Intermediate public economics 6 Public goods Hiroaki Sakamoto

Transcription

1 Intermediate public economics 6 Public goods Hiroaki Sakamoto June 26, 2015

2 Contents 1. Definition and examples 2. Modeling public goods 2.1 Model 2.2 Efficient allocation and equilibrium 3. Lindahl mechanism 3.1 Design 3.2 Efficiency 3.3 Practical relevance 4. VCG mechanism 4.1 Design 4.2 Demand-revealing property 4.3 Inefficiency

3 Private goods, public goods Private goods (as opposed to public goods) Most of the goods have the following properties: rivalry: once consumed, nobody else can consume the same good (simultaneously, at least) excludability: people who do not pay the fair cost can be prevented from enjoying the benefit of the good Public goods Public goods are a particular type of goods which have both (or part) of the following properties: non-rivalry: consumption by one person does not affect the consumption opportunities of the others non-excludability: once provided, no one can be prevented from enjoying the benefits 1.1 Private goods, public goods 3

4 Real-world examples Pure public goods (almost, I should say) Common examples of public goods include: national defense services lighthouse and street lighting environmental services (e.g., clean air) Impure public goods Goods in between private and pure public goods: a) broadcasting services (excludable), b) fishery stock in the high seas (rivalrous), c) national parks and roads (partially rivalrous, excludable at some cost), where goods of type a) is called club goods while b) is called common-pool (or -property) resources 1.2 Real-world examples 4

5 Why public goods? Another type of market failure You can enjoy the benefit of public goods without purchasing them (once purchased by somebody else) Who in the world would pay for public goods then? Incentive to supply public goods is inevitably weak since they are not sufficiently paid for makes it difficult for public goods to be traded in market Free-riding and inefficiency This is what we call the free-riding problem, a consequence of non-rivalry and non-excludability Due to free-riding, public goods are likely to be undersupplied relative to the efficient level Government, as a result, should play a role 1.3 Why public goods? 5

6 Simple example Public broadcasting service Two consumers, A and B Strategies: P (pay fee) or N (not pay fee) High-quality TV program only supplied when both pay Otherwise low-budget poor-quality program supplied Payoff matrix Consider the payoff matrix listed below Assume 2c > g > c > 0 and find the Nash equilibrium Consumer B P N P g c, g c g/2 c, g/2 Consumer A N g/2, g/2 c 0, Simple example 6

7 Preference Modelling public goods n 2 consumers in the economy Utility function is U i (g, x i ), where x i is consumption of private good whereas g is consumption of public good i has endowment x i > 0 of private good (numéraire) x i can be used for contribution c i [0, x i ] for public good: x i + c i = x i (1) Technology Consumers know that public good is produced by g = G( n i=1 c i) (2) Cost of producing g units of public goods is hence C(g) := G 1 (g) (3) 2.1 Modelling public goods 7

8 Characterizing efficient allocation Theorem (Samuelson, 1954) If an allocation (g, x 1,..., x n) is Pareto efficient, then it must be the case that n Ug(g i, xi ) U i=1 x(g i, xi ) = C (g ), (4) which is called the Samuelson condition In other words, sum of MRS should be equal to MRT Proof (sketch) Suppose that (4) does not hold, say LHS > RHS Then, for sufficiently small ε R ++, there must exist (δ 1,..., δ n ) R n ++ such that reallocation to (g + ε, x 1 δ 1,..., x n δ n ) is feasible and Pareto-improving 2.2 Characterizing efficient allocation 8

9 Nash equilibrium Public goods provision game Public goods provision involves strategic interaction among consumers You would have little incentive to make a contribution c i if everybody else s contribution c i is sufficient Considered to be a game Nash equilibrium Profile (ci N ) i=1 n of contribution is a Nash equilibrium if u i (c N i, c N i ) ui (c i, c N i ) c i [0, x i ] (5) for all i {1, 2,..., n}, where u i (c i, c i ) := U i (G(c i + j =i c i ), x i c i ) (6) 2.3 Nash equilibrium 9

