomenon to its logical extreme. We shall examine the theory of the production and consumption of goods whose character is essentially public, rather than private. What do we mean by a public good? Some goods have the property that when one person uses them, all people use them. That is, their use is nonexclusive; if the goods are available to one, they are available to all. There is no practical way for one person to use them alone. Goods that aren't like this, goods that are really private, or exclu- sively used by one person, are easy to think of: a glass of beer, a set of false teeth, a pair of socks, a hamburger. When A is using or consuming one of these things, then, necessarily, B isn't. Goods whose use is neces- sarily nonexclusive are less common, but there are many important ones: Radio and television broadcasts (unless scrambled) are nonexclusive. If A can get the TV signal and B lives nearby, then B can get the TV signal also. (Note that we are talking about the signal, not the TV set, which is a private good. Also note that we are not talking about cable TV, the access to which is again a private good.) There is no practical way to deliver radio waves to A without simultaneously delivering them to B. The signals of a lighthouse are nonexclusive. If a lighthouse is warning ship A to stay away from the shoals, then it is also necessarily warning any nearby ship B to stay clear. The outside of the Washington Monument is nonexclusive. If person A can view it and enjoy it, person 162 WELFARE ECONOMICS AND SOCIAL CHOICE THEORY, 2ND ED. B can do so as well. It wouldn't be practical to screen it and charge to let people inside the screen. On the other hand, it is practically possible to exclude people from the inside of the Washington Monument. The Statue of Liberty, viewed from outside, is a nonexclusive good. If it's available for one to see, it's available to all. National defense is an important nonexclusive good. If the person and property of U.S. resident A are being protected from foreign armies, then the person and property of U.S. resident B are also necessarily being protected from those armies. Another important example is scientific and technological knowledge. Some technological knowledge is patentable and its use can be restricted, but a larger part is not. The technology of the internal combustion en- gine is nonexclusive. If it is available in person A's library, it is probably available in ß's. Medical knowledge is partly nonexclusive. If a cure for infection, like penicillin, is known to A's doctor, then it is known to ß's. When someone discovers a cure for some form of cancer, that cure will ultimately be public knowledge, and nonexclusive in its use. We say ultimately because it will likely be patented and restricted in use initially. In each of these cases, when the good is there for one, it is necessarily there for all. Goods with this property are called public goods. What are the efficiency implications of public goods? Like exter- nalities, public goods undermine optimality in a standard competitive equilibrium. What then should be done? What are the optimal quan- tities of public goods? How should public goods be financed? We shall explore these and similar questions in this chapter. 2. The Public Goods Model In this section we shall develop a rather special, and rather different model to analyze the problems of public goods. A good is public if it is by nature available to all: if one man uses it, everyone can use it. Public goods can be viewed as goods with extreme external effects: if person i's consumption of the good is X, then X appears in each and every person's utility function. However, we won't continue the externality and Pigouvian tax/subsidy analysis of the last chapter: it is more convenient to start anew. The model we use here has both production and consumption, because one principal question we want to answer is this: How much of the public good should be produced? And the answer to the question depends both on people's demand for it, and on the nature of the productive sector of the economy. But in order to avoid notational and analytical complexity, our model will be exceedingly simple. We assume that there are only two PUBLIC GOODS 163 goods, one private, and one public. Also we assume that the productive sector of the economy can transform units of the private good into units of the public good, in the ratio of one to one. And, therefore, we assume the equilibrium prices of the two goods are 1 and 1. Our model will also make a crucial simplifying assumption about the nature of utility functions. We assume that person i's utility is the sum of the quantity of the private good he consumes, plus a well-behaved function of the quantity of the public good produced and available to all, including i. Such a utility function is said to be separable between private and public consumption. Some of the analysis below hinges on this special assumption; some does not. We make the assumption for two reasons: (1) it greatly simplifies all the mathematics, and (2) the discussion of demand revealing taxes breaks down without it. Now let's develop some of the notation. It should be observed that this notation differs slightly from what is used in the exchange and pro- duction models treated above. First, we let X = the quantity of the public good. Note that x is a scalar, not a vector. Also, note that x can be viewed as the quantity (or size) of the public good in physical units, or in dollars, since we assume that the prices of both the public and the private good are one. Second, we let yi = person i's quantity of the private good. Note that yi is a scalar. We assume that person i's utility function ui can be written Ui = Vi{x) +yi. That is, i's utility is the sum of the function vi^ which depends only on X, plus i's quantity of the private good. We also assume that vi is continuous, smooth, monotonic, and concave; that is, it looks like the one in Figure 8.1. (Actually, monotonicity can largely be relaxed. We could assume instead that only some individuals' utility functions are monotonic.) In the figure, the function vi is smooth and concave, that is, it bends downward. The intercept a might be positive or negative, or vi might even be asymptotic to the (negative half of the) vertical axis. At the point P, the ratio Avi/Ax is person i's marginal utility from the pub- lic good, or approximately the amount by which his utility rises if the quantity of the public good is increased by 1, while his private consump- tion remains fixed. Note that for sufficiently small A's, Ai;^/Ax and the slope of the vi function at P are equal. Instead of writing MU of the 164 WELFARE ECONOMICS AND SOCIAL CHOICE THEORY, 2ND ED. Figure 8.1. public good for i, we now use this simpler notation: v[{x) = MU of the public good for i, when the quantity of the public good is x. We also assume that each person starts with an initial endowment of the private good. We let iüi i's initial quantity of the private good. Note that uoi is also a scalar. In order to be feasible., a vector of public and private good consump- tion levels (x, yi, ^25 • • • 5 Vn) must satisfy this condition: X + ^yi = ^^i' i=l i=l (Later in this chapter, we will relax the feasibility condition to x + 3. The Samuelson Public Good Optimality Condition Let's now see how a Pareto optimal output for the public good can be found. In order to do this, we start by considering the inequality v[{x) + V2{x) + . + V^{x) > 1. That is, we consider an output x of the public good, such that the sum of the marginal utilities of all individuals at x exceeds 1. Let's assume for mathematical simplicity that each person's private consumption yi PUBLIC GOODS 165 exceeds his marginal utility from public consumption v[[x). Now suppose we reduce each person's consumption of the private good by an amount v[[x)'^ that is, we define a new amount of private good consumption for each i: If we stop here, then each person's utility must drop by the amount of the private good he loses, v[[x). But we have extracted v[{x) + v'^ix) + ... + <(x) = 1 + A units of private good from the economy, where A is some number greater than zero. Now suppose we take 1 unit of private good from this total (leaving A) and send it to the productive sector of the economy (or the firm) to be transformed into 1 unit of public good. Then we get a new public good output of X = X + 1. But increasing the public good available by a unit increases each person's utility by an amount approximately equal to the marginal utility of the public good, or, for i^v[[x).
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