Intermediate public economics 6Public goods

Hiroaki Sakamoto

June 26, 2015

Contents

1. Definition and examples2. Modeling public goods

2.1 Model2.2 Efficient allocation and equilibrium

3. Lindahl mechanism3.1 Design3.2 Efficiency3.3 Practical relevance

4. VCG mechanism4.1 Design4.2 Demand-revealing property4.3 Inefficiency

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 consumethe same good (simultaneously, at least)

– excludability: people who do not pay the fair cost canbe 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

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) iscalled common-pool (or -property) resources

1.2 Real-world examples 4

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

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 BP N

P g − c , g − c g/2 − c , g/2Consumer A

N g/2 , g/2 − c 0 , 0

1.4 Simple example 6

Modelling public goods

Preference• n ≥ 2 consumers in the economy• Utility function is Ui(g, xi), where xi is consumption of

private good whereas g is consumption of public good• i has endowment xi > 0 of private good (numéraire)• xi can be used for contribution ci ∈ [0, xi] for public

good:xi + ci = xi (1)

Technology• Consumers know that public good is produced by

g = G(∑ni=1 ci) (2)

• Cost of producing g units of public goods is hence

C(g) := G−1(g) (3)2.1 Modelling public goods 7

Characterizing efficient allocation

Theorem (Samuelson, 1954)• If an allocation (g∗, x∗1 , . . . , x∗n) is Pareto efficient, then it

must be the case thatn

∑i=1

Uig(g∗, x∗i )

Uix(g∗, x∗i )

= 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(δ1, . . . , δn) ∈ Rn

++ such that reallocation to(g∗ + ε, x∗1 − δ1, . . . , x∗n − δn) is feasible andPareto-improving

2.2 Characterizing efficient allocation 8

Nash equilibrium

Public goods provision game• Public goods provision involves strategic interaction

among consumers• You would have little incentive to make a contribution ci

if everybody else’s contribution c−i is sufficient• Considered to be a game

Nash equilibrium• Profile (cN

i )ni=1 of contribution is a Nash equilibrium if

ui(cNi , cN

−i) ≥ ui(ci, cN−i) ∀ci ∈ [0, xi] (5)

for all i ∈ {1, 2, . . . , n}, where

ui(ci, c−i) := Ui(G(ci + ∑j =i ci), xi − ci) (6)

2.3 Nash equilibrium 9

Characterizing Nash equilibrium

First-order condition• If (cN

i )ni=1 is an (interior) Nash equilibrium, then it must

be the case thatUi

g(gN , xNi )

Uix(gN , xN

i )= C′(gN), (7)

wheregN := G(∑i cN

i ) and xNi := xi − cN

i (8)

Inefficiency• Notice that (9) implies

n

∑i=1

Uig(gN , xN

i )

Uix(gN , xN

i )> C′(gN), (9)

violating the Samuelson condition (undersupplied)

2.4 Characterizing Nash equilibrium 10

Example

Setup and Nash equilibrium• Ui(g, xi) := γi ln(g) + ln(xi) for some γi > 0

• G(c) := c1/2 and hence C(g) := g2

• At eqm,

γixi − cN

igN = 2gN (10)

• For sufficiently small ε ∈ R++, define δi ∈ R++ by

δi :=

(γi

xi − cNi

gN − 1n

(∑

jγj

xj − cNj

gN − 2gN

))ε (11)

• Then (gN + ε, xN1 − δ1, . . . , xN

n − δn) Pareto-dominatesthe eqm allocation (gN , xN

1 , . . . , xNn )

2.4 Characterizing Nash equilibrium 11

Another example (quasi-linear utility)

Setup and Nash equilibrium• Ui(g, xi) := γi ln(g) + xi for some γi > 0• Assume γ1 > γj for all j = 1

• G(c) := c1/2 and hence C(g) := g2

• In this case, eqm involves corner solution:

cNj = 0 for all j = 1 (12)

andcN

1 = 2−1γ1, gN = G(cN1 ) = (2−1γ1)

1/2 (13)• Only consumer 1, who most highly values the public

good, makes a contribution and everybody elsecompletely free-rides!

