1
Introduction/Review: Public Goods
A good is called public if the consumption by one agent does not prevent from using it. This
is commonly referred to as non-rivalry in use. Often times, it is also built into the definition
that the good is non-excludable. Excludability is not interesting in perfect information
environments (which is what we will look at initially), so the model below will be one where
it is assumed that no agent is excluded from usage (there is no particular reason to do so
even if it is technologically feasible since the additional costs for usage, once the good is
produced, are zero.
1.1
Environment
For simplicity we will focus on the simplest (non-trivial) case with a single private good and
a single public good. We will begin with briefly looking at a few much studied institutional
arrangements (Private Provision, Majority Voting and Lindahl Equilibria) for a public goods
economy consisting of:
• n consumers indexed by i = 1, .., n
• xi denotes agent i’s consumption of private good. We will use x to denote the vector
of private consumptions, i.e. x = (x1 , ..., xn )
• y denotes (common) consumption of public good
2
• Agent i’s preferences described by utility function ui : R+
→ R , which we take to be
differentiable (we will put on more structure later)
• w is the endowment of private good in economy (public good endowment taken to be
zero)
• Public good may be produced from the private according to the differentiable function
f : R+ → R+ , which we assume is strictly increasing. For ease of notation we will use
1
z for the units of private goods that are used as inputs to produce the public good and
write y = f (z) for the corresponding quantity public good produced.
1.2
Optimal Allocations
An allocation is Pareto optimal if there is no way of making an agent better off w/o making
someone else worse off, that is:
Definition 1 An allocation (x, y) is Pareto optimal if:
1.
P
i
xi + z ≤ w and y ≤ f (z) (feasibility)
2. There exists no other feasible allocation (x0 , y 0 ) such that ui (x0i , y 0 ) ≥ ui (xi , y) for all
i and ui (x0i , y 0 ) > ui (xi , y) for some i.
Remark 1 Alternatively we could write the feasibility condition as
X
i
xi + f −1 (y) ≤ w
It follows directly from this definition that the set of Pareto optimal allocations can be
characterized as solutions to the problem
¡
¢
max u1 x1 , y
(1)
(x,y,z)
¡
¢
s.t. ui xi , y − ui ≥ 0 for i = 2, ..., n
X
xi − z ≥ 0
w−
(multiplier γ i )
(multiplier λ)
i
f (z) − y ≥ 0
(multiplier µ)
y ≥ 0, z ≥ 0 and xi ≥ 0 for all i,
where ui are treated as parameters of the problem. Usually the non-negativity constraints
are ignored and for now we will do the same and simply assume an interior solution. One
sees easily (prove!) that all other constraints must be binding so that the Kuhn-Tucker
2
conditions reduce to
∂ui (xi , y)
−λ = 0
∂xi
X ∂ui (xi , y)
−µ = 0
γi
∂y
i
γi
−λ + µf 0 (z) = 0
(xi )
(2)
(y)
(z),
where we have set γ 1 = 1 by convention. Substituting the first n equations and the last
equation into the middle condition we get
X
i
∂ui (xi ,y )
∂y
∂ui (xi ,y)
∂xi
=
f0
1
.
(z)
This condition is referred to as the Samuelson condition, the Lindahl-Samuelson condition, or
sometimes even the Bowen-Lindahl-Samuelson condition and is probably familiar to anyone
who have taken an intermediate course in Public Economics.
Remark 2 Under the assumptions stated so far (2) is only a set of necessary conditions
for optimality. However, under rather standard convexity and boundary assumptions (think
about what you need) (2) are also sufficient.
1.3
Interpretation
To interpret this condition we recall that
∂ui (xi ,y ) ∂ui (xi ,y )
/ ∂xi ,
∂y
the marginal rate of substitution,
should be thought of as the quantity of private good consumer i is willing to give up a (small)
units increase in the level of public good.
f 0 (z) on the other hand is, for small changes in z, the change in the production of public
good associated with each (small) units change in input of private good so (remember the
inverse function theorem) 1/f 0 (z) is the amount of private good required to produce an
additional (small) unit of the public good (aka marginal rate of transformation).
Hence, the Samuelson condition says that any optimal allocation is such that the sum of
the quantity private goods consumers would be willing to give up for an additional unit public
good must equal the quantity actually required to produce the additional unit.
3
1.4
Private Provision of Public Goods
By the Second Welfare Theorem, if y would have been a private good, then any optimal
allocation could be decentralized as a competitive equilibrium. We now imagine that a free
market exists for the public good:
• Let p denote the price of the public good (in terms of the private).
• y i denotes the quantity public good purchased by agent i
• wi denotes agent i’s initial (private good) endowment.
• Without loss of generality we assume that there is a single price-taking profit maximizing firm that operates on the market.
While we think of the exercise to follow as some sort of “competitive” benchmark we do
actually have to consider strategic effects in order for it to be of much interest. The reason
is simply that even if everybody are price-takers, they do care about the aggregate purchase
of the public good.
1.4.1
The Linear Case
Not surprisingly, all the interesting action is in the decisions on how much to contribute
made by individual agents. Therefore, most of the literature assumes that f (z) = z for all
inputs z ≥ 0. Hence, the competitive firm solves
maxy,z py − z
s.t. y = f (z) = z
⇔ max (p − 1) y
y
and the equilibrium price must consequently be p∗ = 1, independenly of the purchases of
public goods. We can thus completely forget about the supply side of the market. We follow
the literature and assume that private provisions are made simultaneously and that nobody
can contribute more than their wealth, so the strategy set is [0, w i ] for i = 1, .., n. For any
4
vector (y 1 , .., y n ) ∈ ×ni=1 [0, wi ] , the utility for agent i is
Ã
!
n
X
¡ 1
¢
i
n
i
i
i
j
v y , ..., y = u w − y ,
y .
(3)
j=1
Thus, we have a fully specified normal form game. For brevety we use the probably familiar
notation y −i = (y 1 , .., y i−1 , y i+1 , ..., y n ) for a vector of the contributions by all agents except
agent i. A Nash equilibrium of the game is then a vector (y ∗1 , ..., y ∗n ) satisfying v i (y ∗i y ∗−i ) ≥
v (y i , y −i∗ ) for all y i ∈ [0, w i ] and all i. Now, this means that for each i
Ã
!
X
y ∗i ∈ arg max ui w i − y i , y i +
y ∗j
i
y ∈[0,1]
m
j6=i
´
³
P
(xi∗ , y ∗ ) = wi + j6=i y ∗j − y ∗ , y ∗ ∈ arg maxxi ,y ui (xi , y)
P
s.t wi − j6=i y i∗ − xi − y ≥ 0
non-negativity
This transformation of the individual maximization problem is a common change of variables that has been used a lot in the analysis of this model. Usually, the reason for the
transformation is that some concepts from standard demand theory can be introduced (the
maximization problem is identical to a standard consumer problem where prices are (1, 1)
³ P
´
and endowments are wi , j6=i y i∗ ). For now however, the main point is that we see directly from the second problem that the maximization problem has a unique solution if ui
is strictly quasi-concave1 .
Now, given any y −i , the unique best response for agent i is fully characterized by the
conditions
−
∂ui (·) ∂ui (·)
+
+ λi − µ i = 0
i
∂x
∂y
λi y i = 0
¡
¢
µi wi − y i = 0
¡ ¢
=
define agent is best response correspondance in the usual way as β i y−i
³
´
£
¤
P
arg maxyi ∈[0,wi ] ui wi − y i , yi + j6=i y j for each y−i ∈ ×j6=i 0, wj . If ui is strictly quasi-concave it fol¡ ¢
lows that β i y−i is singleton-valued (consists of a single element for each y−i ), so we can then without loss
1
Also,
talk about the best response function.
5
First, to build intuition, suppose that there is an equilibrium in which 0 < y ∗i < wi for all i.
Then,
∂ui (∗)
∂y
∂ui (∗)
∂xi
= 1 for all i
⇒
X
i
∂ui (∗)
∂y
∂ui (∗)
∂xi
= n,
where the “´∗” is to indicate that we are evaluating the expressions at (wi − y ∗i ,
P
j
y ∗j ).
Given the technology f (z) = z that we are using the Samuelsson condition simplifies to
P ∂ui ∂ui
i ∂y / ∂xi = 1, so we see that the equilibrium in the model of voluntary contributions can
not be efficient (at least not when everybody is contributing).
The intuition is simple: agent i contributes up to the point where the quantity private
good the agent is willing to give up for a (small) additional unit of the public good equals
the private extra costs in terms of reduced consumption of the private good for the (small)
additional unit. In other words, the agent sets the marginal rate of substitution equal to
the relative price, which is unity. Doing this, the agent does not consider the benefits to
other agents of the public good he finances by his contribution. Thus, there is an externality
that is not internalized by the provision mechanism under consideration. Alternatively put,
there is an incentive to free ride on the provisions made by other which means that we
expect underprovision of the public good (however, there are some subtle issues involved
when formalizing this last statement).
1.4.2
Voluntary Provision with Decreasing Returns-“Backyard Production”
If we would simply assume that (as done in Laffonts book), for any vector of contributions
³P ´
1
n
j
everything is simple. This would
(z , ..., z ) , the output of the public good is f
jz
correspond to a world where the technology is in control by some entity that wants to break
even, so it has a flavor of a production process collectively owned by the participants in the
contribution game.
The strategy sets are as before and the natural payoff functions are then
à n
Ã
!!
X
¡
¢
v i z 1 , ..., z n = ui wi − z i , f
zj
.
j=1
6
The added generality is more or less a vacuous relabeling of the scale that the public good
2
is measured in: define u
ei : R+
→ R as
u
ei (x, z) = ui (x, f (z)) .
The resulting conditions for a Nash equilibrium are then just like in the linear model. The
only issue is that it is a bit unpleasant to make assumptions directly on a derived object (the
curvature in u
ei comes both from technology and preferences), but in this particular instance
this is not much of an issue. Decreasing returns leads to “more convexity” rather than less
ei is (strictly) quasi-concave.
and if ui is (strictly) quasi-concave and f is concave, then u
1.4.3
Voluntary Provision with Decreasing Returns with a Competitive Firm
Ultimately we would want the notion of a decentralized market to be as close to the standard competitive
model as possible. A natural first shot may be the following:
¢
¡
Definition 2 (that doesn’t work) An equilibrium is a price p∗ and a vector of contributions y ∗1 , ..., y ∗n
a level of the public good y ∗ and a quantity z ∗ of the private good used as input for producing the public good
such that
1. y ∗i ∈ arg maxy
i
h
wi
i ∈ 0, p∗
³
´
P
ui wi − p∗ y i , y i + j6=i y∗j
2. (y ∗ , z ∗ ) ∈ arg maxy,z py − z subject to y ≤ f (z)
3. y ∗ =
P
i
y∗i
4. Let the implied consumption of private goods be given by x∗i (= wi − p∗ y i∗ ). It then seems that we
want
X
wi =
i
X
x∗i + z ∗
i
Notice that this is a bit of a hybrid between competitive and game theoretic ideas. The idea is that,
for any fixed price, the contribution decisions are made simultaneously exactly as above, that the firm is
maximizing its profit given the price and that markets clear. Going through the details of the analyis (check!)
one finds that in equilibrium p∗ =
0 < y i∗ <
i
w
p∗ ,
1
f 0 (z ∗ )
=
1
f 0 (f −1 (y ∗ ))
and that, for agents that in equilibrium contribute
the contribution is characterized by
∂ui (∗)
∂y
∂ui (∗)
∂xi
=
7
1
f 0 (z ∗ )
.
At this point one may believe that we are done since the equilibrium conditions look “the same” as above,
so one may tempted to think that we’ve shown that the equilibrium outcome is the same as in the case with
“backyard production”
¢
¡
Unfortunately, this turns out to be nonsense. To see this, suppose that z ∗1 , ..., z ∗n is an equilibrium
in the model where contributions are transformed to public good automatically (that is the “backyard
production case”). The resulting allocation is



