Customer‘s safety is the future trend of the foodservice and hospitality industry

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Public Goods II

Part 1. Cobb-Douglas two consumer example

Economics 313

Cobb-Douglas two consumer example  Assume that good 𝑥𝑥 is a PG, good 𝑦𝑦 is a private good.

 Assume identical Cobb-Douglas preferences, given by 𝑈𝑈𝐴𝐴 = 𝑥𝑥𝑦𝑦 and 𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦

 Each agent is endowed with 60 units of 𝑦𝑦 (the private good) and they can transform one-for-one into the public good.

 The choice is the “contribution” to the public good.

Cobb-Douglas two consumer example  The contribution choice can be modeled as

technology or expenditure

 Viewed as technology, it is represented by the production function

𝑥𝑥 = 𝑓𝑓 𝑦𝑦 = 𝑦𝑦 0 ≤ 𝑦𝑦 ≤ 60

 Recall that the MRT is a measure of the rate at which the goods can be transformed. For this economy

𝑀𝑀𝑀𝑀𝑀𝑀 = 1

Cobb-Douglas two consumer example

 Viewed as an expenditure choice, each consumer is facing a budget constraint. For consumer 𝐴𝐴 this is

𝑝𝑝𝑥𝑥𝑥𝑥 + 𝑝𝑝𝑦𝑦𝑦𝑦 = 𝑝𝑝𝑥𝑥𝜔𝜔𝑥𝑥𝐴𝐴 + 𝑝𝑝𝑦𝑦𝜔𝜔𝑦𝑦𝐴𝐴

where 𝑝𝑝𝑥𝑥 = 𝑝𝑝𝑦𝑦 = 1, 𝜔𝜔𝑥𝑥𝐴𝐴 = 0, and 𝜔𝜔𝑦𝑦𝐴𝐴 = 60

 The slope of the budget line is 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 1

Cobb-Douglas two consumer example  You should convince yourself that these are

equivalent ways to model the public good contributions.

 Let 𝑥𝑥𝐴𝐴 be the contribution of A to the public good and 𝑥𝑥𝐵𝐵 be the contributions of B to the public good.

 Questions: 1. What is the equilibrium level of PG provision? 2. What is the efficient level of PG provision?

Private provision equilibrium  Consumer A sets 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 =

𝑦𝑦 𝑥𝑥

= 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 1

 But because 𝑥𝑥 is a PG:  A consumes their contribution (𝑥𝑥𝐴𝐴) and B’s (𝑥𝑥𝐵𝐵)  So the 𝑥𝑥 that enters A’s utility function (and MRS) is 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵  Good 𝑦𝑦 is a private good, so that the y that A consumes is just

the 𝑦𝑦 that A buys (𝑦𝑦𝐴𝐴)

 So 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 1 ⇒ 𝑦𝑦 𝑥𝑥

= 𝑦𝑦𝐴𝐴 𝑥𝑥𝐴𝐴+𝑥𝑥𝐵𝐵

= 1

Solve for demand  A also needs to be on her budget line:

𝑥𝑥𝐴𝐴 + 𝑦𝑦𝐴𝐴 = 60 => A consumes 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵, but only contributes 𝑥𝑥𝐴𝐴

So to solve for the choice, we have two equations:

1. MRS = px py → 𝑦𝑦

𝑥𝑥 = 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴+𝑥𝑥𝐵𝐵 = 1

2. 𝑥𝑥𝐴𝐴 + 𝑦𝑦𝐴𝐴 = 60

Public Goods: Unpriced benefits  Notice that A’s choice depends on B’s actions in a way

that is not mediated by prices. The relationship comes from preferences: For given contributions, A’s utility is

𝑈𝑈𝐴𝐴 = (𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵)𝑦𝑦𝐴𝐴

 The interdependence is an externality, not captured in the price (which fixed by the technology). B gets no benefit from the spillover; likewise A from B’s use of 𝑥𝑥𝐴𝐴

 Because PGs involve interdependence of decisions we can use the tools of game theory to solve them

Public Goods: Best responses  Solving the two equations:

1. 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴+𝑥𝑥𝐵𝐵 = 1 ⇒ 𝑦𝑦𝐴𝐴 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵

2. 𝑥𝑥𝐴𝐴 + 𝑦𝑦𝐴𝐴 = 60 ⇒ 𝑦𝑦𝐴𝐴 = 60 − 𝑥𝑥𝐴𝐴

1 & 2 ⇒ 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 60 − 𝑥𝑥𝐴𝐴 ⇒ 2𝑥𝑥𝐴𝐴 = 60 − 𝑥𝑥𝐵𝐵 ⇒ 𝑥𝑥𝐴𝐴 = 30 −

1 2 𝑥𝑥𝐵𝐵

 We call this last function the Best Response Function for A (𝐵𝐵𝑀𝑀𝐴𝐴).

-- end of part 1 --

Public Goods II

Part 2. Nash Equilibrium private provision

Economics 313

Best Responses  The best response function tells us the best thing for A to

do in response to B’s choice of PG provision.

 We can also solve B’s utility maximization problem in the same way.

 Symmetry implies ⇒ 𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 30 − 1 2 𝑥𝑥𝐴𝐴

Tells us the utility- maximizing choice of 𝑥𝑥𝐵𝐵, as function of 𝑥𝑥𝐴𝐴.

Best Responses  Note: each consumer’s contribution to the PG is a

decreasing function of the other consumers contribution.

 If A contributes more PG, B contributes less (and vice versa)  They each “free-ride” on the provision of the other.

 But A’s contributions do not crowd out B’s one-for-one.

 If 𝑥𝑥𝐴𝐴↑ by 1 unit, 𝑥𝑥𝐵𝐵↓ by 0.5 units A, so (𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵) ↑ by 0.5 units.

 So overall provision of the PG in increasing in each consumer’s level of provision.

Equilibrium Concept: Nash Equilibrium  We have seen that A’s choices depend on B’s, and vice

versa.

 An equilibrium is where no-one has an incentive to change what they are currently doing, taking as given everything else that is going on (prices, other consumer’ behavior, etc.).

 In other words, we are looking for a Nash Equilibrium where 𝐵𝐵𝑀𝑀𝐴𝐴 crosses 𝐵𝐵𝑀𝑀𝐵𝐵

The Best Response Functions

𝑋𝑋𝐴𝐴 (Contribution of A)

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 30 − 1 2 𝑥𝑥𝐵𝐵

𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 30 − 1 2 𝑥𝑥𝐴𝐴

Utility of A ↑

Utility of B ↑

𝑋𝑋𝐵𝐵 (Contribution of B)

Indifference Curves  It is convenient to draw indifference curves in the (𝑥𝑥𝐴𝐴, 𝑥𝑥𝐵𝐵)

space. These map pairs of contributions that keep utility constant, recognising that when 𝐵𝐵, for example, increases their contribution, 𝐴𝐴 is able to reallocate expenditure to good 𝑦𝑦 (otherwise these would be straight lines with slope equal to 1)

𝑈𝑈𝐴𝐴 = (𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵)𝑦𝑦𝐴𝐴 where 𝑦𝑦𝐴𝐴 = 60 − 𝑥𝑥𝐴𝐴

 Substituting the indifference curve is solves

�𝑈𝑈 = 𝑥𝑥𝐴𝐴 + 𝐼𝐼 𝑥𝑥𝐴𝐴 60 − 𝑥𝑥𝐴𝐴

Indifference Curves  Therefore, the indifference curve is given by

𝐼𝐼 𝑥𝑥𝐴𝐴 = �𝑈𝑈 − 60𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐴𝐴

60 − 𝑥𝑥𝐴𝐴  This has slope

2𝑥𝑥𝐴𝐴 − 60 60 − 𝑥𝑥𝐴𝐴

+ 𝑥𝑥𝐴𝐴 2 − 60𝑥𝑥𝐴𝐴 + �𝑈𝑈

60 − 𝑥𝑥𝐴𝐴 2

 You can verify that the slope of this line is zero at the point the indifference curve crosses the Best Response Function. (Remember to evaluate �𝑈𝑈 too.)

