1. MICROECONOMICS Principles and Analysis Frank Cowell
Prerequisites
Almost essential
Welfare and Efficiency
Public Goods
MICROECONOMICS
Principles and Analysis
Frank Cowell
August 2006

2. Overview... Characteristics of public goods Public Goods The basics
Efficiency
Characteristics of public goods
Contribution schemes
The Lindahl approach
Alternative mechanisms

3. Characteristics of public goods
Two key properties that we need to distinguish:
Excludability
You are producing a good.
A consumer wants some.
Can you prevent him from getting it if he does not pay?
Rivalness
Consider a population of people all consuming 1 unit of commodity i.
Another person comes along, also consuming 1 unit of i.
Will more resources be needed for the ?
These properties are mutually independent
They interact in an interesting way

4. Typology of goods: classic definitions
Rival?
[ Yes ]
[ No ]
[??]
[ Yes ]
pure
private
Excludable?
[??]
pure
public
[ No ]

5. How the characteristics interact
Example: Bread
(E) you can charge a price for bread
(R) an extra loaf costs more labour and flour
Private goods are both rival and excludable
Public goods are nonrival and nonexcludable
Consumption externalities are non-excludable but rival
Non-rival but excludable goods often characterise large-scale projects.
Example:
bread
Example: National defence
(E) you can't charge for units of 'defence‘
(R) more population doesn't always require more missiles
Example:
defence
Example:
flowers
Example: Scent from Fresh Flowers
(E) you can't charge for the scent
(R) more scent requires more flowers
Example: Wide Bridge
(E) you can charge a toll for the bridge
(R) an extra journey has zero cost
Example:
bridge

6. Aggregating consumption:
How consumption is aggregated over agents depends on rivalness characteristic
Also depends on whether the good is optional or not
Private goods nh
xi = S xih
h=1
Pure rivalness means that you add up each person’s consumption of any good i.
Pure nonrivalness means that if you provide good i for one person it is available for all.
Optional public goods
xi = max h ( xih )
Pure nonrivalness means that if one person consumes good i then all do so.
Non-optional public goods
xi = xi1 = xi2 =...

7. Public Goods
Overview...
The basics
Efficiency
Extending the results that characterise efficient allocations
Contribution schemes
The Lindahl approach
Alternative mechanisms

8. Public goods and efficiency
Take the problem of efficient allocation with public goods.
The two principal subproblems will be treated separately...
Characterisation
Implementation
Implementation will be treated later
Characterisation can be treated by introducing public-goods characteristics into standard efficiency model
Jump to “Welfare: efficiency”

9. Efficiency with public goods: an approach
Use the standard definition of Pareto efficiency
Use the standard maximisation procedure to characterise PE outcomes...
Specify technical and resource constraints
These fix utility possibilities
Fix all persons but one at an arbitrary utility level
Then max utility of remaining person
Repeat for another person if necessary
Use FOCs from maximum to characterise the allocation

10. Efficiency: the model
Let good 1 be a public good, goods 2,...,n private goods
Then agent h’s consumption vector is
(x1h, x2h , x3h, ..., xnh)
where x1 is the same for all agents h.
and x2h , x3h, ..., xnh is h’s consumption of good 2,3,...n
Agents 2,…,nh are on fixed utility levels uh
Differentiating with respect to x1 involves a collection of nh terms
good 1 enters everyone’s utility function.

11. Efficiency: the model
Let good 1 be a public good, goods 2,...,n private goods
Then agent h’s consumption vector is
(x1h, x2h , x3h, ..., xnh)
where x1 is the same for all agents h.
and x2h , x3h, ..., xnh is h’s consumption of good 2,3,...n
Agents 2,…,nh are on fixed utility levels uh
Problem is to maximise U1(x1, x21, x31, ..., xn1) subject to:
Uh(x1, x2h, x3h, ..., xnh) ≥ uh, h = 2, …, nh
Ff(qf) ≤ 0, f = 1, …, nf
xi ≤ qi + Ri , i= 1, …, n
Use all this to form a Lagrangean in the usual way…

12. Finding an efficient allocation
Lagrange multiplier for each utility constraint
max L( [x ], [q], l, m, k) :=
U1(x1) + åhlh [Uh(xh)  uh]
 åf mf F f (q f)
+ åi ki[qi + Ri  xi]
where
xh = (x1, x2h, x3h, ..., xnh)
xi = åh xih , i = 2,...,n
qi = åf qi f
Lagrange multiplier for each firm’s technology
Lagrange multiplier for materials balance, good i

