The ability to make good decisions quickly is key, whether you’re a seasoned leader or an entry-level employee. It’s a skill that inspires confidence in your abilities and leads to success. Shelley Row, an expert in efficient decision-making, says the secret is listening to your intuition while incorporating facts, a combination she calls “infotuition.” “Infotuition is a practical leadership tool,” she explains. “It’s a skilled, self-aware, decision-making approach—not a willy-nilly use of the whim.” She shares her advice for listening to your gut, and making smarter decisions as a result. The Skill Set: 5 Ways to Make Better, Faster Decisions CONSIDER INTUITION PART OF YOUR INTELLIGENCE “We honor thinking, but tend to underplay (or even denigrate) feeling,” says Shelley. In reality, neuroscience shows that intuition is an essential part of thinking. “It’s as though your brain stores bits of your life experience in file folders, but the ones you rarely use are in dusty file cabinets in the back,” she explains. “Intuition—a nagging gut feeling—is information from one of those dusty file folders trying to get through. We have to let it in by quieting the loud, logical voice and attending to the quiet, wise voice in our heads.” END YOUR LOVE AFFAIR WITH DATA Or at least put it in perspective. “It is inherently backward-looking,” says Shelley. “If the future differs from the past, data must be viewed with caution.” It’s also subject to bias. “Methods of collection, analysis and interpretation provide opportunities for bias to creep in despite best intentions,” she says. “Don’t get me wrong. Data is an important thing; it’s just not the only thing.” GATHER INFORMATION, THEN TAKE A BREAK It’s not a coincidence that your best ideas occur to you when you’re in the shower or drifting off to sleep. “When we overwhelm ourselves with activity—and who doesn’t?—the brain can’t put all the pieces together, it just gets tired,” Shelley explains. “When faced with a complex decision, review the data, gather input from the right people, particularly those with differing opinions, then take a brain break, something that distracts the mind and lets it wander. The brain continues to work on the problem during the downtime, allowing it to pick up subtle signals from the dusty file folders.” WATCH OUT FOR OVERTHINKING Shelley advises looking out for these phrases: “We’ve been over this again and again,” “This is taking waaay too long” and “We’re making this harder than it has to be.” They’re key indicators that you and your team are overthinking the problem. “When this happens,” advises Shelley, “take a step back and notice the nagging feeling that holds you back.” EXPLORE YOUR GUT FEELINGS Ask yourself, “What’s bugging me?” or “What’s not sitting right?” Shelley explains that these questions coax the brain to identify the underlying gut feel, and that it usually points to either fear or insight. If it’s fear, she suggests talking yourself through it, by asking yourself, “What am I afraid of?” “What would happen if I let go of that fear?” “What would I be freed up to do?” If it’s not fear, it may be insight, which Shelley defines as, “some part of your experience that wants to weigh in on the decision.”
The value of econometrics (1) Testing whether financial markets are weak-form informationally efficient (2) Testing whether the Capital Asset Pricing Model (CAPM) or Arbitrage Pricing Theory (APT) represent superior models for the determination of returns on risky assets (3) Measuring and forecasting the volatility of bond returns (4) Explaining the determinants of bond credit ratings used by the ratings agencies (5) Modelling long-term relationships between prices and exchange rates (6) Determining the optimal hedge ratio for a spot position in oil (7) Testing technical trading rules to determine which makes the most money (8) Testing the hypothesis that earnings or dividend announcements have no effect on stock prices (9) Testing whether spot or futures markets react more rapidly to news (10) Forecasting the correlation between the stock indices of two countries.
Journal of Public Economics 66 (1997) 489–504 On the possibility of efficient private provision of public goods through government subsidies Georg Kirchsteiger*, Clemens Puppe Department of Economics, University of Vienna, Hohenstaufengasse 9, A-1010 Vienna, Austria Received 31 March 1996; accepted 21 February 1997 Abstract This paper investigates the possibility of implementing an efficient provision of a public good through distortionary tax-subsidy policies in a simple one-shot game of voluntary contributions. Within the class of all linear tax-subsidy policies two cases are distinguished. The first is where individual taxes only depend on the sum of all other individuals’ contributions. Although such policies may increase total supply of the public good, it is shown