Provision Of Public Goods

To derive the conditions in which efficient provision of a public good is obtained, let us review private good production. Assume a society in which there are two individuals, Hanzel and Gretal. There are two private goods, gingerbread cookies and bread crumbs. In Figure 6.1a, the quantity of cookies (c) is measured on the horizontal axis, and the price per cookie (Pc) on the vertical axis. Hanzel's demand curve for cookies is denoted as DHc . The demand curve indicates the quantity of cookies that Hanzel would be willing to consume at each price, ceteris paribus. Similarly, DG in Figure 6.1b is Gretal's demand curve for cookies.

To derive the market demand curve for cookies, we simply add together the number of cookies each person demands at each price. In Figure 6.1a, at a price of $2, Hanzel demands one cookie, the horizontal distance between DHc and the vertical

Figure 6.1 Horizontal summation of demand curves.

Cper year

Figure 6.2 Efficient provision of a private good.

Cper year

Figure 6.2 Efficient provision of a private good.

axis. Gretal's demand for cookies at a price of $2 is shown in Figure 6.1b. The total quantity demanded at the $2 price is Hanzel's and Gretal's demand summated. The total quantity demand is therefore three cookies, labeled DH+G in Figure 6.1c.

As we have just shown, the point at which price is $2 and quantity demanded is three lies on the market demand curve. In this way, finding the market demand at a given price on the vertical axis involves the summation of the horizontal distance between each of the private demand curves. This process is referred to as horizontal summation.

To explore the efficient provision of a private good, we need to superimpose Figure 6.1c on the market supply curve, labeled Sc, as illustrated in Figure 6.2. Equilibrium* in the market, noted as E, is the price at which supply and demand are equal. This occurs at the price of $2 for cookies with demand equal to three cookies. At this price, as we illustrated in Figure 6.1, Hanzel is supplied the one cookie he demands while Gretal is supplied the two cookies she demands. Importantly, there is no reason to expect that Hanzel's and Gretal's

* The point at which the marginal benefit and the marginal cost both equal the price of the product. Thus, the marginal benefit equals the marginal cost, which is precisely the condition required for economic efficiency.

consumption levels are equal. Differences in taste, income, and other characteristics affect the individual demand for cookies from both Hanzel and Gretal.

The equilibrium in Figure 6.2 has an important property: the allocation of cookies is Pareto efficient.* Let us explore Pareto efficiency further. In consumer theory, if Gretal is a utility-maximizing individual, she will set her marginal rate of substitution** of cookies for bread crumbs (MRScb) equal to the price of cookies (Pc) divided by the price of bread crumbs (Pb) or MRScb = PJPb. Since only relative prices impact rational choice, the price of bread crumbs can be simply set at Pb = $1. This allows simplification in the mathematics without having any negative impact on the derivation of utility maximization by reducing the condition for utility maximization to MRScb = Pc. The price of cookies therefore measures the rate at which Gretal is willing to substitute cookies for bread crumbs. Gretal's demand curve for cookies (DG)now shows the maximum price per cookie that she would pay at each level of cookie consumption. In this way, the demand curve also shows the MRScb at each level of cookie consumption. Similarly, we could compute Hanzel's MRScb using DH.

Now let us take a look at the supply curve, Sc. This curve shows how the marginal rate of transformation of cookies for bread crumbs (MRTcb) varies with cookie production.*** As shown in Figure 6.2, Hanzel and Gretal both set MRScb equal

* Pareto efficiency is the condition in which a resource allocation has the property that no one can be made better off without someone being made worse off. This was named after the Italian economist and sociologist Vilfredo Pareto (1848-1923). This is what economists normally mean when they talk about efficiency.

** This is the rate at which an individual needs to substitute one commodity for another in order to maintain constant total utility from the commodities taken together.

