Public Goods. Slide 1

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1 Public Goods Pareto Efficiency Market Failure: Competitive markets are not efficient Solutions to the Free-Rider Problem: Clarke Groves Mechanism General Policy Recommendation: Separate funding from who benefits Slide 1

2 Private vs. Public Goods Private goods are excludable: If you don t pay you won t get the good. rival: if you consume a certain amount of the good there is less to consume for others. Public goods are non-excludable: If you don t pay you can still get the good. non-rival: your consumption of the good does not diminish the amount available for others. Slide 2

3 Private vs. Public Goods Remember that rivalness and excludability are the keys to defining public vs. private goods Just because government supplies a good does not make it public e.g. public education is largely a private good Just because a good is not supplied by government does not make it private Many governments are doing nothing to reduce carbon emissions; but reduction is clearly a public good. Slide 3

4 Examples of Public Goods National defense Public safety Clean air Street lights (a very local public good) Child health, happiness (public good primarily to their parents; to a lesser extent to society) Slide 4

5 Examples of Collective Goods These goods are non-rival but excludable Cable TV Websites Technically you could be charged to visit any website (excludable) e.g. MOOCS (massive open online course) Slide 5

6 Examples of Commons Goods These goods are non-excludable but rival Fisheries Well water Other open access resources Slide 6

7 Impure public goods Goods occupying the middle ground between these extremes: i.e., exhibit some excludability or rivalness Roads: Costly to exclude people (given current technology) but as more and more people use it consumption becomes rival (congested). Polar bears: Rival when hunted; congestible when viewed in person; non-rival and non-excludable when enjoyed for pure existence value (i.e. when people get utility from knowing polar bears exist even though they can t see them in person) Slide 7

8 Examples of GE models We will mostly work with simple examples of general equilibrium models. One input, two outputs, two consumers The 2x2 production model with two consumers: 2 goods, 2 inputs, 2 consumers Slide 8

9 Assumptions for the 2x2 production model Two inputs: Capital K, Labor L Two consumption goods: private good x, pure public good G Person 1 and person 2 (consumers): strictly convex indifference curves (decreasing MRS between goods). Producers: strictly convex isoquants (decreasing marginal rate of technical substitution between inputs). We also assume that production of the two goods does not exhibit increasing returns to scale. Slide 9

10 Problems society must solve: How to allocate the existing stock of capital and labor efficiently between the production of good x and the production of good G. How to distribute these goods efficiently among the population once they are produced. Slide 10

11 Consumption Efficiency A distribution of goods is consumption efficient if it is not possible to reallocate these goods and make at least one person in the economy better off without making someone else worse off. Given a fixed amount of goods x and G, both people will always consume the same amount of the public good G, and hence we can only make one person better off by redistributing good x thus making the other person worse off. Slide 11

12 Production Efficiency An allocation of inputs (K and L) is production efficient if it is not possible to reallocate these inputs and produce more of at least one good in the economy without decreasing the amount of some other good that is produced. Slide 12

13 Production Efficiency max L x,k x,λ f x ( ( ) G) ( L,K ) x x + λ f G L L x,k K x Interior solution : f x L x λ f G L G = 0 f x K x λ f G K G = 0 G = f ( G L L x,k K ) x Slide 13

14 Condition for Interior Production Efficiency A given set of inputs available in an economy should be allocated across sectors until the marginal rate of technical substitution for each pair of inputs is equal in each sector. Slide 14

15 The Production Possibility Frontier with 2 Inputs While the set of Pareto efficient allocations in the Edgeworth box of production makes all the efficient input combinations visible, the PPF gives us all the combinations of goods that are production efficient. The PPF is the value function of the problem that gives us production efficient allocations as its solution. Slide 15

16 From Edgeworth Box of Production to PPF capital Good G x=12, G=28 x=24, G=20 x=30, G=10 Good x Good G PPF labor Good x Slide 16