that the implementation of an efficient amount is not possible unless the government has complete information about individual characteristics. In the second case, where taxes depend on the distribution of contributions, the equilibrium supply of the public good is no longer unique. For any efficient interior solution there might also exist inefficient boundary solutions. Moreover, unlike the boundary solutions, the efficient interior solution is in general not stable. ó 1997 Elsevier Science S.A. Keywords: Public goods; Tax subsidy; Voluntary contributions JEL classification: H29; H40; H41 1. Introduction Voluntary contributions to a public good typically entail underprovision of that good. Many authors have therefore considered models in which a ‘government’ *Corresponding author. Tel.: 143 1 401032423; fax: 143 1 5321498; e-mail: georg.kirchsteiger@univie.ac.at 0047-2727/97/$17.00 ó 1997 Elsevier Science S.A. All rights reserved. PII S0047-2727(97)00029-7 490 G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 subsidizes private contributions so as to increase the total supply of the public good. Usually, in these models, government’s subsidy payments are financed by appropriate taxation. While lump-sum transfers typically leave the equilibrium amount of the public good unchanged, distortionary tax-subsidy policies may 1 indeed increase total equilibrium supply of the public good. Given this possibility to influence aggregate supply of a public good, the question arises whether by choosing an appropriately designed tax-subsidy policy a government can implement an efficient amount of the public good. This is the problem addressed by the present paper. The analysis is restricted to the most natural case of linear tax-subsidy policies. Unfortunately, in this case our results are rather negative. Indeed, it is shown that either (i) the government needs to know individual preferences in order to implement an efficient allocation, or (ii) the contribution game admits a multiplicity of equilibria with not all of them corresponding to efficient allocations. Moreover, in the latter case, an efficient equilibrium is in general unstable. The framework for our analysis is the following general tax-subsidy scheme. Each agent’s own contribution is subsidized at some fixed individual subsidy rate. At the same time, each agent faces a tax that is a linear function of all other agents’ contributions to the public good. Subsidy payments and taxes are linked in such a way that the government’s budget is balanced for any possible distribution of individual contributions. In determining the level of her own contribution to the public good, each agent optimizes against all other agents, taking their decisions as given. Aggregate supply of the public good then results from the equilibrium level of individual contributions in this simultaneous one-shot game. Given this general model, two cases have to be distinguished. Firstly, it can be shown that within our framework the model suggested by Andreoni and Bergstrom (1996) corresponds to the case where each agent’s tax only depends on the sum of all other individuals’ contributions (and not on their distribution). In the following, we refer to this case as the case of individually uniform tax rates. In this case, each choice of subsidy rates induces a unique aggregate equilibrium supply of the public good which is increasing with the subsidy rates (see Andreoni and Bergstrom, 1996). Moreover, it is easy to determine subsidy rates that induce an efficient allocation provided that all individual contributions are positive in equilibrium. However, in this paper we prove that given such subsidy rates all individual contributions remain positive in 1Warr (1983) has shown that lump-sum transfers do not alter the equilibrium amount of the public good provided that the set of contributors does not change. Bergstrom et al. (1986) provide a general analysis of income redistributions. Warr’s neutrality result is confirmed in Bernheim (1986) who considers distortionary income taxes. For the possibility to influence aggregate supply of a public good through subsidies to voluntary contributions in a framework with ‘naive’ individuals who ignore the government’s budget constraint, see Roberts (1987), (1992) and Boadway et al. (1989). Non-neutrality of tax financed subsidies with rational individuals who take into account the government budget constraint has been established in the models of Andreoni and Bergstrom (1996) and Falkinger (1996). G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 491 equilibrium only if the resulting effective prices for the individuals are the Lindahl prices. This implies that in order to implement an efficient allocation the government must have complete information about individual preferences (or at least about the individuals’ demand for the public