10 Characterizing Nash equilibrium First-order condition If (ci N ) i=1 n is an (interior) Nash equilibrium, then it must be the case that where Inefficiency Notice that (9) implies U i g(g N, x N i ) U i x(g N, x N i ) = C (g N ), (7) g N := G( i c N i ) and x N i := x i c N i (8) n Ug(g i N, xi N ) i=1 Ux(g i N, xi N ) > C (g N ), (9) violating the Samuelson condition (undersupplied) 2.4 Characterizing Nash equilibrium 10

11 Example Setup and Nash equilibrium U i (g, x i ) := γ i ln(g) + ln(x i ) for some γ i > 0 G(c) := c 1/2 and hence C(g) := g 2 At eqm, γ i x i c N i g N = 2g N (10) For sufficiently small ε R ++, define δ i R ++ by ( ( )) δ i := γ i x i c N i g N 1 n x j c N j γ j g j N 2g N ε (11) Then (g N + ε, x N 1 δ 1,..., x N n δ n ) Pareto-dominates the eqm allocation (g N, x N 1,..., xn n ) 2.4 Characterizing Nash equilibrium 11

12 Another example (quasi-linear utility) Setup and Nash equilibrium U i (g, x i ) := γ i ln(g) + x i for some γ i > 0 Assume γ 1 > γ j for all j = 1 G(c) := c 1/2 and hence C(g) := g 2 In this case, eqm involves corner solution: c N j = 0 for all j = 1 (12) and c1 N = 2 1 γ 1, g N = G(c1 N) = (2 1 γ 1 ) 1/2 (13) Only consumer 1, who most highly values the public good, makes a contribution and everybody else completely free-rides! Inefficiency: g N < g := (2 1 j γ j ) 1/2 2.4 Characterizing Nash equilibrium 12

13 Lindahl mechanism: the idea Personalized tax system Reasons why public goods are undersupplied: strategic incentive marginal value of public goods varies across consumers, which is not taken into account Remove the strategic interaction (among consumers) by transforming the game into market-like setting Personalize the price so that everybody pays a fair share of the cost Stark contrast to private goods Private goods: shared price personalized quantity Public goods: personalized price shared quantity 3.1 Lindahl mechanism: the idea 13

14 Procedure Lindahl mechanism: procedure First government announces a personalized tax-transfer system (τ i, T i ) n i=1 Consumer i s demand for g is determined by (g d i (τ i, T i ), x d i (τ i, T i )) argmax U i (g, x i ) s.t. x i + τ i g = x i + T i Given the feedback (gi d(τ i, T i )) i=1 n from consumers, government adjusts (τi, T i )n i=1 in such a way that and g d i (τ i, T i ) = gd j (τ j, T j ) =: g for all i, j (14) i τ i = C (g ), i T i + C(g ) = i τ i g (15) This tax-transfer system is called Lindahl mechanism 3.2 Lindahl mechanism: procedure 14

15 Lindahl equilibrium (quasi-linear utility) Computing Lindahl tax rate U i (g, x i ) := γ i ln(g) + x i for some γ i > 0 G(c) := c 1/2 and hence C(g) := g 2 Demand function g d i (τ i) is Set τ i and T i by g d i (τ i) = γ i /τ i (16) τ i := γ i (2 1 j γ j ) 1/2, T i := 1 2n ( j γ j ) (17) so that g d i (τ i ) = (2 1 j γ j ) 1/2 =: g for all i Efficiency restored b/c i MRS(g, xi ) = γ i /g = 2g = MRT(g ) (18) i 3.3 Lindahl mechanism: example 15