X
z ∗j  .
(x∗ , y∗ ) = w1 − z ∗1 , ...., wn − z ∗n , f 
j
Suppose that can be supported as an equilibrium in the present model. Note that
X
x∗i + z ∗ =
i
X¡
¢ X ∗i X i
wi − z ∗i +
z =
w,
i
i
i
so the 4th condition is automatically satsfied by choice of the allocation that we are trying to support an
allocation “on the frontier”. However individual contributions must be given by y ∗i = z ∗i /p∗ and by the
third condition we have that
p∗ y ∗ = p∗
X
y ∗i = p∗
i
X z ∗i
i
p∗
=
X
z ∗i = z ∗ ,
i
where the last equality comes from the fact that in order to produce the same output of the public good as
before, the input must be the same. But, this means that the maximized profits p∗ y∗ − z ∗ = maxy,z py − z
subject to y ≤ f (z) are zero, which can not be the case with a strictly concave production function. To see
this let π (z, p) ≡ pf (z) − z and note that
∂π (z ∗ , p∗ )
∂π
z ∗ f 0 (z ∗ )
−1=
y∗
z ∗ f 0 (z ∗ ) − f (z ∗ )
f (z ∗ )
= p∗ f 0 (z ∗ ) − 1 =
=
Now, since f is strictly concave and satisfies f (0) = 0 we have that
0 = f (0) < f (z ∗ ) + f 0 (z ∗ ) (0 − z ∗ )
m
z ∗ f 0 (z ∗ ) < f (z ∗ ) .
Hence
∂π(z ∗ ,p∗ )
∂π
< 0, which means that (y ∗ , z ∗ ) can not be profit maximizing (profits would increase by
decreasing the scale of operations).
The above discussion is not too important in terms of how to think about the economics. The problem
is that there is a “black hole” in the model since the firms must earn positive profits (if production is
strictly positive) and these are not going back to the consumer. One can fix the problem easily by letting the
8
consumers/agents get shares of the profits made by the firm(s) in the same way as in the Lindahl equilibrium
model below. The resulting model wouldn’t produce exactly the same equilibria, but the flavor of the analysis
is more or less the same.
What is more important is that the example shows the danger of looking at first order conditions without
too much thought.
1.5
Majority Voting
An alternative institution to voluntary provision is for the agents to agree to respect the
outcome of a democratic election. We will study majority voting in greater detail later in
the course and this section will be rather informal.
For expositional simplicity we assume that the production function is the identity map,
so that y = f (z) = z for all z ≥ 0. An assumption that is not equally innocuous is that
we assume that the citizens for some reason has agreed that the financing of the public good
is through a lump-sum t that is common to all agents. What is important here is not so
much that the tax is lump-sum, or even that it is equal for all agents, but that the sharing
rule is predetermined, which means that the only decision left for the voters is the level of
the public good, which is a single-dimensional policy variable.
Given any public good level y and equal sharing we have that the implied tax rate is n1 y
and since there is only a single private good the consumer maximization problem is trivial
and we can immediately write down the indirect utility of consumer i (as a function of public
good spending) as
³
y ´
i
v (y) = u w − , y
n
i
i
Now define y ∗i as agent i’s most preferred level of public good given the sharing rule under
consideration, i.e. y ∗i = arg maxy v i (y) . We will also need a restriction on v i in order to
proceed below in order to guarantee existence of voting equilibria. This restriction, singlepeakedness is much used in the applied literature, but it should be observed that, in this
context, stronger assumptions than strict quasi-concavity of ui has to be made in order to
guarantee single-peakedness (for many common parametric utility functions the restriction
9
is satisfied however). Formally we define single-peakedness in the obvious way:
£ ¤
Definition 3 A utility function v i over a single-dimensional policy-space y, y is said to be
single-peaked if there exists some y i∗ such that
£
¤
a. For y, y 0 ∈ y, y i∗ , y > y 0 ⇒ v i (y) ≥ vi (y 0 )
b. For y, y 0 ∈ [y i∗ , y] , y > y 0 ⇒ v i (y) ≤ vi (y 0 )
If anyone is confused, draw a picture. The definition simply says that single-peaked
preferences means that the agents are happier the closer to their ideal point they are.
In the applied literature it is more or less taken for granted that, with single-peaked
preferences, majority voting implies that the most preferred outcome of the median voter is
the outcome of the voting process. We will discuss later on what institutional assumptions
are needed later when we study voting models in more detail, but roughly what is needed
is that we have a two party or two candidate election and that parties/candidates care only
about winning. For now, we simply assert that the “median voter theorem” holds and order
the agents so that y 1∗ ≤ y 2∗ ≤ ... ≤ y n∗ . Given an odd number of agents (which we assume)
we then have that Mr. m = (n − 1) /2 is the decisive voter and that the level of the public
good is y m∗ , which is characterized by the first order condition,
¡
¢
¡
¢
∂um wm − ny , y 1
∂ui wm − ny , y
=
∂xi
n
∂y
As you will verify in your homework, the median voter equilibrium does not, in general,
satisfy, the Samuelson condition. In textbooks (Laffont for example) this is interpreted as
an inefficiency of the voting process. We will discuss this later on, but note that it may
be argued that it’s somewhat “unfair” to call this an inefficiency since the median voter
equilibrium is constrained Pareto efficient relative the set of feasible allocations under the
voting process (verify!). The inefficiency is thus caused by an inappropriate choice of sharing
rule rather than by some failure of the voting process to select the “right” level of the public
good.
10
1.6
Lindahl Equilibria
Both the voluntary provision model and the voting model will in general imply that the
resulting allocation is inefficient. There is however a much studied “market institution” that
in principle could achieve efficiency. The idea is to think of the amount purchased by each
agent as a distinct commodity and have each agent to face a personalized price pi and to
have these prices chosen so that all agents agrees on the level of the public good.
A Lindahl Equilibrium is then defined in the spirit of a standard competitive equilibrium
as a vector of prices and an expanded allocation so that 1) each consumer is willing to buy
“his share” of the public good at his personalized price, 2) firms are willing to supply the level
of public good if paid the sum of all personalized prices×the quantity supplied, 3) markets
clear.
We let si be agent i’s share of the firms profit and define a Lindahl equilibrium as follows:
Definition 4 A Lindahl Equilibrium is a vector p∗ = (p∗1 , .., p∗n ) and an allocation (x∗1 , ..., x∗n , y ∗ )
such that
a. The firm maximizes profits, that is
y ∗ = arg max
y,z
Ã
X
i
!
pi∗ y − z
s.t y ≤ f (z) , non-negativity
b. Each consumer maximizes utility, that is
s.t. wi + si
Ã
X
i
c. Market clears
pi∗ y ∗ − f −1 (y ∗ )
X
i
¡ i∗ ∗ ¢
¡ i ¢
i
u
x ,y
x , y = arg max
xi ,y
!
xi∗ + f −1 (y ∗ ) ≤
11
− xi − pi∗ y ≥ 0
X
i
wi
The literature has mainy considered the case with constant returns, in which profits are
zero and there is no need for consumers to hold shares in the firm.
In our simple two good economy, constant returns means linearity, which by choice of
units means that we can write y = z and that the firms problem is
Ã
!
X
i∗
max
y−z
p
y,z
i
s.t y ≤ z
Hence, any solution satisfies y = z. This means that we can forget about the production
sector and define Lindahl equilibrium simply in terms of utility maximization and market
clearing.
If now ui satisfies standard regularity conditions (quasi-concave and differentiable) a
necessary and sufficient condition for a. (given interior solution) is:
∂ui (xi∗ , y ∗ ) ∗i ∂ui (xi∗ , y ∗ )
p =
for each agent i,
∂xi
∂y
P
where xi∗ = +si (
condition for b is
i
(a’)
pi∗ y ∗ − f −1 (y ∗ )) . Given a convex technology a necessary and sufficient
X
pi∗ =
i
f0
1
1
= 0 −1 ∗
∗
(z )
f (f (y ))
(b’)
And by summing over the n conditions in (a’) and then substituting with (b’) we get
X
i
∂ui (xi∗ ,y∗ )
∂y
∂ui (xi∗ ,y∗ )
∂xi
=
f0
1
1
= 0 −1 ∗ ,
∗
(z )
f (f (y ))
the by now familiar Samuelson condition. Furthermore, all agents budget sets must hold
with equality, which means that market clears with equality (verify). As we have already
argued, the Samuelson condition together with market clearing with equality are sufficient
conditions for Pareto optimality under appropriate convexity assumptions on preferences (go
back and verify), so a Lindahl equilibrium is efficient.
To see the parallel with the Competitive Equilibrium framework it is instructive to look
at a more primitive argument, which more or less is a copy of the textbook proof of the first
welfare theorem. I.e., we want to show:
12
Proposition 5 Any Lindahl Equilibrium is Pareto optimal.
Proof. Let x∗ = (x1∗ , .., xn∗ , y ∗ ) be a Lindahl Equilibrium allocation (in a private ownership economy with shares) with corresponding prices (p1∗ , ..., pn∗ ) and suppose there exist a
feasible allocation x0 = (x10 , ..., xn0 , y 0 ) that Pareto dominates x∗ . That is
¡
¢
¡
¢
ui xi0 , y 0 ≥ ui xi∗ , y ∗ for all i
¡
¢
¡
¢
ui xi0 , y 0 > ui xi∗ , y ∗ for at least one agent i
By revealed preference this implies that (assuming local non-satiation)
xi0 + pi∗ y 0 ≥ xi∗ + pi∗ y ∗ for all i
xi0 + pi∗ y 0 > xi∗ + pi∗ y ∗ for at least one agent i
Hence
X
xi0 + y 0
i
X
X
pi∗ >
i
xi∗ + y ∗
i
X
=
wi +
i
X
à i
X
i
pi∗ =
!
X
wi +
i
X
si
i
"Ã
X
i
!
#
pi∗ y ∗ − f −1 (y ∗ ) =
pi∗ y ∗ − f −1 (y ∗ )
where the first equality follows since each agent has to fulfill her budget constraint with
equality in order to maximize utility.
P
y ∗ maximizes profits given p∗ = i pi∗ , which means that
!
!
Ã
Ã
X
X
pi∗ y 0 − f −1 (y 0 ) ≤
pi∗ y ∗ − f −1 (y ∗ ) .
i
i
Combining with inequality above we get that
Ã
!
Ã
!
X
X
X
X
i0
i∗
∗
−1
∗
−1
0
i
i∗
x +
p
y − f (y ) + f (y ) >
w +
p