*** This can be demonstrated if we remember that under competition firms produce up to the point in which price equals marginal cost. In this way the supply curve Sc shows the marginal cost of each level of cookie production. In giving consideration to the role of welfare economics, MRTci) = MCc/MC6. Because P6 = $1 and price equals marginal cost, then MC6 = $1, and MRTci) = MCc. Thus, we identify the marginal rate of transformation with marginal cost, which is the same as the supply curve.

to two, and the producer also sets MRTc6 equal to two. As a consequence, at equilibrium:

The necessary condition for Pareto efficiency is Equation (6.1). With a competitive marketplace that is functioning properly, the Fundamental Theorem of Welfare Economics* assures us that this condition will hold.

Now that we have reviewed the conditions for efficient production of private goods, let us look at the case of public goods.** We will begin by exploring the efficient conditions through intuitive reasoning before deriving them graphically. Let us say that both Hanzel and Gretal enjoy fireworks displays. Hanzel's enjoyment clearly does not diminish Gretal's enjoyment. We can say that fireworks displays are a public good. We know that individuals do not buy public goods. We can, however, ask a simple question relating the pricing concept for public goods: How much of a public good would be demanded if an individual (either Hanzel or Gretal) had to pay a given amount for each extra unit of the public good (fireworks)? Although this is a hypothetical question, it is not that farfetched, since as expenditures increase on public goods (Fourth of July fireworks displays), so do individuals' taxes. The extra payment that the individual has to make for each additional unit of the public good is termed the tax price. As we proceed, we make the assumption that the government has at its discretion the ability to charge different tax prices to different individuals.

* The Fundamental Theorems of Welfare Economics state that every competitive economy is Pareto efficient and every Pareto-efficient resource allocation can be attained through a competitive market mechanism, with the appropriate initial redistribution.

** Not everything that is good for the public is a public good. For example, education is good for the public. However, an individual benefits from his or her own education. Education is only a public good to the extent that I enjoy "free-rider" benefits when you are educated. If individual incentives are sufficient to achieve the optimal production of something, then it is not a public good.

Let us label this individual tax price as t; accordingly, for each unit of the public good the individual must pay t. We can state the individual's budget constraint in the following manner:

where C is the individual's consumption of private goods, G is the total amount of public goods provided, and Y is the individual's income. The budget constraint is a representation of the combination of goods this individual can purchase, here public and private goods, given that person's income level and tax price. Illustratively, Figure 6.3 shows the budget constraint as the line PP. Looking at the budget constraint, if government expenditures are higher, consumption of private goods decreases. We assume that individuals will maximize utility;* that is, they will obtain the highest utility possible given their budget constraints. Individuals are willing to give up some private goods if they get more public goods. The quantity of private goods a particular individual is willing to give up to get an extra unit of public goods is that person's marginal rate of substitution. As the individual receives more public goods, the amount of private goods he or she is willing to forego to receive an extra unit of public goods becomes smaller; that is, the individual has a diminishing marginal rate of substitution. Graphically, the marginal rate of substitution is the slope of the indifference curve. As a person consumes more of public goods and less of private goods, the indifference curve becomes flatter.

The highest level of utility for an individual, utility maximization, is the point of tangency between the individual's indifference curve and the budget constraint, denoted in

* Economists sometimes refer to the benefits an individual gets from consumption as the utility that person receives from the combination of goods he or she consumes. The concept of utility is only a useful way of thinking about the benefits that an individual gets from consumption. There is no way of measuring utility (other than indirectly through willingness to pay) since we cannot ascertain what "utility" an individual derives from eating a cookie or listening to the radio.

Indifference Curves

Indifference Curves

Consumption of Public Goods
Quantity of Public Goods Figure 6.3 Individual demand curve for public goods.

