Lecture Notes on Pure and Impure Public Goods by Dale Squires and Niels Vestergaard

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1 This version 24 th January 2011 Lecture Notes on Pure and Impure Public Goods by Dale Squires and Niels Vestergaard 1. General Points About Public Goods Goods that are non rival and non excludable in consumption Private markets generally do not guarantee efficient production. External benefits. Undersupplied and underinvestment Free riding a chronic problem Two general types of market failure may occur and undermine economic efficiency of resource management, even when property rights are secure: (1) supply function has external costs and (2) presence of external benefits leads to free riding and private underprovision of public good. With private provision of public goods, market failure occurs because the amount of a public good is privately underprovided, and thus marginal social benefits exceed marginal social costs. In this case, more of the (public) good should be provided, but it is forthcoming only if society subsidizes a private supplier, or provides it publicly, which is a collective action issue. Public bads are an undesirable public good, and are oversupplied 2. Notation: Q pure public good (non rival and non excludable in consumption), received by each individual, excludable in production, produced at a cost and can be sold at a price to anyone y i private good (rival, excludable) consumed by agent i 1,,n. aggregate endowment of private good is used to produce Q money price of unit of acquisition of Q Price of y = 1 1

2 (y i,q) utility function of individual i. U is continuous, strictly increasing, strictly quasi concave, everywhere twice differentiable with respect to and. exogenous money income, used to either purchase units of private good or acquire units of public good. can also be thought of as initial endowment of the private good that can be contributed to provide the public good. number of units of public good acquired by an individual. Differs from quantity in the utility function, since each individual enjoys the contributions of others. where = quantity contributed by individual h. If is contributed to the public good, the total contribution Q can be thought of as producing the final public good Z through a production function with decreasing returns to scale:, 0, 0. is the rest of the community s contribution, 0 is the transformation function between private and public goods, where is the total endowment of the private good. The opportunity cost for an individual in terms of the private good foregone in acquiring is. A specific form of, 0 is: or for two person economy,. This also provides a resource constraint to the utility maximization problem. 3. Some Preliminaries A good is excludable if people can be excluded from consuming it. A good is nonrival is one person s consumption does not reduce the amount available to other consumers. Rival goods are sometimes called diminishable or depletable. Public goods are not excludable and are nonrival. Club goods are nonrival but excludable. Open access resources are rival but not excludable 4. Conditions Necessary for Pareto-Efficient Allocation of Public Goods The problem is to determine the Pareto optimum amounts of a private good and a public good to be provided. The optimum amount of a private good, i.e. is Paretoefficient allocation, is that it should be provided up to the point at which the marginal cost of production equals the price and the quantity produced equals the amount demanded at that price. Simply put, produce and consume where the supply and demand curves intersect. Demand curves for rivalrous private goods are the horizontal summation of the individual demand curves for private goods. 2

3 For nonrival public goods, interest is in the total marginal willingness to pay for specific amounts of the nonrival good, i.e. at a specific price how much of the good will consumers wish to consume. In contrast to rivalrous private goods, all consumers consume the same amount of the good, which is captured by vertical summation of the consumers demand curves. Since in general the marginal willingness to pay for a public good depends on the amount of private consumption, the efficient level of the public good Q will typically depend on the quantities of the private good consumed by consumers,,. The good is produced in the same way, whether or not it is rival in consumption, so that the same supply or marginal cost curve applies to its production. 3

4 4.1. Graphical Representation of Pareto-Provision of Private and Public Goods Graphically, the Pareto efficient provision of rivalrous private good and nonrivalrous public goods is: P, MWTP Demand, nonrival public good (vertical summation) Marginal cost of supply Demand, rival private good (horizontal summation) y Q Note that more of the Pareto optimum nonrival public good Q is produced than the rivalrous private good y, although that is often not the case Formal Development of Pareto-Efficient Allocation of Private and Public Goods Now we turn our attention to a more formal development of the Pareto efficient allocations of private and public goods. An allocation has the form,,. A Pareto efficient allocation maximizes the utility of any one individual, given that the utility of other individual does not decrease. 4

5 A Pareto efficient allocation,, is a solution to 1 :,,, (1A) Subject to,,, 0 0, 0, 0 (1B) (1C) (1D) Introducing Lagrange multipliers and and assuming an interior solution, the first order necessary conditions are:,, 0 (2), 0 (3) where. The last two equations can be written:, 0 (4),, (5) Divide the three terms in (2) by the first, second or the third term I (5) to give:,,,, (6) Equation (6) can also be written as: (7a) For the specialized transformation function or budget constraint Equation (7A) can be written as:, 1 Note that the superscript * indicates a Pareto efficient allocation for both the private and public goods. 5