good). However, if a government does have complete information, there is of course no point in designing a complicated tax-subsidy policy. Indeed, in that case the government has complete control over the state of the economy via much simpler tax policies, e.g. the government could take care of the supply of the public good and impose appropriate lump-sum taxes to cover its expenditures. In order to overcome this difficulty, one has to allow agents’ tax payments to depend on the distribution of the other agents’ contributions to the public good. Indeed, an example of a subsidy policy where the existence of an efficient interior solution is not the exception is the model recently suggested by Falkinger (1996). In this model, the population is divided into subgroups and each agent’s tax only depends on the contributions made by individuals belonging to the same subgroup. If, however, the agents’ tax payments depend on the distribution of the other individuals’ contributions other problems arise. Firstly, in that case the equilibrium amount of the public good is no longer uniquely determined. Specifically, we prove the following result. For any subsidy scheme where tax rates are not individually uniform there exist individual preferences and distributions of incomes such that besides an efficient interior solution there is also a non-efficient boundary solution where at least one individual contributes zero. Moreover, in contrast to the boundary solution the efficient interior solution is not stable in an appropriately defined sense. This result casts some serious doubt on the possibility to implement efficient allocations by linear tax-subsidy policies involving different tax rates for the other agents’ contributions. The plan of the paper is as follows. In Section 2 we present the general framework of our analysis and discuss its relation to the literature. Section 3 is devoted to the case of individually uniform taxation. In Section 4, we consider linear tax rules that are not individually uniform. Concluding remarks are offered in Section 5. 2. The model Consider an economy with n individuals, indexed by i51, . . . , n. Each i individual’s utility is given by a strictly quasi-concave utility function u (c ,G), i where c denotes i’s consumption of a private good and G the consumption of a i purely public good. Throughout, private consumption and the public good are assumed to be strictly normal goods at every level of wealth. Furthermore, we assume that each individual’s utility function is continuously differentiable. Each individual has an initial endowment of m units of the private good. For simplicity, i let the price of the private good be equal to 1. Hence, one may think of m as i 492 G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 consumer i’s income. The public good is produced from private goods at a cost of one unit private good per unit of public good. The public good is supplied by voluntary contributions of the consumers. For each i, denote by g consumer i’s contribution to the public good. Furthermore, let i G :5o g denote the sum of the contributions of all agents different from i. A 2i j±i j common assumption in a model of private provision of a public good is that each individual takes the activities of all other agents as given for her own decision. Consequently, consumer i’s decision problem is i max u (c ,g 1G ) s.t. c , g i i 2i i i (2.1) c 1g 5m and g $0. i i i i A pair of n-tuples (c*, . . . , c*) and ( g*, . . . , g*) that solves (2.1) for all i is 1 n 1 n hence a Nash equilibrium of the corresponding contribution game played by the n individuals. It is well known that without any intervention an equilibrium of the game described so far entails underprovision of the public good. Many authors have therefore considered extensions of this model allowing for the possibility that a government subsidizes private contributions (see, among others, Andreoni (1988); Andreoni and Bergstrom (1996); Boadway et al. (1989); Brunner and Falkinger (1995); Falkinger (1996) and Roberts (1987), (1992)). In its most general form, such a government intervention may be described as follows. Each individual’s private contribution is subsidized at a rate s , where 0#s ,1. Hence, i i if i contributes g she receives a payment of s g . Government expenditure, in turn, i ii is financed by taxes where each agent’s tax payment depends on all other individuals’ contributions. In the present paper, we confine ourselves to the most natural case where each agent’s tax is a linear function of all other agents’ contributions. Denoting by t $0 agent i’s tax rate with respect to agent j’s ij 2 contribution, consumer i’s budget constraint may thus be written as c 1(12s )g 5m 2Ot g , (2.2) i i i i ij j j±i with the additional constraint that g $0. i Remark: the subsidy scheme described by (2.2) is the most general form of