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵 30

𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 30 − 1 2 𝑥𝑥𝐵𝐵

𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 30 − 1 2 𝑥𝑥𝐴𝐴

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

Utility of A ↑

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵 30

𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 30 − 1 2 𝑥𝑥𝐵𝐵

𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 30 − 1 2 𝑥𝑥𝐴𝐴

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

Utility of B ↑

Question  What is the equilibrium contribution?

𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 30 − 1 2 𝑥𝑥𝐵𝐵

𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 30 − 1 2 𝑥𝑥𝐴𝐴

A. 10 B. 20 C. 30 D. 40

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Nash Equilibrium

𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 30 − 1 2 𝑥𝑥𝐵𝐵

𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 30 − 1 2 𝑥𝑥𝐴𝐴

→ 𝑥𝑥𝐴𝐴 = 30 − 1 2

30 − 1 2 𝑥𝑥𝐴𝐴

→ 3 4 𝑥𝑥𝐴𝐴 = 15

𝑥𝑥𝐴𝐴 = 20 = 𝑥𝑥𝐵𝐵20

20 𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Only when 𝑥𝑥𝐴𝐴 = 𝑥𝑥𝐵𝐵 = 20 is A choosing 𝑥𝑥𝐴𝐴 to maximize her utility given B’s provision, while B is choosing the 𝑥𝑥𝐵𝐵 that maximizes his utility given A’s provision (both are playing their BRs)

20 30

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

-- end of part 2 --

Public Goods II

Part 3. Efficient provision

Economics 313

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Is the Nash Equilibrium efficient?

20 30

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

Efficiency

Is the Nash Equilibrium efficient?

Recall: 𝐴𝐴 sets her 𝑀𝑀𝑀𝑀𝑆𝑆 = 𝑝𝑝𝑥𝑥

𝑝𝑝𝑦𝑦 = 𝑀𝑀𝑀𝑀𝑀𝑀 ⇒ 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 1

𝐵𝐵 sets his 𝑀𝑀𝑀𝑀𝑆𝑆 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀 ⇒ 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 1

𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 2 > 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

⇒ together A and B value an extra unit of the PG more than an extra unit costs.

 Too little is purchased in the Nash equilibrium.

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Both people better off than in NE.

20 30

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

Efficient Public Good Provision  A planner would choose to maximise total utility:

max 𝑥𝑥,𝑦𝑦𝐴𝐴,𝑦𝑦𝐵𝐵

𝑥𝑥𝑦𝑦𝐴𝐴 + 𝑥𝑥𝑦𝑦𝐵𝐵

subject to 𝑥𝑥 + 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵 = 120

ℒ 𝑥𝑥, 𝑦𝑦𝐴𝐴, 𝑦𝑦𝐵𝐵, 𝜆𝜆 = 𝑥𝑥𝑦𝑦𝐴𝐴 + 𝑥𝑥𝑦𝑦𝐵𝐵 + 𝜆𝜆(120 − 𝑥𝑥 − 𝑦𝑦𝐴𝐴 − 𝑦𝑦_𝐵𝐵) FOC

𝑥𝑥: 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵 = λ 𝑦𝑦𝐴𝐴: 𝑥𝑥 = λ 𝑦𝑦𝐵𝐵: 𝑥𝑥 = λ 𝜆𝜆: 120 = 𝑥𝑥 + 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵

Efficient Public Good Provision  Solving these gives

𝑥𝑥 = 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵

So 2𝑥𝑥 = 120 or 𝑥𝑥 = 60

Notice that with the efficient amount of 𝑥𝑥 produced, the planner is indifferent to the allocation of the private good: that is, who contributes. (Convince yourself of why this is true…)

Efficient Public Good Provision  Another way to see this is to use the efficiency condition

for public goods: 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 =

𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀

=> Aggregate willingness to pay for additional units of the PG to equal the cost of additional units of the PG, the 𝑀𝑀𝑀𝑀𝑀𝑀

 So we need 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 +

𝑦𝑦𝐵𝐵 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵

= 1

𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀

Equilibrium Public Good Provision  Re-arranging we can solve

𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵

= 1 → 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵

Given the endowments in the economy, we know that

𝑦𝑦𝐴𝐴 = 60 − 𝑥𝑥𝐴𝐴 and 𝑦𝑦𝐵𝐵 = 60 − 𝑥𝑥𝐵𝐵 These equations imply that:

