Microeconomics (Public Goods Ch 36 (Varian)) - web.iitd.ac.inweb.iitd.ac.in/~debasis/Lectures_HUL212(2017)/Public_goods(Varian... ·

  • View
    216

  • Download
    0

Embed Size (px)

Text of Microeconomics (Public Goods Ch 36 (Varian)) -...

  • Microeconomics (Public Goods Ch 36 (Varian))

    Microeconomics (Public Goods Ch 36 (Varian))

    Lectures 26 & 27

    Apr 24 & 27, 2017

  • Public Goods -- Definition

    A good is purely public if it is both nonexcludable and nonrival in consumption.Nonexcludable -- all consumers

    can consume the good.Nonrival -- each consumer can

    consume all of the good.

  • Public Goods -- Examples

    Broadcast radio and TV programs.National defense.Public highways.Reductions in air pollution.National parks.

  • Reservation Prices

    A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of not having the good isU w( , ).0

    w.

  • Reservation Prices

    A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of not having the good isUtility of paying p for the good is

    U w( , ).0w.

    U w p( , ).1

  • Reservation Prices

    A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of not having the good isUtility of paying p for the good is

    Reservation price r is defined by

    U w( , ).0w.

    U w p( , ).1

    U w U w r( , ) ( , ).0 1

  • Reservation Prices; An ExampleConsumers utility is U x x x x( , ) ( ).1 2 1 2 1Utility of not buying a unit of good 2 is

    V w wp

    wp

    ( , ) ( ) .0 0 11 1

    Utility of buying one unit of good 2 atprice p isV w p w p

    pw pp

    ( , ) ( ) ( ) .1 1 1 2

    1 1

  • Reservation Prices; An ExampleReservation price r is defined by

    V w V w r( , ) ( , )0 1I.e. by

    wp

    w rp

    r w

    1 1

    22

    ( ) .

  • When Should a Public Good Be Provided?

    One unit of the good costs c.Two consumers, A and B.Individual payments for providing the public good are gA and gB.gA + gB c if the good is to be provided.

  • When Should a Public Good Be Provided?

    Payments must be individually rational; i.e.

    andU w U w gA A A A A( , ) ( , )0 1

    U w U w gB B B B B( , ) ( , ).0 1

  • When Should a Public Good Be Provided?

    Payments must be individually rational; i.e.

    and

    Therefore, necessarilyand

    U w U w gA A A A A( , ) ( , )0 1

    U w U w gB B B B B( , ) ( , ).0 1

    g rA A g rB B.

  • When Should a Public Good Be Provided?

    And ifand

    then it is Pareto-improving to supply the unit of good

    U w U w gA A A A A( , ) ( , )0 1

    U w U w gB B B B B( , ) ( , )0 1

  • When Should a Public Good Be Provided?

    And ifand

    then it is Pareto-improving to supply the unit of good, so is sufficient for it to be efficient to supply the good.

    U w U w gA A A A A( , ) ( , )0 1

    U w U w gB B B B B( , ) ( , )0 1

    r r cA B

  • Private Provision of a Public Good?

    Suppose and .Then A would supply the good even if B made no contribution.B then enjoys the good for free; free-riding.

    r cA r cB

  • Private Provision of a Public Good?

    Suppose and .Then neither A nor B will supply the good alone.

    r cA r cB

  • Private Provision of a Public Good?

    Suppose and .Then neither A nor B will supply the good alone.Yet, if also, then it is Pareto-improving for the good to be supplied.

    r cA r cB

    r r cA B

  • Private Provision of a Public Good?

    Suppose and .Then neither A nor B will supply the good alone.Yet, if also, then it is Pareto-improving for the good to be supplied.A and B may try to free-ride on each other, causing no good to be supplied.

    r cA r cB

    r r cA B

  • Free-Riding

    Suppose A and B each have just two actions -- individually supply a public good, or not.Cost of supply c = $100.Payoff to A from the good = $80.Payoff to B from the good = $65.

  • Free-Riding

    Suppose A and B each have just two actions -- individually supply a public good, or not.Cost of supply c = $100.Payoff to A from the good = $80.Payoff to B from the good = $65.$80 + $65 > $100, so supplying the good is Pareto-improving.

  • Free-Riding

    -$20, -$35 -$20, $65

    $100, -$35 $0, $0

    Buy

    DontBuy

    BuyDontBuy

    Player A

    Player B

  • Free-Riding

    -$20, -$35 -$20, $65

    $100, -$35 $0, $0

    Buy

    DontBuy

    BuyDontBuy

    Player A

    Player B

    (Dont Buy, Dont Buy) is the unique NE.

  • Free-Riding

    -$20, -$35 -$20, $65

    $100, -$35 $0, $0

    Buy

    DontBuy

    BuyDontBuy

    Player A

    Player B

    But (Dont Buy, Dont Buy) is inefficient.

