Chapter Thirty-Six Public Goods 1 Important Characteristics of Goods u A good is excludable if a person can be prevented from using it. – Excludable: fish tacos, wireless Internet access – Not excludable: FM radio signals, national defense u A good is rival in consumption if one person’s use of it diminishes others’ use. – Rival: fish tacos – Not rival: An MP3 file of Kanye West’s latest single 2 The Different Kinds of Goods Private goods: excludable, rival in consumption Example: food Public goods: not excludable, not rival Example: national defense Street light Public park Common resources: rival but not excludable Example: fish in the ocean Club goods: excludable but not rival Example: cable TV 3 ACTIVE LEARNING Categorizing roads u u 1 A road is which of the four kinds of goods? Hint: The answer depends on whether the road is congested or not, and whether it’s a toll road or not. Consider the different cases. 4 ACTIVE LEARNING Answers u u 1 Rival in consumption? Only if congested. Excludable? Only if a toll road. Four possibilities: Uncongested non-toll road: public good Uncongested toll road: club good Congested non-toll road: common resource Congested toll road: private good 5 Public Goods -- Examples u u u u u Broadcast radio and TV programs. National defense. Public highways. Reductions in air pollution. National parks. 6 Reservation Prices u u u A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it. Consumer’s wealth is w. Utility of not having the good is U ( w,0 ). 7 Reservation Prices u u u u A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it. Consumer’s wealth is w. Utility of not having the good is U ( w,0 ). Utility of paying p for the good is U ( w − p,1). 8 Reservation Prices u u u u u A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it. Consumer’s wealth is w. Utility of not having the good is U ( w,0 ). Utility of paying p for the good is U ( w − p,1). Reservation price r is defined by U ( w,0 ) = U ( w − r ,1). 9 Reservation Prices; An Example Consumer’s utility is U ( x1 , x2 ) = x1 ( x2 + 1). Utility of not buying a unit of good 2 is w w V ( w,0 ) = ( 0 + 1) = . p1 p1 Utility of buying one unit of good 2 at price p is w− p 2( w − p ) V ( w − p,1) = (1 + 1) = . p1 p1 10 Reservation Prices; An Example Reservation price r is defined by V ( w,0 ) = V ( w − r ,1) I.e. by w 2( w − r ) w = ⇒r= . p1 p1 2 11 When Should a Public Good Be Provided? u u u u 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. 12 When Should a Public Good Be Provided? u Payments must be individually rational; i.e. U A ( wA ,0 ) ≤ U A ( wA − gA ,1) and UB ( wB ,0 ) ≤ UB ( wB − gB ,1). 13 When Should a Public Good Be Provided? u Payments must be individually rational; i.e. U A ( wA ,0 ) ≤ U A ( wA − gA ,1) and UB ( wB ,0 ) ≤ UB ( wB − gB ,1). u Therefore, necessarily gA ≤ rA and gB ≤ r B . 14 When Should a Public Good Be Provided? u And if U A ( wA ,0 ) < U A ( wA − gA ,1) and UB ( wB ,0 ) < UB ( wB − gB ,1) then it is Pareto-improving to supply the unit of good 15 When Should a Public Good Be Provided? u And if U A ( wA ,0 ) < U A ( wA − gA ,1) and UB ( wB ,0 ) < UB ( wB − gB ,1) then it is Pareto-improving to supply the unit of good, so rA + r B > c is sufficient for it to be efficient to supply the good. 16 Private Provision of a Public Good? u u u Suppose rA > c and r B < c . Then A would supply the good even if B made no contribution. B then enjoys the good for free; freeriding. 17 Private Provision of a Public Good? u u Suppose rA < c and r B < c . Then neither A nor B will supply the good alone. 18 Private Provision of a Public Good? u u u Suppose rA < c and r B < c . Then neither A nor B will supply the good alone. Yet, if rA + r B > c also, then it is Paretoimproving for the good to be supplied. 19 Private Provision of a Public Good? u u u u Suppose rA < c and r B < c . Then neither A nor B will supply the good alone. Yet, if rA + r B > c also, then it is Paretoimproving for the good to be supplied. A and B may try to free-ride on each other, causing no good to be supplied. 20 Free-Riding u u u u 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. 21 Free-Riding u u u u u 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. 22 Free-Riding Player B Don’t Buy Buy Buy -$20, -$35 -$20, $65 Player A Don’t $80, -$35 Buy $0, $0 If A swich to buy, payoff <0; if B switch to buy, payoff<0, so they both don’t have incentive to change 23 Free-Riding Player B Don’t Buy Buy Buy -$20, -$35 -$20, $65 Player A Don’t $80, -$35 Buy $0, $0 (Don’t’ Buy, Don’t Buy) is the unique NE. 24 Free-Riding Player B Don’t Buy Buy Buy -$20, -$35 -$20, $65 Player A Don’t $80, -$35 Buy $0, $0 But (Don’t’ Buy, Don’t Buy) is inefficient. 25 Free-Riding u u u u 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 = $20 > $0. Payoff to B from the good = $25 > $0. 26 Free-Riding Player B Don’t Contribute Contribute Contribute Player A Don’t Contribute $20, $25 -$60, $0 $0, -$40 $0, $0 27 Free-Riding Player B Don’t Contribute Contribute Contribute Player A Don’t Contribute $20, $25 -$60, $0 $0, -$40 $0, $0 Two NE: (Contribute, Contribute) and 28 (Don’t Contribute, Don’t Contribute). Free-Riding u u u 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. 