16 Vulnerable to cheating Practical relevance In the Lindahl mechanism, we essentially ask consumers to report their preference for public goods In principle, consumers can tell a lie, misrepresenting her demand for public goods Mechanism only works if everybody truthfully reveals their preference Any incentive to tell a lie? Suppose that consumers know how the Lindahl mechanism works Then they will realize that they can be (individually) better off by understating their demand for public goods Incentive to report g d i (τ i) = γ i /τ i for some γ i < γ i 3.4 Practical relevance 16

17 Alternative to Lindahl mechanism Mechanism design In general, a mechanism is said to be demand-revealing if nobody has an incentive to misrepresent their demand Lindahl mechanism is, by design, not demand-revealing Is it possible to design a mechanism that is demand-revealing? Vickrey-Clarke-Groves mechanism One example of demand-revealing mechanisms is Vickrey-Clarke-Groves (VCG) mechanism Under this mechanism, everybody truthfully reveals their preference even if they are not required to do so But this is only achieved at the cost of efficiency 4.1 Alternative to Lindahl mechanism 17

18 VCG mechanism: procedure Quasi-linear utility Consumers preference is represented by U i (g, x i ) := v i (g) + x i (19) Here v i (g) is the true valuation of public goods, which is unknown to government Special case is v i (g) := γ i ln(g) Procedure Government first asks consumers to reveal v i : R + R Consumers report ṽ i : R + R (can be different from v i ) Given the feedback ṽ := (ṽ i ) i=1 N, government then decides the level g(ṽ) of public goods as well as the personalized lump-sum tax c i (ṽ) 4.2 VCG mechanism: procedure 18

19 VCG mechanism: design Designing the mechanism Prior to asking consumers to reveal their preference, government announces that g(ṽ) is determined by ( N ) g(ṽ) argmax ṽ i (g) C(g) (20) g i=1 and c i (ṽ) is determined by ( ) c i (ṽ) := t i (ṽ) ṽ j (g(ṽ)) C(g(ṽ)) j =i for some t i (ṽ) R (which can depend on ṽ := (ṽ i ) N i=1 ) Consumer i s utility is then (21) u i (ṽ i, ṽ i ) := v i (g(ṽ)) + x i c i (ṽ) (22) 4.3 VCG mechanism: design 19

20 VCG mechanism: Nash equilibrium Truth-telling is a dominant strategy Suppose that everybody else truthfully reveals their preference, i.e., ṽ j = v j j = i (23) Then, consumer i s utility is u i (ṽ i, v i ) = v i (g(ṽ i, v i )) + x i c i (ṽ i, v i ) (24) v i (g(ṽ i, v i )) + v j (g(ṽ i, v i )) C(g(ṽ i, v i )), j =i where g(ṽ i, v i ) argmax g ( ) ṽ i (g) + j =i v j (g) C(g) Reporting ṽ i = v i (truth-telling) is the best response! (25) 4.4 VCG mechanism: Nash equilibrium 20

21 VCG mechanism: feasibility Designing the lump-sum tax Mechanism is only feasible if i=1 N c i(ṽ) C(g(ṽ)) for any ṽ := (ṽ i ) i=1 N, which is equivalent to ( N N ) t i (ṽ) (N 1) ṽ i (g(ṽ)) C(g(ṽ)) (26) i=1 i=1 or N { ( t i (ṽ) i=1 ṽ i (g(ṽ)) N 1 N Feasibility guaranteed by setting ( t i (ṽ) := max g j =i )} C(g(ṽ)) 0 (27) ṽ j (g) N 1 N C(g) ) (28) 4.5 VCG mechanism: feasibility 21

22 VCG mechanism: inefficiency Government s budget surplus VCG mechanism successfully encourages people to reveal their preference Government s budget, on the other hand, is not balanced in general Budget surplus: N i=1 c i(ṽ) C(g(ṽ)) 0 Cost of information Surplus cannot be transferred back to consumers because that would provide an incentive to tell a lie Surplus, if any, has to be wasted by design Pareto efficiency needs to be given up if we want to make the mechanism demand-revealing Can be interpreted as the cost of information 4.6 VCG mechanism: inefficiency 22