y ∗ − f −1 (y ∗ )
i
i
i
X
m
xi0 + f −1 (y 0 ) >
i
X
i
wi
i
Thus, x0 is not feasible.
There are analogous of the second welfare theorem as well (See Foley (1970)).
13
1.7
Is Lindahl Equilibrium a Reasonable Market Mechanism?
We will now look at a simple example that illustrates the incentives to misreport preferences
if the planner/policymaker has to elicit information from the agents in order to compute the
Lindahl equilibrium.
Example 6 Consider the case with 3 agents i = 1, 2, 3 and utility functions ui (xi , y) =
ln xi + αi ln y. We suppose that each agent has a endowment of the private good wi = 1 and
no public good. Furthermore we restrict ourself to the linear technology so that y = f (z) = z
for all z ≥ 0. The problem for agent i is to solve
max ln xi + αi ln y
(4)
s.t xi + pi y ≤ wi = 1
non-negativity
(5)
We note that the market clearing requirement simplifies considerably with the linear technology since
X
xi + f −1 (y) =
i
X
xi + y =
i
X
wi
i
Now, each agents budget constraint must be satisfied with equality at an optimum, which
means that
X
wi =
i
X
i
xi + y =
X
i
X
xi + y
i
xi + y
X
i
pi ⇔
X
pi
i
X
⇓
pi = 1
i
The solution to (4) is fully characterized by its first order condition, so the Lindahl equilibrium ( it will be unique) is fully characterized as the solution to
αi
pi
=
for i = 1, 2, 3
1 − pi y
y
p1 + p2 + p3 = 1
14
Algebra on the individual FOC gives
αi
pi y =
⇒ y∗ =
1 + αi
Ã
!
X
X
pi y ∗ =
i
i
αi
1 + αi
Plugging back into the FOC we then get the Lindahl prices
αi
1+αi
P αj
j 1+αj
i∗
p =
and private consumption
xi∗ =
1
.
1 + αi
Suppose for concreteness that the true preference parameters are αi = 1 for i = 1, 2, 3 The
Lindahl equilibrium is then pi∗ = 13 , xi∗ =
agent i is
1
2
and y ∗ = 32 ,so the equilibrium utility level or
µ ¶
µ ¶
1
3
+ ln
ln x + α ln y = ln
2
2
i∗
i
∗
Now make the following thought experiment: Suppose Mr. 2 and 3 report truthfully that their
type as αi = 1, but that Mr. 1 lies and claims that α1 = 0. Furthermore, the “auctioneer”
or planning agency that computes the Lindahl price believes both agents and implements the
corresponding Lindahl equilibrium, with is p1 = 0, p2 = p3 = 12 , x1 = 1, x2 = x3 =
1
2
and
y = 1. The utility of Mr. 1 would then be
2 ln (1) = 0
We note either by brute force (calculator) or by concavity of the logaritm that Mr. 1 is indeed
better off lying. I.e. we note that since 1 = λ 12 + (1 − λ) 32 for λ =
1
2
and since the logaritm
is strictly concave we have
µ ¶
µ ¶
1
1
1
3
ln (1) > ln
+ ln
,
2
2
2
2
which directly gives the result.
The example is special, but the logic is perfectly general. If agents have to report preferences (or wealth) they will take into consideration that under-reporting means a lower price,
15
so the “free-riding” problem we discussed in connection with voluntary provision equilibrium
applies. To see that there would always be misreporting one implicitly differentiates the Lindahl equilibrium conditions to get the Lindahl price and level of public good as a function
of the report by an agent (assuming everyone else is truthfully revealing) and verifies that
at the true αi , utility is decreasing in the report.
1.8
Testing the Lindahl Equilibrium Model
Due to the reasoning along the lines above (and since we don’t observe personalized lump sum
taxes) efficient provision of public goods is typically seen more as a normative prescription
rather than a positive description of how public goods are allocated. However, there is
also a “Coasian view” on these matters that suggests that the agents in society should be
able to reach an efficient allocation by bargaining (provided that “transactions costs” are
negligible). As Snyder (1999) puts it: it is ultimately an empirical question whether a
particular allocation is efficient or not.
Snyder proposes a non-parametric test of the model (actually an extension with several
private and public goods) requiring very weak assumptions on preferences and technology and
only data that can easily be observed. The test uses an extension of revealed preference type
arguments, so functional form assumptions are not necessary. However, the one downside
of the test is potentially quite damaging: one has to observe an equilibrium of the model
more than once! Hence either one would use time series data and make the bold assumption
that we observe a sequence of static Lindahl equilibria or one could use a cross section if one
would assume that preferences are drawn from the same distribution in each “group” (which
could be families, county, country or something else depending on the context).
1.9
Lindahl Equilibrium and the Core
In an economy without externalities, Debreu and Scarf (1963) proved that the core of the economy shrinks to
the set of competitive equilibria as the economy is appropriately enlarged. Sometimes the core is interpreted
as the set of outcomes that could be acheived by collectively rational social decision mechanisms. For
16
this reason one may view the “Core Convergence Theorem” of Debreu and Scarf as a justification of the
competitive hypothesis: i.e., for large economies the competitive model predicts the outcome of any “rational”
social decision rule. Hence, even if one doesn’t believe that the competitive mechainism is “reasonable” in
itself, one could view it as an approximation of what any “reasonable” mechanism could achieve (that is, if
one subscribes to the idea that allocations outside the core are unreasonable predictions).
Now, one could argue that if the core would shrink to the set of Lindahl equilibria in the case with public
goods, then the Lindahl equilibrium concept would rest on the same (firm?) foundations as the competitive
equilibrium concept in the standard Arrow-Debreu model.
The idea with the core is simple. An allocation (outcome) is said to be in the core if no coalition (=subset
of the agents in the economy) can do better for all its members with its own resources. Formally, this may
be (in the constant returns case) be written,
Definition 7 An feasible allocation (x, y) is in the core if there exists no allocation (x0 , y 0 ) and no coalition
C ⊂ I = {1, ..., n} such that
1.
P
i∈C
xi0 + y 0 ≤
P
i∈C
wi
¡
¢
¡
¢
2. ui xi0 , y 0 > ui xi , y for all i ∈ C
Note that we don’t require (x0 , y 0 ) to be feasible. What is required is that for the “defecting coalition”
´
³¡ ¢
´
³¡ ¢
C, the |C| + 1 dimensional vector xi0 i∈C , y 0 is feasible given endowment wi i∈C , 0 .
Foley (1970) proved in a more general case than this that Lindahl equilibria must be in the core. In the
proof below I’ve made use of the assumption that preferences satisfy local non-satiation (I don’t believe this
is essential).
Proposition 8 (Foley) Any Lindahl equilibrium is in the core.
Proof. Let (x, y) be a Lindahl equilibrium allocation with corresponding prices p and suppose that the
¡
¢
¡
¢
conclusion is false. Then there exists a coalition C and an allocation (x0 , y 0 ) such that ui xi0 , y 0 > ui xi , y
P
P
for all i ∈ C and i∈C xi0 + y0 ≤ i∈C wi . By the same revealed preference argument as in the proof of
Pareto efficiency of Lindahl equilibria it follows that
xi0 + pi y 0 > xi + pi y = wi ,
where the equality comes from the fact that the budget constraint must hold with equality if local nonsatiation holds (⇐monotonicly increasing). Summing over C gives
X
i∈C
xi0 + y 0
X
i∈C
pi >
X
xi + y
i∈C
17
X
i∈C
pi =
X
i∈C
wi
But
P
i∈C
pi < 1 (
coalition C)
Pn
i
i=1 p
= 1), implying that (use is made of fact that deviation must be feasible within
X
wi >
i∈C
X
i∈C
xi0 + y0 ≥
X
wi ,
i∈C
which is a contradicion.
1.9.1
Core Convergence?
Despite the issue with truthful revelation of preferences, the Lindahl equilibrium concept would probably
appeal to many (also as a positive model) if there would be some parallell with the Debreu and Scarf result.
This is not quite the case:
1. Muench (1973) wrote down an example where the Lindahl equilibrium is unique, but where the Core
with a continuum of agents turns out to contain a larger set of allocations. This was (at least for
some time) basically viewed as the last word on the subject.
2. However, Conley (2000) notes that it may make more sense to study sequences of replicas of a given
finite economy. He then provides some somewhat reasonable looking conditions for when the Core
converges (and also some conditions for non-convergence).
3. On top of this, there is an issue with exactly how to “create replicas” of a public good economy.
1.9.2
An Example
Assume that we start of with an economy with a single agent with utility function u and endowment w (in
which there is no distinction between a private and a public goods) with a constant returns production of
the “public good”.
Lindahl Equilibrium A Lindahl equilibrium of replication n is the a vector of Lindahl prices p∗ (n)
with
Pn
∗
i=1 pi (n)
= 1 and an allocation (y∗ , x∗1 , .., x∗n ) such that
s.t. w − xi −
(x∗i , y∗ )
∈
p∗i (n) y
≥ 0
arg max u (xi , y)
(xi ,y)
Assuming local non-satiation, the constraint must bind and we conclude that
y ∗ ∈ arg
max
y∈[0,w/p∗
i (n)]
18
u (w − p∗i (n) y, y)
for all agents. But, if we assume strict concavity and that goods are normal (meaning that demands of both
goods increase in income⇒downward sloping demand curves) we conclude that if p∗i (n) < p∗j (n) then
arg
max
y∈[0,w/p∗
i (n)]
u (w − p∗i (n) y, y) = yi∗ > yj∗ = arg
max
y∈[0,w/p∗
j (n)]
which is inconsistent with a Lindahl equilibrium. We conclude:
¡
¢
u w − p∗j (n) y, y ,
Example 9 If all agents have identical strictly concave preferences and goods are normal, the unique Lindahl
equilibrium in economy n is given by:
1. p∗i (n) =
1
n
¡
¢
2. y ∗ (n) = arg maxy u w − ny , y
3. x∗i (n) = w − n1 y∗ (n) for i = 1, ..., n.
Now suppose that
¤− 1
£
u (x, y) = x−ρ + αy −ρ ρ .
If my algebra is right, this generates a Lindahl equilibrium level of the public good given by
1
(nα) ρ+1
∗
y (n) =
1+
1
(nα) ρ+1
w=
1
1
1
(nα) ρ+1
n
+
1
n
w,
∗
so y (n) → ∞ given that ρ > −1. Hence,

