5.2 Definitions. pure public good. pure private good. commons: tendency of overusing. nonexcludability. nonrivalry. excludable & rivalry

Transcription

1 5. Public Goods

2 5. Definitions pure public good nonexcludability if the public good is supplied, no consumer can be excluded from consuming it nonrivalry consumption of the public good by one consumer does not reduce the quantity available for consumption by any other pure private good excludable & rivalry commons: tendency of overusing

3 rivalrous nonexcludable nonrivalrous excludable private good club good common property resource public good 3

4 5.3 Private Provision model: two consumers, two goods motive for free-riding (public g, private x) prices: given strategic interaction: public good: benefit from the purchases of others incomes: M, M both goods have a price of budget constraint: M h =x h +g h total consumption of public good: g +g utility fn: U h (x h,g +g ) 4

5 5 standard Nash assumption each consumer takes the purchase of the other as given when they make their decision : choose g to max utility given g ---> choice of is the best reaction to g Nash equilibrium max U (M -g,g +g ) ---> indifference curves 0 = = + + g U x U dg dg dg g U dg g U x U

6 g : small enough dg U U = = MRS dg x g < 0 g : large enough dg = MRS > 0 dg by diminishing marginal rate of substitution best-response: g ĝ Nash reaction function (best-response fn) g = r ( g ), g = r ( g ) 6

7 g budget constraint ĝ ĝ g 7

8 g budget constraint ĝ g g 8

9 g g =r (g ) ĝ g =r (g ) ĝ g 9

10 welfare property of Nash equilibrium s indifference curve: horizontal s indifference curve: vertical shaded area = set of Pareto improvements tangency point = Pareto optimal (efficient) when individuals privately choose the quantity of the public goods, the outcome is Pareto-inefficient total consumption of public goods is too low <--- strategic interaction and free-riding necessity of gvn intervention or voluntary cooperation 0

11 g locus of Paretoefficient allocation ĝ set of Pareto improvements ĝ g

12 5.4 Efficient Provision Samuelson rule tangency of indifference curves of both gradients of them are the same calculating the derivatives since, const U const U dg dg dg dg = G x G G G x U U U U U U =,, = x G x MRS G MRS h x h G h x G U U MRS =,

13 finally MRS G, x + MRSG, x = (Samuelson rule or condition) MRS = marginal benefit of another unit of pb gd MC = one unit of private gd efficient allocation <--- total MRS = MC = comparison private goods MRS i, j = MRSi, j = MRTi, j public goods: extra unit of pb gd increases the utility of all consumers social benefit is sum of the marginal benefits gvn intervention? ---> lump-sum taxes 3

14 5.5 Voting alternative allocation mechanism => majority voting cost of pb gd is equally shared G units of pb gd: cost of each consumer is G/H private gd M h -G/H ---> pg s effective price is /H utility U h (M h -G/H,G) median voter theorem assume H (>) is odd, single-peaked preference G h : max h s utility G <G <...<G H 4

15 median voter theorem median voter m=(h+)/ G m is the winner in the majority voting nonmanipulable by misrepresentation of pref. efficiency of the outcome max U m (M m -G/H,G) first-order condition: MRS m =/H Samuelson rule h= G m is efficient only if H MRS = voting does not generally achieve efficiency h MRS m = H h= MRS H h 5

16 private good utility public good G G m G H public good 6

17 5.6 Personalized Prices private market & voting ---> inefficient consumers face incorrect incentives DM take account only of the private benefit extended pricing mechanism = personalized align private and social benefits individual valuation of pb gd price quantity private good same different public good different same 7

18 Lindahl mechanism personalized pricing gvn announces the shares of cost of pb gd each consumer states the demand of pb gd if the same level, then it is supplied if not, the shares are adjusted and the process repeats until all consumers demand the same level lower demand ---> lower cost share lower private cost ---> higher demand the shares match the individual valuations Lindahl equilibrium 8

19 mathematical model of two consumers share of h: τ h, τ +τ = budget constraint: x h +τ h G h =M h Lindahl equilibrium {τ,τ } <--- G =G indifference curves: each consumer prefers more of pb gd but dislikes an increased share utility max given share ---> vertical indi. curves : NW, : NE ---> Lindahl reaction functions Lindahl equilibrium: intersection of two Lindahl reaction functions 9

20 G G G* G* reaction of τ τ reaction of 0

21 G G e L e M τ τ

22 efficiency result utility U m (M m -G/H,G) first-order condition h h UG h MRSG x = = τ, h h U, = x summing these conditions U U G x U, G + = MRS G, x + MRSG, x = τ + τ U x = Samuelson rule Lindahl mechanism (personalized pricing) ---> efficiency