  • Free-Riding

    Now allow A and B to make contributions to supplying the good.E.g. A contributes $60 and B contributes $40.Payoff to A from the good = $40 > $0.Payoff to B from the good = $25 > $0.

  • Free-Riding

    $20, $25 -$60, $0

    $0, -$40 $0, $0

    Contribute

    DontContribute

    ContributeDontContribute

    Player A

    Player B

  • Free-Riding

    $20, $25 -$60, $0

    $0, -$40 $0, $0

    Contribute

    DontContribute

    ContributeDontContribute

    Player A

    Player B

    Two NE: (Contribute, Contribute) and(Dont Contribute, Dont Contribute).

  • Free-Riding

    So allowing contributions makes possible supply of a public good when no individual will supply the good alone.But what contribution scheme is best?And free-riding can persist even with contributions.

  • Variable Public Good Quantities

    E.g. how many broadcast TV programs, or how much land to include into a national park.

  • Variable Public Good Quantities

    E.g. how many broadcast TV programs, or how much land to include into a national park.c(G) is the production cost of G units of public good.Two individuals, A and B.Private consumptions are xA, xB.

  • Variable Public Good Quantities

    Budget allocations must satisfyx x c G w wA B A B( ) .

  • Variable Public Good Quantities

    Budget allocations must satisfy

    MRSA & MRSB are A & Bs marg. rates of substitution between the private and public goods.Pareto efficiency condition for public good supply is

    x x c G w wA B A B( ) .

    MRS MRS MCA B ( ).G

  • Variable Public Good Quantities

    Pareto efficiency condition for public good supply is

    Why?MRS MRS MCA B ( ).G

  • Variable Public Good Quantities

    Pareto efficiency condition for public good supply is

    Why?The public good is nonrival in consumption, so 1 extra unit of public good is fully consumed by both A and B.

    MRS MRS MCA B ( ).G

  • Variable Public Good Quantities

    SupposeMRSA is As utility-preserving compensation in private good units for a one-unit reduction in public good.Similarly for B.

    MRS MRS MCA B ( ).G

  • Variable Public Good Quantities

    is the total payment toA & B of private good that preserves both utilities if G is lowered by 1 unit.

    MRS MRSA B

  • Variable Public Good Quantities

    is the total payment toA & B of private good that preserves both utilities if G is lowered by 1 unit.Since , making 1 less public good unit releases more private good than the compensation payment requires Pareto-improvement from reduced G.

    MRS MRS MCA B ( )G

    MRS MRSA B

  • Variable Public Good Quantities

    Now suppose MRS MRS MCA B ( ).G

  • Variable Public Good Quantities

    Now supposeis the total payment by

    A & B of private good that preserves both utilities if G is raised by 1 unit.

    MRS MRS MCA B ( ).GMRS MRSA B

  • Variable Public Good Quantities

    Now supposeis the total payment by

    A & B of private good that preserves both utilities if G is raised by 1 unit.This payment provides more than 1 more public good unit Pareto-improvement from increased G.

    MRS MRS MCA B ( ).GMRS MRSA B

  • Variable Public Good Quantities

    Hence, necessarily, efficient public good production requires

    MRS MRS MCA B ( ).G

  • Variable Public Good Quantities

    Hence, necessarily, efficient public good production requires

    Suppose there are n consumers; i = 1,,n. Then efficient public good production requires

    MRS MRS MCA B ( ).G

    MRS MCii

    nG

    1( ).

  • Efficient Public Good Supply -- the Quasilinear Preferences Case

    Two consumers, A and B.U x G x f G ii i i i( , ) ( ); , .A B

  • Efficient Public Good Supply -- the Quasilinear Preferences Case

    Two consumers, A and B.

    Utility-maximization requires

    U x G x f G ii i i i( , ) ( ); , .A BMRS f G ii i ( ); , .A B

    MRS pp

    f G p ii Gx

    i G( ) ; , .A B

  • Efficient Public Good Supply -- the Quasilinear Preferences Case

    Two consumers, A and B.

    Utility-maximization requires

    is is public good demand/marg. utility curve; i = A,B.

    U x G x f G ii i i i( , ) ( ); , .A BMRS f G ii i ( ); , .A B

    MRS pp

    f G p ii Gx

    i G( ) ; , .A B

    p f GG i ( )

  • Efficient Public Good Supply -- the Quasilinear Preferences Case

    MUA

    MUB

    pG

    G

  • Efficient Public Good Supply -- the Quasilinear Preferences Case

    MUA

    MUB

    MUA+MUB

    pG

    G

  • Efficient Public Good Supply -- the Quasilinear Preferences Case

    pG

    MUA

    MUB

    MUA+MUB

    MC(G)

    G

  • Efficient Public Good