1
n
1 ∗
y (n) = 1 −
1
n
+
1
(nα) ρ+1


1
w
= 1 −
n
+1
1
x∗i (n) = w −
1
n

w
(nα) ρ+1
Consider first the case with ρ = 0. Then,
x∗i (n)
·
= 1−
1
α
¸
¸
·
1
1
w.
w=
1+α
+1
This we could have figured out without any algebra (ρ = 0 corresponds with the Cobb-Douglas case with
constant income shares). Instead, let −1 < ρ < 0. Then
n
(nα)
while for ρ > 0
=
1
ρ+1
1
α

1
ρ+1
x∗i (n) = 1 −
n
(nα)
1
ρ+1
=
1
α
1
ρ+1
ρ
n ρ+1 → 0 as n → ∞ ⇒
1
n
1
(nα) ρ+1
ρ
+1

 w → 0,
n ρ+1 → ∞ as n → ∞
19
and, consequently

1
x∗i (n) = 1 −
n
+1
1
(nα) ρ+1

 w → w as n → ∞.
The Core Let V (c) be defined as
V (c) = c max u (x, y)
(6)
x,y
s.t cx + y
≤ cw
The interpretation is that V (c) /c is the highest utility a coalition of size c can guarantee all its members
(the function V (c) may be interpreted as a “characteristic function” in the cooperative game theory sense).
We now claim:
¡
¢
¡ © ªn ¢
P
Example 10 If u is concave, then every allocation y, xi i=1 that satisfies i∈C u xi , y ≥ V (|C|) for
any coalition C ⊂ {1, ..., n} is in the Core.
Proof. (if) Efficiency directly implies that the“grand coalition” will not block. Now, suppose that
¡ i ¢
P
i
i∈C u x , y ≥ V (|C|) holds for all coalitions, but that there nevertheless exists a blocking coalition.
Then, there is some C and some (x0 , y0 ) satisfying
u (x0i , y 0 ) > u (xi , y) for all i ∈ C
X
xi0 + y0 ≤ cw
i∈C
This implies that
X
u (x0i , y0 ) >
i∈C
i∈C
{(x0i , y0 )}i∈C
given by
! Ã
!
X1
X1
x0 ,
y0 =
x0 , y 0
c i
c
c i
Consider the convex combination of
Ã
X1
i∈C
By concavity
u
i∈C
Ã
X1
i∈C
Implying that
cu
Ã
X1
i∈C
and that
c
x0i , y0
X ¡
¢
u xi , y ≥ V (|C|)
!
c
≥
c
i∈C
x0i , y 0
X
!
≥
u (x0i , y 0 ) >
i∈C
X ¡
¢
u xi , y ≥ V (|C|)
i∈C
"
X1
i∈C
1X
u (x0i , y 0 ) ,
c i∈C
c
x
i0
#
+ y0 ≤ cw.
This contradicts the definition of the function V since we have concluded that
¡P
1 0
0
i∈C c xi , y
solution to (6) which generates a higher value of the objective function than the solution.
20
¢
is a feasible
Special Case 1 Consider the limiting case with ρ = 0, where we set α = 1 for simplicity. This
corresponds to utility function u (x, y) = ln x + ln y and the solution to (6), which is also the Lindahl
equilibrium allocation, is y (c) =
cw
2 , x (c)
=
w
2,
so the characteristic function is
³ cw ´i
h ³w´
+ ln
V (c) = c ln
2i
h
³2 w ´
+ ln c
= c 2 ln
2
Consider the following allocation for economy n (even number):
• Let y (n) =
nw
2
• For i = 1, ..., n2 , let xi (n) solve
ln (xi (n)) + ln
Hence, xi (n) =
³ nw ´
2
³w´
³n w´
2 ³n´
V
= ln
+ ln
n ³ 2´
2 2
³ nw2 ´
w
+ ln
= ln
4
2
=
w
4
• Let xi (n) = 34 w for i > n2 .
ª
©
ª
©
¡
¢
P
By construction, the coalition 1, ..., n2 and every subset of 1, ..., n2 satisfies the condition i∈C u xi , y ≥
V (|C|) . Exchanging the lower index agents with higer index agents increases the sum of utilities, so
¡ i ¢
P
n
i∈C u x , y ≥ V (|C|) holds for all coalitions with less than or equal to 2 agents. For larger coaliª
©
tions, the sum of utilities is minimized by including everyone in 1, ..., n2 .We note that V (c) is convex,
which implies that for a coalition of size c >
n
2
we have that
µ
¶
2 (n − c)
2 (n − c) ³ n ´
V
+ 1−
V (n)
n
2
n
µ
¶
2c − n
(n − c) ³ n ´
+
V
V (n)
=
n
2
n
2
³ n ´ µ 2c − n ¶
.
n.
< V
+
n−c<
V (n)
2
2
n
,
,
¡ i ¢
¡n¢
P n2
X ¡
¢
i=1 u x , y = V 2
u xi , y
<
¡ i ¢ ¡ 2c−n ¢
P
V (n)
i∈C
i∈W u x , y >
n
V (c) <
Hence, for each n, the proposed allocation is in the Core. We conclude that the Core does not converge.
Special Case 2 (To be added )Work out a case with ρ > 0 [I think Core convergence will be the result].
21
2
Private Provision of Public Goods (More Complete
Treatment)
We have already discussed that public goods will in general be undersupplied by voluntary
contributions. Still, voluntary contributions of public goods constitute a large chunk of
available resources in the economy (approximately 2% of GDP are private donations to
charity).
2.1
A Simple Model of Private Contributions
This section follows Bergstrom et al (1986).
Adding firms to the model as we did in Section 1.4 doesn’t really add anything to the
model, so we assume that the private good can be turned into a public by any agent.
No additional insights are gained by assuming that there is curvature in the production
technology, so we consider the case where the output of public good is given by the total
voluntary contributions (linear technology). The notation is as follows:
• We have a set of consumers N = {1, .., n}
• wi denotes i’s (exogenous) wealth
• xi denotes i’s consumption of private goods
• gi denotes i’s contribution towards the public good. For ease of notation we let G =
P
P
j gj and G−i =
j6=i gj
• Consumer i has utility function ui (xi , G) , increasing in both arguments.
Optimal behavior implies that the budget constraint must hold with equality, wi = gi +xi ,
so we can eliminate private consumption as a choice variable. Assuming that agents move
simultaneously a strategy is then simply a contribution gi ∈ [0, wi ] . A Nash equilibrium in
22
the game is then a vector of contributions (g1∗ , ..., gn∗ ) such that
¡
¢
gi∗ ∈ arg max ui wi − gi , gi + G∗−i
gi ∈[0,wi ]
for all i. While this is a compact and nice expression, it will be useful to note that (g1∗ , ..., gn∗ )
is an equilibrium if and only if
G∗ ∈ arg
max
G∈[G∗−i ,wi +G∗−i ]
¡
¢
ui wi + G∗−i − G, G
for all i, or even more explicitly, G, xi solves
max ui (xi , G)
xi ,G
s.t. xi + G = wi + G∗−i .
G ≥ G∗−i
The point of writing be condition for i playing a best response in this more complicated way
is that it should be clear that this is just like any ordinary consumer maximization problem,
where relative prices equals unity and wealth is given by wi + G∗−i .
Warr (1983) considered this model and showed by implicitly differentiating the first order
condition that if all agents contribute, then small changes in the distribution of income will
leave the allocation unchanged. Bergstrom et al (1986) extended this argument by a noncalculus approach and could then give a more complete description of how equilibria are
affected by changes in the wealth distribution.
Proposition 11 Suppose ui is quasi-concave for each i ∈ N and let (gi∗ )ni=1 be the initial
equilibrium. Consider a redistribution of income among contributing consumers such that no
consumer loses more than his initial contribution. Let wi is the before and wi0 be the after
redistribution wealth. Then there is a new equilibrium (gi∗∗ )ni=1 after the redistribution such
that