Figure 6.3 as E. This point defines the point at which the slope of the indifference curve and the slope of the budget constraint are identical. Intuitively, the slope of the budget constraint indicates how much in private goods the individual must give up in order to realize a gain of one more unit of public goods, which is simply equal to the individual's tax price. The slope of the indifference curve tells us how much the individual is willing to give up to receive one more unit of public goods. We can then use this information to arrive at point E, which is the individual's most preferred point and an indicator of the amount that the individual must be willing to give up to receive one more unit of the public good. As illustrated in Figure 6.3, as the price of the public good (the tax price) is lowered, the individual realizes a shift in the budget constraint from PP to PP', with the individual's preferred point moving from E to E'. As shown in Figure 6.3, this leads to an increase in the individual's demand for public goods.

To trace out the demand curve for public goods, we can lower and raise the tax price. The lower graph in Figure 6.3 shows the quantity of public goods demanded at tax prices PG1 and PG2, which correspond to points E and E'. We could continue this process by shifting the budget constraint further.

Now that we have seen the trade-off between public and private goods through the use of a budget constraint, how do we know how many fireworks to display in total? To derive this result, we will say that Hanzel and Gretal really enjoy fireworks; in fact, both prefer more fireworks to fewer fireworks, other things being equal. We know that the fireworks display contains 29 fireworks and to expand the fireworks display costs an additional $5 per firework. Hanzel says he would be willing to pay an additional $3 for another firework added to the display. Gretal says that she is willing to pay $7 for an additional firework added to the display. Is it efficient to increase the number of fireworks in the display?

To assess this efficiency, we must compare the marginal cost to the marginal benefit. In calculating the marginal benefit, we must remember that this is a nonrival good. Both Hanzel and Gretal can consume the 30th firework added to the display. We can say that given the property of nonrivalry in consumption, the marginal benefit of the 30th firework is the sum of what both Hanzel and Gretal are willing to pay, which is $10. Since we know that the marginal cost of adding a firework is $5, it pays to add the 30th firework to the display. We can generalize this example by saying that if the sum of individuals' willingness to pay for an additional unit of the public good is more than the marginal cost, efficiency requires that the additional unit be supplied. If the marginal costs exceed the sum of the marginal benefits to the individuals, the unit should not be supplied. Efficient provision of the public good requires that the sum of each person's marginal valuation (benefit) for the last unit be equal to the marginal cost of producing that unit.

To graphically show this intuitive result, consider Figure 6.4. The figure shows both Hanzel's demand for fireworks (DHf) and Gretal's demand for fireworks (DGf). The graphical representation shows the price on the vertical axis and the number of fireworks on the horizontal axis. Note that the price that Hanzel is willing to pay ($3) and the price that Gretal is willing to pay ($7) for the 30th firework in the display are both indicated on the vertical axis. Recall that to find the collective demand curve for cookies — the private good — we summated the horizontal axis for each person's demand. Horizontal summation allowed Hanzel and Gretal to consume different quantities of cookies at the same price. For the private good, horizontal summation is fine. In the case of a public good, the services made available by the fireworks must be consumed in equal amounts. If Hanzel consumes a 30-firework display, Gretal must also consume a 30-firework display. Thus, it is not practical to try to summate the quantities of a public good that each individual would consume at a given price. So how do we find the collective willingness to pay for the 30th firework — the public good? We add the prices that both Hanzel and Gretal would be willing to pay for the 30th firework. The bottom graph in Figure 6.4 shows their collective demand (Dh+g ) and the summation of the prices each individual was willing to pay ($10) for the 30th firework to be added to the display.* Vertical summation is appropriate since a pure public good is necessarily provided in the same amount to all

* DH+G is not a conventional aggregated (collective) demand schedule since it does not indicate the quantity demanded at each price. However, for uniformity with the private good case, this notation is useful.

individuals. Rationing is not feasible or desirable, since Hanzel's viewing of the public good does not detract from Gretal's enjoyment of the public good (the fireworks display).