17 The Slope of the PPF With good x on the horizontal axis and good G on the vertical axis, the absolute value of the slope of the PPF represents the Marginal Rate of Transformation (MRT) of good G for good x; it indicates how many units of good G the economy would have to sacrifice (by transferring inputs from the production of good G to the production of good x) in order to produce 1 more unit of good x. Slide 17

18 Units of x forgone to produce one more unit of G Similarly, the marginal rate of transformation (MRT) of good x for good G, indicates how many units of good x the economy would have to sacrifice (by transferring inputs from the production of good x to the production of good G) in order to produce 1 more unit of good G. Slide 18

19 Product Mix Efficiency Which combination of goods will give us Paretoefficiency? We can have efficiency in production and in consumption (trivially satisfied with one private and one public good), and yet there is still room for a Pareto improvement, because we are producing too much of one good and not enough of the other. Product mix efficiency puts together both sides, consumers and producers. Slide 19

20 Solving for Pareto Efficient Allocation maxu ( x,g) 1 1 s.t. u ( x,g) = u s.t. x + x = f ( L,K ) 1 2 x x x s.t. G = f ( L,K ) G G G s.t. s.t. L x + L G = L K x + K G = K Slide 20

21 Using PPF, the problem is maxu ( x,g) 1 1 s.t. u ( x,g) = u s.t. s.t. x 1 + x 2 = x G = G( x) max x 1,x,φ u 1 ( ) ( x 1,G(x) ) + φ u x x,g x 2 1 ( ( )) u 2 Slide 21

22 Pareto Efficiency with PG First Order Conditions are necessary and sufficient u 1 u φ 2 x 1 (x x 1 ) = 0 u 1 G G x + φ u 2 (x x 1 ) + u 2 G G x = 0 Slide 22

23 Samuelson Condition (1954) Combining both equations from previous slide u 1 G u φ 2 (x x 1 ) + u 2 G u 2 (x x 1 ) Since u 1 u φ 2 x 1 (x x 1 ) = 0 and (x x 1 ) = x 2 = x G u 1 G u 1 x 1 + u 2 G u 2 x 2 = x G Slide 23

24 The Samuelson Condition u 1 1 s MRS G + u 1 x 1 u 2 G u 2 x 2 2 s MRS = MRT of x for G x G MRT tells us the marginal cost of producing the public good to society in terms of the units of the private good society must sacrifice. MRS of person i tells us the marginal benefit of G expressed in units of good x; person i is willing to give up MRS units of good x for one more unit of G Slide 24

25 The Samuelson Condition Efficient provision of public goods requires that the sum of the marginal rate of substitution of the private good for the public good across all individuals is equal to the marginal rate of transformation of the private good for the public good. Slide 25

26 Pareto-efficient conditions for an economy with public goods 1. Allocate private goods until the point at which the marginal rate of substitution between any two private goods is equal and equal to the marginal rate of transformation between these goods. 2. For efficient production, the marginal rate of technical substitution of the inputs to production of all goods must be equal. 3. Wherever public goods exist, the sum of the marginal rates of substitution (for all people in society) of private for public goods must equal the marginal rate of transformation between these goods. Slide 26

27 More on efficiency In the case of an economy with only private goods, the benefit to society of the last unit of a private good provided (expressed as the willingness to forgo units of another good) is equal to the benefit of the one person in society who receives this last unit. If there are some people who receive a higher benefit from the last unit than others, we don t have Pareto efficiency. Hence, the marginal benefit of a private good must be the same across all people for the allocation to be efficient. However, in the case of public goods, everybody is forced to consume the same amount of the public good, but the marginal benefit for each person of consuming this amount may differ at an efficient allocation. The marginal benefit to society of providing this amount of public good is equal to the sum of the marginal benefits received by all people. This implies that the marginal rate of substitution of different members of society for a public good (in terms of a private good) need not be the same for efficiency to hold! Slide 27

28 Exercise Suppose there are 2 consumers, Ara and Bahar, and two goods, good x and good G. Ara s utility function is given by u A (x A, G) =x A G and Bahar s utility function is given by u B (x B, G) =x 1/4 B G 3/4. Both goods are produced with labor and capital and the PPF is given by G(x) = 8 - x. Slide 28

