The Demand and Supply of Public Goods

Simple Exchange in a World of Equals

In this chapter we shall examine the demand and the supply of public goods in the simplest of models, one in which there are only two persons and two goods, one public and one private. To make simplicity absolute, we assume initially that the two persons are identical, both as to productive capacity and as to tastes. For convenience, we shall name these two persons Tizio and Caio, adding a touch of Italian flavor to the analysis. We may think of these two persons as being the only inhabitants of an island in the tropics. This allows us to use coconuts as the purely private good. Coconuts are available to each person upon a specific outlay of time spent in gathering them, and this outlay per coconut gathered remains constant over relevant quantities. Mosquito repelling is the other good (service), and this is purely public or purely collective. That is to say, the death of one mosquito benefits each man simultaneously, and is thus equally available to each man. The service of mosquito repelling is also continuously variable, and specific quantities can be secured by certain outlays of time on the part of either person. The cost per unit of output remains constant over relevant quantities.

Our purpose is to examine the process through which equilibrium in the demand and the supply of both the private and the public good is attained, and to define the characteristics of this outcome which will tend to emerge from the simplified two-person exchange process.

Independent Adjustment

Examine first the situation in which the two persons act independently, which would be the case if neither Tizio nor Caio recognizes that mosquito repelling activity exhibits publicness. Each would then consider this activity, along with that of gathering coconuts, as purely private, and under the conditions we have assumed (equal tastes, equal productive capacities, constant returns) there would be no incentive to engage in trade. Each man would proceed to reach a wholly private position of equilibrium without trading with the other. The preliminary position sought for by each person would be equivalent to that which would be attained in the one-person world.

Figure 2.1

Each man’s preferences for the two goods can be depicted on an orthodox indifference map which is derived from a standard utility function. This construction for one man is shown in Figure 2.1, on which units of the public good are measured along the vertical axis and units of the private good along the horizontal axis. The opportunities open to the individual are limited by his capacity to locate coconuts on the one hand and his capacity to repel mosquitoes on the other. These opportunities are summarized in the transformation function, which by our simplified assumptions is linear, drawn in Figure 2.1 as
PP. The individual will initially seek to attain position
E. He will fail to reach this point, however, because in his calculus he does not, by our assumption, take into account the publicness of the one good. Because of this publicness, the activities of the two persons will necessarily be interdependent.

In attempting to attain position
E, the person will actually reach
F because his fellow will be making an outlay on mosquito repelling precisely equivalent to his own. Since, by definition, the public good or service is equally available to both persons, no matter by whom produced, the individual will find himself with a bundle that contains double the amount of the public good that he anticipated in making his initial decision to commit resources.

F is not a position of final equilibrium, however, except under the highly restrictive condition where the income elasticity of demand for the private good is zero. Finding himself at
F, the individual will consider it advantageous to change his plans. He will treat the newly found public good as a simple increase in his opportunities, in his real income, although the rate at which he can change one good into the other will not be modified. In making new plans, the individual will try to adjust to his apparent transformation curve
P’P.

Normally, he will reduce somewhat his production of mosquito repelling and expand his production of the private good, coconuts. In the extreme case where the income elasticity of the public good is zero, he would seek to attain an adjusted position at
G. If both goods exhibit positive income elasticity, the second sought-for position will fall somewhere between
F and
G. For simplicity, assume that the income elasticity of the public good (and, in the two-good model, for the private good also) is unitary. In this case, the second-round objective under wholly independent adjustment would be shown by
H.

This position will not be attained and for the same reason that
E was not attained; the activities of the two persons are interdependent. Adjustments will continue to take place until a position at
E* is reached, which will represent one of final equilibrium under wholly independent behavior. Note that, geometrically,
E* is located where
BC is equal to
CE*. By our assumption of unitary income elasticity the position of equilibrium is located along the ray
EE*
H.

In describing the adjustment toward this final equilibrium we have assumed that the publicness of the one good remains wholly hidden from the individuals in the model. This insures that there is no strategic behavior in the adjustment process. Tizio does not recognize that Caio’s efforts provide him with benefits; therefore, he has no incentive to modify his own behavior in the hope of securing more of the external economies.

Under the conditions assumed and with the utility function as depicted in Figure 2.1, the introduction of strategic behavior on the part of one or both of the persons will not modify the location of the final equilibrium position. This is insured by the fact that the position of equilibrium,
E*, lies on a higher utility level than
G, the extreme position that might be sought, and potentially attained, by one of the two individuals who behaves strategically. As the construction makes clear, however, this ordinal relationship between
E* and
G need not be present, even in the two-person model. If this relationship is reversed, and if one of the persons succeeds in reaching
G while the other remains in
E, a nonsymmetrical equilibrium of sorts is achieved. Although the active strategist will not be in full marginal adjustment, he will recognize that some concealing of his true preferences remains optimal.