government subsidies through a change of relative prices when taxes are linear. Firstly, observe that there is no rationale to let individual i’s tax depend on other individuals’ private activities, i.e. their private consumption c . Clearly, individual j i’s tax may depend on her own private consumption c . However, any reasonable i form of such a dependence must be linear in c . Since all that matters are relative i prices, such a dependence is already incorporated in the s s. By a similar argument i 2Our analysis is completely general with respect to the distribution of income. Without loss of generality, we therefore neglect lump-sum transfers in our model. G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 493 one may also assume without loss of generality that agent i’s tax does not depend on her own contribution g to the public good. i Note that if agents optimize against each other according to (2.2) there is a problem of bankruptcy, since some configurations of strategies (i.e. choices of (c , i g )) may entail negative net-income for some consumers. In this case there would i be no well-defined payoff. Consequently, in analyzing the corresponding contribution ‘game’, we will assume throughout that in equilibrium individual contributions ( g*, . . . , g*) satisfy the following condition.3 1 n NB (No bankruptcy). For all i and i i 0,s ,1. Furthermore, for all i and all j, t 5s b, hence consumer i’s total tax i ij i burden is s bG . Rearranging individual i’s budget constraint yields i 2i c 1(12b )g 5m 2s bG. (2.4) i i i i Hence, each consumer’s contribution g is subsidized at a rate of b, and at the i same time the consumer is taxed for a fixed share s of total government i expenditure on subsidies. As a consequence, consumer i’s effective price for the public good is 12b1s b. Clearly, the government’s budget is balanced if and i only if o s 51. Notice that each individual is taxed for each unit of the public i i good provided by any other agent at a constant rate, i.e. each individual’s tax payment only depends on the sum of all other agents’ contributions. Hence, the Andreoni/Bergstrom model belongs to the class of subsidy schemes with individually uniform taxation. In this model, Andreoni and Bergstrom (1996) prove that for each b and any family s , . . . , s with 0,s ,1 and o ]]] is the absolute value of the marginal rate of i i -u (c ,G) /-c i substitution between the public good and private consumption. Summing up over all individuals thus gives n OMRSi 5n2b(n21). i51 Consequently, if all individual contributions are positive in equilibrium — an assumption which will be shown to be extremely restrictive — an efficient provision of the public good would require b51. Indeed, with b51 one obtains i 4 o MRS 51, the condition characterizing efficient allocations. Although also for i b51 there is a unique aggregate equilibrium supply of the public good, individual equilibrium contributions are no longer unique for that particular value of b (see 5 Section 3). Nevertheless, one may ask if by choosing b sufficiently close to 1 one can implement an amount of the public good arbitrarily close to an efficient amount as the result of a unique Nash equilibrium. This question is addressed in Section 3. Another special case of the subsidy scheme described by (2.2) and (2.3) is the model recently proposed by Falkinger (1996). In his model, the population is partitioned into subgroups and each individual is rewarded or penalized on the basis of the average contribution of the subgroup to which the individual belongs (cf. Section 4). Note that, since individual taxes only depend on the contributions made within the same subgroup, taxation is not individually uniform in that model. 3. Efficient allocations with individually uniform tax rates In this section, we investigate existence of efficient allocations for the case of individually uniform tax rates. Hence, assume that in (2.2) each individual i is taxed at a constant rate for each contribution made by another individual, i.e. assume that for all i and all j, t 5t for some t $0. The analysis is substantially ij i i simplified by the observation that any subsidy scheme of the form of (2.2) with that property can be rewritten as in (2.4). Hence, the Andreoni/Bergstrom model described by (2.4) exactly corresponds to the case of individually uniform taxation. Indeed, it can be shown that given the governments budget constraint (2.3) the assumption of individually uniform taxation implies that for all i, j, t 1s 5t 1s . Defining b:5t 1s and s :5t /b, the budget constraint (2.2) then i i j j i i i i 4By strict normality, private consumption and total supply of the public good are always positive in equilibrium. Hence, efficiency in equilibrium is always characterized by the standard Samuelson rule. 5This seems to be the reason why Andreoni and Bergstrom (1996) explicitly exclude the case b51 in their analysis. G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 495 6 easily transforms into (2.4). Also, it can be checked that o s 51. The case b#1 i i n corresponds to the case where o s #n21. Of course, this is the only case of j51 j interest, since with os .n21 one would obtain ‘overprovision’ of the public j good. Firstly, we analyze the case of interior equilibrium of the contribution game corresponding to (2.4). It has already been observed in the previous section that efficiency in an interior equilibrium requires b51. In order to characterize the equilibrium of contribution game for b51, consider the following closely related problem for individual i. i max u (c ,G) s.t. i ci,G (3.1) c 1s G5m and G$0. i i i The solution G˜ (m , s ) to this problem is individual i’s demand for the public i i i good provided that its price is s and that no other individual contributes to the i public good. Consequently, we refer to G˜ (m , s ) as individual i’s stand-alone i i i contribution. Now, compare (3.1) to individual i’s maximization problem given the subsidy scheme (2.4) for the value b51. i max u (c ,g 1G ) s.t. c , g i i 2i i i (3.2) c 1s G5m and g $0. i i i i Obviously, the only relevant difference to the problem (3.1) is the nonnegativity constraint. Denote by M the set of those individuals with maximal stand-alone contribution, i.e. M:5h j g 2] ( g 1g ),0J, i j k l i 2 j 4 k l where j belongs to the same subgroup, whereas k and l together form the other subgroup. Assuming that all individuals contribute a positive amount, the 9Note that this requires at least three consumers in the economy. Indeed, with only two individuals taxes are automatically individually uniform. G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 499 following equilibrium contributions can be derived, g*5g*55m /823m /8 and 1 2 I II g*5g*55m /823m /8. Consequently, there is an efficient interior solution if 3 4 II I and only if 3/5,m /m ,5/3. I II However, in addition to the efficient interior equilibrium there is a variety of inefficient boundary solutions. For instance, if 3/5,m /m ,4/3 there are two I II equilibria where exactly one individual from subgroup I contributes zero whereas all other agents make a positive contribution. Similarly, if 3/4,m /m ,5/3 there I II are two equilibria where exactly one individual from subgroup II contributes zero. Hence, whenever an efficient interior solution exists there are also inefficient boundary equilibria. In fact, it can be checked that there are further boundary equilibria in addition to those described. In any of these equilibria the nobankruptcy condition is satisfied, i.e. each agent’s private consumption is positive. Besides the multiplicity of equilibria in this example there is another problem of instability of the interior equilibrium. Indeed, as can be seen from Fig. 2, the reaction curves of two individuals belonging to the same subgroup intersect with Fig. 2. Non-uniqueness and instability of the interior solution. 500 G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 the ‘wrong’ slope. This implies instability of the interior equilibrium with respect to any dynamics where individuals adjust their contribution in direction of their best responses. The difficulties with non-uniform tax rates are by no means specific to the particular example just discussed. Specifically, one has the following result. Theorem 2: For any linear tax-subsidy scheme with property (4.1) there exist preference and income distributions such that besides the efficient interior equilibrium there is also an inefficient boundary equilibrium for the corresponding contribution game. Before we proceed to the proof of Theorem 2, we need the following preliminary result. Observe first that an efficient interior solution can only exist if subsidy rates satisfy n Os 5n21. (4.2) i i51 Let t˜ denote individual i’s average tax rate, i.e. ˉt :5o t /(n21). i ij±i ij Lemma 4.1: Suppose that subsidy rates satisfy (4.2). Either there exists an individual i whose average tax rate t is larger than her effective price 12s , 0 i 0 i 0 or, for all i, tˉ 512s . i i Proof : Assume by way of contradiction that, for all i, tˉ #12s with strict i i inequality for some i. Summing over i one would obtain n n OOt ,(n21)O(12s ). (4.3) ij i i51j±i i51 However, differentiating the government’s budget constraint (2.3) with respect to g yields s 5o t for all i. Using this and interchanging the order of i i j±i ji n summation in (4.3) one could conclude o s ,n21. However, this is in j51 j contradiction to (4.2). The following proof of Theorem 2 is based upon the case distinction described in Lemma 4.1. Firstly, assume that there is an individual i whose average tax rate 0 is larger than her effective price for the public good. Suppose that the valuation for the public good is sufficiently