29 Exercise Cont d Suppose Ara receives one unit of good x. How many units of good x must Bahar receive and how many units of G should be produced for the allocation to be Pareto efficient? Slide 29

30 Answer We need to ensure that the Samuelson condition is satisfied. MRS Ara x for G + MRS Bahar x for G =MRT x for G We know that x B = x 1, G = 8 - x x A /(8-x) + 3x B /(8-x) = 1 1/(8-x) + 3(x-1) /(8-x) = x=8-x x= 2.5, G = 5.5, x B = 1.5. Slide 30

31 Exercise Cont d How does the Pareto efficient product mix change as we let Ara reach higher levels of utility? Slide 31

32 Answer maxu ( x,g) = x 1/ 4 G 3 / 4 B B B s.t. s.t. s.t. x A G = u A x A + x B = x G = 8 x max x 1/ 4 8 x B x B,x,φ ( ) 3 / 4 + φ x x B (( ) 8 x ) ( ) u A Slide 32

33 Answer We need to ensure that the Samuelson condition is satisfied. MRS Ara x for G + MRS Bahar x for G = MRT x for G We know G = 8-x, and u A = x A (8-x) x A /(8-x) + 3x B /(8-x) = 1 u A /(8-x) 2 + 3((8-x)x-u A ) /(8-x) 2 = 1-2u A +24x - 3x 2 =(8-x) 2 Slide 33

34 Answer We find the Pareto efficient amount of x as a function of u A. Implicitly given by -2u A +24x(u A ) - 3x(u A ) 2 =(8-x(u A )) 2 And G(u A ) = 8 x(u A ). Note that the highest utility Ara could get is by maximizing his own utility subject to G = 8 x. This leads to x = 4. Slide 34

35 Answer Ct d From last two equations (totally differentiated with respect to x and u A ) dx/du A - 6xdx/du A = -2(8-x) dx/du A dx/du A = 1/(20-4x) > 0, for x 5 dg/du A = - 1/(20-4x) <0, for x 5 As we want to achieve a higher utility for Ara, the product mix shifts in favour of more of the private good but never more than 4 units and less of the public good. Slide 35

36 Efficient Provision of Public Goods with Q-linear Prefs Let s assume that both consumers have quasi-linear utility functions of the form ( ) = x i + γ i G u i x i,g γ i ' > 0,γ i ''< 0. ( ) Slide 36

37 The Samuelson Condition u / G 1 + u / G 2 = x u / x u / x γ 1 ' G ( ) + γ 2 ' G ( ) = x G G Slide 37

38 Demand for the public good Note that if we were to set-up the standard utility maximization problem for each consumer, we find γ 1 ' ( G) = p G p x γ 2 ' G ( ) = p G p x Slide 38

39 Vertically adding demand Suppose p x = 1. Then p G = γ i (G) For a given y, p G is equal to the amount of good x the individual is willing to give up for one more unit of good G. The Samuelson Condition on LHS in the case of quasi-linear preferences is adding individual demands for the public good vertically. Slide 39

40 Price of G/ Marginal Willingness to pay γ 1 (G 1 ) + γ 2 (G 1 ) γ 1 (G 1 ) γ 2 (G 1 ) Social Marginal Benefit Curve Demand 2 Demand 1 G 1 Public good Slide 40

41 Efficient Provision of Public Good We can draw the MRT in the above graph. It tells us the marginal cost of producing the public good to society in terms of the units of the private good society must sacrifice. With strictly concave PPF this marginal cost is increasing; with linear PPF this marginal cost is constant. Slide 41

42 Price of G/ Marginal Willingness to pay Social Marginal Benefit Curve Social Marginal Cost Curve Efficient G Demand 2 Demand 1 Public good Slide 42

