04 - Club Goods and Local Public Goods

Club Goods and Local Public Goods

Scotchmer. Chapter 29 in "Handbook of Public Economics." Vol. 4, 2002, 1997-2042 * Should read this paper for qualifying exam

Summary - if market equilibrium exists for club good, it's typically efficient; this is more of a general equilibrium thing (not public economics), but there are similarities with local public goods

Club Good - people come together to save costs, but agents cause externalities on each other (congestion); excludable, congested public good; no spatial element like a local public good (i.e., don't need to live near the club)

Classic Model - by Buchanan (1965)

- Only 1 club

- n = # of members - Club provides a service (could be a vector)

- c(n, ) = cost function to supply club good to all members

- x = numeraire private good (scalar); think of this as money so when we measure the amount

of the club good in terms of the private good, we're can just use dollars

- Identical utility for all individuals: u(x, n, )

- Identical endowment of the private good for all individuals: w

- Equal Cost - cost of membership is evenly split among members; this is "clearly efficient"

because everyone is identical cost to an individual is c(n, ) / n

- Indirect Utility - V (n) max u w - c(n, ) , n, ... Note: x = w - c(n, )

n

n

*(n) arg max u w - c(n, ) , n, ... given n members, *(n) is optimal level of club

n

good to provide to maximize utility for each member

n* arg max V (n) ... choose number of members to maximize the optimal utility (based on

n

optimal amount of club good)

- Small Membership - usually assume n* is finite (congestion causes optimum to be finite), but

sometimes go further and assume sometimes assume n* is "small" (relative to the number

of people in society); support for n* being small:

Congestion - at some point un < 0

Rising Cost - cn > 0

Anonymous Crowding - congestion has nothing to do with characteristics of individuals Non-Anonymous Crowding - can identify differences in members (e.g., gender) and attribute

those differences for the congestion

Equilibria - 4 types of equilibria that have been studied; "that statement might not be very clear" (1) Core Allocations - simple core... "that's sort of a scary phrase" (2) Competitive Equilibrium - can get complicated for non-anonymous crowding; also called price-taking equilibrium; Scotchmer says it assumes the following:

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- Commodity space is defined independently of the set of agents (i.e., don't let character of population dictate types of clubs); "I don't want to try to defend that."

- Price system should be complete (gotta be able to think about pricing goods that don't exist)

- Prices should be anonymous (i.e., no price discrimination); Romano's note: if you force this with non-anonymous crowding the equilibrium will not be efficient

- Agents optimize independently (i.e., don't form coalitions) Apply to Classic Model -

Commodity Space - any club with ( , n) is feasible for and n {1,2, } Minor issue: Romano prefers n {1,2, , N} (capped by population)

Price System - q( , n) ; this will equal average cost in the classic model, but that's a

result, not an assumption "We're way out in theoretical Neverland."

Existence - only clubs with ( *, n*) can exist with q( *, n*) = c(n*, *) / n*

Proof: (sort of)

Must have q( *, n*) = c(n*, *) / n* because of free entry (zero profit)

q( *, n*) > c(n*, *) / n* other club can enter and take all the business

q( *, n*) < c(n*, *) / n* club is not viable (operating at a loss)

Suppose a club exists with ( 0 , n0 ) (with at least one 0 > * )

This club's price must be q( 0 , n0 ) c(n0 , 0 ) / n0

But ( *, n*) maximizes utility for all individuals (recall they're identical) so no one

wants to use the ( 0 , n0 ) club

Properties of Equilibrium - (if it exists)

- Efficient

- It will not exist generally; equilibrium only exists if N / n * is an integer value

Integer Problem - don't have market clearance; similar to

$/q

result in perfect competition model where demand at min

ATC

ATC has to be divisible by output for optimum sized firm

(i.e., quantity that minimizes ATC)

D

Approximate Equilibrium - as N , it's easier to have a

solution (more likely to have an integer); "I know this is not very satisfying"

q q* 2q* 3q*

General - not much in the way of results (if equilibrium exists, it's efficient), but it is

very general (not many assumptions)

(3) Nash Equilibria - not a price-taking equilibrium; can help with existence, but is difficult to

define the strategy space; Scotchmer (with co-authors) is main contributor in this area)

(4) Free Mobility Equilibria - similar to local public goods (Epple & Romer paper)

Schools - clubs of non-anonymous crowding; to be efficient, have to have price discrimination (Epple & Romano AER 1998)... more on the theoretical side

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Local Public Goods - "I wish I had another week"

Model - Economy divided into jurisdictions (or regions or communities) - Each region supplies a local public good

Excludability - have to live in the region to consume the public good Provision - several mechanisms for provision of public good (see below) - Free Mobility - people will choose which region to live in; once in the region, they have to buy/rent land and abide by the laws in the region (i.e., pay taxes) Land - implicit price of jurisdiction membership; taxes add to this price and are a cause of

inefficiency - Peer Effects - may have them (non-anonymous crowding) or not (anonymous crowding)

Differences Between Local Public Goods and Club Goods - Prices & payment determined by non-market process - Need for land to live in district (vs. no spatial element to club goods) - Agents only live in one region (equivalent to saying people can only belong to one club) - Agents restricted to looking at regions that exist; this actually limits the commodity space and

helps the non-existence problem

Differences Between LPG Models - Character of public good - Character of public choice (e.g., majority choice, land developer, benevolent dictator

Land Developer - basic idea is to maximize land value (e.g., gated communities); "These models don't turn me on as much"

Malevolent Dictator - brought up by Len since it's more common... problem is there are too many different ways to be malevolent, but "I don't want to pooh-pooh that idea."