low for all individuals but i , and consider an 0 income distribution such that there is nevertheless an interior equilibrium in which all individuals contribute the same amount. Since i ’s average tax is higher than 0 her effective price, and since moreover, i ’s valuation for the public good is high 0 compared to the other agents, there is an additional (inefficient) boundary equilibrium in which only i supplies the public good. Next, consider the case 0 where average tax rate equals effective price for all individuals. Then, since taxation is not individually uniform, there must exist two individuals, i and j 0 0 such that i ’s tax rate for j ’s contribution is larger than i ’s effective price for the 0 0 0 public good. From this, the existence of an inefficient boundary equilibrium in G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 501 addition to the efficient interior equilibrium can be inferred by a similar argument as in the first case. Proof of Theorem 2 : Assume that all individuals are endowed with Cobb- Douglas type preferences ui(c , G)5caiG12ai. In this case, agent i’s (unrestricted) i i reaction function can be calculated as 12a 12a i i g 5]]m 2 ]] O S t 1a D g . (4.4) i 12s i 12s ij i j i j±i i Consider a family of ‘efficient’ subsidy rates s , . . . , s satisfying (4.2). We 1 n distinguish two cases according to Lemma 4.1. Case 1. There exists i such that tˉ .12s . Without loss of generality, 0 i 0 i 0 assume that i 51. Fix a ]m . 1 12s 1 1 On the other hand, since (1, 1, . . . , 1) is a solution one obtains, again by (4.4), 12a1 l t1j ]]m 511(12a ) O]]1a (n21). 12s 1 1 12s 1 1 j±1 1 Since by assumption tˉ .12s , it follows that g**.n. Now consider all 1 1 1 individuals different from 1. Since the vector (1, 1, . . . , 1) is an interior solution, l and since the a converge to 1 one obtains from (4.4) i l 12a i l ]]m ®n if l®`. i 12 s i Hence, since g**.n, agent i’s unrestricted reaction function becomes negative 1 for sufficiently large l provided that all agents j±1,i contribute zero. Conse- quently, i’s best response to g 5g** and g 50 for j±1,i is g**50 for 1 1 j i sufficiently large l. This shows that for large l, ( g**, 0, . . . , 0) is an additional 1 equilibrium. Observe that this equilibrium cannot be efficient. Indeed, at the 1 equilibrium allocation one has MRS 512s . However, all other individuals’ 1 unrestricted reaction function becomes strictly negative in equilibrium. Hence, 502 G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 j MRS ,12s for j52, . . . , n. Together with (4.2) this immediately implies j inefficiency. Also observe that the no-bankruptcy constraint is satisfied in the boundary equilibrium. Case 2. Suppose now that for all i, tˉ 512s . Since the tax-subsidy scheme i i satisfies (4.1), there must exist i and j ±i such that t .12s . Without loss 0 0 0 i 0 j 0 i 0 of generality, assume that i 51 and j 52. The proof in this case is similar to the 0 0 proof in Case 1. Again, fix a so that in the end the no-bankruptcy condition is 1 satisfied for individual 1, and consider for each i52, . . . , n a strictly increasing l l sequence (a ) converging to 1. For each l, choose the distribution (m , . . . , i l ],1,]], . . . ,]]D is an interior equilibrium. By (4.4) this implies that n21 n21 n21 for all l ]m .2 and ]]m 52. 12s 1 12s 2 1 2 Furthermore, for each i53, . . . , n, l 12ai l ]]m ®2 if l®`. i 12 s i This implies by the same arguments as in Case 1 that, for sufficiently large l, an additional equilibrium is given by 12a1 l ( g**,g**, . . . ,g**)5S]]m ,0, . . . ,0D. 1 2 n 1 12 s1 Again, this equilibrium is inefficient and satisfies the no-bankruptcy constraint if l is large enough. The preferences constructed in the proof of Theorem 2 might seem rather extreme. Note, however, that this is due to the great generality of Theorem 2 since it applies to arbitrary linear tax-subsidy schemes satisfying (4.1). In specific examples — such as the one considered above — much less extreme preference distributions yield similar conclusions. Also note that the instability of the interior equilibrium uncovered in the example is a general phenomenon. Suppose, for instance, that individual preferences are of Cobb-Douglas type. By Lemma 4.1, if a tax-subsidy scheme is not individually uniform, there exist individuals i and j ±i such that t .12s . 0 0 0 i 0 j 0 i 0 If a is sufficiently close to 1 this implies that the reaction curves of individuals i j 0 0 and j intersect qualitatively as shown in Fig. 2. 