LECTURE 3* Flashcards
<p>How do club goods differ from public goods?</p>
<p>- Partly rivalrous: congestion after a certain point.- Excludable: club membership</p>
<p>Local public good =+ example</p>
<p>a public good that is only accessible to individuals in a restricted geographical area = excludable to those outside the area. Non-rival/partly rivalrous within the area. Such as local library.</p>
<p>In Buchanan's club model, what assumption do we make about the population</p>
<p>homogenous - identical tastes and preferences</p>
<p>2 choices of club</p>
<p>1. How much of the good to supply2. How many members to admit.</p>
<p>Utility function in club model</p>
<p>U(x, G, n)</p>
<p>U'(x)</p>
<p>> 0</p>
<p>U'(G)</p>
<p>> 0</p>
<p>U'(n)</p>
<p>< 0 congestion =0 no congestion</p>
<p>Individual BC BCM + 2 assumptions on it.</p>
<p>M = x + C(G)/nEach individual pays a fixed share of the cost of providing the public good.Assume Px=1.</p>
<p>What do the club do for optimisation?</p>
<p>Max utility of representative consumer s.t. the individual's BC by choosing x, G and n.</p>
<p>Do we need Lagrange for BCM?</p>
<p>NO - sub BC into U to eliminate x = unconstrained.</p>
<p>BCM FOC wrt G + interpretation</p>
<p>N*MRS(g,x) = C'(G)Sum of MRS across consumers = MC of public goodSAMUELSON RULE.</p>
<p>is the club provision PE? Explain.</p>
<p>YES - FOC wrt G satisfies Samuelson rule</p>
<p>BCM FOC wrt n + interpretation</p>
<p>MRS(n, x) = - C(G)/n^2MU cost of additional member in terms of private good = extent to which additional n reduces cost per person.</p>
<p>What is optimum is U'(n)=0</p>
<p>No congestion = non-rivalrous.Optimal club size infinite.MRSn,x=0 so n--> infinity from foc.</p>
<p>What is optimum is U'(n)<0</p>
<p>Congestion = partly rivalrous.Membership should be restricted.</p>
<p>To determine optimum n* in BCM, what is necessary?</p>
<p>CONGESTION U'(n) < 0 otherwise n-->infinity.</p>
<p>What allows the club to achieve efficiency?</p>
<p>The fact that when consumers join the club they reveal their preferences by how additional members affect existing member's utility.</p>
<p>What's variable utilisation?</p>
<p>Previously, assumed number of visits per member fixed.Now: allow variable in frequency of visits by members.</p>
<p>What is congestion parameter in variable utilisation?</p>
<p>V = nv = total number of visits</p>
<p>Utility function variable util</p>
<p>U(x, G, v, V)</p>
<p>U'(v) =</p>
<p>> 0 - higher utility more u visit the club</p>
<p>U'(V)=</p>
<p>< 0 - lower utility as more congested.</p>
<p>Individual BC variable</p>
<p>M = x + C(G, nv)/n</p>
<p>In variable util, we assume the cost of the public good depends on...</p>
<p>Not only on the level of G, but also the total number of visits.</p>
<p>3 decisions in variable util for club</p>
<p>Efficient level of club good to provide.efficient level of membershipEffective number of visits per member to allow.</p>
<p>Variable FOC wrt G</p>
<p>n MRSx,G = C'(G)Sum MRS = MC of public goodSAMUELSON RULE</p>
Variable FOC wrt n + interp
vU’(V)/U’(x) = - C(G, nv)/n^2 + vC’(V)/n
MU cost of additional member due to congestion = reduction in cost as n rises + increased cost of servicing additional visits
Variable FOC wrt v + interp
U’(v)/U’(x) = C’(V) + nU’(V)/U’(x)
MB of additional visit = marginal maintenance cost + marginal congestion cost
What’s a two part tariff? Why is it beneficial?
Previously: individuals pay fixed share of C(G) - so individuals only accounted for private cost of additional v, leading to over usage.
Now charge fixed fee F and variable fee p = price per visit.
The best solution for a club w variable utilisation is…
2 PART TARIFF
Individual BC for 2 part tariff
M = x + F + pv
What’s the break even constraint for 2 part tariff?
nF + npv = C(G, nv)
For 2 part tariff, which FOCs are the same as variable?
wrt G
wrt v
For 2 part tariff, which FOC is different to variable and how?
Same, but then sub in C(G, nv) = nF + npv
FOC: F + pv = C’(V)v - nvU’(V)/U’(x)
Efficient allocation in public good provision is attained by…
Consumers separating themselves into a series of efficient clubs.
