Public Goods

Definition Public good: purely public if nonexcludable $ nonrival in consumption Nonexcludable: all consumers can consume the good Nonrival: each consumer can consume all of the good

Ex. – broadcast radio & TV programs- national defense- reductions in air pollution- national parks

Reservation Prices consumer’s reservation price: max willingness-to-pay for unit of good

o wealth: wo utility of not having good: U(w,0)o utility of paying ‘p’ for good: U(w – p, 1)o reservation price ‘r’:

U(w,0) = U(w – r,1)

When Should a Public Good be Provided? 1 unit good costs ‘c’ 2 consumers A & B individual payments for providing public good gA & gB

o if good is to be provided: gA + gB > c payments = individually rational

o each individ pays no more than their reservation $o if individs are better off paying for and having the public good

Pareto-Improving to supply the unit of good “provision of public good” to be Pareto-efficient

Private Provision of a Public Good? Suppose rA > c & rB < c

o A would supply the good even if B made no contributiono B enjoys good for free; free-riding

Suppse rA < c & rB < co Neither A nor B supply the good aloneo If rA + rB > c also, Pareto-improving for the good to be suppliedo A & B may try to free-ride on each other no good to be supplied

Variable Public Good QuantitiesEx. how many broadcast TV programs, how much land to include in national park

c(G) = production cots of G units of public good individuals A & B private consumption: xA, xB

budget allocation must satisfy:xA + xB + c(G) = wA + wB

MRSa & MRS b are A’s & B’s MRS between private and public goodso Pareto-efficiency condition for public good supply (Samuelson condition)

o Public good is nonrival in consumption; extra 1 unit is fully consumed by A & B Suppose MRSa + MRSb < MC(G)

o MRSa = A’s utilities-preserving consumption in private good units for 1 unit reduction in public good o Same with Bo MRSa + MRSb total payment to A & B of private good that preserves both utilities of G is lowered by 1 unito Making 1 less public good unit releases more private good than the compensation payment requires pareto-

improvement from reduced G Suppose MRSa + MRSb > MC(G)

o MRSa + MRSb total payment by A & B of private good that preserves both utilities if G is raised by 1 unit pareto-improvement from increased G

Hence

‘n’ consumer efficient public good production requires

*MANY external Benefits – Collective Consumption Goods (“Public Goods”) good each unit of which simultaneously provides benefit for several people potentially each square kilo of park can provide benefit the different people do NO need separate, private areas

o ALL of them can “collectively consume” same area of park Park ex of a good that is “non-rivalrous” in consumption “collective consumption”

2 questions:1. should park be made available? (social perspective)

some quantity of park such that the SUM os all benefits > cost of making oark available YES – that quantity

2. how much parkland? What is socially optimal (pareto-efficeint) quantity of parkland to provide? What is optimal amount of a good that provides a +ve externality (spillover benefit)? Add benefits to ALL CONSUMERS and then find that quantity of park where total benefit > total cost by

largest amount Want to find Qpark to maximize total surplus

Collective Consumption example Cost of provision of park

o Opp cost of NOT clearing this land and using it for something else Total cost of parkland per time period

TC(L) = 2000 + 1000L + 200L^2 L = unit (km^2); 2000 per period Total benefit per period to a single individual ‘i’

TBi(L) = 25000L – 100L^2 Single individ optimal amount of park to provide quantity at which surplus total benefit – total variable cost is

maximized

Optimal amount of parkland is found by maximizing choice Lo Slope = 0 OR Marginal benefit = MC & find L

Suppose there are 2 people:o Same benefito to find optimal L

slope set = 0 set MB = MC & find implicit L

3 people:

socially-efficient amount of park to provideo add benefits to all users and then find amount of park that maximizes

surpluso = to finding amount of park which SUM od individ MB intersect the MC

functiono the more users greater the amount of park that is provided

Collective Consumption Goods: Exclusion

park consumed by anyone, whether or nor consumer has contributed anything its cost 2 problems:

