PPT Microeconomic Theory

PUBLIC GOODS

Public Goods

• Public goods are nonrival

– the use of the good does not prevent others from using it e.g. knowledge

• Pure Public goods are nonexclusive

– once they are produced, they provide benefits to an entire group

– it is impossible to restrict these benefits to the specific groups of individuals who pay for them

Attributes of Public Goods

• A good is nonrival if consumption of additional units of the good involves

zero social marginal costs of production

Attributes of Public Goods

• A good is exclusive if it is relatively easy to exclude individuals from benefiting

from the good once it is produced

• A good is nonexclusive if it is

impossible, or very costly, to exclude individuals from benefiting from the good

Attributes of Public Goods

Exclusive

Yes No

Yes Hot dogs, cars, houses

Fishing grounds,

clean air Rival

No Bridges, swimming

pools

National defense, mosquito

control

• Some examples of these types of goods include:

Public Good

• A good is a pure public good if, once

produced, no one can be excluded from benefiting from its availability and if the good is nonrival -- the marginal cost of an additional consumer is zero

Public Goods and Resource Allocation

• We will use a simple general equilibrium model with two individuals (A and B)

• There are only two goods

– good y is an ordinary private good

• each person begins with an allocation (y^A and y^B)

– good x is a public good that is produced using y

x = f(y_s^A + y_s^B)

Public Goods and Resource Allocation

• Resulting utilities for these individuals are

U^A[x,(y^A - y_s^A)]

U^B[x,(y^B - y_s^B)]

• The level of x enters identically into each person’s utility curve

– it is nonexclusive and nonrival

• each person’s consumption is unrelated to what he contributes to production

• each consumes the total amount produced

Public Goods and Resource Allocation

• The necessary conditions for efficient resource allocation consist of choosing

the levels of y_s^A and y_s^B that maximize one person’s (A’s) utility for any given level of the other’s (B’s) utility

• The Lagrangian expression is

L = U^A(x, y^A - y_s^A) + [U^B(x, y^B - y_s^B) - K]

Public Goods and Resource Allocation

• The first-order conditions for a maximum are

L/y_s^A = U₁^Af’ - U₂^A + U₁^Bf’ = 0

L/y_s^B = U₁^Af’ - U₂^B + U₁^Bf’ = 0

• Comparing the two equations, we find

U₂^B = U₂^A

Public Goods and Resource Allocation

• We can now derive the optimality condition for the production of x

• From the initial first-order condition we know that

U₁^A/U₂^A + U₁^B/U₂^B = 1/f’

MRS^A + MRS^B = 1/f’

• The MRS must reflect all consumers because all will get the same benefits

Failure of a

Competitive Market

• Production of x and y in competitive markets will fail to achieve this allocation

– with perfectly competitive prices p_x and p_y, each individual will equate his MRS to p_x/p_y

– the producer will also set 1/f’ equal to p_x/p_y to maximize profits

– the price ratio p_x/p_y will be too low

• it would provide too little incentive to produce x

Failure of a

Competitive Market

• For public goods, the value of producing one more unit is the sum of each

consumer’s valuation of that output

– individual demand curves should be added vertically rather than horizontally

• Thus, the usual market demand curve will not reflect the full marginal valuation

Inefficiency of a Nash Equilibrium

• Suppose that individual A is thinking about contributing ^y_s^A of his initial ^yA endowment to the production of x

• The utility maximization problem for A is then

choose y_s^A to maximize U^A[f(y_s^A + y_s^B),y^A - y_s^A]

Inefficiency of a Nash Equilibrium

• The first-order condition for a maximum is

U₁^Af’ - U₂^A = 0

U₁^A/U₂^A = MRS^A = 1/f’

• Because a similar argument can be

applied to B, the efficiency condition will fail to be achieved

– each person considers only his own benefit

The Roommates’ Dilemma

• Suppose two roommates with identical

preferences derive utility from the number of paintings hung on their walls (x) and

the number of chocolate bars they eat (y) with a utility function of

U_i(x,y_i) = x^1/3y_i^2/3 (for i=1,2)

• Assume each roommate has $300 to

spend and that p = $100 and p = $0.20

The Roommates’ Dilemma

• We know from our earlier analysis of

Cobb-Douglas utility functions that if each individual lived alone, he would spend 1/3 of his income on paintings (x = 1) and 2/3 on chocolate bars (y = 1,000)

• When the roommates live together, each must consider what the other will do

– if each assumed the other would buy paintings, x = 0 and utility = 0

The Roommates’ Dilemma

• If person 1 believes that person 2 will not buy any paintings, he could choose to purchase one and receive utility of

U₁(x,y₁) = 1^1/3(1,000)^2/3 = 100

while person 2’s utility will be

U₂(x,y₂) = 1^1/3(1,500)^2/3 = 131

• Person 2 has gained from his free-riding position

The Roommates’ Dilemma

• We can show that this solution is inefficient by calculating each person’s MRS

x y y

U

x

MRS U ⁱ

i i

i

i / 2

/ 







 