When is the optimal allocation across clubs simple?
When total population = number of clubs * efficient club size i.e. N = D x n*
Allow n* consumers into each club.
In model of 2 clubs, aggregate welfare W(n) =
W(n) = nU(n) + (N-n)U(N-n)
If the optimal club size is too small…
2n* < N
Efficient allocation is EQUAL division: N/2
If the optimal club size is too large…
2n* > N
May be desirable to put more than N/2 in the bigger club where they get higher utility.
How are local public goods excludable?
Restricted to a particular geographical area - must be a resident and pay taxes in this area to access.
Localities compete for population by determining… (2)
- Level of public good to provide.
2. Taxation
Which locality do individuals chose?
The one giving them the highest utility based on G and tax.
In location choice model, we assume the public good is…(2)
a FIXED resource
Only income e.g. welfare benefits
How does an increase in local population benefit and cost existing residents?
Benefit = reduced per resident cost of public good. Cost = reduced public good per person since a fixed resource.
Location model charge per person for public good =
G/h
How does M(h) chage with h?
M’(h) < 0 - decreases with local population
Location model consumer BC
M(h) = x + G/h
Is income exogenous in location model?
NO - depends on local population.
What do we assume about how localities choose G?
They choose G optimally, given their populations.
FOC wrt G (h as given) for location model
h U’(G)/U’(x) = 1
sum MRS = MC
Samuelson rule
If not stated otherwise, what do we assume MRT is?
1 i.e. 1 unit of private good can be transformed into 1 unit of public.
What’s the next step for location model after finding FOC wrt G?
Find how utility depends on the location population U’(h) and make G(h).
U’(h)=
U’(h) = U’(x)M’(h) + U’(x)G/h^2
SO: what’s the sign of U’(h)?
AMBIGUOUS
2 assumptions on U(h) we impose for location model since sign of U’(h) is ambiguous.
- U(h) N shaped
2. U(H) > U(0) - greater utility from entire population than zero.
Why is U(H) > U(0)
If small population, too expensive to provide public good.
If large, income not reduced too much to offset lower public good cost per capita = net gains.
If each U(h) peaks BEFORE H/2 i.e. h* < H/2, what are the stable equilibria?
h1 = 0, h2 = H
h1 = h2 = H/2
h1=H, h2=0
If each U(h) peaks AFTER H/2 i.e. h* > H/2, what are the stable equilibria?
h1 = 0, h2 = H
h1=H, h2=0
- equal populations is now UNSTABLE.
Optimal fiscal federalism is the question of…
which activities should take place at which level of government
Fiscal decentralisation =
delegation of spending and tax to lower levels of government = opposite of fiscal centralisation.
Tiebout insight of 2 factors missing in public goods market compared to private
- shopping - consumers need options so they can threaten to leave.
- competition - providers of public goods must compete.
When do shopping and competition arise in public good provision? Why?
when they’re provided at the LOCAL level because consumers can vote with their feet and threaten to move to another town –> induces efficiency.
Does samuelson think there’s a market based solution to provide public goods efficiently?
NO
In Tiebout model, how many families are there, how many homes, how much income?
2N families
2 towns with N homes each
Families have identical income Y
In Tiebout, are the public goods pure?
YES
2 types of families in Tiebout ]. How do their U differ?
N families with kids: U(C, G)
N elderly: U(C) - only value private consumption.
2 CONDITIONS FOR ALLOCATION OF FAMILIES ACROSS TOWNS TO TIEBOUT EQUILIBRIUM.
- G decided by median voter in each town & financed equally be residents.
- NO 2 families want to exchange locations.
Individual BC in Tiebout
Y = C + G/N
If G decided by median voter in Tiebout, what does this mean for the 2 towns?
If majority elderly, G=0
If majority families, G=G* which maximises U(Y-G/N, G)
TIEBOUT THEOREM PART 1
In equilibrium, families sort themselves into towns according to their tastes for the public goods –> 1 town with elderly only, 1 with families with kids only.
Proof of why timeout theorem part 1 holds
- If elderly dominate town 1 initially, G1=0
- This means family dominate town 2, G=G*
- Any families in town 1 and elderly left in town 2 want to switch = not an equilibrium.
TIEBOUT THEOREM PART 2
in each town, the level of public good provision is EFFICIENT. In elderly town, G=0 is efficient since G not valued; In family, G* satisfies Samuelson rule.
3 main messages from Tiebout
- people can vote with their feet based on tax and level of G.