1. if someone can consumer ‘public good’ w/o paying anything for it “free-riders” 2. we have no independent revelation of people’s values of benefit from public goods b/c consumers do not have

to purchase good before consuming it

Collective Consumption Goods: Free Riding individuals care about PRIVATE net benefits

o individually rational actions (that max expected private net benefit) result in reduction in social net benefit (collective net benefit)

o if individs can ‘free-ride’ amount of public good provided can be collectively too little PAYOFF matrix “prisoners dilemma)

Payoff matrixo 2 playerso 50 A and 50 B 50 separate but identical parcel of parkland

preserves anonymity but still allow us to use 2 x 2 matrix

Game Theory amount of park size 5km (L = 5), or no park provided if anyone states that they want the park, park will be provided cost of provision will be collected pro rata (in proportion) from those stating that they want park

o 1 player states a want for park entire cost will be collected from that playero both state they want the park half cost will be collected from each

determine net benefits of each:1. player A state want the park; B stat doesn’t want A pays ALL of the cost2. player B stats want the park ; A states doesn’t want B pays ALL of the cost3. both players state they want the park cost is split evenly b/w them4. neither states a preference for park park NOT provided and neither has to pay anything

compute total benefit of a 5km^2 to single player & compare to total cost of provision:

o net benefit to single player paying entire cost of park = 10,000 – 12000 = -2000 cost shared equally b/w 2 players

o total benefit of 5km^2 park to single player & compare HALF the total cost of provision:

o net benefit to each person (collectively consumed) = 10000 – 6000 = $4000

nash equilibrium: socially sub-optimal since social surplus is higher when both state what they “want”

Free Riding: Inefficient Outcomes

What should each player reveal as preference for park?- A: depends no only on A’s actions but also B- (stated preference) also (pref reveleation)- reward each is affected by action of other- each player determines best action

conditional on what the other player does (reveal/ or states)

free-riding leads to social inefficient quantities of collective-consumption goodso even when a +ve quantity is actually providedo Wilderness park 3 potential users

All have identical “willingness to pay” Issue:

What are individ net benefits and social surplus if these are not al revealed honestly Total benefit of all 3: Total cost: Optimal L from before = 6.5 km^2

Cost of provision divided evenly surplus/person more than $7000o Net benefit per person (after paying $2000) = $6375

1 person would lie:o TBi same as before

o

1/3 lies about value of park worth nothing sum of STATED benefit = sum of only 2 benefit

functions STATED social MB:

MC = MB 5 units

2 who reveal benefit (willingness to pay) honestly are charged with half the cost of park (before 12000 so honest pays 6000)

net benefit = 4000 (honest people net benefit of liar = total benefit (no cost)

if 3rd person reveled benefit honesly shared net benefit = 6375

Efficient Public Good Supply – the Quasilinear Preferences Case 2 cosnumers

utility-max requires:

i’s public good demand/marg utility curve; i = A,B

Free-Riding Revisited when is free-riding individually rational?

o Can contribute positively to public good supply nobody can lower the supply levelo Individual utility-max may require lower public good level

Given A contributes gA units of public good, B’s problem :

subject to x – private goodw = wealth

Demand Revelation Scheme that makes it rational for individuals to reveal truthfully their private valuations of a public good is a

revelation mechanism

Ex. Groves-Clarke taxation scheme N individuals All have quasi-linear preferences Vi = individual i’s ture (private) valuation of public good Individ i must provide Ci private good units of the public good is supplied Ci = contribution of individ i to public good

Ni = Vi – Ci net value for i = 1… N- pareto-imporivng to supply the public good if:

if sum of net values (total values – cost) is +Ve

then individual ‘j’ is pivotal (changes the supply decision)

what loss does a pivotal individual j inflict on others?

o all others together value public good at less than the amounts they would have to pay for it forcing good to be provided imposes losses on everyone else

o

all others value the public good at more than their respective shares cost individ i prevents the goods provision, individ causes all others to lose a net benefit