The construction of Figure 2.1 can be used to demonstrate that the independent-adjustment equilibrium is nonoptimal in the Pareto sense. Both persons adjust to the apparent production-possibility curve through
E* parallel to
PP. Under genuine joint or cooperative behavior, the actual production-possibility curve faced by each person is shown by
PP**. Although the individual cannot act independently on the basis of this production-possibility set, simultaneous action on the part of both persons will allow each to move along
PP**, finally attaining the optimal position,
E**. The next section discusses the attainment of this full equilibrium under exchange agreements.

Trading Equilibrium

The characteristics of any equilibrium depend upon the institutions under which the private behavior of individuals takes place. In the initial model, all behavior was assumed to be independent; no exchange or trade, no mutual agreement, no negotiation or bargaining, were allowed. If these restrictions are dropped and the rules or institutions changed so as to allow personal interaction, the position attained under wholly independent adjustments will not remain one of equilibrium.

Each man will now recognize that mosquito repelling is a genuinely collective activity, and that there exist unexploited mutual gains from some trading arrangements that insure a larger total outlay on the provision of this service. Simple two-person, two-commodity trade is impossible, however, since both men enjoy identical quantities of the public good. What can be traded or exchanged here is some
agreement on the part of each man to contribute working time (labor) toward the production of the collective good, in this example, mosquito control. Tizio can “buy” Caio’s agreement to kill mosquitos (1) by agreeing to kill mosquitos himself, and/or (2) by transferring to Caio a quantity of coconuts, the purely private good. The two alternatives will be wholly indifferent to both men under the simplified conditions postulated. If the two men should differ in productive capacity, however, or if there should be returns to scale in the production of either good, comparative advantage in the ordinary sense would determine the efficient trading arrangements. Should Tizio be relatively more efficient in locating coconuts, he would spend all of his time in this way, and then he would “purchase” the public good solely through maintaining Caio’s private-goods consumption. Should Caio, by contrast, be relatively more efficient in coconut gathering, he would provide some private-goods subsistence for Tizio, while the latter carries out the public activity of killing mosquitoes.

Figure 2.2

The process through which trading equilibrium comes to be established may be shown in Figure 2.2, which is an Edgeworth-box diagram converted for current purposes. Here we measure Tizio’s labor time spent in gathering coconuts
for his own consumption on the horizontal axis, and Caio’s time spent in gathering coconuts for his own consumption on the vertical axis. We assume that each man has available a fixed quantity of labor time to devote to the production of goods, public or private, and that this time is identical for each man. In effect, we assume that leisure is not a variable in the model. In the orthodox sense, the origin for Tizio is at 0 , that for Caio at 0′ .

We now define point
A as that position attained under the wholly independent adjustment process previously discussed. Hence, 0
A is the amount of time that Tizio spends in gathering coconuts in the private-adjustment equilibrium; similarly, 0′
A is the time Caio spends on the same activity. Confronted with the private production-possibility curves indicated by
P, both persons are in equilibrium at
A.

With this construction it becomes possible to generate an indifference map for each man that will indicate his tastes for the public good and the private good, but in such a way that exchange can be analyzed. The individual’s evaluation of the public good can be considered as an indirect evaluation of his fellow’s labor time spent in producing the good. Any upward vertical movement in Figure 2.2 represents, for Tizio, an increase in the quantity of the public good supplied because, as Caio gives up gathering coconuts for his own use, he must either (1) devote his time to mosquito control, or (2) gather coconuts for Tizio’s use. Similarly, any leftward horizontal movement on Figure 2.2 represents, for Caio, an increase in the quantity of the public good that is supplied to him (as well as to Tizio). Mutually beneficial exchange can obviously take place so long as the movement from
A is in the general northwesterly direction. The position of trading equilibrium will be located at some point along the contract locus,
JK, in Figure 2.2. At this final equilibrium, both Tizio and Caio will be giving up a specific amount of their own time to the production, directly or indirectly, of the public good. And more of the public good and less of the private good will be supplied than at
A.

Bargaining strength and luck may, of course, determine the shares of the two men in public-goods production, within limits. Since mutual gains are secured in the shift from the no-trade position at
A to a position on the contract locus, there exist many possible distributions of these gains over inframarginal ranges. This may, because of income effects, generate slight differences in the quantity of the public good, but these can be neglected here.