60 − 𝑥𝑥𝐴𝐴 + 60 − 𝑥𝑥𝐵𝐵 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 ⇒ 120 = 2(𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵) ⇒ 60 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵

Efficient provision of the public good equals 60

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Along red line: efficient amount of public good: the planner is indifferent between these allocations

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

The Best Response Functions

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

This allocation gives all the surplus to consumer B. Consumer A contributes 𝑥𝑥A = 33

1 3

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

Equilibrium Public Good Provision

 Again, note that efficiency requires that 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 60, but doesn’t tell us about for 𝑥𝑥𝐴𝐴 or 𝑥𝑥𝐵𝐵, just the total.

 That is, we have the efficient total level of PG, but efficiency alone doesn’t tell us who should provide what quantities.

-- end of part 3 --

Public Goods II

Part 4. Bargaining

Economics 313

Bargaining for Public Good Provision

 Can we have a Coase Theorem type result here?  Can we have individual negotiations lead to efficient provision? A. Yes B. No

 What are the transaction costs that might matter?

Bargaining for Public Good Provision  Suppose that each agree to provide half the efficient

quantity (A pledges to buy 30 units as well as B)

 Will this scheme work?  In particular, do A and B have an incentive to keep their

promises? What allows them to enforce it?

 Let’s model this as a simple game…

Bargaining for Public Good Provision  If each keeps their promise, so that 𝑥𝑥𝐴𝐴 = 𝑥𝑥𝐵𝐵 = 30, then it

will also be true that 𝑦𝑦𝐴𝐴 = 𝑦𝑦𝐵𝐵 = 30.

 Then 𝑈𝑈𝐴𝐴 = 𝑥𝑥𝑦𝑦 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 𝑦𝑦𝐴𝐴 = 60 30 = 1800 and similarly 𝑈𝑈𝐵𝐵 = 1800 (better than the NE U = 1600)

 However neither A nor B is on their BR.

 Because 𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 30 − 1 2 𝑥𝑥𝐵𝐵, if 𝑥𝑥𝐵𝐵 = 30, A should set

𝑥𝑥𝐴𝐴 = 15. Then 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 45, 𝑦𝑦𝐴𝐴 = 45, so 𝑈𝑈𝐴𝐴 = 𝑥𝑥𝑦𝑦 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 𝑦𝑦𝐴𝐴 = 45 45 = 2025 > 1800

Bargaining for Public Good Provision  Makes A better off, but at the expense of B.  Given 𝑥𝑥𝐵𝐵 = 30, 𝑥𝑥𝐴𝐴 = 15, & so 𝑦𝑦𝐵𝐵 = 30, 𝑈𝑈𝐵𝐵 = 1350 < 1800

 By symmetry, if 𝑥𝑥𝐴𝐴 = 30, B should best respond by setting 𝑥𝑥𝐵𝐵 = 15⇒ 𝑈𝑈𝐴𝐴 = 1350, & 𝑈𝑈𝐵𝐵 = 1800

 Recall: if they are both on their best responses (i.e., if we are at the Nash equilibrium) then 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 40, 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵 = 40

 So 𝑈𝑈𝐴𝐴 = 𝑈𝑈𝐵𝐵 = 40 40 = 1600

Not a best response

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Both consumers have the incentive to deviate from this agreement.

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

(30,30)

𝑈𝑈𝐴𝐴 = 1600

𝑈𝑈𝐴𝐴 = 2025

Public Good Game  This tells us that it will be difficult (impossible?) to have

individual negotiations for efficient provision  Each has an incentive to cheat on the deal  Each knows that the other has an incentive to cheat

 Neither can credibly commit to provide the agreed upon level, so the deal unravels

 The Coase theorem assumptions are violated.  In particular, some transaction cost prevents

commitment….

Take it or leave it offers?  What if we give all the bargaining power to one

consumer (say A) and allow her to make a take-it-or- leave-it offer. This is clear commitment

 Unfortunately, consumer B cannot commit. So this doesn’t solve the problem.

 What should A contribute?

