Efficient Provision of Local Public Goods and Services

Efficient Provision of Local Public Goods and Services Motivation ▶ Cities need to provide public goods and services to realize their full potentials. ▶ Important services include the provision of public education, the enforcement of property rights, public safety, the protection from fire and other natural disasters, and the provision of infrastructure. ▶ Hence, cities need to have the competencies to pursue their own fiscal policies, which requires a decentralized organization of government. ▶ If a city lacks these services or if these services are expensive to provide, ▶ firms in the city will be at a comparative disadvantage in the world economy; ▶ the city will also have problems attracting skilled and mobile households who will prefer to live elsewhere. Agenda 1. We establish conditions that must be satisfied if we are to consider government performance to be efficient. 2. These conditions will define a “gold standard” against which we should measure all public sector activities – from dog catching to national health care programs! 3. We study mechanisms that allow us to implement the efficient level of public good provision. 4. We discuss some practical problems associated with implementation. Defining Local Public Goods A pure local public good satisfies the following three conditions: 1. Once the public good is provided, the additional cost of another person consuming the good is zero – consumption is non-rival. 2. To prevent anyone from consuming the good is impossible – consumption is non-excludable. 3. Households that live outside a well-defined geographical area do not benefit from the provision of the public good. This definition does not imply that everybody values the public good the same. Not everybody likes to listen WRTI. A Model ▶ We consider a city with only two individuals. ▶ Each individual has an income, or endowment of the private good, equal to m. ▶ Consider an urban economy with two types of goods: – a public good denoted G , – a private good denoted z. ▶ The costs of providing the public good is given by: C (G ) = c G , where c are the marginal costs. Note that costs do not depend on the number of users. Preferences ▶ Each person has Cobb-Douglas utility over the public good and the private good that are given by: U1 (G , z1 ) = α ln(G ) + (1 − α) ln(z1 ) (1) U2 (G , z2 ) = α ln(G ) + (1 − α) ln(z2 ) (2) and ▶ α ∈ (0, 1) represents the utility weight on, or preference for, public goods. ▶ What might be a reasonable ball park estimate for α? Efficient Provision of Public Goods To compute Pareto efficient allocations, we need to solve the following planer’s problem: max α ln(G ) + (1 − α) ln(z1 ) (3) c G + z1 + z2 = 2m (5) s.t. α ln(G ) + (1 − α) ln(z2 ) = Ū2 (4) Equation (4) is the reservation utility constraint and equation (5) is the aggregate budget constraint. The Lagrange Function To solve the constrained optimization problem we use a Lagrange Function approach: L = α ln(G ) + (1 − α) ln(z1 ) (6) + λ1 (Ū2 − α ln(G ) − (1 − α) ln(z2 )) + λ2 (2m − c G − z1 − z2 ) The variables λ1 and λ2 are called the Lagrange Multipliers. They measure the shadow value of relaxing the relevant constraints. First Order Conditions The first order conditions require that the derivative of the Lagrangian with respect to each of its arguments is equal to zero: ∂L ∂G ∂L ∂z1 ∂L ∂z2 Note that setting constraints. ∂L ∂λi 1 1 − λ1 α − λ2 c = 0 G G 1 = (1 − α) − λ2 = 0 z1 1 = −λ1 (1 − α) − λ2 = 0 z2 = α = 0 gives us back the two binding (7) Efficiency ▶ The FOC’s simplify to α/G α/G + =c (1 − α)/z1 (1 − α)/z2 (8) ▶ Note that the marginal rate of substitution is α/G = MRSGi ,zi (1 − α)/zi (9) which measures the marginal willingness to pay for another unit of the public good in terms of private good consumption. ▶ Optimality thus requires: MRSG1 ,z1 + MRSG2 ,z2 = c (10) The Samuelson Condition ▶ Consider the case of arbitrary utility functions Un (G , zn ) and N individuals. ▶ Convince yourself that the first-order condition for Pareto efficient provision of public goods is given by: N X MRSGn ,zn = c (11) n=1 ▶ The sum of the marginal rates of substitution is equal to the marginal cost of providing the good. ▶ If we measure the outside good as money, then the left hand side of the equation is the total marginal willingness to pay for an increase in public good provision. Efficient Public Good Provision ▶ Multiply both sides of this expression by (1 − α)G /α : z1 + z2 = (1 − α) cG α (12) ▶ Insert the social resource constraint (z1 + z2 = 2m − c G ) into the left-hand side of this expression: 2m − c G = (1 − α) cG α (13) ▶ Hence, the unique efficient level of public goods is given by: GE = 2mα c (14) ▶ The efficient provision level is increasing in individual income m and preferences for the public good α, but is declining in the marginal cost of provision c. Congestion ▶ Most local public goods are subject to congestion. ▶ Examples are parks, roads, community centers, environmental quality, and law enforcement. ▶ To accommodate these congestion effects, we can write utility as a function of the number of users N: Ui = Ui (G , N, zi ) (15) ▶ Utility increases in G and zi , but decreases in N. ▶ Moreover, the costs of producing the public good may also depend on the number of users (public education). ▶ We can treat G and N as choice variables. Optimal Public Good Provision and Optimal City Size ▶ We obtain two types of optimality conditions. ▶ First, efficient levels have to satisfy the standard Samuelson condition. ▶ Second, there is a different optimality condition for city size. ▶ Here we face the trade-off that adding another household to the city increases congestion and thus makes everybody worse off, but it also increases the tax base which creates a positive financial spillover. ▶ The marginal congestion costs of adding another household to the city must be equal to the financial benefits of adding another household to the city. ▶ See chapter 4 of the textbook for a formal derivation of these conditions. The Implementation Challenge Suppose we want to implement the efficient level of public good provision in practice. We face two serious problems: 1. Can we construct a mechanism so that individuals will truthfully reveal their preferences to a planer? 