Public Goods

1. Public Goods Chapter 22

2. Introduction • Previous chapters generally considered only private goods • Commodities consumed individually by consumers • One consumer’s consumption of such a commodity precludes other consumers’ consumption • This chapter considers nonrival commodities • One consumer’s consumption of a commodity does not preclude other consumers’ consumption • Called public goods • Goods where there is no congestion • For example, one consumer’s satisfaction from breathing clean air does not rival (compete with) other consumers gaining pleasure from also breathing the air

3. Introduction • We define public and private goods in terms of their rivalry and exclusive characteristics • We investigate exclusive but nonrival commodities and condition for renting instead of selling nonrival commodities • We address free-rider problem associated with public goods in a game-theory framework • Develop Pareto-efficient conditions for allocating public goods • We discuss how to obtain Pareto-efficient allocation when markets can be developed to establish prices for public goods (called Lindahl prices) • We discuss Clarke tax • Provides a second-best Pareto-efficient mechanism for allocating public goods

4. Introduction • Aim in chapter is to understand distinguishing characteristics between private and public goods • Why a free market (decentralized control) will not result in a Pareto-efficient allocation of public goods • Leaves applied economists with task of developing various mechanisms for determining optimal allocation of resources to production of public goods

5. Rivalry and Exclusive-Good Characteristics • Public good: nonrival commodity • One consumer’s consumption does not reduce amount available to other consumers • Exists when marginal cost of another consumer’s consuming commodity is zero • Private good: rival commodity • Depletable or diminishable commodity • Each additional unit consumed by one consumer results in less of commodity available for other consumers • For a rival commodity, congestion is so severe only one consumer can consume commodity • Both public and private goods are further classified based on their exclusiveness • Exclusive commodity • Other consumers can be excluded from consuming the commodity • Nonexclusive commodity • Either it is illegal to exclude other consumers from consuming the commodity or cost of exclusion is prohibitive

6. Rivalry and Exclusive-Good Characteristics • Pure private goods are distinguished from private goods by their exclusiveness • Private goods are all rival commodities • Pure private goods have additional criterion of being exclusive • Similarly, public goods are all nonrival commodities • Pure public goods are also nonexclusive • If consumption of a nonexclusive commodity also does not deplete commodity for other consumers (a nondepletable commodity) • Commodity is a pure public good

7. Rivalry and Exclusive-Good Characteristics • Pure public goods are a specific type of externality that affects all consumers in an economy • If one consumer is to consume a certain amount of a pure public good • Then all consumers will consume that same level • Examples of commodities possessing characteristics of rivalry and exclusiveness are provided in Table 22.1 • An exclusive commodity, such as food, with high congestion costs associated with rivalry is a pure private good • Cost of an additional unit of commodity is nonzero • In a free-market economy, such pure private commodities are generally provided by firms

8. Table 22.1 Rivalry and Exclusiveness in Commodities

9. Rivalry and Exclusive-Good Characteristics • Table 22.1 also shows that a private good, such as fire protection, can also be a nonexclusive commodity • In contrast, public goods with zero congestion costs are nonrival • An additional consumer does not add any additional cost for providing commodities • For exclusive public goods, consumers can be excluded unless, for example, they pay an entrance price (subscription fee, toll, or ticket) • And marginal cost of an additional subscriber is zero • Exclusion can also be based on some nonprice criteria • Such as gender, race, national origin, or social status • However, in U.S. such criteria are generally illegal • In free markets, public goods can be privately produced by firms (such as movie theaters) • Or publicly produced by government agencies (such as toll roads and bridges) • Nonexclusive pure public goods (such as street lights) are generally provided solely by government agencies

10. Rivalry and Exclusive-Good Characteristics • Distinctions among rival, nonrival, exclusive, and nonexclusive commodities are in terms of degree to which a commodity falls in one category or another • For example, fire and police protection could be considered pure public goods for a community • If protection is at a level where marginal cost of an additional consumer is near zero • As number of consumers increases with an associated increase in marginal cost • Fire and police protection would then become more rival commodities • Increased highway congestion at some point will raise marginal cost of highway commuting • Increases degree of rivalry on roads (Labor Day Weekend) • Finally, consumer preferences for a public commodity are not always positive • Bad public commodities (called public bads) also exist • Examples are environmental degradation (including air and water pollution), global warming, and species extinction