Max subject to 𝐵𝐵𝑀𝑀𝐵𝐵

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

Solves max𝑥𝑥𝐴𝐴(𝑥𝑥𝐴𝐴 + 𝐵𝐵𝑀𝑀 𝑥𝑥𝐴𝐴 )(60 − 𝑥𝑥𝐴𝐴)

Verify that this is max 𝑥𝑥𝐴𝐴

1 2

(3600 − 𝑥𝑥𝐴𝐴 2)

So 𝑥𝑥𝐴𝐴 = 0.

𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

𝑈𝑈𝐴𝐴 = 1800

-- end of part 4 --

Public Goods II

Part 5. Government Provision

Economics 313

Government Provision  How could we achieve efficiency?

 Recall in our example that the equilibrium level of provision is 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 40, while the efficient level is 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 60

 What if the government provides the shortfall of 20 units?

 Also recall Good x is a PG, good y is a private good. 𝑈𝑈𝐴𝐴 = 𝑥𝑥𝑦𝑦, 𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦, 𝑝𝑝𝑥𝑥 = 𝑝𝑝𝑦𝑦 = 1. Each person is endowed with 60 units of y (the private good)

Government Provision  Also recall from the individual optimization problem:

 We have two equations

1. MRS= px py → 𝑦𝑦

𝑥𝑥 = 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴+𝑥𝑥𝐵𝐵 = 1

2. BC: 𝑥𝑥𝐴𝐴 + 𝑦𝑦𝐴𝐴 = 60

 And using them we got the best response of A �𝑥𝑥𝐴𝐴 =

Government Provision  Let 𝑥𝑥𝐺𝐺=government provision=20  If consumer A knows that 𝑥𝑥𝐺𝐺 = 20, then  𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 =

𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦 → 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴+𝑥𝑥𝐵𝐵+𝑥𝑥𝐺𝐺 = 1

→ 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 + 20 = 1 → 𝑦𝑦𝐴𝐴 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 + 20

 From A’s BC, we know that 𝑦𝑦𝐴𝐴 = 60 − 𝑥𝑥𝐴𝐴 60 − 𝑥𝑥𝐴𝐴 = 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 + 20 → 𝐵𝐵𝑀𝑀𝐴𝐴: 𝑥𝑥𝐴𝐴 = 20 −

1 2 𝑥𝑥𝐵𝐵

 Similarly, 𝐵𝐵𝑀𝑀𝐵𝐵: 𝑥𝑥𝐵𝐵 = 20 − 1 2 𝑥𝑥𝐴𝐴

The New Level of Provision

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

20 30 𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

The New Level of Provision

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

40/3

40/3 30 𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B)

𝑥𝑥𝐵𝐵 = 20 − 1 2 𝑥𝑥𝐴𝐴 and 𝑥𝑥𝐴𝐴 = 20 −

1 2 𝑥𝑥𝐵𝐵 → from this

we can plug one into the other and solve.

Doing this yields: 𝑥𝑥𝐴𝐴 = 40 3

and 𝑥𝑥𝐵𝐵 = 40 3

The New Level of Provision

𝐵𝐵𝑀𝑀𝐴𝐴

𝐵𝐵𝑀𝑀𝐵𝐵

40/3

40/3 30 𝑋𝑋𝐴𝐴 (Contribution of A)

𝑋𝑋𝐵𝐵 (Contribution of B) At the new equilibrium level of provision of the public good is

𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 + 𝑥𝑥𝐺𝐺 = 40 3

+ 40 3

+ 20 = 46.67 < 60

Government Provision  Government provision of the PG crowds out private

provision  Now each consumer can free-ride on the government’s

provision, as well as the other consumer’s provision

 If we want to ensure efficiency we could have the government provide all 60 units of the PG, and then tax each consumer according to how they value the PG

 Of course, this leads to the question of how the government would know consumer values of the PG.

-- end of part 5 --

Public Goods II

Part 6. Vickery -Clarke-Groves

Economics 313

Efficiency and asymmetric information  As we saw last lecture, to determine the efficient level of

public good provision a government must know both costs and benefits. Costs are perhaps clear, but the benefits are harder to assess.