We may now examine carefully the characteristics of the position of full trading equilibrium; that is, any point on the contract locus,
JK in Figure 2.2, where trade has stopped and all further prospects for mutual gains are eliminated. By the standard geometry, we know that the indifference curves of the two traders are tangent; in this respect the position is similar to that reached when trade takes place in purely private goods. This tangency condition indicates that the marginal rates of substitution between the two items traded are equal for the two persons. Let us define these marginal rates of substitution precisely.

Tizio is giving up units of private good, coconuts, in exchange for units of public good, mosquito repellent, as the latter is reflected in Caio’s willingness to “supply” the second good, either through his own labor or through providing Tizio with subsistence. Caio is in a similar position on the other side of the exchange. There seems to be something wrong here, however, since
both men value the public good, and
both must adjust to the
same quantity, by definition. Something different from simple two-person, two-commodity trade must be taking place. The mystery here, if indeed there is one, is resolved when we recognize that all exchange is two-sided. If there is a demander there must also be a supplier. Hence, one or both of the two traders in our model must be supplying the public good or service to the other who is demanding it. Let us continue, for now, to assume that there is no comparative advantage, that each man produces an equal share of the public good that is jointly consumed by both. Each man, therefore, is “supplying” units of the public good to the other, in exchange for a similar supply on the part of his trading partner. Equilibrium is defined by the standard equivalence between marginal rates of substitution. But what this definition masks, in its simple form, is the evaluation that each man himself places on the public good that he himself supplies to the other. In this setting, Tizio is supplying Caio with units of public good, but in the process, he is also supplying himself. His marginal rate of substitution is a summation of two separate components. He must consider his own marginal evaluation of the public good, purely as a consumption item, plus his negative marginal evaluation of the same good as this arises from his share of the supply or production cost.

This somewhat particularized interpretation of trading equilibrium is made necessary by the publicness of one of the traded goods. The analysis may be clarified if we assume that one of the two traders does possess a comparative advantage over the other in producing the public good. Let us suppose that Caio can produce mosquito repellent at a relative advantage over Tizio. The trading process will then lead to Caio supplying all of the public good and receiving from Tizio a certain quantity of the private good in order to maintain his own consumption of the latter. In full trading equilibrium, Tizio’s standard marginal rate of substitution in consumption between the two goods will be equated to Caio’s marginal rate of substitution
in exchange. The latter will include two components, Caio’s own marginal rate of substitution in consumption between the private and the public good, and his own marginal rate of substitution between the two in production. This point will be further clarified in the simple algebraic treatment of the model in the next section.

Algebraic Statement of Trading Equilibrium

The simple Tizio-Caio model of two-person, two-good trade when one of the two goods is purely public can now be discussed with elementary algebraic tools. Any complexities that arise in this section will be clarified in subsequent discussion. Essentially the same formal analysis introduced here is again presented for the more general case at the end of Chapter 4.

Tizio’s utility function is defined as,

(1)

where
X₁ is the private good (coconuts) and
X₂ is the public good (mosquito repellent). Superscripts designate the person who produces the goods in question, directly or indirectly. Caio’s utility function is defined in the same way as,

(2)

For simplicity, we continue to assume that each man devotes a fixed amount of labor input to total goods production, public and private.

Each man will confront a transformation function indicating the rate at which the private good can be converted into the public good, and
vice versa, through his own behavior. These transformation functions are,

(3) (4)

If each man acts independently, and no trade takes place, equilibrium will finally come to be attained when the conditions indicated in (5) and (6) below are met. In writing these conditions, we adopt the convention of using lower-case
u‘s and
f ‘s to indicate the partial derivatives of the utility and transformation functions respectively, with goods noted in the subscripts and persons in the superscripts. Thus,

is written as

	(5) (6)

These conditions are the standard ones for individual marginal adjustment; each person modifies his own behavior so long as the marginal rate of substitution in consumption differs from his marginal rate of transformation.

We now want to see why trade takes place, how it takes place, and what equilibrium will tend to emerge.

We know from the definition of a public good that a unit of
x₂ produced and consumed by Tizio is valued by Caio to the same extent that he values a unit of his own production. Similarly, for Tizio’s evaluation of a unit produced and consumed by Caio. This guarantees that, in the no-trade equilibrium, Tizio’s activity in producing the public good exerts a Pareto-relevant external economy on Caio, whereas Caio’s activity in producing the public good exerts a similar externality on Tizio. Each person values the producing activity of the other at some value greater than zero in the no-trade equilibrium. No value will be placed by either man on the production of private goods by the other. In algebraic language, these conditions may be stated,

(7) (8)

Each person places a positive value on the marginal extension of public-goods production by the other. Each will, therefore, be willing to “pay for” this extension, and, in response, each will stand willing to extend his own production for any receipt above zero. Trade will, of course, take place under such conditions, and will continue until (9) and (10) below are satisfied.