• Inefficiency: gN < g∗ := (2−1 ∑j γj)1/2

2.4 Characterizing Nash equilibrium 12

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

Lindahl mechanism: procedure

Procedure• First government announces a personalized

tax-transfer system (τi, Ti)ni=1

• Consumer i’s ‘demand’ for g is determined by

(gdi (τi, Ti), xd

i (τi, Ti)) ∈ argmax Ui(g, xi) s.t. xi + τig = xi +Ti

• Given the feedback (gdi (τi, Ti))

ni=1 from consumers,

government adjusts (τ∗i , T∗

i )ni=1 in such a way that

gdi (τ

∗i , T∗

i ) = gdj (τ

∗j , T∗

j ) =: g∗ for all i, j (14)

and

∑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

Lindahl equilibrium (quasi-linear utility)

Computing Lindahl tax rate• Ui(g, xi) := γi ln(g) + xi for some γi > 0

• G(c) := c1/2 and hence C(g) := g2

• Demand function gdi (τi) is

gdi (τi) = γi/τi (16)

• Set τ∗i and T∗

i by

τ∗i := γi(2−1 ∑j γj)

−1/2, T∗i := 1

2n (∑j γj) (17)

so that gdi (τ

∗i ) = (2−1 ∑j γj)

1/2 =: g∗ for all i

• Efficiency restored b/c

∑i

MRS(g∗, x∗i ) = ∑i

γi/g∗ = 2g∗ = MRT(g∗) (18)

3.3 Lindahl mechanism: example 15

Practical relevance

Vulnerable to ‘cheating’• 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 gd

i (τi) = γi/τi for some γi < γi

3.4 Practical relevance 16

Alternative to Lindahl mechanism

Mechanism design• In general, a mechanism is said to be

demand-revealing if nobody has an incentive tomisrepresent 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

VCG mechanism: procedure

Quasi-linear utility• Consumers’ preference is represented by

Ui(g, xi) := vi(g) + xi (19)

• Here vi(g) is the true valuation of public goods, which isunknown to government

• Special case is vi(g) := γi ln(g)

Procedure• Government first asks consumers to reveal vi : R+ → R• Consumers report vi : R+ → R (can be different from vi)• Given the feedback v := (vi)

Ni=1, government then

decides the level g(v) of public goods as well as thepersonalized lump-sum tax ci(v)

4.2 VCG mechanism: procedure 18

VCG mechanism: design

Designing the mechanism• Prior to asking consumers to reveal their preference,

government announces that g(v) is determined by

g(v) ∈ argmaxg

(N

∑i=1

vi(g)− C(g)

)(20)

and ci(v) is determined by

ci(v) := ti(v)−(

∑j =i

vj(g(v))− C(g(v))

)(21)

for some ti(v) ∈ R (which can depend on v := (vi)Ni=1)

• Consumer i’s utility is then

ui(vi, v−i) := vi(g(v)) + xi − ci(v) (22)

4.3 VCG mechanism: design 19

VCG mechanism: Nash equilibrium

Truth-telling is a dominant strategy• Suppose that everybody else truthfully reveals their

preference, i.e.,vj = vj ∀j = i (23)

• Then, consumer i’s utility is

ui(vi, v−i) = vi(g(vi, v−i)) + xi − ci(vi, v−i) (24)∝ vi(g(vi, v−i)) + ∑

j =ivj(g(vi, v−i))− C(g(vi, v−i)),

where

g(vi, v−i) ∈ argmaxg

(vi(g) + ∑j =i vj(g)− C(g)

)(25)

• Reporting vi = vi (truth-telling) is the best response!

4.4 VCG mechanism: Nash equilibrium 20

VCG mechanism: feasibility

Designing the lump-sum tax• Mechanism is only feasible if ∑N

i=1 ci(v) ≥ C(g(v)) forany v := (vi)

Ni=1, which is equivalent to

N

∑i=1

ti(v) ≥ (N − 1)

(N

∑i=1

vi(g(v))− C(g(v))

)(26)

orN

∑i=1

{ti(v)−

(vi(g(v))− N − 1

NC(g(v))

)}≥ 0 (27)

• Feasibility guaranteed by setting

ti(v) := maxg

(∑j =i

vj(g)− N − 1N

C(g)

)(28)

4.5 VCG mechanism: feasibility 21

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 ci(v)− C(g(v)) ≥ 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