• At the allocations described,

MRS₁ = 1,000/2 = 500 MRS₂ = 1,500/2 = 750

The Roommates’ Dilemma

• Since MRS₁ + MRS₂ = 1,250, the

roommates would be willing to sacrifice 1,250 chocolate bars to have one

additional painting

– an additional painting would only cost them 500 chocolate bars

– too few paintings are bought

The Roommates’ Dilemma

• To calculate the efficient level of x, we must set the sum of each person’s MRS equal to the price ratio

20 . 0

100 2

2 2

2 1

2 1

2

1   









y x

p p x

y y

x y x

MRS y MRS

• This means that

y₁ + y₂ = 1,000x

The Roommates’ Dilemma

• Substituting into the sum of budget constraints, we get

0.20(y₁ + y₂) + 100x = 600 x = 2

y₁ + y₂ = 2,000

• The allocation of the cost of the paintings depends on how each

roommate plays the strategic financing

The Roommates’ Dilemma

• If each person buy 1 painting Person 1’s utility is

U₁(x,y₁) = 2^1/3(1,000)^2/3 126 Person 2’s utility is

U₂(x,y₂) = 2^1/3(1,000)^2/3 126





The Roommates’ Dilemma

126,126 100,131 131,100 0,0

Person 2

Buy Not Buy

Not Buy Person 1

Buy

Lindahl Pricing of Public Goods

• Swedish economist E. Lindahl

suggested that individuals might be willing to be taxed for public goods if

they knew that others were being taxed

– Lindahl assumed that each individual would be presented by the government with the proportion of a public good’s cost he was expected to pay and then reply with the level of public good he would prefer

Lindahl Pricing of Public Goods

• Suppose that individual A would be quoted a specific percentage (^A) and asked the level of a public good (x) he would want given the knowledge that this fraction of total cost

would have to be paid

• The person would choose the level of x which maximizes

utility = U^A[x,y^A*- ^Af ^-1(x)]

Lindahl Pricing of Public Goods

• The first-order condition is given by

U₁^A - ^AU₂^B(1/f’)=0 MRS^A = ^A/f’

• Faced by the same choice, individual B would opt for the level of x which satisfies

MRS^B = ^B/f’

Lindahl Pricing of Public Goods

• An equilibrium would occur when

^A+^B = 1

– the level of public goods expenditure favored by the two individuals precisely

generates enough tax contributions to pay for it

MRS^A + MRS^B = (^A + ^B)/f’ = 1/f’

Shortcomings of the Lindahl Solution

• The incentive to be a free rider is very strong

– this makes it difficult to envision how the information necessary to compute

equilibrium Lindahl shares might be computed

• individuals have a clear incentive to understate their true preferences

Important Points to Note:

• Externalities may cause a

misallocation of resources because of a divergence between private and social marginal cost

– traditional solutions to this divergence includes mergers among the affected parties and adoption of suitable

Pigouvian taxes or subsidies

Voting

• Voting is used as a social decision process in many institutions

– direct voting is used in many cases from statewide referenda to smaller groups and clubs

– in other cases, societies have found it more convenient to use a representative form of government

Majority Rule

• Throughout our discussion of voting, we will assume that decisions will be made by majority rule

The Paradox of Voting

• In the 1780s, social theorist M. de Condorcet noted that majority rule voting systems may not arrive at an equilibrium

– instead, they may cycle among alternative options

The Paradox of Voting

• Suppose there are three voters (Smith, Jones, and Fudd) choosing among

three policy options

– we can assume that these policy options represent three levels of spending on a

particular public good [(A) low, (B) medium, and (C) high]

– Condorcet’s paradox would arise even without this ordering

The Paradox of Voting

Smith Jones Fudd

A B C

B C A

C A B

• Preferences among the three policy options for the three voters are:

The Paradox of Voting

• Consider a vote between A and B

– A would win

• In a vote between A and C

– C would win

• In a vote between B and C

– B would win

• No equilibrium will ever be reached

Single-Peaked Preferences

• Equilibrium voting outcomes always occur in cases where the issue being voted upon is one-dimensional and where voter preferences are “single- peaked”

Single-Peaked Preferences

Quantity of Utility





 Smith

We can show each voters preferences in terms of utility levels





 Jones





 Fudd

For Smith and Jones, preferences are single- peaked

Fudd’s preferences have two local maxima

Single-Peaked Preferences

Quantity of public good Utility

A B C





 Smith





 Jones

Option B will be chosen because it will defeat both A and C by votes 2 to 1

If Fudd had alternative preferences with a single peak, there would be no paradox





 Fudd

The Median Voter Theorem

• With the altered preferences of Fudd, B will be chosen because it is the

preferred choice of the median voter (Jones)

