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Economics 151 The Economics of the Public Sector: Expenditure

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

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  1. Economics 151The Economics of the Public Sector:Expenditure Professor Nora Gordon Fall 2004 Lecture 10

  2. 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)?

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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?

  11. 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

  12. 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

  13. 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

  14. 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.

  15. 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

  16. 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

  17. 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

  18. A and B live together, move in at same time LA LA(LB) 20/3 LB(LA) LB 20/3

  19. 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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