11. Nonrival but Exclusive Commodities (Public Goods) • Nonrival characteristics of certain commodities allow • Public libraries to share books and electronic media • Video outlets to share (for some rental fee) videos and electronic games • Equipment rental firms to rent a variety of items from backhoes to party supplies

12. Nonrival but Exclusive Commodities (Public Goods) • Consider case of a firm offering commodity assuming no sharing exists • Let p(q) be inverse demand function with associated constant marginal and average cost of c • Firm’s profit-maximizing problem is • If consumers rent commodity instead of purchasing it • Level of consumption, y, will be greater than level of production • For example, more videos will be watched than produced

13. Nonrival but Exclusive Commodities (Public Goods) • Let  be number of times each commodity is shared by a consumer • Level of consumption is y = q • Assuming all commodities are only rented, rental price would be p(y) • Along with some positive transactions cost, ct, of renting versus owning commodity • Purchasing a commodity reduces transaction costs involved with renting • Major reason why many households purchase rather than rent lawn mowers • Marginal consumers, with zero consumer surplus, would be willing to pay p(y) to rent commodity minus this transactions cost, p(y) - ct

14. Nonrival but Exclusive Commodities (Public Goods) • Further assuming that there are  of these marginal consumers sharing one commodity • Per-unit price firm receives is  times p(y) - ct • [p(y) – ct] = [p(q) – ct] • Firm’s profit-maximizing problem for renting commodity is

15. Nonrival but Exclusive Commodities (Public Goods) • Comparing this profit-maximizing problem for producing and then renting output versus maximizing problem associated with producing and then selling output • Only difference is in marginal costs • Marginal cost for selling commodity is c • Marginal cost associated with producing commodity for renting is [(c/) + ct] • Profits will be higher for renting if marginal cost for production associated with renting is lower than marginal cost for selling • c/ + ct < c • Illustrated in Figure 22.1 • Assume demand for renting is same as for purchasing • Both can be represented by same demand curve • For profit maximization, firm will equate MR to MC

16. Figure 22.1 Renting versus selling a product

17. Nonrival but Exclusive Commodities (Public Goods) • Marginal cost for renting is lower than marginal cost for selling • Firm will set a rental price of pR*and y* of commodity will be consumed • If firm sold product, price and quantity sold would be p* and q* • If marginal cost for renting is less than marginal cost for selling • Profits from renting π[(c/) + ct], are higher than for selling, π(c)

18. Nonrival but Exclusive Commodities (Public Goods) • The more a product is characterized as nonrival • The higher will be the number of times each product is shared with consumers,  • As  increases, marginal cost of renting falls relative to selling • Enhances profitability of renting versus selling product • As  approaches infinity, condition for renting instead of selling commodity is ct < c • Firm will then rent commodity instead of selling it • When marginal cost of production is greater than consumers’ transactions costs of renting instead of owning • For example, transactions cost for newly released videos is relatively low compared with marginal cost • Consumers generally will rent new releases • In contrast, marginal selling cost of used videos (salvage value) is relatively low • Consumers may instead purchase video rather than rent it

19. Free-Rider Problem • Nondepletable and nonexclusive characteristics of a pure public good (such as PBS) • Result in each consumer’s purchase of commodity providing utility • Not only directly to this consumer but also to all other consumers • If consumers only consider their own utility in purchasing commodity and not effects such purchases have on all other consumers • Externalities are present • Consumers have an incentive to let other consumers purchase public good and receive utility from it without any cost • Called free-rider problem

20. Free-Rider Problem • A free rider is a consumer who cannot be excluded from receiving benefits of a nondepletable commodity • But is unwilling to pay his portion of cost (a non-PBS member viewing a PBS program) • By not cooperating and paying his portion of cost associated with public good, free rider gains • Another form of Prisoners’ Dilemma game • As an illustration, assume two roommates with public good of a clean kitchen (Table 22.2) • If they cooperate and both agree to share in cleaning, their payoffs are 50 each • Payoffs could be in monetary units or some other measurement • However, by not cooperating, becoming a free rider, and letting the other roommate clean • Free rider can increase her payoff from 50 to 120 • Nash-equilibrium result is both attempting to be free riders (not cooperating) • Results in a dirty kitchen