43 Summary (Interior) Efficient provision of the public good requires the Samuelson condition to hold; the sum of marginal rates of substitution of the private good for the public good must equal the marginal rate of transformation. With quasi-linear preferences, this condition boils down to adding individual demand for the public good vertically and finding its intersection with the social marginal cost curve. Slide 43

44 Competitive Equilibrium Now that we understand Pareto efficiency in a general equilibrium model with production and a public good, we want to find out if the competitive equilibrium in this model is Pareto efficient. First discuss what happens in the competitive equilibrium, then determine whether it s Pareto efficient. Slide 44

45 What is the correct assumption about consumer behaviour? However, there is a problem, when we discuss public goods. What is the correct assumption about consumer behaviour? Naïve consumer is not aware of the nature of public goods and hence treats the good like a private good. A more sophisticated consumer understands the nature of the public good and wonders how many units to buy, given that units of the good bought by other consumers can also be consumed. Slide 45

46 Naïve Consumer In the first approach, the answer is straightforward. Each consumer treating both goods as private will choose to buy the goods where MRS = price ratio. If consumers are price takers, they all face the same prices and therefore all their marginal rates of substitution will be equal in equilibrium. Slide 46

47 Production Efficiency Firms maximize profits and hence minimize costs. From profit maximization, MRTS=factor price ratio (because cost minimization is a necessary condition for profit maximization). If all firms are price takers in the factor markets, then all firms have equal MRTS. The competitive equilibrium satisfies production efficiency. Slide 47

48 Samuelson Condition Violated! Firms maximize profits. From profit maximization of competitive firms (if the number of firms is sufficiently large) P=MC. For any two goods, consumers set MRS=p x /p G. From profit max a firm produces an amount of x where p x =MC x, and a firm produces an amount of G where p G =MC G. This implies p x /p G =MC x /MC G, but MC x /MC G =MRT and therefore MRS=MRT, not sum of MRS=MRT! The competitive equilibrium is not efficient. Slide 48

49 Private Contribution to the Public Good Game Next, we consider more sophisticated consumer behaviour. Consumers realize that units of the public good purchased by other consumers will be available for their consumption as well as units purchased by themselves. We start with an example in which both consumers have identical Cobb-Douglas utility functions. Slide 49

50 Exercise: Private Provision vs. Efficient Provision In this exercise we contrast the private provision of the public good with the efficient public good provision. We can think of the private provision as a game: each person decides how much to contribute to the public good. In the end all the contributions collected are used to purchase the public good (as many units as the contributions buy). Slide 50

51 The model Suppose we have two people, person 1 and person 2 with the following utility function over a private good (x) and a public good (G) u ( x,g) = x α G 1 α i i i Both people face prices of p x, p G and have incomes of I 1, and I 2. Slide 51

52 Efficient Provision (a) Write down the conditions that describe the set of Pareto efficient allocations. The optimality condition is given by: u 1 / G + u / G 2 = u 1 / x 1 u 2 / x 2 ( 1 α )x 1 αg + ( 1 α )x 2 p G p x αg = p G p x p x (x 1 + x 2 ) + p G G = I 1 + I 2 Slide 52

53 Efficient Provision Cont d Solve for the efficient quantity of the public good G. We need the optimality condition and the joint budget constraint: ( 1 α )x 1 αg + ( 1 α )x 2 αg = p G ( ) = αp GG p x x 1 + x 2 ( 1 α ) p x p x (x 1 + x 2 ) + p G G = I 1 + I 2 ( ) ( ) I 1 + I 2 ( ) p GG ( 1 α ) = I 1 + I 2 G * = 1 α,x * = α I + I 1 2 p G p x Slide 53

54 Equilibrium Private Provision of the Public Good Now suppose both agents try to maximize their utility given the contribution of the other agent to the public good. That is, we want to find each agent s best response function. max x i,g i,λ x i α ( ) ( G j + G ) 1 α i + λ I i p x x i p G G i α FOCs αx 1 ( i G j + G ) 1 α i λp x = 0 ( 1 α )x α ( i G j + G ) α i λp G = 0 Slide 54