gi∗∗ − gi∗ = wi0 − wi .
Hence, the allocation is unchanged.
23
Proof. Suppose that
gj∗∗ − gj∗ = wj0 − wj
for all j 6= i. Then, the after the change budget constraint is
xi + G =
wi0
+
G∗∗
−i
= wi +
n
X
j=1
= wi + G∗−i
wj0
−
n
X
j=1
wjj + G∗−i =
The feasible set is thus unchanged, except that G ≥ G∗−i − wi0 + wi . However, by assumption
gi∗ ≤ wi0 − wi , so the old level of the public good is feasible. In fact i cannot do better than
this. Let ∆wi = wi0 − wi . We see that if ∆wi < 0, then the budget set is smaller than it
was before the change. Still, the old bundle is feasible, so it follows that G∗ is still optimal.
Second, if ∆wi > 0 and there is a better choice G0 , that is
ui (x0i , G0 ) > ui (x∗i , G∗ )
then it follows by strict quasi-concavity that
ui (λx0i + (1 − λ) x∗i , λG0 + (1 − λ) G∗ ) > ui (x∗i , G∗ )
for any λ ∈ (0, 1) . Since i is among the initial contributors, x∗i < wi , so for λ small enough the
convex combination is feasible before the wealth redistribution. It follows that G∗ couldn’t
be optimal initially, contradicting that (gi∗ )ni=1 is an equilibrium.
The logic extends to the case with a concave production technology as well
2.2
Characterization of the set of Nash equilibria
Consider the problem
max ui (xi , G)
s.t. xi + G = W
This is just a standard consumer optimization problem. Substituting out private consumption we get a single variable decision problem and assuming strict quasi-concavity we will
24
have a unique solution fi (W ) , which in consumer theory language is the demand for the
(public good G). Comparing with the problem that determines i’s best response
max ui (xi , G)
xi ,G
s.t. xi + G = wi + G−i .
G ≥ G−i
we see that the best response function (single-valued due to strict q-concavity) is
max {fi (wi + G−i ) , G−i }
or, if we take the strategic variable to be gi rather than G we have
β i (G−i ) = max {fi (wi + G−i ) − G−i , 0}
Existence of pure strategy equilibria follows since β : ×i [0, wi ] → ×i [0, wi ] is continuous (this
can be verified either from Berges maximum theorem, which in the special case when ui is
strictly concave says that solutions have to be continuous functions rather than upper-hemicontinuous correspondences), so Brouwers theorem applies. In fact, this is a well known
result due to Debreu and Glicksberg. Given an equilibrium (g1∗ , ..., gn∗ ) let
C ∗ = {i ∈ N |gi∗ > 0}
It’s immediate that
G∗ =
X
i
∗
G
=
X
i
¡
¢
gi∗ = fi wi + G∗−i for i ∈ C ∗
¡
¢
/ C∗
gi∗ ≥ fj wi + G∗−j for j ∈
Assuming that the public good is a normal good, fi is strictly increasing and has inverse φi .
Using this inverse we can write the Nash equilibrium condition as
φi (G∗ ) = wi + G∗−i
= wi + G∗ + gi∗ for i ∈ C ∗
25
summing we get
X
φi (G∗ ) =
i∈C ∗
X
i∈C ∗
Rearrange and define F (C ∗ , G∗ ) as
F (C ∗ , G∗ ) =
X
i∈C ∗
wi + (|C ∗ | − 1) G∗
φi (G∗ ) − (|C ∗ | − 1) G∗
we have that for any equilibrium (gi∗ )ni=1 the following relation holds
F (C ∗ , G∗ ) =
X
wi
(7)
i∈C ∗
The importance of the function F is that it is monotonic, which directly implies that for a
fixed C ∗ , there is a unique solution G∗ to the equation above. Monotonicity is however nonobvious since the first term is increasing and the second term is decreasing, so this requires
a proof.
Lemma 12 If both goods are normal, then F is monotonically increasing in G.
Proof. We have that
F (C, G) =
X
i∈C
where c = |C| . The partial w.r.t G is
φi (G) − (c − 1) G,
∂F (C, G) X 0
=
φi (G) − (c − 1)
∂G
i∈C
Now, by the assumption that both goods are normal we have that
fi0 (w) > 0 (demand for public good increasing in income)
1 − fi0 (w) > 0 (demand for private good increasing in income)
Hence fi0 (w) ∈ (0, 1) for all w. By definition of an inverse
φi (fi (w)) = w
φi is differentiable (by the inverse function theorem), so we can just differentiate both sides
to get
φ0i (fi (w)) fi0 (w) = 1
26
Since 0 < fi0 (w) < 1 it follows that φ0i (fi (w)) > 1. Thus
X
∂F (C, G) X 0
=
φi (G) − (c − 1) >
1 − (c − 1) = c − c + 1 = 1 > 0
∂G
i∈C
i∈C
Hence F is monotonically increasing in G.
Proposition 13 There is a unique Nash equilibrium given any distribution of wealth
Proof. Suppose there are two Nash equilibria with total contributions G∗ and G∗∗ , where
we w.l.g assume G∗ ≥ G∗∗ . We have
© ¡
¢ ∗∗ ª
G∗∗ = max fi wi + G∗∗
−i , G−i
and applying φi we get for that
∗∗
wi + G∗∗
−i ≤ φi (G ) for all i ∈ N
summing over all agents in C ∗ (this is important, I was confused here in class)
X
i∈C ∗
wi ≤
X
i∈C ∗
∗
∗∗
φi (G∗∗ ) − |C ∗ − 1| G∗∗
−i = F (C , G )
But (equation (7)) tells us that
F (C ∗ , G∗ ) =
X
wi ,
i∈C ∗
so
F (C ∗ , G∗ ) =
X
i
∗∗
⇒ G
wi ≤ F (C ∗ , G∗∗ )
≥ G∗ (since F is monotonically increasing)
But by hypothesis, G∗ ≥ G∗∗ , so G∗ = G∗∗ .
To finish the proof we need to verify that individual contributions are the same. To see
this we look again at
G∗ =
X
i
∗
G
=
X
i
¡
¢
gi∗ = fi wi + G∗−i for i ∈ C ∗
¡
¢
gi∗ ≥ fj wi + G∗−j for j ∈
/ C∗
27
Suppose that C ∗ 6= C ∗∗ , i.e. C ∗ ∩ C ∗∗ 6= Φ Then there exists some j such that j ∈ C ∗ , but
j∈
/ C ∗∗ , but then
¡
¢
G∗ = fj wj + G∗−j
G∗ ≥ fj (wj + G∗ )
since fj is strictly increasing and G∗ > G∗−j we get that
¡
¢
G∗ = fj wj + G∗−j < fj (wj + G∗ ) ≤ G∗ ,
a contradiction. Finally, we need to rule out the possibility that the composition of contributions within the contributing group changes. To do this, suppose gi∗ > gi∗∗ for some
i ∈ C ∗ = C ∗∗ . Then G∗−i < G∗∗
−i , implying
again a contradiction.
¡
¢
¡
¢
∗
G∗ = fi wi + G∗−i < fi wi + G∗∗
−i = G ,
As a consequence from the fact that
F (C ∗ , G∗ ) =
X
wi
i
in any equilibrium where the set of contributing agents is C ∗ we see that any redistribution
that doesn’t change the set of contributing will leave the allocation unchanged. That is, of
course, as long as the aggregate endowment is fixed.
In general, one can show that
Proposition 14 Consider a wealth redistribution for a fixed aggregate endowment. Then,
1. the equilibrium supply of the public good will increase or be unchanged if the aggregate
wealth of current contributors is unchanged.
2. the equilibrium supply of the public good will increase if the aggregate wealth of current
contributors is increased.
3. if a redistribution among current contributors increases the supply of public good, there
must be strictly fewer contributors after the redistribution
28
2.3
Does Public Provision Crowd Out Private Provision?
It seems intuitively rather obvious that it should be possible to translate these results to