13. FOCs å lhUjh (x1, x2h, x3h, ..., xnh) = k1
For any good i=2,…,n differentiate Lagrangean w.r.t xih.
If xih is positive at the optimum then:
lhUih (x1, x2h, x3h, ..., xnh) = ki
But good 1 enters everyone’s utility function. So, differentiating w.r.t x1: nh
å lhUjh (x1, x2h, x3h, ..., xnh) = k1
h=1
Differentiate Lagrangean w.r.t qif. If qif is nonzero at the optimum then:
mfFif(qf) = ki
Likewise for good j:
mfFjf(qf) = kj
MU to household h of good i
shadow price of good i
Sum, because all are benefited
shadow price of good 1

14. Sum of marginal willingness to pay
Another look at the FOC...
For private goods i, j = 2,3,..., n :
Ujh(xh) kj Fjf(qf)
——— = — = ——
Uih (xh) ki Fif(qf)
Condition when good 1 is public and good i is private
Sum of marginal willingness to pay nh
åh=1
U1h(xh) k1
——— = —
Uih (xh) ki
An important rule for public goods:
Sum over households of marginal willingness to pay = shadow price ratio of goods = MRT

15. Overview... Private provision of public goods? Public Goods The basics
Efficiency
Private provision of public goods?
Contribution schemes
The Lindahl approach
Alternative mechanisms

16. The implementation problem
Why is the implementation part of the problem likely to be difficult in the case of pure public goods?
In the general version of the problem private provision will be inefficient
We have an extreme form of the externality issue
We run into the Gibbard-Satterthwaite result

17. Example Good 1 - a pure public good Good 2 - a pure private good
Two persons: A and B
Each person has an endowment of good 2
Each contributes to production of good 1
Production organised in a single firm

18. Public goods: strategic view (1)
If Alf reneges [–] then Bill’s best response is [–].
If Bill reneges [–] then Alf’s best response is [–].
Alf
[+]
2,2
0,3
Nash equilibrium
[–]
3,0
1,1
[+]
[–]
Bill

19. Public goods: strategic view (2)
If 1 plays [–] then 2’s best response is [+].
Alf
[+]
2,2
1,3
If 2 plays [+] then 1’s best response is [–].
A Nash equilibrium
By symmetry, another Nash equilibrium
[–]
3,1
0,0
[+]
[–]
bill

20. Which paradigm?
Clearly the two simplified +/– models lead to rather different outcomes.
Which is appropriate? Will we inevitably end up at an inefficient outcome?
The answer depends on the technology of production.
Also on the number of individuals involved in the community.

21. A Voluntary Approach (1)
Consider in detail the implementation problem for public goods
Logical to view the way individual action would work in connection with public goods
Begin with a simple contribution model
Take the case with nh persons.
Then see what the “classic” solution would look like

22. A Voluntary Approach (2)
Each person has a fixed endowment of (private) good 2:
R2h
And makes a voluntary contribution of some of this toward the production of (public) good 1:
zh = R2h – x2h
This is equivalent to saying that he chooses to consume this amount of good 2:
x2h

23. A Voluntary Approach (3)
Contribution of all households of good 2 is: nh
z = S zh
h=1
This produces the following amount of good 1:
x1 = f(z)
So the utility payoff to a typical household is:
Uh(x1 , x2h)

24. A Voluntary Approach (4)
Suppose every household makes a “Cournot” assumption: nh
S zk =`z (constant)
k=1
kh
Given this and the production function agent h perceives its optimisation problem to be:
max Uh(f(`z + R2h – x2h ) , x2h)
This problem has the first-order condition:
U1h(x1 , x2h) fz(`z + R2h – x2h ) – U2h(x1 , x2h) = 0

25. A Voluntary Approach (5)
The FOC yields the condition:
U1h(x1 , x2h)
———— = —————
fz(Sh zh ) U2h(x1 , x2h)
MRT = MRSh
However, for efficiency we should have:
U1h(x1 , x2h)
———— = Sh —————
fz(Sh zh ) U2h(x1 , x2h)
MRT = Sh MRSh
Each person fails to take into account the “externality” component of the public good provision problem

26. Outcomes with public goodsx2
Production possibilities
Efficiency with public goods
Contribution equilibrium x ^
MRT = MRS
Myopic rationality underprovides public good... x*
MRT = SMRS x1

27. Graphical illustrations
We can use two of the graphical devices that have already proved helpful.
The contribution diagram:
Nash outcomes
PE outcomes
The production possibility curve