Let us think about the public good demand curve. If we remember that this is each person's willingness to pay for the public good, the demand curve can be thought of as a "marginal willingness to pay" curve.* The public good demand curve says how much the person is willing to pay for an extra unit of the public good. In our fireworks example, Hanzel was willing to pay $3 and Gretal was willing to pay $7 for the additional firework in the display. The vertical summation is just the sum of their willingness to pay or the total amount that both Hanzel and Gretal together are willing to pay for an additional firework to be added to the fireworks display. This is equivalent to finding the total marginal benefit provided by the additional unit of public good because each point on the demand curve of an individual represents that person's marginal rate of substitution at that level of government expenditure. By adding the demand curves vertically, we simply obtain the sum of the marginal rates of substitution (the total marginal benefit provided by the extra unit of public goods). This result is the collective demand curve illustrated in Figure 6.4.

To assess the efficiency of public goods provision, we can add the supply curve as we did in our illustration of the private good (cookies in Figure 6.2). In Figure 6.5, the supply curve has been added to the collective demand curve illustrated in Figure 6.4. Figure 6.5 shows that for each level of output, the price represents how much of the other goods must be foregone to produce one more unit of public goods. At the output level where the collective demand equals the supply, Ec, the sum of the marginal willingness to pay (sum of the marginal rates of substitution) is specifically equal to the marginal cost of production of the public good (marginal rate of transformation).

* Professor Joseph E. Stiglitz offers this interpretation of the public good demand curve.

Fireworks per year
Fireworks per year

I 10

re Fi

Fireworks per year rk

I 10

re Fi

Figure 6.4 Vertical summation of public good demand curves.

Figure 6.5 Efficient production of public goods. This can be demonstrated if we remember that under competition firms produce up to the point in which price equals marginal cost. In this way the supply curve Sc shows the marginal cost of each level of cookie production. In giving consideration to the role of welfare economics, MRTc6 = MCc/MC6. Because P6 = $1 and price equals marginal cost, then MC6 = $1, and MRTc6 = MCc. Thus, we identify the marginal rate of transformation with marginal cost, which is the same as the supply curve.

Figure 6.5 Efficient production of public goods. This can be demonstrated if we remember that under competition firms produce up to the point in which price equals marginal cost. In this way the supply curve Sc shows the marginal cost of each level of cookie production. In giving consideration to the role of welfare economics, MRTc6 = MCc/MC6. Because P6 = $1 and price equals marginal cost, then MC6 = $1, and MRTc6 = MCc. Thus, we identify the marginal rate of transformation with marginal cost, which is the same as the supply curve.

It is at this point, where the sum of the marginal rates of substitution equals the marginal rate of transformation (the intersection of the collective demand and the supply curve), that a public good is Pareto efficient.

Throughout our discussion of public goods provision, the assumption is that we know Hanzel's and Gretal's demand curve for public goods. This assumption is analogous with our construction of the private demand curve, but some distinctions between the two need to be made. We know that market equilibrium occurs at the point where the demand and supply curves intersect. With the public good equilibrium, we have offered little reasoning as to why the supply of public goods should occur at Ec. We know that if this were the production level of public goods it would be Pareto efficient, but we know very little about how this decision (to supply this level of public goods) would occur. Decisions about the provision of public goods are made by governments and not at the individual level. Therefore, the level of production of the public good is predicated on a political process and not on the individuals' desire as in private goods production. In a competitive market for private goods, all individuals face the same price, with the amount of consumption (the quantity desired) reflecting individual preferences for that good. This differs from the pure public good since provision of a public good is at the same quantity to all affected individuals, with our hypothesis that each individual faces a different tax price for access to the public good. Intuitively, let us assume that we could tell everyone what his or her share of the costs of public goods would be. We could say that Hanzel is to bear 5% of the costs, while Gretal is to bear 2% of the costs. Thus, Hanzel would pay $.50 and Gretal would pay $ .20 for an item that costs the government $10.00. This characterization of the Pareto-efficient level of expenditures on public goods corresponds to a specific distribution of income. But the property of nonexcludability introduces a new problem, the free-rider problem.

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