21. Table 22.2 Free-Rider Problem for a Clean Kitchen as the Public Good

22. Free-Rider Problem • Generally, in case of a small number of agents with limited or no transaction costs • Coase Theorem applies and this externality problem is resolved • However as number of agents increases, a Coasian solution is generally not possible • With a large number of agents, it is generally easy to be a free rider

23. Pareto-Efficient Conditions for Pure Public Goods • Efficient allocation of a pure public good • Sum of each agent’s willingness-to-pay is equal to cost of public good • Recall that a Pareto-efficient allocation condition for consumer 1 considering purchasing commodities x1 and y is • MRPT denotes marginal rate of product transformation • MRS is marginal rate of substitution

24. Pareto-Efficient Conditions for Pure Public Goods • Letting x1 and y be a private good and a pure public good, respectively • MRS1(x1 for y) is how much consumer 1 is willing to sacrifice of the private good, x1, for one more unit of pure public good, y • MRS1(x1 for y) is consumer 1’s maximum willingness-to-pay, or reservation price, for pure public good • However, condition does not consider externalities associated with pure public good • With these externalities, society’s maximum willingness-to-pay (MRSS) is higher • Given that pure public good y provides same positive benefits to other consumers • MRPT(x1 for y) = MRS1(x1 for y) < MRSS(x1 for y) • Level of pure public good provided in a perfectly competitive market is below socially efficient solution

25. Pareto-Efficient Conditions for Pure Public Goods • Can develop Pareto-efficient condition for a pure public good supplanting inefficient condition, MRPT = MRS1 • By considering a two-consumer economy with purchasing decisions of x1 and y • Let y be amount of pure public good and x1 and x2 be amounts of private good associated with consumers 1 and 2, respectively • There is no subscript on y • Both consumers consume same amount of y, and y is nondepletable • However, they may consume different amounts of private commodity • So x1 + x2 = Q • Where Q is total amount of commodity produced

26. Pareto-Efficient Conditions for Pure Public Goods • Will describe technological possibilities of this economy by production possibilities frontier, f(Q, y) = 0 • Welfare-maximization problem is • Where U1 and U2 are utility functions for consumers 1 and 2, respectively • Ц is some social-welfare function • Forming the Lagrangian

27. Pareto-Efficient Conditions for Pure Public Goods • F.O.C.s are • Q/x1 = Q/x2 = 1, given Q = x1 + x2

28. Pareto-Efficient Conditions for Pure Public Goods • Solving for Lagrangian multiplier and equating yields • The last equality establishes

29. Pareto-Efficient Conditions for Pure Public Goods • Marginal gain in welfare associated with additional consumption of commodity Q by a consumer must be equal for all consumers • If this is not the case, it would be possible to reallocate Q among consumers in a way that increases social welfare • Cross-multiplying first equality yields

30. Pareto-Efficient Conditions for Pure Public Goods • First term is consumer 1’s MRS1(x1 for y) • Second is consumer 2’s MRS2(x2 for y) • Term on right-hand side is MRPT(Q for y) between public and private good • Thus, condition for Pareto efficiency is • MRS1 + MRS2 = MRPT

31. Pareto-Efficient Conditions for Pure Public Goods • Instead of perfectly competitive condition, MRS1 = MRS2 = … = MRSn = MRPT for n consumers • Pareto-efficient condition is • Sum of willingness-to-pay (MRSS) equated to cost (MRPT) results in a Pareto-efficient allocation of pure public good • An example is if a home theater system costs $5000 and 100 sorority sisters are each willing to pay $50 • Individually, no one sister would purchase the system • But collectively Pareto-efficient response would be to purchase it • MRSS is sum of individual consumers’ MRS • Accounts for benefits all consumers receive from pure public good • Equate MRSS to MRPT to determine Pareto-efficient level of resource allocation • Individual consumers’ MRS(xj for y) are each consumer’s reservation price • Maximum willingness-to-pay for pure public good