28. Outcomes of contribution game
Alf’s ICs in contribution space zb
Alf’s reaction function
Bill’s ICs in contribution space
Bill’s reaction function
ca(·)
Cournot-Nash equilibrium
Efficient contributions
Alf assumes Bill’s contribution is fixed
Likewise Bill’
cb(·)
Cournot-Nash outcome results in inefficient shortfall of contributions. za

29. Overview... “Personalised” taxes? Public Goods The basics Efficiency
Contribution schemes
The Lindahl approach
Alternative mechanisms

30. A solution?
Take the standard efficiency result for public goods: Sj MRSj = MRT
This aggregation rule has been used to suggest an allocation mechanism
The “Lindahl solution” is tax-based approach.
However, it is a little unconventional.
It suggests that people pay should taxes according to their willingness to pay
The sum of the taxes covers the marginal cost of providing the public good.

31. An example Good 1 - a pure public good Good 2 - a pure private good
Two persons: Alf and Bill
Simple organisation of production: A single firm

32. Willingness-to-pay for good 1
Plot Alf’s MRS as function of x1
WTP by Alf for x1
Bill’s MRS as function of x1
Ua(•)/Ua(•)
WTP by Bill for x1
MRS21(x1) a x1
the more there is of good 1 the less Alf wants to pay for extra units x1
Bill is less willing to pay for good 1 than Alf
Ub(•)/Ub(•)
MRS21(x1) b
Use this to derive efficiency condition x1 x1

33. Efficiency  1/fz x1 Ua(•)/Ua(•) 1 2 x1 Ub(•)/Ub(•) 1 2
MRS21(x1)
a * 
MRS for Alf and for Bill
Sum of their MRS as function of x1 x1
MRT as function of x1
Efficient amount of x1
Ub(•)/Ub(•)
MRS at efficient allocation.
MRS21(x1)
b *
Consider these as demand curves for good 1 
For a public good we aggregate demand “vertically”
1/fz
ShUh(•)/Uh(•)
Can we use these WTP values to derive an allocation mechanism?
ShMRS21(x1)
h * * x1  x1

34. But what of individual rationality?x1
Lindahl solution
Ua(•)/Ua(•) pa 
Efficient allocation of public good x1
Willingness-to-pay at efficient allocation.
Charge these WTPs as “tax prices “
Ub(•)/Ub(•) pb
The “ Lindahl solution” suggests that people pay should taxes according to their willingness to pay
Combined “tax prices” pa + pb just cover marginal cost of producing the amount x1* of the public good 
1/fz
ShUh(•)/Uh(•)
But what of individual rationality?
pa + pb * x1  x1

35. The Lindahl Approach
let ph is the “tax-price” of good 1 for person h, set by the government.
The FOC for the household’s problem is:
U1h(x1, x2h)
———— = ph
U2h(x1, x2h)
For an efficient outcome in terms of the allocation of the two goods: nh
S ph = ——
h= fz(z)
Conditions 1,2 determine the set of household-specific prices { ph}
Sh MRSh = MRT

36. The Lindahl Approach (1)
But where does the information come from for this personalised tax-price setting to be implemented?
Presumably from the households themselves
In which case households may view the determination of the personalised prices strategically.
In other words h may try to manipulate ph (and thus the allocation) by revealing false information about his MRS

37. The Lindahl Approach (2)
Take into account this strategic possibility
Then h solves the utility-maximisation problem:
choose (x1, x2h) to max Uh(x1, x2h) subject to
the budget constraint:
phx1 + x2h  R2h
the following perceived relationship:
x1 = f(c + phx1)
But here ph is endogenous:
So this becomes exactly the problem of voluntary contribution

38. The Way Forward
Given that the Lindahl problem results in the same suboptimal outcome as voluntary contribution (subscription) what can be done?
Public provision through regular taxation
Change the problem
Change perception of the problem

39. Overview... Truth-revealing devices Public Goods The basics Efficiency
Contribution schemes
The Lindahl approach
Alternative mechanisms

40. A restricted problem
One of the reasons for the implementation problem is that one invites selection of a social state qQ, where Q is large.
Sidestep the problem by restricting Q.
We would be changing the problem
But in a way that is relevant to many situations
Suppose that there is an all-or nothing choice.
Replace the problem of choosing a specific amount of good 1 from a continuum …
…by substituting the choice problem “select from {NO-PROJECT, PROJECT} ”

41. The Clark-Groves approach
Imagine a project completely characterised by
the status-quo utility,
the payment required from each member of the community if the project goes ahead
the utility to each person if it goes ahead.
For all individuals
utility is separable and
income effect of good 1 is zero:
Uh(x1 , x2h) = y(x1) + x2h

42. The C-G method (2) Person h has endowment of R2h of private good 2.
The project specifies a payment zh for each person conditional on the project going ahead.
Total production of good 1 is f(z) where
z := Sh zh
Social states states Q = {q0 , q1} where
q0 : f(0) = 0
q1 : f(z) = 1
Measure the welfare benefit to each person by the compensating variation CVh .