55 Solving for best response function Want to know what is the optimal amount of public good purchased by person i if person j contributes G j So we are looking for a solution G i (G j ) Use FOCs and b.c. to get rid of x i. αp x i = G ( G ( 1 α )p j + G ) i x p x αp G G ( 1 α )p j + G i x ( ) + p G G i = I i G ( i G ) j = 1 α ( )I i αp G G j p G Slide 55

56 Efficient and Private Provision Draw a diagram with person 1 s contribution to the public good on the x-axis and person 2 s contribution to the public good on the y-axis. We will see in the diagram why the Nash Equilibrium is not efficient. Think about the Cournot Duopoly game. Slide 56

57 Nash Equilibrium First present analytical results Then draw a graph using a symmetric setup Slide 57

58 Nash Equilibrium Nobody has an incentive to deviate. G 1 = G 2 = G 1 = ( 1 α )I αp G 1 G 2 p G ( 1 α )I 2 αp G G 1 p G ( 1 α ) I αi 1 2 ( ) 1 α 2 G 2 = I αi 2 1 ( 1 + α )p G ( ) ( )p G = I αi 1 2 ( ) p G 1 + α ( ) Slide 58

59 Nash Equilibrium is not efficient Sum up private contributions in NE and you ll see they are lower than the efficient G. ( ) ( ) G 1 + G 2 = I 1 αi 2 + I 2 αi 1 ( 1 + α )p G 1 + α G 1 + G 2 = ( 1 α ) ( I + I ) α ( ) p G < G ( ) * = ( 1 α ) I + I 1 2 p G ( ) p G Slide 59

60 The best response functions Price of x =1, price of G = 2, Income of each person = 200, alpha =1/2 Contribution of person Best response of person 1 Nash Equilibrium Best response of person Contribution of person 1 Slide 60

61 Contribution of person 2 Diagram Along red line: efficient amount of public good Utility of person 1 increases Both people better off than in NE Utility of person 2 increases Contribution of person 1 Slide 61

62 Neutrality result of inefficient provision of G What would happen in the Nash equilibrium if we were to change the distribution of income between the two people, leaving their combined income the same? We have seen that the the sum of contributions in the NE is given by G 1 + G 2 = ( 1 α ) ( I 1 + I ) 2 ( 1 + α ) p G A redistribution of income holding combined income constant does not change the amount of public good provided. Slide 62

63 Extra $ all spent on public good How come the level of public good does not change? Surely the contribution of both people changes. We can find out by totally differentiating each person s equilibrium strategy with respect to the changes in each person s income, such that di 1 +di 2 = 0 Slide 63

64 Poorer contributor reduces, richer contributor increases contribution ( ) dg 1 = di αdi α = di + αdi 1 1 = 1 di ( ) p G ( 1 + α ) p G p 1 G ( ) dg 2 = di αdi α ( ) p G = 1 p G di 2 Since di 1 = - di 2, the increase in contribution due to the increase in income by one person is offset by the decrease in the equilibrium contribution by the other person in the exact same amount. Total level of the public good is unchanged. Slide 64

65 Neutrality Result of Inefficient Provision It can be shown that this neutrality result holds even if people have different preferences. They don t have to be C-D either. For any utility functions, as long as a redistribution of income results in both people contributing privately to the public good, the total amount of the public good purchased remains unchanged. Bergstrom et al. (1986). Slide 65

66 Increasing the number of contributors What happens to the level of provision if the number of contributors increases? It is straight forward to analyze a special case of the previous set-up. Let s assume we have n identical individuals facing identical budget constraints. Then the Nash Equilibrium is symmetric: everybody contributes the same amount. Slide 66

67 Increasing the number of contributors To put this more formally, G ( i j i G ) j = 1 α G 1 * = G 2 * =... = G n * G i * = G i * = ( )I αp G i G j i j p G ( 1 α )I * i αp G (n 1)G i p G ( 1 α )I i 1 + α(n 1) ( ) p G Slide 67