0 G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 503 5. Conclusion In this paper, we have argued that linear tax-subsidy policies in a simple one-shot, simultaneous move game of voluntary contributions to a public good are not an appropriate tool for implementing efficient allocations. In designing such a policy, the central planner (the ‘government’) faces a dilemma. Either the government chooses a policy where each individual’s tax only depends on the sum of all other individuals’ contributions, i.e. an individually uniform tax-subsidy scheme, or an incentive scheme where some individuals’ tax depends on the distribution of contributions. In the first case, an efficient interior equilibrium only exists if the government can implement the Lindahl prices through subsidy rates. However, this requires knowledge that the government is assumed not to have. It is worth noting that this problem can be solved in a different framework which has been suggested in the literature. Consider, for instance, the following two-stage game proposed by Danziger and Schnytzer (1991) (see also Althammer and Buchholz (1993); Varian (1994)). In the first stage individuals announce appropriate subsidy rates by which they will subsidize other agents’ contributions to a public good. Given these subsidy rates, individual contributions are then simultaneously determined in a second stage. In this two-stage game, it can be shown that the Lindahl subsidies indeed form the unique subgame perfect equilibrium. Given the difference in the informational structure of the two games, the difference in the results is of course not surprising. In both models, individual preferences are common knowledge to any potential contributor. Consequently, in the Danziger/Schnytzer game all players have complete information. On the other hand, in the model considered in the present paper subsidy rates are set by a central planner who has no information about individual preferences. Clearly, either of the two models implies extremely restrictive informational assumptions on the part of potential contributors. Taking these assumptions for granted, we believe that the model considered here has a priori much more practical appeal, in particular, if the number of agents is large. In the second case, when a subsidy policy is chosen where individuals tax payments depend on the distribution of the other agents’ contributions, it has been shown that uniqueness of the equilibrium is no longer guaranteed. Moreover, even if an efficient interior equilibrium exists, it is in general not stable, and typically there exist additional stable and inefficient boundary equilibria. Our overall conclusion, that linear tax-subsidy policies are not appropriate for implementing efficient allocations, bears some resemblance to negative results obtained in the very different framework of mechanism design models (see e.g. Green and Laffont (1979)). In our context, an interesting open question is whether the multiplicity of equilibria can be avoided by designing suitable non-linear taxation rules. On the other hand, it seems to us that the problem of instability would persist also under more complicated tax policies. 504 G. Kirchsteiger, C. Puppe / Journal of Public Economics 66 (1997) 489 –504 Acknowledgements We are grateful to Josef Falkinger, Konrad Podczeck and two anonymous referees for most valuable comments. References Althammer, W., Buchholz, W., 1993. Lindahl-equilibria as the outcome of a non-cooperative game. European Journal of Political Economy 9, 399–405. Andreoni, J., 1988. Privately provided public goods in a large economy: the limits of altruism. Journal of Public Economics 35, 57–73. Andreoni, J., Bergstrom, T., 1996. Do government subsidies increase the private supply of public goods. Public Choice 88, 295–338. Bernheim, D., 1986. On the voluntary and involuntary provision of public goods. American Economic Review 76, 789–793. Bergstrom, T., Blume, L.,Varian, H., 1986. On the private provision of public goods. Journal of Public Economics 29, 25–49. Boadway, R., Pestieau, P., Wildasin, D., 1989. Tax-transfer policies and the voluntary provision of public goods. Journal of Public Economics 39, 157–176. Brunner, J., Falkinger, J., 1995. Non-neutrality of Taxes and Subsidies for the Private Provision of Public Goods, Working Paper No. 9519. University of Linz. Danziger, L., Schnytzer, A., 1991. Implementing the Lindahl voluntary-exchange mechanism. European Journal of Political Economy 7, 55–64. Falkinger, J., 1996. Efficient private provision of public goods by rewarding deviations from average. Journal of Public Economics 62, 413–422. Green, J., Laffont, J.J., 1979. Incentives in Public Decision Making. North-Holland, Amsterdam. Roberts, R.D., 1987. Financing public goods. Journal of Political Economy 95, 420–437. Roberts, R.D., 1992. Government subsidies to private spending on public goods. Public Choice 74, 133–152. Varian, H., 1994. A solution to the problem of externalities when agents are well-informed. American Economic Review 84, 1278–1293. Warr, P., 1983. The private provision of a public good is independent of the distribution of income. Economics Letters 13, 207–211.
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