43. Should project go ahead?
Project payoffs x2 a
Consumption space for Alf and Bill
Endowments and preferences q° R2 a
Outcomes if project goes ahead
Alf
The elements of Q
R2 – z
a a q′
Compensating variation for Alf, Bill
Alf would like the project to go ahead.
Bill would prefer the opposite. x1 1 x2 b
Bill R2 b q°
CV is positive for Alf...
...negative for Bill
But sum is positive
R2 – z
b b q′
Should project go ahead? x1 1

44. A criterion for the project
Let CVh be the compensating variation for household h if the project is to go ahead.
Then clearly an appropriate criterion overall is nh
S CVh > 0
h=1
Gainers could compensate losers
But how do we get the right information on CVs?
Introduce a simple, powerful concept

45. Use announced information
Approve the project only if this is positive
nh 
S CVh > 0
h=1
If person k is pivotal, then impose a penalty of this size
S CVh
hk
Theorem: a scheme which
approves a project if and only if announced CVs is non-negative, and
imposes the above penalty on any pivotal household
will guarantee that truthful revelation of CVs is a dominant strategy.

46. The pivotal person Pick an arbitrary person h.
What would be the sum of the announced CVs if he were eliminated from the population?
If this sum has the opposite sign from that of the full sum of the CVs, then h is pivotal. Adding him swings the result.
We use this to construct a mechanism.
Consider the following table

47. Public goods: revelation
Everyone else says:
Two possible states
Agent h decision
[Yes]
[No]
Payoff table
Decision
[Yes]
Nil
S costs imposed on others
[No]
S forgone gains of others
Nil
An example

48. Example: model Amount of public good is 0 or 1
if public good is produced cost is shared equally
population of size N each pay 1/ N of total
Agents’ valuations differ
valuation of h is net of contribution to public good
vh = a + [ h − 1] [b – a ] / [N − 1] , h = 1,2,…,N
assume b > 0 > a
Mean valuation is ½[a + b]
project should go ahead if a + b > 0
assume, however, that a + b < 0
Define zh := ½N[a + b] − vh
measures the sum-of-valuations if h is excluded.
Suppose v1 < v2 < 0 and that z1 > 0, z2 < 0
both agents 1 and 2 would prefer no project
agent 1 is pivotal if reports truthfully (z1 is opposite sign to a + b )
agent 2 is not pivotal if reports truthfully (z2 is same sign as a + b )

49. Example: choices If agent 1 declares…
v = v1 then outcome is no project
reverses sign of willingness to pay – so must pay penalty
gets payoff of –z1
v < v1 then outcome and payoff are as above
v > v1 then
if v − v1 is small, outcome and payoff are as above
if v − v1 is large, project goes ahead and payoff is v1
If agent 2 declares…
v = v2 then outcome is no project and gets payoff of 0
v < v2 then outcome and payoff are as above
v > v2 then
if v − v2 is small, outcome and payoff are as above
if v − v2 is large, outcome reversed and payoff is v2 + z2

50. Example: outcomes Payoff to agent 1 is… –z1 if declares v1
–z1 or v1 otherwise
but z1 + v1 = ½N[a + b] < 0 so that –z1 > v1 …
… so declaring v1 is optimal
Payoff to agent 2 is…
0 if declares v2
0 or v2 or v2 + z2otherwise
but v2 < 0 and z2 < 0 …
… so declaring v2 is optimal
Overall outcome
Each has incentive to report truthfully
More resources are paid (in penalties) than are necessary to produce the public good

51. The C-G model – assessment
Strengths:
only uses announced information
elicits truth-telling
Drawbacks:
Restriction to Ziff preferences
Does not ensure budgetary balance
The tipping mechanism can be used as the foundation for more interesting design problems.

52. Summary
A big subject. A few simple questions to pull thoughts together:
What is the meaning of “market failure”?
Why do markets “fail”?
What’s special about public goods?

53. Public goods: summary Implementation problem:
replace the MRS = MRT rule by S MRS = MRT
Characterisation problem:
The Lindahl "solution" may not be a solution at all if people can manipulate the system.
Implementation problem:

54. Public goods
The externality feature of public goods makes it easy to solve the characterisation problem
Implementation problems are much harder.
Intimately associated with the information problem.
Mechanism design depends crucially on the type of public good and the economic environment within which provision is made.