68 Increasing the number of contributors Comparative Statics: change n for n 2 : ( 1 α )I G * i = i ( 1 + α(n 1) ) p G * G i n = αp G ( 1 α )I i < 0 ( 1 + α(n 1) ) 2 p G n( 1 α )I G * = ng * i = i ( 1 + α(n 1) ) p G G * n = ( 1 α )I ( 1 + α(n 1) )p G αnp G i 1 + α(n 1) ( ) 2 p G 2 = 1 α Individual contributions decrease with n ( 1 α )p ( )I G i 1 + α(n 1) ( ) 2 p G 2 > 0 Total contributions increase with n Slide 68

69 BBV and increasing n Note that BBV look at cases where they increase n at the same time as they leave total wealth in society the same. In this case, it can be shown that more equal distribution of wealth is lowering the public good provision (Theorem 5). We considered the case of increasing the number of agents and at the same time adding I i to the wealth of the economy with each agent that we added. Slide 69

70 BBV and government provision BBV also look at the possibility of government providing some amount of the public good, while individuals still contribute voluntarily Govt taxes individuals to finance public good purchases. Question of crowding out: Are private contributions reduced by the same amount as government contributes? Slide 70

71 Crowding out Theorem 6. Assumptions: taxes are collected from contributors and non-contributors. As government provides some amount of the public good, there is less than a dollar for dollar reduction in private contributions partial crowding out. Slide 71

72 Theorem 6 in detail Complete crowding out: taxes collected from people who contributed before and taxes are smaller than contribution. Partial crowding out if: a) some of the contributors are taxed more than their private contribution b) non-contributors are taxed as well as contributors Slide 72

73 Conclusion In the presence of public goods, private provision of public goods is often inefficient. We have also encountered the problem of freeriding: People try to get out of paying for a public good knowing that if it is paid for by others, they will still be able to consume the public good. Note that while equilibrium contributions may be greater than zero, we generally expect them to be too low. We will next talk about solutions to the provision of public goods. Slide 73

74 Achieving an Efficient Provision of Public Goods The Demand-Revealing Mechanism or Groves-Clarke Mechanism Exercise to see how the mechanism works Weaknesses of the Groves- Clarke Mechanism Slide 74

75 Theoretical Solutions to the Public Good Problem People have an incentive to lie about their true preferences in order to get out of paying for the public good while they can still benefit from its consumption. We would therefore need a scheme that will give people an incentive to reveal the truth about their public good preferences to the government. To find such a scheme we will now take a look at the work of Groves and Clarke. Slide 75

76 Edward H. Clarke Discovered the demand revealing process as a University of Chicago graduate student during the late 1960s. He has written extensively on the application of demand revealing processes which were noted in the 1996 Nobel Prize awards in economics. He is currently a senior economist with the Office of Management and Budget in the area of government regulatory management. Slide 76

77 The Demand-Revealing Mechanism A demand-revealing mechanism creates the incentive for people to reveal their public good preferences in a truthful manner. Here is how the demand-revealing mechanism works. Slide 77

78 5 easy steps 1. A distribution of cost shares is announced. 2. Given these cost shares each person reports the net benefit from consuming a certain amount of the public good (net benefit is equal to total benefit minus cost share). People may or may not tell the truth. Slide 78

79 More steps 3. Based on their reports in 2, the level of public good that maximizes the sum of reported net benefits of all people is provided. 4. Based on their reports in 2, a tax for person 1 is calculated as follows: Find the sum of net benefits for each quantity of the public good without person 1. If the level of public good at which the sum of net benefits without person 1 is maximized changes from the level determined in 3, person 1 s tax is equal to the difference of the sum of net benefits at the new optimal level and the level determined in 3 without taking person 1 s net benefit into account for any of the quantities of the public good. 5. Repeat step 4 for all the other people. Slide 79

80 Example Consider a community comprising three individuals: Alice, Brenda, and Chip. They have quasi-linear preferences with the private good entering their utility functions linearly. A benevolent planner is undertaking the provision of a pure public good. The preferences of community members are given in the following table. Slide 80