2. Can households correctly state their preferences so that we can implement this mechanism in practice? Heterogeneous Preferences ▶ We can generalize the model above and assume that individuals are heterogeneous in their preferences for public goods. ▶ Thus preferences are now given by: Ui = αi ln(G ) + (1 − αi ) ln(zi ) (16) ▶ Heterogeneous preferences imply that α1 ̸= α2 . Hence, there is potential conflict among the two individuals. The Lindahl Mechanism ▶ Let’s consider a mechanism that is due to the Swedish economist Erik Lindahl (1919). ▶ Lindahl proposed a clever mechanism that mimics an auction which determines a sharing rule for the costs of paying for the public good. ▶ We will show that this mechanism yields efficient provision of public goods as along as individuals are not very strategic. Cost Shares ▶ The government announces individual prices or a sharing rule (s1 , s2 ), which represent cost shares for the public good: s1 + s2 = 1 (17) ▶ Thus, individual contributions to the public good, denoted by g1 and g2 : g1 = s1 c G (18) g2 = s2 c G (19) ▶ s1 c and s2 c are “personalized prices” for purchasing G . Lindahl Prices and Cost Sharing ▶ Given these Lindahl prices, individuals announce their most preferred level of public good. ▶ Denote G1L as player 1’s most preferred level and G2L as player 2’s most preferred level. ▶ Player 1’s most preferred level is determined by maximizing the following utility function: α1 ln (G ) + (1 − α1 ) ln (m − s1 cG ) (20) Individual Strategies ▶ In this setup, player 1’s most preferred provision of public goods is given as follows: G1L = α1 m s1 c (21) ▶ The analogous expression for individual 2 is given as follows: G2L = α2 m α2 m = s2 c (1 − s1 ) c (22) The Unanimity Requirement ▶ If the two players do not agree on a level of public good provision, i.e. if G1L ̸= G2L , the government announces a different sharing rule. ▶ This process continues until the government announces a sharing rule and the two players agree on the level of the public good provision given the joint sharing rule. ▶ This will be the case if G1L = α2 m α1 m = = G2L s1 c (1 − s1 )c (23) ▶ The following figure illustrates the two demand functions and the resulting equilibrium! G G1 (s1 ) G2 (1 s1 ) GL sL1 s1 The Lindahl Equilibrium ▶ Solving the equation above, we obtain the following sharing rule: α1 (24) s1 = α1 + α2 and s2 = α2 α1 + α2 (25) ▶ It is then easy to see that both individuals agree on the following level of public good provision G L = G1L = G2L = (α1 + α2 )m c (26) Efficiency of the Lindahl Mechanism ▶ The government sets Lindahl prices such that the player with the stronger preference for the public good pays the most. ▶ That is, if α1 > α2 , then s1 > s2 . ▶ Also, note that the Lindahl equilibrium implements the Pareto efficient allocation of public goods, G L = G E . ▶ The Lindahl mechanism thus overcomes the free-rider problem. Truthful Revelation of Preferences? ▶ Individuals will recognize that their share of the costs depends on the stated preferences for public goods. ▶ They will have an incentive to report lower valuations for the public good than they have. ▶ If individuals do not truthfully reveal their preferences for public goods, the Lindahl mechanism will fail to provide an efficient allocation. ▶ Vickrey, Clark, and Groves developed a different mechanism in the 1970’s in which truth-telling is a dominant strategy. Can Individuals Correctly State Preferences for Public Goods? ▶ Contingent valuation is a survey-based economic technique for the valuation of non-market resources, such as environmental preservation or the impact of contamination. Here is an example. ▶ Penn spent almost $40 million on electricity alone in 2005. ▶ In 2016 Penn was purchasing 50 percent of its energy from green or renewable sources. Let’s assume that renewable energy is 10-20 percent more expensive than non-renewable energy. ▶ There are roughly 10,000 undergraduates at Penn. ▶ How much of an increase in tuition are you be willing to pay for a switch to green energy? Revealed Preferences: What’s the Optimal Class Size? ▶ We need to balance the benefits of a smaller class size against the costs of reducing class size. ▶ Let us simplify our model by assuming that parents only care about the income of their children and that school quality depends on class size. ▶ To measure the benefits of class size reductions, we need to measure the causal relationship between class size and test scores, and then translate the gains in test scores into increases in earnings. ▶ In the textbook, I go through a lengthy calculation that suggests that the optimal average SAT score is approximately 1160, which translates into a class size of approximately 20 students per teacher. ▶ Most private schools and suburban schools seem to have it right! Conclusions ▶ Implementing the efficient level of public good provision is rather challenging. ▶ The Lindahl mechanism may work within small cohesive units (families, fraternities & sororities, etc.). ▶ Individuals may tell the truth even if it may not always be in their interest. ▶ Other mechanisms are too complicated to implement. ▶ We need to rely on more practical approaches to determine fiscal policies.