29 Variable Public Good Quantities u E.g. how many broadcast TV programs, or how much land to include into a national park. 30 Variable Public Good Quantities u u u u 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. 31 Variable Public Good Quantities u Budget allocations must satisfy xA + xB + c (G ) = wA + wB . 32 Variable Public Good Quantities u u u Budget allocations must satisfy xA + xB + c (G ) = wA + wB . MRSA & MRSB are A & B’s marg. rates of substitution between the private and public goods. Pareto efficiency condition for public good supply is MRS A + MRS B = MC (G ). 33 Variable Public Good Quantities u u Pareto efficiency condition for public good supply is MRS A + MRS B = MC (G ). Why? 34 Variable Public Good Quantities u u u Pareto efficiency condition for public good supply is MRS A + MRS B = MC (G ). Why? The public good is nonrival in consumption, so 1 extra unit of public good is fully consumed by both A and B. 35 Variable Public Good Quantities u u u Suppose MRS A + MRS B < MC (G ). MRSA is A’s utility-preserving compensation in private good units for a one-unit reduction in public good. Similarly for B. 36 Variable Public Good Quantities u MRS A + MRS B is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit. 37 Variable Public Good Quantities u u MRS A + MRS B is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit. Since MRS A + MRS B < MC (G ), making 1 less public good unit releases more private good than the compensation payment requires ⇒ Paretoimprovement from reduced G. 38 Variable Public Good Quantities u Now suppose MRS A + MRS B > MC (G ). 39 Variable Public Good Quantities u u Now suppose MRS A + MRS B > MC (G ). MRS A + MRS B is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit. 40 Variable Public Good Quantities u u u Now suppose MRS A + MRS B > MC (G ). MRS A + MRS B is 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 ⇒ Paretoimprovement from increased G. 41 Variable Public Good Quantities u Hence, necessarily, efficient public good production requires MRS A + MRS B = MC (G ). 42 Variable Public Good Quantities u u Hence, necessarily, efficient public good production requires MRS A + MRS B = MC (G ). Suppose there are n consumers; i = 1,…,n. Then efficient public good production requires n ∑ MRS i = MC ( G ). i =1 43 Efficient Public Good Supply -the Quasilinear Preferences Case u u Two consumers, A and B. U i ( xi , G ) = xi + fi (G ); i = A , B . 44 Efficient Public Good Supply -the Quasilinear Preferences Case u u u u Two consumers, A and B. U i ( xi , G ) = xi + fi (G ); i = A , B . MRSi = − fi′(G ); i = A , B . Utility-maximization requires pG ⇒ fi′(G ) = pG ; i = A , B . MRSi = − px 45 Efficient Public Good Supply -the Quasilinear Preferences Case u u u u Two consumers, A and B. U i ( xi , G ) = xi + fi (G ); i = A , B . MRSi = − fi′(G ); i = A , B . =MU(G) Utility-maximization requires pG ⇒ fi′(G ) = pG ; i = A , B . MRSi = − px u pG = fi′(G ) is i’s public good demand/marg. utility curve; i = A,B. 46 Efficient Public Good Supply -the Quasilinear Preferences Case pG MUB Equals to MRS MUA G 47 Efficient Public Good Supply -the Quasilinear Preferences Case pG MUA+MUB MUB MUA G 48 Efficient Public Good Supply -the Quasilinear Preferences Case pG MUA+MUB MUB MC(G) MUA G 49 Efficient Public Good Supply -the Quasilinear Preferences Case pG MUA+MUB MC(G) MUB MUA G* G 50 Efficient Public Good Supply -the Quasilinear Preferences Case pG MUA+MUB MC(G) MUB pG* MUA G* G 51 Efficient Public Good Supply -the Quasilinear Preferences Case pG * pG = MU A (G*) + MUB (G*) MUA+MUB MC(G) MUB pG* MUA G* G 52 Efficient Public Good Supply -the Quasilinear Preferences Case pG * pG = MU A (G*) + MUB (G*) MUA+MUB MUB MC(G) pG* MUA G* G Efficient public good supply requires A & B 53 to state truthfully their marginal valuations. Free-Riding Revisited u When is free-riding individually rational? 54 Free-Riding Revisited u u When is free-riding individually rational? Individuals can contribute only positively to public good supply; nobody can lower the supply level. 55 Free-Riding Revisited u u u u When is free-riding individually rational? Individuals can contribute only positively to public good supply; nobody can lower the supply level. Individual utility-maximization may require a lower public good level. Free-riding is rational in such cases. 56 Free-Riding Revisited u Given A contributes gA units of public good, B’s problem is max UB ( xB , gA + gB ) xB , gB subject to x B + gB = wB , gB ≥ 0. 57 Free-Riding Revisited G B’s budget constraint; slope = -1 gA xB 58 Free-Riding Revisited G B’s budget constraint; slope = -1 gB > 0 gA gB < 0 is not allowed xB 59 Free-Riding Revisited G B’s budget constraint; slope = -1 gB > 0 gA gB < 0 is not allowed xB 60 Free-Riding Revisited G B’s budget constraint; slope = -1 gB > 0 gA gB < 0 is not allowed xB 61 Free-Riding Revisited G B’s budget constraint; slope = -1 gA gB > 0 gB = 0 (i.e. free-riding) is best for B gB < 0 is not allowed xB 62 Skip 36.7 – 36.11 63