32. Pareto-Efficient Conditions for Pure Public Goods • Can relate concept of MRS(xj for y) as a consumer’s reservation price for y to market price for y, py • By letting price of Q be a numeraire, so pQ = 1 • For utility maximization consumer sets MRS(xj for y) = py/pQ • For pQ = 1, a consumer’s reservation price is equal to market price, MRS(xj for y) = py • Thus, for a pure public good, summing reservation prices yields total per-unit price society is willing to pay for pure public good y • Equating total per-unit price to cost of supplying one more unit, MRPT(Q for y), yields a Pareto-efficient allocation

33. Pareto-Efficient Conditions for Pure Public Goods • Consumers paying their reservation price per unit for pure public good is one Pareto-efficient outcome • Yields a marked distinction for efficiency between private and pure public goods • For a private good, all consumers consume different amounts of commodity but pay same market price • For a pure public good all consumers consume same amount of commodity (say, national defense) but pay different prices • Illustrated in Figures 22.2 and 22.3

34. Figure 22.2 Horizontal summation of the demand curves for a private good

35. Pareto-Efficient Conditions for Pure Public Goods • Market demand curve for a private good is horizontal summation of individual consumers’ demand curves for private good • Treating reservation price, MRS(y for xj), as price of private good Q, Pareto-efficient allocation is for both consumers to pay the same price • MRS1(y for x1) = MRS2(y for x2) = MRPT(y for Q) • From Figure 22.2, at this price, 10 and 8 units of Q are demanded by consumers 1 and 2, respectively • Yields a total market demand of 18 • Individual demand curves are based on preference orderings of consumers • Represented by indifference curves

36. Pareto-Efficient Conditions for Pure Public Goods • For a public commodity, each consumer consumes same amount of commodity but at a different price • Derive market demand curve by vertically summing individual consumers’ demand curves • Shown in Figure 22.3 • Consumer 1’s MRS1(x1 for y) is 4 and consumer 2’s MRS2(x2 for y) is 1 • Pareto-efficient allocation is where • MRS1(x1 for y) + MRS2(x2 for y) = MRPT(Q for y) • If price of Q is numeraire, pQ = 1, then ratio py/pQ = 4 = MRS1(x1 for y) and py/pQ = 1 = MRS2(x2 for y) • Consumer 1 pays $4 per unit for pure public good and consumer 2 pays $1 • But they each consume the same amount

37. Figure 22.3 Vertical summation of the demand curves for a pure public good

38. Pareto-Efficient Conditions for Pure Public Goods • One solution for inefficiency of perfectly competitive markets in providing for pure public goods • Establish another market that will account for externalities associated with public goods • Offered such a solution in Chapter 21 • Market for permits could be established to yield a second-best Pareto-efficient allocation • Solution may or may not be feasible, depending on nature of inefficiency • A market solution works well when a commodity can be segmented • Proportion of property rights for commodity can be transferred from public to a private agent • Market-permit system is one case where this market solution can be feasible • Permits transfer a proportion of a common property commodity to a private agent • However, it is not as attractive for correcting public-goods allocation problem

39. Pareto-Efficient Conditions for Pure Public Goods • Nonrival and nonexclusive characteristics of a pure public good prevent segmenting of commodity • For example, an individual household cannot purchase a proportion of national defense • To establish such a market (a Lindahl market) for a pure public good • Each consumer would have to voluntarily reveal and pay their reservation price (their Lindahl price) per unit for a pure public good • Summing reservation prices and equating sum to MRPT would determine efficient allocation of pure public good • Such markets generally are not feasible • Mainly as a consequence of free-rider problem

40. Pareto-Efficient Conditions for Pure Public Goods • Consumers’ dominant strategy • Understate their preferences (by discounting their Lindahl prices) and rely on other consumers to pay a larger share for pure public goods • Nonexclusive characteristic of pure public goods fosters this free-rider strategy • One solution to free-rider problem • For government to impose a per-unit tax on each consumer equivalent to their respective Lindahl prices • However, unlike reservation prices for private goods, Lindahl prices are not revealed in market • Consumers cannot adjust their level of consumption unilaterally • Destroys possibility of a market for pure public goods • Government agency has no feasible mechanism for determining each consumer’s willingness-to-pay • To impose such a tax government agency must perfectly price discriminate among consumers • Such systems are difficult to achieve

