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Economics 151 The Economics of the Public Sector: Expenditure. Professor Nora Gordon Fall 2004 Lecture 10. Outline for today. How does the optimal level of public goods provision compare with that for private goods?

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Economics 151 the economics of the public sector expenditure l.jpg

Economics 151The Economics of the Public Sector:Expenditure

Professor Nora Gordon

Fall 2004

Lecture 10


Outline for today l.jpg
Outline for today

  • How does the optimal level of public goods provision compare with that for private goods?

  • What level of public goods will the market produce without government intervention (voluntary contributions only)?


Optimal provision of private goods l.jpg
Optimal provision of private goods

  • Two consumers: Leah and Tara

  • Two private goods: books and cupcakes

  • Price of cupcakes=$1 (cupcakes are the numeraire good)

  • We derive individual demand curves from consumer’s utlity maximization problem. For Leah:

    MRSbcL = MULb/MULc = pb/pc = pb/1= pb

    Similarly for Tara: MRSbcT = MUTb/MUTc = pb

  • We derive supply curve from producer’s profit max problem:

    MCb = pb (and MCc = pc = 1)

  • Equilibrium: MRSbc = MCb/MCc

  • MRSbc = MSB absent mkt failure

  • MCb/MCc = MCb=MSC absent mkt failure


Horizontal summation of demand for private goods l.jpg
Horizontal summation of demand for private goods

Leah and Tara’s summed D for books

Leah’s D for books

Tara’s D for books

Pb

Pb

Pb

# books

# books

# books


Horizontal summation of demand for private goods5 l.jpg
Horizontal summation of demand for private goods

Leah and Tara’s summed D for books

Leah’s D for books

Tara’s D for books

Pb

Pb

Pb

5

5

# books

# books

1

2

3

# books

Comes from summing what Leah and Tara each demand at Pb=5


Market equilibrium pareto optimum for private good l.jpg
Market equilibrium=Pareto optimum for private good

Leah’s D for books

Tara’s D for books

Market for books

Pb

Pb

Pb

Sb

5

5

4

# books

# books

1 1.5

2 3

3 4.5

# books


Vertical summation of d for public good l.jpg
Vertical summation of D for public good

Pm

6

Leah’s D for rockets

DLr

# rockets

20

Pm

Tara’s D for rockets

4

DTr

# rockets

Pm

10

Leah & Tara’s D for rockets

# rockets

20


Pareto efficient level of public good l.jpg
Pareto-efficient level of public good

Pm

6

Leah’s D for rockets

DLr

# rockets

20

Pm

Tara’s D for rockets

4

DTr

# rockets

Pm

Sr

10

Leah & Tara’s D for rockets

DT+Lr

# rockets

20


Conditions for pareto optimality l.jpg
Conditions for Pareto optimality

  • MSB = MSC for public and private goods

  • Private goods:

    MRSLbc = MRSTbc = MSB

    MRTbc= MSC

    so MRSLbc= MRSTbc = MRTbc

  • Public goods:

    MRSLbc + MRSTbc = MSB

    MRTbc= MSC

    so MRSLbc+ MRSTbc = MRTbc


Example 1 efficient allocation of public vs private good l.jpg
Example 1: Efficient allocation of public vs. private good

  • Let MCc=1, so MRTbc=MCb and MRTrc=MCr

  • Let MRSLbc = MRSTbc = 1

    • They each are willing to trade one cupcake for the next book

  • Let MRSLrc = MRSTrc = 1

    • They each are willing to trade one cupcake for the next rocket

  • Let MRSLrc = MRSTrc = 1

  • Let MCr =1.9, and MCb = 1.1

  • Should we produce another book? Another rocket?


Example 2 finding efficient allocation of public good l.jpg
Example 2: Finding efficient allocation of public good

  • Two roommates (A and B), identical preferences over carrots (pvt good) and lava lamps (pub good)

  • U(C,L) = ln(C) + ln(L)

  • Identical budget constraints:

    Each has income=100

    PC=5, PL=5.

    Total BC: 5(CA + CB) + 5L = 200

  • Efficient allocation requires:

    MRSALC+MRSBLC = MRTLC


Example 2 cont l.jpg
Example 2 cont.

  • MRSALC+MRSBLC = MRTLC

  • MRSALC=MUAL/MUAC

    Recall: U(C,L) = ln(C) + ln(L)

    MUAC=1/C, MUAL=1/L

    MRSALC=CA/L

  • MRSBLC=CB/L

  • MRTLC = PL/PC = 1

  • Pareto optimum: CA/L+CB/L= 1

  • BC: 5(CA+CB)+5L = 200

    Rewrite BC: CA+CB=40-L

  • Subst BC into PO condition to solve for L:

    (CA+CB)/L= 1

    (40-L)/L=1

    40-L=L

    L=20, CA=CB=10


Voluntary contribution equilibrium for public good l.jpg
Voluntary contribution equilibrium for public good

  • “Free riding”  underprovision of the public good

  • When roommate A spends on lava lamps, this yields a positive externality for roommate B


A and b live apart l.jpg
A and B live apart

  • Imagine A and B lived in separate apartments. A would max U(C,L) = ln(C)+ln(L) s.t. BC: 5C+5L=100

  • BCC=20-L

  • Choose L to max U=ln(20-L)+ln(L)

  • FOC: 1/(20-L)+1/L=0

  • 20-L=L  L=10

  • Lava lamps are treated as a private good here.


A and b live together b moves in first l.jpg
A and B live together, B moves in first

Lava lamps A can consume

100/5=20

Budget constraint if A lives with B, who moved in first with 10 lava lamps

10

100/5=20

30

carrots A can consume

Budget constraint if A lives alone


A and b live together b moves in first16 l.jpg
A and B live together, B moves in first

  • Recall A and B have identical preferences

  • B moves in first, with his 10 lava lamps (privately optimal quantity).

  • How many does A buy now?

  • Now max U=ln(20-LA)+ln(10+LA)

  • FOC: -1/(20-LA) + 1/LA = 0

  • LA+10=20-LA

  • 2LA = 10 LA = 5


A and b live together move in at same time l.jpg
A and B live together, move in at same time

  • Think of A’s decision as a response to B’s

  • Now max U=ln(20-LA)+ln(LB+LA)

  • FOC:

  • This is A’s “best response function” to B.

  • We know A and B are identical, so B’s best response to A is: LB =10-LA/2

  • Substitute to find eq’m where both conditions hold:

  • LA = 10 – (10-LA/2)/2

  • LA = 5+LA/4

  • ¾LA = 5  LA= 20/3=LB


A and b live together move in at same time18 l.jpg
A and B live together, move in at same time

LA

LA(LB)

20/3

LB(LA)

LB

20/3


A and b live together move in at same time19 l.jpg
A and B live together, move in at same time

  • LA= 20/3=LB; CA=CB=40/3

    • Is this efficient?

    • Check if ΣMRS=MRT

  • Remember MRSALC=CA/Land MRSBLC=CB/L

  • [Tip: MRSxy=MUx/MUy=f/g, meaning would trade f units of good y for g units of good x.]

  • MRSALC=

 A would trade 2 carrots for a lava lamp.

ΣMRS=2+2=4  together A and B would trade 4 carrots for one lava lamp

MRT=1  a lava lamp only costs one carrot

So level of lava lamps is inefficiently low.


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