2 Aspects to Efficiency 1) Do people sort efficiently between regions 2) Given sorting, are local public goods efficiently provided within regions Third Aspect - are borders drawn in the right places; most models assume borders are fixed so this area is much less studied

"You're not going to be mad if I keep you late, right?"

Example 5 - Scotchmer is terse in her presentation and leaves out lots of detail Basic Results - based on the efficiency aspects above: 1) Many tax systems will lead to inefficient choices of residences because of fiscal externalities (i.e., not efficient between regions) 2) Majority choice "will sometimes be not too inefficient" with respect to levels of public goods (i.e., efficient within regions) Model - 2 regions ( i = 1,2 ); each with land area 1 (fixed area like Epple & Romer paper which gives upward sloping supply... "Let me shut up and just get on with it.") - Agents differ by income with y ~ Uniform [0,1]

- Let J i [0,1] denote partition of agents into regions (so J1 J 2 = [0,1] )... this will be clear when we see the income stratification result below

- Agents have identical utility: u y (x, z, s) = x + b(z, y) + f (s) x = numeraire private good (money)

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z = amount of local public good (assumed to be uncontested within the region) s = land consumption (similar to housing in E&R)

Structure - set up so demand for housing not dependent on income; only difference in utility caused by income is preference for public good

b1 > 0 b12 > 0 b11 < 0 f '> 0

(more public good is better) (higher income types have stronger willingness to pay for the public good) (diminishing returns to the public good) (more land is better)

- Majority choice within region decides local income tax used to provide public good

Local Budget Constraint - tiYi = zi

Yi = aggreaget income in region i ("It's trivial, just integrate.")

ti = tax rate in region i - Individual chooses region, land, x , and voting simultaneously

Myopic Voting - don't worry about mobility; individuals assume voting doesn't impact the number of residents in the region (get same problems as E&R)

Results - take the first two as given (we won't prove)

Equal Land - each agent living in region i consumes 1/ Ni of the land area (where Ni

is the number of residents in the region; recall 1 is the amount of land); "I'll take that

as obvious" (follows from utility function where contirbution from land has nothing to

do with income)

J1

J2

Income Stratificaiton - regions split the income spectrum:

y

0

ye

1

Land Price - pi = f '(1/ Ni ) is equilibrium price of land in region i (i.e., price = MU of

land)... "Maybe it's a little bit more than 'Sure, it's right.'"

Voting Equilibrium - voter in region i solves: (looking at efficiency within regions)

max ti

y(1 - ti ) -

pi

1 Ni

+ b(tiYi , y) + f

1 Ni

Only parts with ti matter (circled)

x

Amount spent on

public good

Single Peaked - it's "easy" to verify voting preferences are single peaked (we did it

in both sets of majority choice notes)

Median Voter Theorem - applies so equilibrium exists and guy with median

preference determines the level of tax... spontaneous notation: median voter is

also called the pivotal voter so use "piv" subscript

FOC - "simple derivative" wrt ti and evaluate at median: - ypiv + Yib1 (tiYi , ypiv ) = 0

ti*( y) solves this FOC (as function of income y ; plug in ypiv to get majority

choice equilibrium)

Monotonic - if b2 is small, we can verify dti*( y) / dy < 0 (i.e., monotonic with

higher income individuals having lower preference for tax Tradeoff - higher income types prefer more public good, but prefer lower tax (recall

utility function has all individuals valuing money equally so at same percentage tax, the higher income type is hurt more by the tax)

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Sort-of Efficient -

Mean = Median - uniform distribution is symmetric so median income = mean

income; rewrite the FOC: - ymean + Yib1 (tiYi , ymean ) = 0

Rewrite this to address efficiency:

b1 (tiYi , ymean ) =

y mean Yi

=

1 Ni

Recall

ymean

=

1 Ni

y=

yJ i

Yi Ni

(def'n of mean and aggregate income)

Translation - marginal value of equilibrium value of public good for average

guy is 1/ Ni

Efficiency - Samuelsonian condition: MB = MC

ye ymean 1

In this case: b1dy = 1 (1 is MC of public good because it's measured in

Ji

b1

terms of the numeraire [dollars])

Translation - average marginal value = average marginal cost

These conditions are the same if the mean income type's marginal utility is the

same as average marginal utility... this happens if b12 is constant (which

y

happens if b1 is linear in y )... this condition seems pretty strict, but reality

usually isn't far off from this "not too inefficient"

Between Regions - Scotchmer shows when people choose regions, equilibrium will be inefficient; the price of admission into a region should include any and all costs; land price is proper (P = MC) to ensure efficient sorting, but income tax distorts the price of living in the region (results in too many people moving to the low tax region) Solution - recall the public good has MC of 0 so region can't efficiently charge for it with an income tax; one alternative is to use a land tax... read Scotchmer to understand

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