23 difficulties of personalized pricing practical difficulty ---> announcing and adjusting shares incentive problem ---> reveal true demands? manipulation (cheating) assume that acts honestly, knows this and knows s reaction function if is honest, e L is LE (intersection of both reaction curves) if manipulates by dotted reaction curve, e M is realized efficient if both consumers act honestly 3

24 5.7 Mechanism Design 5.7. Examples of Preference Revelation Example : False Understatement G=0 (not produced), G= (produced), cost C= benefit v =v =, social benefit v +v =>C report: r h false: r h =0, true: r h = collective decision rule G= if r +r C=, G=0 otherwise announcements and payoffs c h = if r h = and r h =0 c h =/ if r h = and r h = c h =0 if r h =0 and r h =0 or 4

25 net benefit U h =v h -c h if r +r =0 otherwise r h =0 is weakly dominant strategy Nash equilibrium (r,r )=(0,0) both players understate their valuation of pb gd player 0 player ½ 0 ½ 5

26 Example : False Overstatement assume gross payoffs when pb gd is provided are v =0 < v =3/4 announcement player 0 announcement player 0 0 ¼ 0 -½ ¼ ¼ -½ -½ 6

27 social benefit v +v =3/4 < C = possible announcement: r =0 or, r =3/4 or uniform charge c h =/ if r +r C= c h =0 otherwise net benefit U h =v h -c h if r +r =0 otherwise r =0 is weakly dom. str. ---> r = is best resp. Nash equilibrium (r,r )=(0,) player overstate his valuation of pb gd 7

28 5.7. Clarke-Groves Mechanism construction of mechanism truth telling efficient provision of pb gd two consumers case net benefit: v, v, report: r h pb gd provided if r+r 0, if G=0, v =v =0 side payment = reported net benefit of the other consumer consumer : v +r : v +r internalize the net benefit of pb gd for the other by the side payment 8

29 player - + player v - 0 v + v + v + v - v + pl. pl pl pl v = - v = + 9

30 Clarke-Groves Mechanism provide no incentive to make a false announcement assume true and report: either - or + focus on pl. v=-: r=- is weakly dominant r=- ---> greater r=+ ---> equal v=+: indifferent between r=+ and r=- no incentive to provide a false announcement problem of this mechanism = side payment if pb gd is supplied and v=v=+, total side payments are equal to = total benefit obtaining the truth is possible but costly 30

31 5.7.3 Clarke Tax side payment ---> information (costly) reduce the side payments ---> modify only if the announcement change social decision pivotal agent <--- Clarke tax tax the pivotal agent for the cost inflicted on the other agents through the changed social decision game with two players v if r +r 0and r 0 v t if r +r 0 and r <0, with t = r >0 t if r +r <0 and r 0, with t =r 0 0 if r +r <0 and r <0 3

32 is pivotal ---> second and third cases tax: t = r >0 for cost of imposing provision t =r 0 for cost of stopping provision Clarke taxes induce truth-telling, guarantee efficient provision suppose pb gd is socially desirable v +v 0, but v <0 pl. honest: r =v ---> to prevent provision r < r pl. must pay t =r =v v pl. is better off telling the truth ---> pl. is also better off telling the truth (pl. is pivotal ---> pay tax t = r = v ) 3

33 5.7.4 Further Comments mechanism design reveal the truth make correct social decision rationality and precise calculation nonmonetary benefit ---> sufficient incentive market activities ---> reveal the valuation private gd = total value of characteristics one related to pb gd ---> reveal the valuation experimental implementation 33

34 5.8 More on Private Provision 5.8. Neutrality and Population Size effect of redistributing income transfer: income from consumer to income: M -, M + s utility is unchanged U (M -g,g +g )=U ([M - ]-[g - ],[g - ]+[g + ]) ---> indifference curves and best-response function moves s utility is unchanged U (M -g,g +g )=U ([M + ]-[g + ],[g - ]+[g + ]) 34

35 g ĝ + ĝ ĝ ĝ g 35

36 g ĝ + ĝ ĝ ĝ g 36

37 transfer D from to s purchase decrease by, s purchase increases by private good consumptions unchanged total public good consumption unchanged income distribution invariance policy neutrality utility levels unchanged utility uqalization 37

38 variations in the number of households H assume all consumers are identical in terms of preference and income total provision of pb gd h h h h U = U ( M g, G) = U ( M g, G + symmetric equilibrium g h =g for all h g G = H h= ray through the origin with gradient H eq. is given by the intersection of a ray with the bestreaction function large population ---> no contribution G = H g h g h ) 38

39 G H=3 H= reaction function Paretoefficient g 39

40 5.8. Experimental Evidence private purchase ---> too little of pb gd free-riding on others purchases experiment tokens ---> invest in private gd or pb gd max their profits private benefit social benefit private good 5 5 public good 0 40