(9) (10)

As stated in (9) and (10), the conditions are fully general for two-person, two-good exchange, and these same statements encompass any degree of externality or “publicness” in
x₂. For example, suppose that
x₂ has been erroneously labeled as being purely public when, in fact, both Tizio and Caio consider it to be purely private. In this case, the left-hand terms in (9) and (10) become zero; the no-trade position is restored. Trade will not emerge under the restricted conditions of this example where the two persons are identical with respect to tastes and productive capacities and where production functions are constrained. As a second example, suppose that
x₂ is only partially public; that is to say, Tizio values his own mosquito repelling activity more than he does the similar activity of Caio, although he places some positive valuation on the latter. Conditions (9) and (10) are not modified; they remain those that must be met in full trading equilibrium. Or, to take a less familiar variation, suppose that
both x₁ and
x₂ are purely collective. Conditions (9) and (10) continue to define equilibrium in the two-person, two-good case. However, as we shall introduce at a later point, the generalization here to three or more persons becomes different from that in models where at least one purely private good exists.

If we postulate at the outset that one of the two goods is purely public, as we have done in this chapter, it becomes possible to simplify greatly the statement of the necessary conditions for equilibrium. This simplification has been implicit in most of the statements made by those scholars who have been instrumental in developing the modern theory of public goods. When
x₂ is known to be purely public, these necessary conditions can be reduced to (9) alone if the assumption is made that only one of the two persons produces the public good. Suppose that Caio actually produces this good, and that Tizio pays him for the appropriately determined share through a transfer of private goods. This allows us to transpose and to drop the now unnecessary sub- and superscripts to get,

(9A)

which can be readily recognized as the familiar definition of the conditions for public-goods optimality, as presented by Paul A. Samuelson and others. The summed marginal rates of substitution between the public good and the private good must be equal to the marginal rate of transformation, or, somewhat loosely, marginal cost.

Note that, as these have been discussed here, the conditions (9), (9A) and (10) have not been explicitly connected with “optimality” or “efficiency.” These conditions are presented as those which allow us to define the characteristics of an equilibrium position, one that will tend to emerge from a two-person trading process. Until and unless these are satisfied, mutual gains from further trade can be shown to exist. In such situations trade will take place, provided that we ignore, as we shall throughout most of the elementary analysis, the costs of negotiating market agreements themselves.

Some Marshallian Geometry

One of Professor Frank Knight’s favorite quotations is from Herbert Spencer’s Preface to the
Data of Ethics: “Only by varied iteration can alien conceptions be forced on reluctant minds.” Since the analysis attempted here qualifies as alien, at least to some degree and to some students, I shall heed Spencer’s advice, even at the expense of redundancy. Having presented the theory of simple exchange in one of the most sophisticated of the economist’s several languages, I shall now discuss the same material with more mundane tools. Some rigor is necessarily lost in the process, and the logic becomes imperfect in its details. Elementally, however, the principles that emerge are not modified, and considerable gain may be registered toward genuinely intuitive understanding of the exchange process.

Figure 2.3

The tools that are most familiar to traditional micro-economists are the geometrical constructions of Marshallian demand and supply, and these can be employed here in analyzing trade or exchange in the mixed world that contains both private and public goods. For expositional simplicity, it is necessary to neglect income-effect feedbacks on individual marginal evaluations of the public good. We assume continuous variation in the quantities of the two goods. Under these conditions, it becomes possible to derive a single marginal evaluation schedule or curve for the public good, measured in units of the private good, a schedule or curve that will not shift as a result of changes in the distributions of the net gains-from-trade in the public good. Such a curve is plotted as
E in Figure 2.3. Because of our assumption that Tizio and Caio are identical with respect to both tastes and productive capacities, the construction is simplified greatly. This allows us to utilize the same marginal evaluation curve for each person. We can also draw in a curve that measures the marginal cost of producing the public good. For simplicity, we assume this to be uniform over varying quantities; this is drawn as
MC in Figure 2.3.

In complete independence of the other person’s activity, Tizio and Caio would each aim initially at reaching the position shown at
A. In trying to do so, however, each would find himself at
A”, where double the amount of public good anticipated is available for his own consumption. At this juncture each person will have a strong incentive to cut back on his own production of the public good. This is because, at the consumption margin, the marginal evaluation placed on the good falls below the marginal cost of producing it. If action takes place instantaneously, costlessly and simultaneously, we could expect both persons to cease production, each expecting the other to provide the public good in the desired quantity. Under these extreme conditions, we should expect a cyclical pattern of behavior, between no production of the public good and an excessive amount. It seems reasonable here to make the model somewhat less restrictive by assuming that there will be some departure, however slight, from absolute simultaneity in adjustment.