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Pure Public Goods. GE with Production Example. A and B each supply 10 hours of labor a day. With one hour of labor A can catch 3 fish or gather 1 coconut, B can catch 1 fish or 2 coconuts per hour. A and B form a company and agree to share the profits equally. What is the PPF and the MRT?

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ge with production example
GE with Production Example

A and B each supply 10 hours of labor a day. With one hour of labor A can catch 3 fish or gather 1 coconut, B can catch 1 fish or 2 coconuts per hour.

A and B form a company and agree to share the profits equally.

What is the PPF and the MRT?

F = 40 -.5 C if C < 20 and F = 90 – 3C if C ≥ 20

MRT = .5 C if C < 20 and 3C if C ≥ 20

Let A and B have identical preferences

ge with production example1
GE with Production Example

Pareto optimality requires MRT = MRSWhat is the PPF and the MRT?

Case 1: MRS = .5 => F = C/4

Is there a feasible production plan?

C/4 = 40 – C/2

C = 160 – 2C

C = 160/3 not feasible

Case 2: MRS = 3 => F = 3C/2

Is there a feasible production plan?

3C/2= 90 – 3C

C = 20 and F = 30 feasible!

competitive equilibrium
Competitive Equilibrium

Let the price of fish equal 1.

Intuition; A and B sharing profits equally should have the same income. And with identical preferences the should have the identical allocations. So they should split the total production levels equally. Each consumes 10 coconuts and 15 fish. And the ratio of price should equal the MRS, thus the price of coconuts is 2.

Let’s verify: demand (C*,F*) = (2I/3pc, I/3)

I = ½(pc20 + 30)

Substitute in Pc =2

externalities and public goods reading varian chapter 34 and 36
Externalities and Public Goods(Reading: Varian Chapter 34 and 36 )
  • In general, when external effects are present, competitive equilibria are not Pareto optimal.

Two critical properties:

  • It is impossible to exclude individuals from enjoying the benefits of the goods (NON-EXCLUDABILITY)
  • The marginal cost of an additional individual enjoying the good is zero (NON-RIVAL CONSUMPTION). It is undesirable to exclude individuals from enjoying the benefits of the goods, since their enjoyment of these goods does not detract from the enjoyment of others.
theory of pure public goods
Theory of Pure Public Goods
  • n consumers indexed by i
  • 2 commodities, x the pure public good and y the private good
  • Utility functions ui(x,yi)
  • yi0 is i's endowment of the private good.
  • zi: amount of the private good idevotes to production of the public good
  • x = f(z) the production function for the public good
pareto efficiency
Pareto Efficiency
  • Maximize Social Welfare
  • Subject to the constraints
  • and
  • First Order Kuhn-Tucker Conditions
  • x:
  • z:
  • yi:
pareto efficiency constant returns to scale
Pareto Efficiency – Constant Returns to Scale
  • 1) Now let’s assume the public good is produced with a constant returns to scale techonology
  • This gives rise to constant marginal costs
  • Now the Pareto Optimal conditions for an interior solution is

x = 1/B

special case b 0
Special Case B = 0
  • Example: I am willing to give up y units to have a lower temperature.
  • FOC
  • If n=2, then MRS1 + MRS2= 0
  • Pareto Optimal still occurs at the IC tangencies. But will not hold if n > 2







provision of public goods alternative institutions
Provision of Public Goods: Alternative Institutions
  • Competitive Market Outcome B > 0

MRS1(y1,x1) = p = B

If person 2 takes 1’s decision as given

there two possible cases. Case 2 can

be a CE, but not Case 1. Why?

Free Rider Problem!



Case 2: MRS2<p, x2=0

Case 1: MRS2=p, x2>0











competitive market provision of public good
Competitive market provision of public good

Utility Maximization requires for each person i

p = MC(xi) = B = MRSi(xi,yi0-Bxi) or

p = MC(xi) = B > MRSi(the sum of all other xj,yi0-Bxi)

So in the competitive market allocation p=B and

only the person with the highest MRS purchases his

utility maximizing amount of the public good

competitive market example
Competitive Market Example
  • Three agent economy, each agent has an initial endowment of 10 units of the private good and the utility function and B = 2
  • ui(x,yi) = yi + ailn(x) andMRSi = ai/x
  • a1 =20, a2 =10, a3 = 6
  • In a competitive equilibrium p = B, and p = ai/x . So we have

x1 =10, x2 = x3 = 0 => market provision of the public good is 10 .

  • The Pareto optimal allocation is

or x= 16

voluntary contribution mechanism
Voluntary Contribution Mechanism
  • Let everyone simultaneously make an offer to contribute.
  • Well if my MRS is less than the price of x, if will always contribute zero.
  • If my MRS at my endowment if greater than the MC what will I do?
    • If I know everyone else’s MRS than I don’t contribute if there is someone with higher MRS
    • If I think I have the highest MRS I purchase the amount that maximizes my utility
  • The outcome is the same as the competitive outcome
voluntary contribution mechanism class experiment
Voluntary Contribution Mechanism Class Experiment
  • Endowed with 10 units of the private good
  • What is your utility function?

u(yi,x) = yi +x

What is the production function?

½ x= z

C.E. everyone consumes their endowment because MRS < MC

P.O. is everyone contributes if there are more than 2 in the group

  • N is a large number
  • Let’s assume everyone has a unique level of the public good they like the best
  • What does everyone vote if we take the mean level of the vote?
  • What if we take the Median?
  • Group going out to dinner and share the bill problem

Clearly when the outcome depends substantially on the actions of others we need a new set of tools. As we have seen truth revealing and behaving strategic are endemic.