41 0 participants, each has 0 tokens unit of private gd costs token, provides a benefit of 5units unit of pb gd costs token, provides a benefit of unit to all participants (social benefit =0) one-shot game Nash E = to purchase only private gd total return to each = 50 socially efficient outcome = to purchase only pb gd = 00 to each 4

42 experiment average value = 30 ~ 90% of tokens to pb gd most observation: 40~50% lowest for participants studying economics fall with the number of years of economics taken purchase of pb gd significantly >0 repeated decision learn about free-riding ---> optimal strategy? free-riding is not completely supported some attempted to establish cooperative eq. those not trained in GT cannot derive optimal strg. 4

43 group size increasing size leads to increased divergence from optimality when accompanied by decrease in marginal return from pb gd in total experiments do not provide great support for private-purchase eq. in one-shot games in repeated situations neither a strategic or a learning hypothesis is confirmed participants is initially guided more by a sense of fairness than by Nash behavior free-riding is not actually observed in practice 43

44 5.8.3 Modification experimental evidence <---> theoretical model two assumptions utility depends only on the consumption of pb gd and total supply of pb gd consumers act noncooperatively and according to NE conjectural variation each forms an opinion as to how others are affected modified preferences ---> warm glow consumer derives utility from the contribution social interaction ---> nonindividualistic reciprocity ---> contribution of others 44

45 5.9 Fund-Raising Campaigns fund-raising game target level of funds must be achieved before a pb gd ca be provided infinite horizon solicitation for donations can continue until the goal is met one pb gd: production cost = C two identical players X and Y benefit B from pb gd ---> total benefit B same discount rate δ, 0<δ< 45

46 players alternate in making contributions sequential contribution: (...,x t-,y t,x t+,...) game ends and pb gd is provided only when the total contributions cover the cost of pb gd no benefit before completion of the fund-raising marginal contribution yields no return until the cost is met incentive to wait for the other to contribute must be balanced against the cost of delaying pb gd is socially desirable C<B but B<C contribution campaign subscription campaign 46

47 5.9. The Contribution Campaign contributions are sunk at the time they are made lack of commitment impossible to raise the money two players ---> asked to make a contribution suppose X s turn at the final round T X is indifferent between making up the difference to get B x T and waiting Y will do in the next period with payoff δb x T =[ δ]b the contribution is equal to benefit of speeding up completion of the project 47

48 [-δ]b<c (cannot cover the full cost) backward ---> Y s turn at T- Y anticipates that up to C-x T at T-, X will complete the project the next period Y is indifferent between bringing contribution to get δby T- and waiting to make himself the final x T to get δ [BxT] y T- =δ[-δ ]B X s turn at T- X is indifferent between bringing contribution to get δ [BxT]-x T- and waiting to delay completion with a payoff δ 3 B-δ y T- x T- =δ 3 [-δ ]B 48

49 Y s turn at T-3 y T-3 =δ 5 [-δ ]B X s turn at T-4 x T-4 =δ 7 [-δ ]B summing these contributions [-δ]b+δ[-δ ]B+δ 3 [-δ ]B+δ 5 [-δ ]B +δ 7 [-δ ]B+...=B the total potential for contributions never exceeds the individual benefit from the project B<C ---> impossible to raise sufficient contributions for a successful campaign 49

50 total contribution C x T y T- x T- y T-3... x T-4 T time 50

51 5.9. The Subscription Campaign agents alternate in making donation pledges and bear the cost only when and if enough contributions are pledged to complete the project conditional commitment possible to raise an amount equal to total valuations two players ---> asked to make a contribution suppose X s turn at the final round T X is indifferent between making up the difference to get B-xT and waiting Y will do in the next period with payoff db xt=[-d]b the contribution is equal to benefit of speeding up completion of the project 5

52 [-δ]b<c (cannot cover the full cost) backward ---> Y s turn at T- Y anticipates that up to C-x T at T-, X will complete the project the next period Y is indifferent between bringing contribution to get δby T- and waiting to make himself the final x T to get δ [BxT] y T- =δ[-δ ]B X s turn at T- X is indifferent between bringing contribution to get δ [BxT]-x T- and waiting to delay completion with a payoff δ 3 B-δ y T- x T- =δ 3 [-δ ]B 5

53 Y s turn at T-3 y T-3 =δ 5 [-δ ]B X s turn at T-4 x T-4 =δ 7 [-δ ]B summing these contributions [-δ]b+δ[-δ ]B+δ 3 [-δ ]B+δ 5 [-δ ]B +δ 7 [-δ ]B+...=B the total potential for contributions never exceeds the individual benefit from the project B<C ---> impossible to raise sufficient contributions for a successful campaign 53