results about what happens when the government provides the good publicly, financed by
taxes on the citizens. Let
• g0 be the public provision of the good
• ti be the (lump-sum) tax on individual i
Obviously we will require budget balance, g0 =
P
i ti .
From the single consumers point of
view it is immaterial whether the good is provided by the government or other consumers, so
g0 affects best responses just like any other voluntary contribution. Also, the lump sum tax
ti is just a reduction in the endowment for i. These two facts suggests that taxing everyone
less than they contribute and publicly providing the good would not change anything, which
is what we show next.
Proposition 15 Let (g1∗ , ..., gn∗ )be the (unique) equilibrium in the model with no public provision and consider a policy (g0 , t1 , ..., tn ) such that ti ≤ gi∗ for all i ∈ N. Then, there is a
unique equilibrium in the model with policy (g10 , ..., gn0 ) satisfying gi0 = gi∗ − ti for all i ∈ N
(which means that the allocation is unchanged).
Proof. Let’s first verify that gi0 = gi∗ − ti is an equilibrium with the policy. Suppose
gj0 = gj∗ − tj for all j 6= i. Agent i’s best response is then a (the) solution to
max
gi ∈[0,wi −ti ]
¡
¢
ui wi − ti − gi , gi + G0−i + g0
or, letting G = gi + G0−i + g0 , the solution to
max ui (xi , G)
xi ,G
s.t G ≥ g0 + G0−i
xi + G = wi − ti + g0 + G0−i
29
Now
g0 + G0−i = g0 +
= g0 +
X
gj0 = g0 +
j6=i
X
j6=i
X
j6=i
gj∗ −
n
X
j=1
gj∗ −
X
tj =
j6=i
tj + ti = g0 + G∗−i − g0 + ti = G∗−i + ti
so the problem is equivalent to
max ui (xi , G)
xi ,G
s.t G ≥ G∗−i + ti
xi + G = wi + G∗−i
¡
¢
Remember that G∗ , wi − G∗ − G∗−i solves
max ui (xi , G)
xi ,G
s.t G ≥ G∗−i
xi + G = wi + G∗−i
¡
¢
If G∗ , wi − G∗ − G∗−i is not a best response with public provision it cannot be feasible any
¡
¢
more. But, gi∗ = G∗ − G∗−i ≥ ti by hypothesis⇔ G∗ ≥ G∗−i − ti , so G∗ , wi − G∗ − G∗−i is
feasible and must therefore be a best response. Since i was arbitrarily chosen this ends the
proof that gi0 = gi∗ − ti for all i is an equilibrium.
To prove that no other equilibria exists we can use the same reasoning as in the case with
no policy. With F defined exactly as before we get
F (G, C) =
X
i∈C
wi −
X
t i + g0
i∈C
and proceed as before
2.4
Voluntary Provision and Strategic Insurance
An interesting consequence of the voluntary provision model is that it can result in a situation
where risk averse agents would turn down fair insurance, which is a point explored in a paper
by Julio Robledo (JPubEcon 1999). The intuition is as follows:
30
• Risk aversion creates a precautionary motive for saving for a rainy day. Hence in any
public good decision problem (or any other consumption problem) which is undertaken
prior to the uncertainty being resolved the consumer would provide less if there is
uncertainty than if they would insure the uncertainty away at a fair rate.
• The crucial aspect of it being a public goods problem is that other agents will pick up the
slack. Hence, the reduced public goods provision will to some extent be compensated
by others. This in turn means that there is a strategic advantage with uncertainty that
can overtake the risk taking disadvantage.
It can be shown that while this is a gain for the agent who refuses to insure herself, this
gain is dominated by losses by others, so the strategic risk-taking is a loss for society. Hence,
the model provides:
1. An efficiency rationale for in-kind provision of insurance. There are other models of this
(Sinn (1982) and Coate (1995), both relying on ideas by Buchanan), but these models
rely on altruistic models, where individuals are bailed out if they occur a serious loss
(the Samaritans Dilemma). This story has nothing to do with altruism, but is purely
based on egoistic agents.
2. The model can also with some stretching be thought of as a model of global pollution
emissions by different countries. Then the model would say that there is an advantage
of having an uncertain future because this induces other countries to reduce emissions
more.
The model is the following:
• Two individuals.
• Utility U i (G, xi ) , where G = g1 +g2 and xi is the private consumption. Both goods are
assumed normal⇒unique Nash equilibria in voluntary provision game under certainty
(& also with uncertainty if normality is suitably defined)
31
• Income wis ∈ [w, w] for states s ∈ S-random income
• Budget constraint: xis + gi ≤ wis
• Prices of both goods normalized to one (choice of units)
Timing:
1. First gi is chosen by each i
2. Then s ∈ S is realized and the resulting consumption is xis = wis − gi
Since the second period choice is trivial we can then analyze this as a static game, where
the best response is the solution to
max EU i (wis − gi , g1 + g2 ) ,
gi
so the first order condition is just like the standard static model with an expectation operator
in frony,
−E
∂U i (wis − gi , g1 + g2 )
∂U i (wis − gi , g1 + g2 )
=0
+E
∂xi
∂G
We can now show:
Proposition 16 If player 1’s uncertainty increases in the sense of a mean preserving spread
i
i
and if Uxxx
− UGxx
> 0 then:
• Player 1 decreases his provision
• Player 2 increases her provision
• The total level of the public good decreases
Proof. First order conditions are
¶
Z µ
∂U i (w − gi , g1 + g2 ) ∂U i (wis − gi , g1 + g2 )
H (θ, g1 , g2 ) =
−
+
fi (w, θ) dw
∂xi
∂G
i
32
where fi (w, θ) is the distribution of income and Fi (w, θ) is the cdf. A mean preserving
spread is a change of the income distribution so that
Z w
∂Fi (w, θ)
dw = 0
∂θ
w
Z w
∂Fi (w, θ)
dw ≥ 0 for w < w < w
∂θ
w
Now, change player 1s disctribution in accordance to a mean preserving spread an keep it fix
for player 2. Let g2 (g1 ) denote the best response for player 2 and differentiate the first order
condition for player 1 with respect to θ and g1 . Since the first order condition (together with
g2 (g1 )) gives a single equation in two variables we can then in principle (and also by messy
calculations in practice) express the change in g1 as a function of θ.
¶
µ
∂H 1 (θ, g1 , g2 )
dg1 ∂H 1 (θ, g1 , g2 ) ∂H 1 (θ, g1 , g2 ) dg2 (g1 )
=−
+
dθ
∂g1
∂g2
dg1
∂θ
Computing the stuff and integrating by parts twice
Z
¡ 1
¢ ∂f (w, θ)
∂H 1
=
UG − Ux1
dw =
∂θ
∂θ
Z
£¡ 1
¢
¤w
¡ 1
¢
1
1
=
UG − Ux Fθ (w, θ) w −
UGx − Uxx
Fθ (w, θ) dw
|
{z
}
=0 according to because
F (w,θ)=0 and F (w,θ)=1
⇒Fθ (w,θ)=Fθ (w,θ)=0
½
Z
¡ 1
¢
1
= − UGx − Uxx
< 0
|
w
Fθ (x, θ) dx
w
{z
¾w
}
=0 by MPS
It is then straightforward to show that
+
w
dg2 (g1 )
dg1
Z
w
w