 Non-rivalry means that efficiency demands everyone who positively values the good should get it, so the problem for an all-knowing government is simple.

 However, the “free rider” problem due to non-excludability makes assessment a challenge for government.

Efficiency and asymmetric information  Typically, consumer valuations are private information. To

determine these, the government could simply ask. This information has two possible uses 1. To decide how much of the good to provide 2. To allocate the payment across consumer/taxpayers

 Holding payment constant, consumers are tempted to overstate valuations to ensure the good is provided. Holding the level of provision constant, taxpayers might choose to understate valuations to limit tax liability.

Efficient mechanisms  Truthful revelation of values is helped by mechanisms that

break the connection between provision and payment.

 We have seen this already in auctions, with the example where a Vickery second price auction lead people to bid their valuations

 As mentioned, something similar can be made to work (imperfectly) for public good provision by a benevolent government when taxpayer/consumers have private information about their values

An example  For clarity and simplicity, I will go through a simple

example of an externality

 Assume consumer and firm, with an (negative) externality ℎ ∈ {0, �ℎ} that goes from the firm to the consumer.

 The “provision” question is whether to permit ℎ = �ℎ or restrict output so ℎ = 0.

 Efficiency requires that �ℎ iff the gain to the firm outweighs the harm to the consumer.

An example

 Let 𝜋𝜋 ℎ, 𝜃𝜃 be the profits for externality ℎ when the firm is type 𝜃𝜃, and 𝑏𝑏 𝜃𝜃 = 𝜋𝜋 �ℎ, 𝜃𝜃 − 𝜋𝜋 0, 𝜃𝜃 > 0

 Let 𝑢𝑢 ℎ, 𝜈𝜈 be the utility of the consumer for ℎ if the type is ν, and 𝑐𝑐 𝜈𝜈 = 𝑐𝑐 �ℎ, 𝜈𝜈 − 𝑐𝑐 0, 𝜈𝜈 < 0

 Assume that the government asks for reports on 𝑏𝑏 and 𝑐𝑐. ⇒ more generally report on type, assumed to be private information.

An example

 Reporting valuations amounts to reporting the “unknown” types, θ and 𝜈𝜈. Why should agents tell the truth?

 Ignoring payments, the consumer who is harmed should report the type with the greatest harm. The firm in contrast should report the type with the greatest benefit.

Vickery – Clarke- Groves  Rule:  Ask for reports, �𝑏𝑏 and �̂�𝑐  Set ℎ = �ℎ if and only if �𝑏𝑏 > �̂�𝑐  If ℎ = �ℎ then tax the firm �̂�𝑐 and transfer to the consumer �𝑏𝑏

 The higher is the reported benefit, the more likely the externality is generated; the higher the reported cost the less likely.

 Weakly dominant strategies to report �𝑏𝑏 = 𝑏𝑏 𝜃𝜃 and �̂�𝑐 = 𝑐𝑐(𝜈𝜈)

Vickery – Clarke- Groves

𝑐𝑐(𝜈𝜈)

�̂�𝑐0 �𝑏𝑏0

�̂�𝑐1

�𝑏𝑏1

Assume firm truthful. �𝑏𝑏 = 𝑏𝑏(𝜃𝜃)

The line is the social loss when project should happen and doesn’t, and is borne entirely by the consumer for the misreport of �̂�𝑐0 > 𝑐𝑐(𝜈𝜈)

The line is the social loss when project shouldn’t happen and does, and is borne entirely by the consumer for the misreport of �̂�𝑐1 < 𝑐𝑐(𝜈𝜈)

The government runs a deficit, which can be eliminated at the cost of having a surplus for some realizations. Balanced budget are “second best”

-- end of part 6 --

Public Goods II

Part 7. Summary

Economics 313

Summary  Public goods are a particular form of multilateral non-

depletable positive externality.

 Non-rivalry means that individual contributors ignore the positive spillovers

 Non-excludability means it is hard to prevent free riding

 These can be viewed as specific transaction costs.

What I Expect You To Know  How to solve a public goods provision problem like those

covered

 Be able to find:  The efficient level of provision  The level of provision of the market  With partial government provision

 Discuss bargaining and partial government provision

-- end of part 7 --