6 (7B) Equation (7A) is the fundamental necessary condition for Pareto efficiency allocation/provision of pure public goods, and is referred to as the Samuelson condition. The two person problem of Equation (1) can be rewritten for an n person economy as: n n max {y 1,...,y n,q} {U1 y 1,Q y i p Q Q I i,u h y h,qu h y h*,q * h i}. (8) i1 Manipulation of the first order necessary conditions gives the Samuelson condition: i1 U 1 Q U2 Q U 1 1 y U 2 y U n Q 2 U n y p n Q (9) 1 MRS Qy MRS 2 n Qy MRS Qy p Q. (10) Thus, the provision of a public good occurs to the level at which the marginal rate of transformation between the private and public goods equals the sum over all individuals of the marginal rate of transformation between the private and public goods. measures the marginal cost of providing the public good in terms of the amount of private good foregone. The benefits from the public good are not exclusive to any one individual, so that (aggregate) marginal benefit is obtained by summing over all individuals, giving the left hand side of Equation (9) Relationship between Individual s Provision of Public Good and Rest of Community s Provision of Public Good Individual behavior with public goods can be examined in terms of the relationship between the individual s provision of q and the rest of the community s provision of the public good,. Before starting, note that the reaction function for agent (firm, consumer) i gives the amount that agent i wants to contribute as a function of the other agents contributions (output, consumption levels). The utility function is next written as a function of, to generate a family of indifference curves in, space implied by the individual s preferences. Rearranging the individual s budget constraint to give, 6

7 implying that if the individual provides q of the public good, only of money is available for the private good. Substituting for y in the individual s utility function, allows utility to be defined as a function of the two quantities, :,,, ;, (11) The variables,, while affecting utility, can be specified as exogenously fixed. The individual must decide how much q to consume. The choice depends on how much of the public good is provided by everyone else,. A family of indifference curves in, space for the utility function, is implied by the individual s preferences and is given in Figure 6.1 on page 146 of Cornes and Sandler. 7

8 Q N B Q 3 Q 2 Q 1 q If we assume that preferences are such that for any utility level the consumer wants to spend money on both private goods and public goods, then indifference curves must be U shaped, defining a convex area above each indifference curve. Higher values of imply more preferred allocations. More is needed to compensate for the extremes of all private good consumption or all public good consumption, to keep utility constant. At high levels of public goods provision by others,, it is entirely plausible that no privately provided public goods are necessary. Accordingly, increases to keep utility constant as q rises from zero and indifference curves will be upward sloping in q. The indifference map is truncated by the vertical line B, which corresponds to the value of q that would, by itself, completely exhaust the consumer s budget. The individual s choice of q is determined by the value of. The simplest and most common assumption is that the individual is a quantity taker with respect to. For any given value of a horizontal line can be drawn (not drawn in the above diagram due to the limitations of Word) that represents the feasible set facing the individual. Thus, in the above diagram, there would be three horizontal lines, one corresponding to, one corresponding to, and on to. The tangency between the horizontal line and an indifference curve provides a necessary condition for this tangency point to be an interior optimum, and given the convexity of preferences (indicated by the U shaped indifference curve), this is also a sufficient condition. Parametric variation of generates a locus of best responses to, the curve N in the above figure, and called the individual s reaction curve. The downward slope to the reaction curve reflects the idea that the higher the expected contribution by the rest of the community, the lower the individual s own contribution. 8

9 4.4. Algebraic Characterization of Individual s Indifference Cure to Give Points on the Reaction Curve The actual reaction curve is derived below using a simpler method than the indirect utility function approach of Cornes and Sandler. Movements along any given indifference curve imply unchanged utility. Utility can be expressed in terms of either, or UI p q q,q Q. Therefore, along any indifference curve taking the total differential of the utility function and setting it equal to zero (since utility is unchanged) gives: 0 (12) The slope of an indifference curve is therefore: (13) Quantity taking behavior implies an allocation for which this expression is zero. An increment of q, with Q given, yields a marginal benefit to the individual of. The marginal cost is the quantity of the numeraire good ( y ) that has to be given up. The utility cost of this sacrifice is. The optimum entails equality of marginal cost and marginal benefit, or, which equates the right hand side of Equation (13) to zero. 5. Private Provision of a Public Good: Pareto Inefficiency of a Private Market for a Public Good This section demonstrates that private provision of public goods leads to a suboptimal Nash Cournot equilibrium, so that the corresponding amount of the public good privately provided is less than the Pareto efficient provision of the public good. A general cost benefit approach demonstrates this. The price p Q measures the marginal cost of providing the public good in terms of the amount of private good forgone, i.e. in terms of the public good s opportunity cost. With the Pareto optimal provision of public goods, since the benefits of the public good are not exclusive to any one individual, the marginal benefit is obtained by summing the marginal benefits over all individuals, giving the left hand side of Equation (9) or (10). Equations (9) or (10) thus provide the Pareto optimum quantity of the public good, where the marginal benefit equals the marginal cost. 9

