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Lindahl Pricing and Equilibrium – Proof of Pareto Optimality
A Lindahl equilibrium is a method for finding the efficient level of provision for public goods.
Recall that for public goods, in equilibrium all agents consume the same quantity but may face
different prices1. As it is framed in our textbook, the Lindahl equilibrium occurs when the perunit price paid by each agent sums to the total per unit cost of the public good.
The Graph
We start with a good ol’ fashioned demand curve for a public good. The lower the price of the
good, the more Person 1 wants to consume. Now imagine that the dashed horizontal line is the
full price of the good. At this point, the demand curve makes it look like Person 1 will demand
very little. But what if rather than the price dropping, the percentage of the price he have to pay
goes down? As far as Person 1 is concerned, this is equivalent to the price he sees going down,
so he’ll demand more.
Full
price
Price * 50%
Price * 25%
Price * 0%
D1
Qfull price QPrice * 50%
QPrice * 25%
Q
Now lets look at another demand curve (Person 2). This person sees the vertical axis flipped the
other way around, with the full price on the bottom and percentage decreasing as one moves
upward. Like Person 1, Person 2 will demand more as her observed price goes down.
Price * 0%
D2
Price * 50%
Full
price
Qfull price
QPrice * 50%
Q
1
This differs from equilibrium of private goods, which instead has all agents viewing the same price with the
possibility to consume different quantities.
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2006
Again, note that here Person 2’s observed price going down means we move further up the
vertical axis.
Equilibrium is when both of these people demand the same amount of the public good. This
happens when the two demand curves intersect each other. If we draw a line over to the price
axis from that point of intersection, we get the percentage share for each agent that is required to
get that price.
Person 2
P*55%
D2
Person 1
P*45%
D1
Full price = P
Q*
Q
So Person 1 is paying P*45% per unit, Person 2 is paying P*55% per unit, and the economy
produces Q* units.
Here we have a Lindahl equilibrium, and the corresponding prices are called Lindahl prices. But
is it a Pareto Optimal (PO) equilibrium? Well, recall from the last couple of chapters that a PO
allocation occurs with public goods when the sum of the marginal rates of substitution equals the
marginal rate of transformation. So if we can show that holds true in a Lindahl equilibrium, we
know it is PO.
The Math
Let’s say there are two goods: a public good, and “everything else”. Call the price of the public
good Ppublic and the price of everything else Pelse.
Under the current system, no one actually sees the full price of the public good – they just see the
percentage that’s been allocated to them in equilibrium. In the graph above, that turned out to be
P*45% for Person 1 and P*55% for Person 2. In general lets call the percentage that Person 1
pays α, and the percentage that Person 2 pays (1-α). If Person 1 is a utility maximizer, we know
that he’ll choose a bundle where
" * Ppublic
= MRSPerson1
Pevery
(1)
This is just the usual price ratio/marginal rate of substitution deal . . . the only change is that we
multiply Ppublic by α to allow for the price adjustment to the public good. Similarly, Person 2 will
! their bundle such that
choose
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2006
(1" # ) * Ppublic
= MRSPerson 2
Pevery
(2)
So now we have both agents utility maximizing . . . so far, so good. We know that in a
competitive equilibrium, it must be the marginal cost ratio (price ratio) is equal to the marginal
! transformation, or
rate of
MC public Ppublic
=
= MRT
MCevery
Pevery
(3)
What we set out to show was that the Lindahl equilibrium is PO. That would require that
!
MRSPerson1 + MRSPerson 2 = MRT
Using (1), (2), and (3)
!
MRSPerson1 + MRSPerson 2 =
" * Ppublic (1# " ) * Ppublic
+
Pevery
Pevery
=
" * Ppublic # " * Ppublic + 1* Ppublic
Pevery
=
Ppublic
Pevery
= MRT
As they say in mathematical circles, Q.E.D.2
!
The Issues
As with all good ideas in economics, there are some issues with real world applications of the
Lindahl equilibrium. For one, it assumes that we know each individuals preferences. What if
people intentionally hide their true willingness to pay? This starts getting into the “free rider”
problem issue again, where people decide to let others cover the costs, but reap the benefits
themselves.
Even if we DID know exactly what everyone’s preferences were, things would still get
significantly more complicated as there were more and more people involved in the discussion.
It’s one thing to get two people to agree on some provision of a public good, but getting a city of
100,000 people to do so is just plain nuts3.
2
It’s an abbreviation for the Latin phrase “quod erat demonstrandum”, or “that which was to be demonstrated”. It
basically means the proof is done . . . use it to impress your friends and family.
3
Basically, that means we’re dealing with up to 100,000 prices for just one good. This makes finding an equilibrium
much, much harder.
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2006