81 Total Benefit of Public Good Person Quantity Alice Brenda Chip The numbers in the table represent the total benefits accruing to the person named at left for the unit numbered above. So for example, Brenda would receive a total benefit of 120 were two units of the public good produced instead of one unit. All terms are in dollars. Slide 81

82 Question a) Assume that the good can be provided at a constant marginal cost of $120. What is the socially efficient level of the good? Slide 82

83 Answer a) The socially efficient level of the public good maximizes social net benefit, that is, total social benefit minus total cost. Because we have only four choices of quantities, we can directly calculate the social net benefit (SNetB )and then pick the quantity of the public good that yields the highest SNetB. More generally, SNetB is maximized, where the sum of marginal benefits equals the marginal cost. If we cannot use fractions, then pick the quantity that has a social marginal benefit closest to but still greater than marginal cost. Slide 83

84 Answer a) Finding highest SNetB. Quantity P e r s o n Alice Brenda Chip SB Total Cost SNetB SNetB highest, therfore optimal quantity is 3 Slide 84

85 Answer a) Another way of determining social optimum Table gives marginal benefit for each person and social marginal benefit (add up marginal benefit at each quantity). At a quantity of 3 we have SMB > MC but at 4 units we have SMB<MC. Therefore the efficient amount of the public good is 3 units. Slide 85

86 Question b) Suppose we announce that all three members have to share the cost of the public good equally, that is, everybody pays $40 per unit but there is an extra tax on a person whenever this person s net benefit changes the group decision. Calculate the net benefit of each person (total benefit minus cost for each person) and fill in the table below. Slide 86

87 Answer b) Quantity P e r s o n Alice Brenda Chip SNetB units of public good will be picked given these reports of the three people Slide 87

88 Question c) Calculate the Groves-Clarke taxes for each person. Slide 88

89 Answer c) In the absence of Alice, three units of the public good would be provided. Because Alice s valuation of the public good does not change the group choice, her tax would be 0. Quantity P e r s o n Brenda Chip SNetB units of public good will be picked given these reports of Brenda and Chip Slide 89

90 Answer c) If we do not count Brenda s marginal benefit, the quantity of the public good would be four since this maximizes the sum of net benefit from Alice and Chip. Brenda therefore needs to pay a tax in the amount of the difference in the sum of net benefits of Alice and Chip if the quantity of the public good is changed from 4 to 3 units. That is, Brenda s tax is equal to = 10. Quantity P e r s o n Alice Chip SNetB units of public good will be picked given these reports of Alice and Chip Slide 90

91 Answer c) Finally, without Chip the quantity would be 2 not 3. Chip needs to pay a tax of 70-50=20. Quantity P e r s o n Alice Brenda SNetB units of public good will be picked given these reports of Alice and Brenda Slide 91

92 Table with Tax Quantity P e r s o n Tax Alice Brenda Chip SNetB Overall net benefit with Groves-Clarke taxes for each person is for Alice: 30, for Brenda: 20-10=10, for Chip: =130. Slide 92

93 Question d) Brenda is the one who prefers two units to three units under this payment scheme. Has she an incentive to lie? Quantity P e r s o n Alice Brenda Chip SNetB Slide 93

94 Answer d) Suppose she changes the choice from three to two units by stating the following net benefit schedule. Quantity P e r s o n Alice Brenda b 1 b 2 b 3 b 4 Chip SNetB b 1 b 2 b 3 b 4 Slide 94

95 We assume 150+b 2 is greater than any of the other sums of net benefits. Then only two units will be supplied. Brenda is now paying a tax for changing the quantity from 4 to 2 units: 4 units would be supplied without her and the difference in net benefit of the other two is = 40. This means Brenda s net benefit from getting two units is 40 but then she would have to pay a tax of 40, which implies an overall net benefit of 0. This is lower than if she tells the truth and 3 units of the public good are provided. Then she receives a net benefit of 20 minus her tax of 10, that is, 20 10=10. Slide 95