10 The Nash Cournot equilibrium for the private provision of the public good is suboptimal (the amount of public good provided is less than the Paretoefficient allocation), since the benefits of the public good are not exclusive to any one individual. Individuals adjust their own contributions to the public goods independently of their neighbors which adds to the public good only up to the point at which their private marginal rates of transformation, p Q, equal their private marginal rates of substitution. The individual provider of q does not take into account the external benefits flowing to others as a result of an increment of contribution to the public good, so that the private suboptimal amount falls short of the Pareto optimal amount. If the individual could be sure that when they raise their q, everyone else matches them by raising their qs, then consumers see all the advantages of raising and lowering q. Note that the Nash Cournot noncooperative provision of the public good is lower than the Pareto efficient provision, though still positive. The basic reason for this is that when each person chooses q noncooperatively, he/she sees some benefits from buying/providing q, but too much of the benefits occur to others and are ignored in deciding how much to consume. Thus, the market may well provide some public goods, but left to its own devices, the market typically under provides public goods. The price system gives Pareto efficiency for private goods because it ensures that the marginal rate of substitution of every individual equals the marginal rate of transformation. A competitive price system does not yield Pareto efficiency for public good allocations, since the necessary conditions are different. The inefficiency of the private market for a public good can be demonstrated by formalizing it as a noncooperative game and finding the Nash Cournot equilibrium points. A few points to remind you about the Cournot equilibrium. A Cournot equilibrium is the set of production quantities that have the property that no individual firm has an incentive to change its own production level if other firms do not change theirs. Quantities, not prices, are the individuals choice variables. A Cournot equilibrium is not a single price that coordinates supply and demand. Rather, a Cournot equilibrium is the complete set of production levels, one for each firm, that balance their profit interactions in the market. 10

11 Cournot equilibrium. A set of output (consumption) levels {y 1, y 2,..., y n } constitutes a Cournot equilibrium if for each i 1,2,...,n the profit (utility) to firm (consumer) i cannot be increased by changing y i alone. A few points to remind you about a Nash equilibrium. In a Nash equilibrium, no agent s own reward can be increased by a unilateral strategy change. That is, if all other agents strategies are fixed, it does not increase the payoff (utility,profit) to change one s own strategy. This can be viewed as a generalization of the defintion of equilibrium introduced in the context of oligopoly by Cournot. With public goods and two individuals, a Nash equilibrium is a set of contributions {q i,q j },i, j A,B;i j such that each agent contributes an optimal amount, given the contribution of the other agent. The private market structure can be formalized as a noncooperative game and finding the Nash equilibrium points. Consider an economy consisting of two consumers, each of whom may differ in their preferences and income levels, but who face the same price for the public good. Assume that there is one private good y, whose price is again unity. Each individual has the problem: Maximize {y i,q i } {Ui y i,q i q j y i p Q q I i },i, j A,B;i j. (14) There are two implied utility functions,, ;, ), represented by indifference curves in Figure 6.4 on page 154 of Cornes and Sandler. The pair of quantities q ˆ A, q ˆ B is a Nash equilibrium if:, U, for all feasible,,,;. (15) Thus each individual s chosen contribution is a best response to the others. The private market structure can be formalized as a noncooperative game and finding the Nash equilibrium points through solving for the reaction function of agent i. Consider a game with two players, where each player i has utility U i y i,q,i A,B,i j and some endowment of the private good I i. The individual must decide how much to contribute to the public good. If individual i decides to contribute q i, that individual will have y i I i q i of private consumption, Here we have set p Q 1, so that the budget line has a slope of 1. 11

12 Agent 1 s maximization utility maximization problem may be written as: max q1,y 1 U 1 q 1 q 2, y 1 (16) such that q 1 y 1 I 1 and q 1 0. Using the fact that Q q 1 q 2, the problem can be rewritten as: max, U 1 Q y 1 (17) Qy, 1 such that Q y 1 I 1 q 2 and Q q 2 (after adding q2 to the right hand side of both constraints). The second formulation, Equation (17), effectively says that agent 1 is choosing the total amount of the public good subject to his/her budget constraint, and the constraint that the amount he/she chooses must be at least as large as the amount provided by the other person. The budget constraint says that the total value of his/her consumption must equal the value of his/her endowment. 12