Demand Revelation for efficiency a pivotal agent must face the full cost/benefit of her action GC tax scheme makes pivotal agents face the full stated costs/benefits of their actions in a way that makes these

statements truthful GC Tax Scheme:

o Assign cost Ci to each individo Each agent states public good net valuation Si (may not be truthful)o Public good supplied if:

otherwise not

Pivotal person ‘j’ who changes the outcome from supply to not supply Pays a tax of Pivotal person ‘j’ who changes the outcome from not supply to supply Pays tax of

o *taxes aren’t paid to other individuals, but to some other agent outside market

Demand Revelation, Clarke Tax Ex 3 people, A,B,C valuations of public good: $40 A, $50 B, $110 C cost of supplying good = $180 $180 < 40 + 50 + 110 = $200 efficient to supply good

assign c1 = 60 , c2 = 60, c3 = 60 B & C’s net valuations sum to

o (50 – 60) + (110 – 60) = $40 > 0 A,B & C’s net valuations sum to

o (40 – 60) + 40 = 20 > 0 A is not pivotal

If B & C are truthful what net valuation Sa should A state?o If Sa > -40 A makes supply of public good, & loss of $20 to him (net payoff of -$20)o A prevents supply by becoming pivotal

Sa + (50 – 60) + (110 – 60) < 0 ex. A must state Sa < -40o Then A pays Clarke tax of

-10 + 50 = $40 A’s net payoff when lying = -40 A’s net payoff when truthful = -20 A cant do better than state the truth Sa = -20

assign c1 = 60 , c2 = 60, c3 = 60 A & C’s net valuations sum to

o (40 – 60) + (110 – 60) = $30 > 0 A,B & C’s net valuations sum to

o (50 – 60) + 30 = 20 > 0 B is not pivotal

what net valuation Sb should B state?o If Sb > -30 B makes supply of public good, & loss of $10 to him (net payoff of -$10)o B prevents supply by becoming pivotal

Sb + (40 – 60) + (110 – 60) < 0 ex. B must state Sb < -30o Then B pays Clarke tax of

-20 + 50 = $30 B’s net payoff when lying = -30 B’s net payoff when truthful = -10 B cant do better than state the truth Sb = -10

assign c1 = 60 , c2 = 60, c3 = 60 A & B’s net valuations sum to

o (40 – 60) + (50 – 60) = -30 > 0 (don’t want public good) A,B & C’s net valuations sum to

o (110 – 60) - 30 = 20 > 0 C IS pivotal

what net valuation Sc should C state?o Sc > 30 changes nothingo C stays pivotal & pays Clarke tax of

-(40 – 60) – (50 – 60) = $30 (=losses imposed on A & B) net payoff

o (110 – 60) – 30 = 20o Sc < 30 less likely that the public good will be supplied

No public good C saves $60 cost C loses the $110 benefit of public good

C’s net payoff now = $0 (<$20 payoff if truthful) C cant do better than state of truth Sc = $50

Clarke tax scheme implements due to taxes removing private good (money) from pivotal individualso Ri = voter ‘i’ reservation priceo Voters equally share the cost ‘c’ of a discrete public goodo Each voter states a net value Si

o Public good is provided otherwise it is not provided Voter is pivotal if changes the vote by stating a net value which is

o > in abs value & o of opposite sign to

sum of all other voters’ states values pivotal voter pays Clarke tax = to abs value of all voter’s stated value

o result: all voters reveal honestly by statingSi = Ri – c/n

public good is provided if an only if it is efficient to do so

ex.

Another Clarke Tax example assumptions:

o 4 voters each has an income of 500o Ui = Yi + AiG

A1 = 30, a2 = 40, a3 = 45, a4 = 100o 2 levels of public good provision : G = 0, G = 1o reservation prices: Ri = Aio cost of public good: c = 200o voter share of cost c/4 = 50

Problems with Clarke Tax some people are made worse off as a result of Clarke tax decision (persons 1,2,3 in prev ex) Clarke tac ant be allocated to people voting level of private good consumption is not at Pareto efficient level