Z
¡

¢ w
 1

1
Fθ (x, θ) dx dw
 UGxx − Uxxx
|

{z
} w
{z
}
<0 by assumption |
>0 by MPS
< 0 from the FOC of the other agent and one
can grind out that
∂H 1 (θ, g1 , g2 ) ∂H 1 (θ, g1 , g2 ) dg2 (g1 )
+
<0
∂g1
∂g2
dg1
|
{z
} |
{z
}| {z }
−
−
−
¯
¯
¯ 2 (g1 ) ¯
by using that ¯ dgdg
¯ < 1 and that the terms are the same in both expressions.We thus
1
conclude that
dg1
dθ
< 0. The decrease in aggregate provision is verified from a contradiction
argument on the first order conditions.
33
Thus, the effect of the incraesed uncertainty is:
1. Total provision goes down
2. More uncertainty about private good consumption for agent 1
3. Higher provision by agent 2. Hence, this is an income transfer from the other agent
when uncertainty incraeses.
4. Agent 2 is unambigously worse off
5. Agent 1 may or may not be worse off. Can construct examples where the agent is
better off⇒may refuse to buy fair insurance.
6. If so, a Pareto improvement would be for agent 2 to subsidize the insurance of agent
1 sufficiently so that 1 is willing to buy. Efficiency rationale for compulsory insurance
schemes.
2.5
Dynamic Models of Voluntary Contributions
Even before the literature on private provision of public goods started, Schelling had the
following insightful(?) comment on the issue:
If each party agrees to send a million dollar to the red cross on condition the other
does, each may be tempted to cheat if the other contributes first, and each ones
anticipation of the other cheating will inhibit agreement. But if the contributions
are divided into consecutive small contributions, each can try the other’s good
faith at a small price. Furthermore, since each can keep the other on short tether
to finish, no one ever need risk more than a small contribution at a time.
The idea that dividing the contributions into small sums may help was surprisingly
enough not analyzed until the late 80s.
34
2.6
Admati & Perry
The most well-known paper on the subject is probably Admati & Perry (1991) who considered a model where the public good is a discrete “project”. The idea is that the players
contribute sequentially, but no utility of the good is received until the project is complete.
In their model players take turns to make contributions until the project is complete. They
show
1. For a large set of parameter values, there is a unique subgame perfect equilibrium.
2. It is possible that the project is not completed although it is socially desirable to do
so (inefficient level of public good)
3. Even if project is completed, there is an inefficiency due to delay. While they don’t
make this connection it is clear that possibilities of splitting contributions in parts still
helps since the project would not be completed in a one-shot voluntary contribution
game.
Consult the paper for details.
2.7
Marx & Matthews
A more recent paper is Marx & Matthews (2000). Their findings are roughly:
1. Allowing for contributions to be split up in small pieces may help in the sense that equilibria where a project is implemented exists in the dynamic game for parametrizations
where the unique equilibrium in the static game has nobody contributing.
2. There is still an inefficiency due to delay.
3. However, this inefficiency may be rather inconsequential if the time period between
contributions are small.
35
The main difference in their setup is that they don’t force the agents to move sequentially,
but the differing assumptions on the technology also plays a role to explain the differences
in terms of results. Let
• N = {1, ..., n} be the set of agents, where n ≥ 2 to avoid trivialities.
• Time is discrete. Periods are indexed by t = 0, 1, ...
• Players can contribute towards a public project in each period. Let zi (t) denote the
P
contribution by player i in period t. We let z (t) = (z1 (t) , ..., zn (t)) , Z (t) = j zj (t)
P
and Z−i (t) = j6=i zj (t) .
• We denote i’s cumulative contribution by xi (t) =
P
lative contribution by X (t) = j xj (t)
Pt
τ =1 zi
(t) and the aggregate cumu-
• Let δ = e−rl be the discount factor ( r is the discount rate and l is the length of a
period)
• The agents are supposed to receive utility from the public good in each period. For
total cumulative contribution X and period contribution zi we assume that the period
utility is (1 − δ) f (X) − zi 2 . Thus, all agents have the same quasi-linear preferences
over public and private consumption.
Given s sequence of contributions z = {zi (t)}∞
t=0 , player i’s payoff is
Ui (z) =
∞
X
t=0
δ t ((1 − δ) f (X (t)) − zi (t))
Note, the normalizing factor (1 − δ) is multiplied to per period payoff for convenience in
front of the benefits function. The reason why this is useful is that for X (t) = X for all
P
t
1
t we get ∞
t=0 δ f (X) = 1−δ f (X) . In the repeated game language, the multiplication by
2
One natural interpretation is that the instantenous utility over public and private consumption is
u (X, c) = v (X) + ci . Then define f (Z) =
v(Z)
1−δ
and note that the utility if the agent has income w and
contributes zi is (1 − δ) f (X) + (w − zi ) . Since w enters additively we can just drop it without affecting the
maximization problem.
36
(1 − δ) turns benefits into average benefits. The interpretation is that f (X) is the discounted
present value of the benefit flow if the capital in the project were equal to X in every period.
To translate this back into flow payoffs we must multiply the present value by (1 − δ) .
It is assumed that the benefits function f has some particular structure, namely

 λX for X < X ∗
f (X) =
,
 V for X ≥ X ∗
£
¤
where λ ∈ 0, XV∗ . Special cases are:
• λ = 0; the project is only valuable after completion (bridge)
• λ=
V
;
X∗
“no jump”
¡
¢
For λ ∈ 0, XV∗ there is a “jump” at completion. The idea is that when building say
a road network, the last road makes all the other roads much more valuable. The jump is
interpreted as the additional utility gained from the last cents worth of contributions and
the notation b = V − λX ∗ for the size of the jump will be useful later on. The authors claim
without proof that the analysis would be qualitatively similar when the benefit function is
continuous if there are strong increasing returns towards completion of the project.
Maintained Assumptions
• V >
X∗
n
(Project is socially valuable)
• X ∗ < V (Nobody willing to complete project alone)
• b = V − λX ∗ ≥ 0 (“jump” positive). Observe that this implies that λ ≤
V
X∗
<1
Given a contribution sequence {z (t)}∞
t=0 and the associated sequence of aggregate cumu-
lative contributions {X (t)}∞
t=0 we refer to T as the completion date if it is the smallest t
for which X (t) ≥ X ∗ . We call a contribution sequence wasteless if X (t) ≤ X ∗ for all t (all
additional contributions above X ∗ are wasteful since no additional gains are created). Given
37
a wasteless contribution sequence we can write the payoffs as
Ui (z) =
=
T −1
X
t=0
T
−1
X
t=0
T
δ ((1 − δ) λX (t) − zi (t)) − δ zi (T ) + (1 − δ)
∞
X
δtV =
t=T
δ t ((1 − δ) λX (t) − zi (t)) − δ T zi (T ) + δ T V
P
Z (t) for all t and V = b + λX ∗ = b + λX (t) = b + λ Tt=0 Z (t) , so
Ã
Ã
!
!
T −1
t
T
X
X
X
Ui (z) =
δ t λ (1 − δ)
Z (t) − zi (t) − δ T zi (T ) + δ T b + λ
Z (t) =
Now X (t) =
Pt
t
τ =0
t=0
τ =0
= (1 − δ)
∞
X
t=0
t
δ λZ (0) + (1 − δ)
∞
X
t=1
t
δ λZ (1) + ... + (1 − δ)
= λZ (0) + δλZ (1) + .... + δT Z (t) + δ T b −
=
T
X
t=0
T
X
t=0
∞
X
t
t=T
T
δ λz (T ) + δ b −
(8)
T
X
t=0
δ t zi (t)
t=0
δ t (λZ (t) − zi (t)) + δ T b
Let the contributing horizon be T ≤ ∞, meaning that T is the last period contributions can
be made. The extensive form has following rules:
1. contributions can be made in periods t = 0, .., T
2. the players contribute simultaneously
3. at the start of period t, player i observes only his own and aggregate past contributions,
that is the history for player i has form {zi (τ ) , Z (τ )}tτ =0 .
2.8
The Static Game
If T = 0 the game reduces to a standard static voluntary contribution game. Rather than
indexing all variables by zi (0) , Z (0) , we drop time indices all together and write zi , Z. A
strategy for a single agent is then to choose a contribution zi in [0, w] , where w is the total
income. We assume that w is large enough so that the constraint zi ≤ w is never binding
(for ex w > X ∗ would be more than enough).
38
δ t zi (t) =
For each contribution vector z = (zi , z−i ) we have that the payoff for agent i is