– Jones’s preferences are between the preferences of Smith and the revised preferences of Fudd

The Median Voter Theorem

• If choices are unidimensional and

preferences are single-peaked, majority rule will result in the selection of the

project that is most favored by the median voter

– that voter’s preferences will determine what public choices are made

A Simple Political Model

• Suppose a community is characterized by a large number of voters (n) each with income of y_i

• The utility of each voter depends on his consumption of a private good (c_i) and of a public good (g) according to

utility of person i = U_i = c_i + f(g)

where f > 0 and f < 0

A Simple Political Model

• Each voter must pay taxes to finance g

• Taxes are proportional to income and are imposed at a rate of t

• Each person’s budget constraint is

c_i = (1-t)y_i

• The government also faces a budget constraint

n A

i tny

ty

g 





A Simple Political Model

• Given these constraints, the utility function of individual i is

U_i(g) = [y^A - (g/n)]y_i /y^A + f(g)

• Utility maximization occurs when

dU_i /dg = -y_i /(ny^A) + f_g(g) = 0 g = f_g^-1[y_i /(ny^A)]

• Desired spending on g is inversely

A Simple Political Model

• If G is determined through majority rule, its level will be that level favored by the median voter

– since voters’ preferences are determined solely by income, g will be set at the level preferred by the voter with the median level of income (y^m)

g* = f_g^-1[y^m/(ny^A)] = f_g^-1[(1/n)(y^m/y^A)]

A Simple Political Model

• Under a utilitarian social welfare criterion, g would be chosen so as to maximize the sum of utilities:

 ^{g ny g nf}  ^g

y f y n

y g U

SW ⁿ _i ^A _Aⁱ   ^A  



 



  



 



 





 

1

• The optimal choice for g then is

g* = f_g^-1(1/n) = f_g^-1[(1/n)(y^A/y^A)]

– the level of g favored by the voter with

Voting for Redistributive Taxation

• Suppose voters are considering a lump- sum transfer to be paid to every person and financed through proportional

taxation

• If we denote the per-person transfer b, each individual’s utility is now given by

U_i = c_i + b

Voting for Redistributive Taxation

• The government’s budget constraint is

nb = tny^A b = ty^A

• For a voter with y_i > y^A, utility is maximized by choosing b = 0

• Any voter with y_i < y^A will choose t = 1 and b = y^A

Voting for Redistributive Taxation

• Note that a 100 percent tax rate would lower average income

• Assume that each individual’s income has two components, one responsive to tax rates [y_i (t)] and one not responsive (n_i)

– also assume that the average of n_i is zero, but its distribution is skewed right so n_m < 0

Voting for Redistributive Taxation

• Now, utility is given by

U_i = (1-t)[y_i (t) + n_i] + b

• The individual’s first-order condition for a maximum in his choice of t and g is now

dU_i /dt = -n_i + t(dy^A/dt) = 0 t_i = n_i /(dy^A/dt)

• Under majority rule, the equilibrium condition will be

The Groves Mechanism

• Suppose there are n individuals in a group

– each has a private and unobservable valuation (u_i) for a proposed taxation- expenditure project

The Groves Mechanism

• The government states that, if

undertaken, the project will provide each person with a transfer given by







i i

i v

t

• If the project is not undertaken, no transfers are made

The Groves Mechanism

• The problem for voter i is to choose his or her reported net valuation so as to maximize utility













i i i

i

i t u v

u utility

The Groves Mechanism

• Each person will wish the project to be undertaken if it raises utility

• This means

 0





 i i

i v

u

• A utility-maximizing strategy is to set v_i = u_i

The Clarke Mechanism

• This mechanism also envisions asking individuals about their net valuation of a public project

– focuses on “pivotal voters”

• those whose valuations can change the overall evaluation from positive to negative of vice versa

The Clarke Mechanism

• For these pivotal voters, the Clarke mechanism incorporates a Pigouvian- like tax (or transfer) to encourage truth telling

– for all other voters, there are no special transfers

Important Points to Note:

• Public goods provide benefits to

individuals on a nonexclusive basis - no one can be prevented from

consuming such goods

– such goods are usually nonrival in that the marginal cost of serving another user is zero

Important Points to Note:

• Private markets will tend to

underallocate resources to public

goods because no single buyer can appropriate all of the benefits that such goods provide

Important Points to Note:

• A Lindahl optimal tax-sharing scheme can result in an efficient allocation of resources to the production of public goods

– computing these tax shares requires substantial information that individuals have incentives to hide

Important Points to Note:

• Majority rule voting may not lead to an efficient allocation of resources to public goods

– the median voter theorem provides a useful way of modeling the outcomes from majority rule in certain situations

Important Points to Note:

• Several truth-revealing voting

mechanisms have been developed

– whether these are robust to the special assumptions made or capable of

practical application remain unresolved questions