41. Pareto-Efficient Conditions for Pure Public Goods • Even if it were feasible to determine consumers’ Lindahl prices and perfectly price discriminate • Consumers may object to paying differentially per unit for pure public goods • May be more inclined to support funding for pure public goods based on ability to pay rather than willingness-to-pay • Many public health and housing agencies base fees and rents on ability to pay • In general, pure public goods are financed by taxes based on income and wealth • As opposed to decentralized control for allocation of private goods • Some type of centralized control is required for public-goods allocations • Determination of types, amounts, and funding for pure public goods may then be based on some mechanism design

42. Pareto-Efficient Conditions for Pure Public Goods • In general, such mechanism designs attempt to determine intensities of individual and group desires • And formulate a mechanism composed of policies and rules for group choice and actions • Clarke tax is one such mechanism • Under some rather restrictive conditions, tax provides incentives for consumers to reveal their true preferences for a social choice

43. Clarke Tax • Eliciting truthful preferences for pure public goods can mitigate misallocation of governments’ taxing, spending, and regulatory authorities • By overcoming free-rider problem • Proponents of Clarke tax mechanism claim, based on a second-bid auction, tax mechanism will not completely cure free-rider problem by yielding a Pareto-efficient allocation • But has potential to treat the symptoms

44. Clarke Tax • As an illustration, consider a group of consumers jointly deciding on purchase of a pure public good • If purchased, every consumer will pay a predetermined amount for purchasing the good • An example is an appliance such as a microwave oven in a dormitory • Let cj be this predetermined amount for consumer j • Summing over all consumers equals cost of pure public good • Consumers will then state how much they are willing to pay • Difference between consumer j’s willingness-to-pay, WTPj, and predetermined amount is net benefit, NBj • NBj = WTPj - cj • If sum of net benefits over all consumers is positive • Then pure public good should be purchased

45. Clarke Tax • Problem is designing a mechanism that provides an incentive for consumers to reveal their true net benefit • Instead of revealing an exaggerated figure in an attempt to influence this social choice • However, an exaggeration is only of concern if it affects social choice • For example, say consumer j attempted to be a free rider by stating a value of zero • Yielding a net benefit of -cj • If sum of net benefits over all consumers is still positive • Free rider does not influence social choice, so it is of no concern • Only consumers whose exaggeration will affect social choice are of concern • Such consumers are called pivotal consumers • Their net benefit determines whether sum of net benefits is positive or negative • In the extreme, all consumers could be pivotal consumers or none could be pivotal

46. Clarke Tax • It is possible that any one consumer could be pivotal • Ensuring that all potential pivotal consumers have the right incentives corresponds to ensuring that all consumers reveal their true preferences • When a social choice is changed by a pivotal consumer • Adversely affects other consumers • For example, if other consumers have positive net benefits for a pure public good and pivotal consumer’s negative net benefit resulted in not purchasing good • All other consumers are made worse off • A measure for how much other consumers, in aggregate, are worse off • Sum of net benefits excluding pivotal consumer, say, consumer 1

47. Clarke Tax • If other consumers have negative net benefits for pure public good and pivotal consumer’s positive net benefit resulted in purchasing commodity • All other consumers are again made worse off • Measure for how much other consumers, in aggregate, are worse off is the negative of the sum of net benefits excluding pivotal consumer, say, consumer 1 • Analogous to imposing a Pigouvian tax on negative externalities, pivotal consumer is taxed by amount he or she harms other consumers, 1 • Called a Clarke tax • Which is paid by all pivotal consumers • Results in these consumers having incentive to reveal their true preferences for pure public good

48. Clarke Tax • Clarke tax mechanism is a second-bid, sealed bid auction for a pure public good • A pivotal consumer’s tax is equal to second-highest valuation • Sum of all other consumers’ net benefits • Benefits from tax revenue cannot be distributed to other consumers in such a manner that it influences other consumers’ net benefit for pure public good • Consumers facing a higher tax rate relative to others may not respond well to tax-discriminating nature of Clarke tax • One problem with Clarke tax is that it is not necessarily Pareto efficient • Predetermined payment may result in some cases where a group of consumers has negative net benefits for pure public good • Even when sum of net benefits is positive • Purchasing pure public good will harm these consumers • Not result in a Pareto improvement • Some type of compensation principle would be required to justify social-welfare benefits of a Clarke tax