P

V − zi
if Z = i zi ≥ X ∗
³
´
ui (zi , z−i ) =
 λ z + P z − z if Z = P z < X ∗
i
j
i
j6=i
i i
To find i’s best response to z−i , fix Z−i =
zi < zi0 < X ∗ − Z−i and note that
P
j6=i zj
< X ∗ . Then consider two alternatives
ui (zi0 , z−i ) − ui (zi , z−i ) = (λ − 1) (zi0 − zi ) .
Since λ < 1 this means that the best choice of zi ∈ [0, X ∗ − Z−i ) is to set zi = 0. If
X ∗ − Z−i ≤ zi < zi0 on the other hand
ui (zi0 , z−i ) − ui (zi , z−i ) = −zi0 + zi > 0,
so the best choice of zi ∈ [X ∗ − Z−i , ∞) is to set zi = X ∗ − Z−i . Thus, if Z−i ≥ X ∗ , the best
response is to set zi = 0. If Z−i < X ∗ we only need to compare the payoff of 0 versus the
payoff of X ∗ − Z−i . We see that
ui (0, z−i ) − ui (X ∗ − Z−i ) = λZ−i − (V − X ∗ − Z−i ) ≤ 0
m
(1 − λ) Z−i ≥ V + X ∗
For interpretation it is useful to rewrite this as
∗
X − Z−i
V − λX ∗
≤
= c (0) ,
1−λ
where c (0) has the interpretation of being the critical level of how much is left of completing
the project to make it worth the while for a single agent to do so. The set of pure best
responses is thus
β i (z−i ) =
0
if X ∗ − Z−i > c (0)
{0, X ∗ − Z−i }
if X ∗ − Z−i = c (0)
min {X ∗ − Z−i , 0} if X ∗ − Z−i < c (0)
Note that this means that:
39
1. independenly of what the other are doing, no player will ever want to contribute more
than c (0) , which is the maximal contribution in the static game
2. c (0) is decreasing in λ.
3. Z = X ∗ in any Nash equilibrium.
4. z = (0, 0, ...., 0) is always an equilibrium. To see this, suppose that zj = 0 for all j 6= i
and note that
X ∗ > V ⇒ X ∗ − λX ∗ > V − λX ∗ ⇔ X ∗ >
V − λX ∗
= c (0)
1−λ
Hence β i (0, 0, .., 0) = 0, so z = (0, ..., 0) is a Nash equilibrium.
5. If there is a symmetric Nash where the project is completed, zi =
X∗
n
for all i. For this
to be Nash it must be that
X∗ −
n − 1 ∗ X∗
V − λX ∗
X =
≤
n
n
1−λ
m
µ
¶
1 + (n − 1) λ
X∗
≤V
n
We see that for λ = 0 this holds, while for λ =
V
X∗
the condition can never hold.
6. It is intuitive (and straightforward to show) that if
X∗
n
<
V −λX ∗
,
1−λ
then there exists
many other equilibria as well (just increase one agents contribution with and decrease
someone elses with
other hand, if
X∗
n
≥
and you see that the equilibrium condition still holds). On the
V −λX ∗
,
1−λ
there is no way for an asymmetric equilibrium to exist.
Summing this up as a proposition:
+
Proposition 17 (0, ..., 0) is always an equilibrium. z ∈ R+
is an equilibrium if and only if
Z = X ∗ and zi ≤ c (0) for all i. Efficient equilibria exists if and only if c (0) ≥
¡ ∗
∗¢
unique symmetric efficient equilibrium is Xn , ..., Xn .
40
X∗
n
and the
2.9
The Dynamic Game
Keep in mind that the interesting question is if allowing for contributions in multiple periods
may help, that is, is there equilibria where the project is completed for some T > 0 when it
can not be completed in equilibrium of the static game.
In spite of its simplicity, solving for “credible” equilibria in the game is rather complicated. The reason is that individual contributions are assumed to be non-observable, which
means that there are no subgames. For this reason, equilibrium concepts like sequential equilibrium or perfect Bayesian equilibrium must be used to get rid of equilibria where agents
are behaving like idiots off the equilibrium path.
However, as a first step, the authors consider what is sustainable as Nash equilibria of
the game. While this exercise is by no means trivial, at least one does not have to think
about what beliefs off the equilibrium path can sustain the strategies used.
We (as well as Marx & Matthews) restrict attention to symmetric equilibria where zi (t) =
P
∗
g (t) for each i and t. The project is completed if and only if t g (t) ≥ Xn . If the project
is completed in finite time, define the completion period given a symmetric action profile
(g, ..., g) , T (g) as the integer satisfying
T (g)−1
X
t=0
T (g)
X∗ X
g (t) <
≤
g (t) ,
n
t=0
otherwise let T (g) = ∞.
It is pretty clear that
1. No contributions can ever occur in equilibrium after T (g) . Given that the project is
already complete it is a strictly dominated strategy to contribute.
2. If the project is completed, it is contributed wastelessly. The reason is simply that
otherwise a player could deviate in period T (g) by giving exactly the amount that is
sufficient to complete the project.
For g to be sustainable as a Nash equilibrium outcome it must be that no player wishes to
deviate given that deviations are punished by the harshest possible punishment. Clearly, any
41
players payoff is strictly increasing in the giving of all the others, so the maximum punishment
is for all other players to stop contributing forever whenever a deviation is detected. For this
reason we consider grim or trigger strategies of the form

 g (t) for all histories h (t) such that Z (τ ) = ng (τ ) for τ < t
zi (h (t)) =
 0
otherwise
Note that the strategy is only conditioned on what is common knowledge, the aggregate
giving in each period.
Let VL (X) be the lowest value of the game (same for all agents) after an history with
P
total contributions X. As we remarked above, VL (X) = λX = λn tτ =0 g (τ ) . One would
then be led to comparing the payoff of a unilateral deviation by i at time t, zi , which would
be
Ã
−zi + VL n
t
X
τ =0
g (τ ) − g (t) + zi
!
Ã
= −zi + λ n
t
X
τ =0
g (τ ) − g (t) + zi
!
with the payoffs of following the candidate equilibrium strategy which could be derived from
(8). However, besides the fact that its sort of a mess to write out the right continuation
valuation given that everyone follows the candidate equilibrium strategies, this is simply not
good enough. There is no one-shot deviation principle for Nash equilibrium, so we
must consider deviations of all possible forms. Hence, we derive the ex ante payoffs given
the grim profile {z1t , .., znt }Tt=0 , where zit : Ht → R+ is given by the grim strategy discussed
above. The payoff for an agent i if all agents are following the candidate equilibrium strategy
is
U
eq
= (λn − 1)
T (g)
X
δ τ g (τ ) + δ T (g) b
τ =0
There are three different types of deviations to consider:
1. Completing the project immediately.
2. Deviate with an amount zi in period t that is not sufficient to complete the project
and then never contribute again
3. Deviate with an amount that is not sufficient to complete the project and contribute
again at later stages.
42
You are asked to rule out the last deviation on your problem set. A deviation of the
second type yields payoff
1
U (zi , t) = (λn − 1)
t−1
X
τ =0
δ τ g (τ ) + δ t (λ (n − 1) g (t) − (1 − λ) zi ) .
Since λ < 1, the best deviation of this type is to set zi = 0. Hence, nobody has an incentive
to deviate to “non-completing one-shot deviation strategies” if and only if U 1 (0, t) ≤ U eq
for all t ≤ T (g) − 1, or
t
T (g)
X
t
δ (λ (n − 1) g (t)) ≤ δ (λn − 1) g (t) + (λn − 1)
δ τ g (τ ) + δ T (g) b
(9)
τ =t+1
m
T (g)
(λn − 1) X τ −t
δ T (g)−t b
g (t) ≤
δ g (τ ) +
1 − λ τ =t+1
1−λ
If on the other hand i finishes the project immediately by deviating to
∗
zi = g (t) + X − n
t
X
g (τ )
τ =0
the payoff is
U 2 (t) = (λn − 1)
t−1
X
τ =0
Ã
Ã
δ τ g (τ ) + δ t (λn − 1) g (t) − (1 − λ) X ∗ − n
t
X
g (τ )
!!
τ =0
+ δtb
so in order for these deviations not to be profitable U 2 (t) ≤ U eq for all t ≤ T (g) − 1 or
δ
t
Ã
Ã
∗
b − (1 − λ) X − n
∗
t
X
!!
g (τ )
τ =0
X −n
≤ (λn − 1)
m
t
X
τ =0
g (τ ) ≥
T (g)
X
δ τ g (τ ) + δ T (g) b
(10)
τ =t+1
³
´
T (g)−t
1−δ
b
1−λ
T (g)
(λn − 1) X τ −t
+
δ g (τ )
1 − λ τ =t+1
Observe that this constraint is only relevant just before completion. While this limits the
possibilities for punishments somewhat, this constraint on g (t) does not impose any severe restrictions on what can be done. After proving that deviations of the third type are
dominated (problem set) we can conclude:
43
Proposition 18 A (symmetric) sequence of contributions {g (t)} can be supported as a Nash
equilibrium outcome if and only if (9) and (10) are satisfied for each t.
It is easy to make sure that (10) holds (noting that g (t) + n
t together with (9) is sufficient.
2.9.1
PT (g)
τ =t+1
g (τ ) ≥
b
1−λ
for each
Credible Equilibria
As you will see in the problem set, the grim strategies cannot be supported as a perfect
Bayesian equilibrium. However, Marx & Leslie shows that the strategies can be modified
so that the modified strategies together with some consistent beliefs constitutes a perfect
Bayesian equilibrium. For the construction, see the paper.
2.9.2
Completing Equilibria
The main point with the paper is that the project may be completed even under parameters
where it would not in the static setting. In particular:
• If b > 0 the project may be completed in finite time (this is on problem set)
• If b = 0, the project may be asymptotically completed.
• The inefficiency disappears in the limit when time between contributions goes to zero.
44