96 No incentive to lie Brenda cannot make herself better off if she lies. Note that this is true no matter which lie she tells as long as the reported values b 1, b 2, b 3, b 4 result in a group choice of 2 units. How much Brenda benefits depends on her true net benefits for the different units and the reports of all the other people. Slide 96

97 Exercise Generalize above argument and show that given the other people s reports nobody has an incentive to lie. Quantity P e r s o n Alice a 1 a 2 a 3 a 4 Brenda b 1 b 2 b 3 b 4 Chip c 1 c 2 c 3 c 4 SNetB a 1 +b 1 a 2 +b 2 a 3 +b 3 a 4 +b 4 +c 1 +c 2 +c 3 +c 4 Slide 97

98 Weaknesses of the Groves- Clarke Mechanism 1. The Groves-Clarke tax doesn t really generate a Pareto efficient outcome. The level of the public good will be optimal, but the private consumption could be greater. We are taking away money from the people by imposing the tax that they otherwise could spend on private consumption (see Brenda s and Chip s situation). These taxes have to be taken away completely, i.e. the money collected through Groves-Clarke taxes would have to be destroyed. On the other hand, the more people are involved in the decision making process the less likely it is that their valuation of the public good will change the group decision. In this case Groves-Clarke taxes will only rarely be collected. Slide 98

99 Weaknesses of the Groves- Clarke Mechanism There are equity and efficiency tradeoffs. While it is efficient to supply the level of the public good determined by the Groves- Clarke mechanism, there are some people who might be strictly worse off with this level of public good than with a different amount (Brenda and Chip). The Groves- Clarke mechanism implements a solution that is potentially a Pareto improvement over the private provision of public goods, but we cannot compensate the losers to achieve an actual Pareto-improvement. Slide 99

100 Weaknesses of the Groves- Clarke Mechanism Another weaknesses of the Groves- Clarke mechanism is that it only is strategy-proof (people have incentive to report truthfully) if they have quasi-linear utility functions. Slide 100

101 How does the government decide on the level of Public Good? General elections, public goods funded out of general revenues which are raised independently of how people feel about public goods levels (by and large) Slide 101

102 Falkinger, Fehr, Gächter, Winter- Ebmer A Simple Mechanism for the Efficient Provision of Public Goods: Experimental Evidence

103 Pivotal Mechanism Elegant theoretical solution Difficult to explain to people [ ] considerations such as simplicity and stability to encourage trust, goodwill and cooperation, will have to be taken into account Laffont (1987, HB of PE p. 567) Just because mechanism does well in theory does not mean it does well in experimental settings Factors like confusion may impede efficient outcome.

104 Experimental Setting Use mechanism proposed by Falkinger (1996) Have a control and treatment group Control group plays voluntary contribution to the public good game Treatment group faces Falkinger s mechanism All other features are the same Vary group size, pay off functions NE sometimes at the boundary of strategy space, sometimes in the interior

105 Falkinger Mechanism Each individual gets a reward or pays a penalty depending on the deviation of his/ her contribution from the mean contribution If contribution is above mean reward If contribution is below mean penalty Theoretical result: NE is efficient. Result does not depend on quasi-linear prefs, either.

106 Experimental Set-up

107 More on the experiment Individuals in first experiment played 10 times. Some groups were always control group, others always treatment group and some were first control and then treatment group Individuals did not know who the other players in their group were (ever!) After each round, players were informed of the average contribution of the other group members, their own income (consumption of the private good), as well as their income from the public good.

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109 Conclusion In each of the implemented economic environments the mechanism caused an immediate and large shift towards an efficient solution. Little practical experience with the mechanism is necessary to induce a significant and large increase in contributions. Long-run behavior: In the control group there is a steady decline in contributions over time and towards the end the free-riding equilibrium exerts a strong drawing power. In the treatment group contributions are relatively stable over time with more contributions in later periods. There is no over-contribution in treatment group.