- This puts competitive pressure on towns to reflect taste of median voter = Samuelson rule.
- Individuals sort themselves into communities.
Tiebout sorting outcome:
Heterogeneity across communities
Homogeneity within communities
4 institutional limitations to Tiebout
- Local gov political so profit max may not be achieved.
- Barriers to entry - new towns cannot freely form (unlike clubs)
- > 2 types of individuals = need > 2 towns to allow sorting so it’s an equilibrium.
- Perfect instruments to adjust size of community such as advertising and zoning do not exist.
4 unrealistic assumptions of Tiebout
- Assumes income exogenous - migration actually affects income.
- Assumes no externalities of public goods across communities
- Assumes 0 mobility cost and only care about G and C.
- Assumes perfect information about G and taxes in different towns.
What’s a well established empirical result on Tiebout sorting?
More choice –> more sorting.
So within each community, homogenous, across = heterogenous.
Do we see Tiebout sorting leading to greater efficiency empirically e.g. more productive schools?
Not proven - controversial result.
Do we see Tiebout sorting becoming more efficient over time due to declining mobility costs in the data?
Should see more homogenous within communities, greater heterogeneity across in terms of preferences for and levels of public goods and taxes. BUT: we see less variation across communities in terms of policies AND preferences.
Tibeout suggests there is no need for central governments. But what argument (2) is there for central governments?
Local governments cannot redistribute from rich to poor as then would see rich leaving and it would break down. Also, hard to vote with your feet in reality.
How can migration of the rich be avoided at a local level when increasing taxes?
Tax-benefit linkages.
Intergovernmental grants are what kind of form of redistribution?
INDIRECT
direct = taxes and spending.
When we analyse intergovernmental grants, whose ICs and BC do we look at and why?
Assume local communities choose G and tax according to median voter, so G* and X* is where the IC of the median voter is tangent to their BC.
State 3 types of grants
- Matching
- Block
- Conditional block
Define matching grant
The amount of the grant is tied to the local communities spending on the public good only. Usually 1 to 1.
How does a matching grant affect IC/BC diagram?
1 to 1 matching = halves the cost of the public good = BC pivots outwards (not shift). This BC is tangent to a higher IC.
Impact of matching grant ob public and private spending.
Both increase.
But public good cost for local community decreases (don’t double spending) which they use on the private good.
Define block grant
Fixed amount with no mandate on how to spend it.
How does a block grant affect IC/BC diagram?
Transfer of income = BC SHIFTS OUT PARALLEL.
No change in price.
Matching vs block grant IE and SE.
Matching: IE increases both, SE increase public good.
Block: IE increases both - NO SE since no change in price.
On diagram to compare matching and block, how do we shift out the BC for block to make them comparable?
Shift out BC so that it cuts through the matching new equilibrium so that this is affordable (yet is isn’t chosen).
Matching vs block which gives higher utility? Which increases G more?
BLOCK HIGHER UTILITY.
Matching increases G more.
Define conditional block grant
Fixed amount WITH a mandate on how to spend it.
How does a conditional block grant affect IC/BC diagram?
KINKED IC: flat portion at low G so that a certain amount must be spent on G and it makes a certain part of BC unfeasible. After this, normal trade off between x and G as in block grant.
A conditional grant has no effect over a block grant if…
If under a block, optimise where G > x* and x* is the point of the kink under conditional block.
does a conditional block have IE and SE?
NO SE - only IE.
Based on standard economic grant models, money is said to be …. what does this mean in practice?
FUNGIBLE: an X increase in private income has the same effect as an X increase in block grant (both shift BC the same). An X increase in grant for schools has same effect on schools and private good as X increase in grant for police. Only total resources matter, doesn’t matter where the money comes from or what it’s intended for with block grant.
The flyer paper effect is an anomaly to money fungibility. What is it?
When a $1 of exogenous grants leads to a significantly greater effect on public spending than $1 in citizen income (tax): money sticks where it hits i.e. money stays in gov sector.
What’s the opposite effect to flypaper i.e. that standard economics shows.
Crowd out: state spending crowded out by federal block grants so public spending doesn’t increase by full amount - what we saw in the graphs.
school finance equalisation =
laws that mandate redistribution of funds across communities in a state to ensure more equal financing os schools.
In America, local Govs fund schools how?
With property taxes.
Tax price =
the amount of revenue a local district has to raise to gain $1 of spending.
When is tax price = 1
No equalisation so get to keep while $1 raises.
When is tax price infinity.
Perfect equalisation so have to pool all of revenue.
