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Journal of Public Economics 71 (1999) 403–414 Strategic risk taking when there is a public good to be provided privately * Julio R. Robledo ¨ ¨ Institut f ur offentliche Finanzen und Sozialpolitik, Fachbereich Wirtschaftswissenschaft, Freie ¨ Universitat Berlin, Boltzmannstraße 20, D-14195 Berlin, Germany Received 31 August 1997; received in revised form 30 May 1998; accepted 2 June 1998 Abstract We describe a situation where a risk averse individual has a preference for risk taking. In the literature, we find this strategic risk behaviour in an altruistic framework, where the individual actually benefits from his noninsurance only in the loss outcome. In our model, all agents are perfectly selfish. When a public good is to be provided privately after the insurance decision, the player facing greater uncertainty can expect an income transfer from the other individuals through the commitments to the public good. This ex-ante income transfer is not conditional on the loss. 1999 Elsevier Science S.A. All rights reserved. Keywords: Risk taking; Strategic commitment; Private provision of public goods; Insurance demand JEL classification: H41; D81 1. Introduction A risk averse player buys full insurance if someone offers fair insurance. This is one of the most robust results in insurance economics and is almost tautological, given the definition of risk aversion. This paper shows that even in a perfect information context the full insurance result may not hold if the player interacts strategically with other players in the future. If the player knows he will belong to * Tel.: 100-49-30-838-3743; Fax: 100-49-30-838-3330; E-mail: [email protected] 0047-2727 / 99 / $ – see front matter 1999 Elsevier Science S.A. All rights reserved. PII: S0047-2727(98)00075-9

Strategic risk taking when there is a public good to be provided privately

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Journal of Public Economics 71 (1999) 403–414

Strategic risk taking when there is a public good to beprovided privately

*Julio R. Robledo¨ ¨Institut f ur offentliche Finanzen und Sozialpolitik, Fachbereich Wirtschaftswissenschaft, Freie

¨Universitat Berlin, Boltzmannstraße 20, D-14195 Berlin, Germany

Received 31 August 1997; received in revised form 30 May 1998; accepted 2 June 1998

Abstract

We describe a situation where a risk averse individual has a preference for risk taking. Inthe literature, we find this strategic risk behaviour in an altruistic framework, where theindividual actually benefits from his noninsurance only in the loss outcome. In our model,all agents are perfectly selfish. When a public good is to be provided privately after theinsurance decision, the player facing greater uncertainty can expect an income transfer fromthe other individuals through the commitments to the public good. This ex-ante incometransfer is not conditional on the loss. 1999 Elsevier Science S.A. All rights reserved.

Keywords: Risk taking; Strategic commitment; Private provision of public goods; Insurancedemand

JEL classification: H41; D81

1. Introduction

A risk averse player buys full insurance if someone offers fair insurance. This isone of the most robust results in insurance economics and is almost tautological,given the definition of risk aversion. This paper shows that even in a perfectinformation context the full insurance result may not hold if the player interactsstrategically with other players in the future. If the player knows he will belong to

*Tel.: 100-49-30-838-3743; Fax: 100-49-30-838-3330; E-mail: [email protected]

0047-2727/99/$ – see front matter 1999 Elsevier Science S.A. All rights reserved.PI I : S0047-2727( 98 )00075-9

404 J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414

a group of players who make voluntary contributions to a public good, it may be astrategic advantage for him to stay uninsured. He may elicit higher contributionsfrom others and may be expected to make lower contributions to the public good ifhis income prospects are uncertain. This advantage can overturn his risk takingdisadvantage. Consider the following situation where it can pay to face uncertain-ty. A group of n players contribute to financing the provision of a (continuous)public good. Intuitively, if everybody knows that player i is prudent and facesuncertainty, i will be expected to reduce his contribution ‘precautionarily’. Theother players will anticipate this behaviour and will increase their contributionsaccordingly. In the Nash equilibrium, the player with, say, uncertain incomeenjoys a strategic advantage.

In their seminal paper on the private provision of a public good, Bergstrom et al.(1986) show that the resulting equilibrium level of the public good is unique andin general lower than the optimal Samuelson level. No player takes account of thepositive effect of his contribution on the other players’ utility.

This paper considers uncertain income and derives the full comparative staticsfor symmetric and asymmetric situations. There are two players and the secondplayer’s income is more risky in a mean preserving spread sense than the incomeof the first player. The focus of this paper, however, is on the ‘dynamics’. Weconsider a two-stage game:

1. In the first stage, both players are offered insurance.2. In a second stage, the individuals play a game of private provision of a public

good in a Bergstrom et al. (1986) setting.3. All income random variables are realized and observed by both agents.

A risk averse individual may rationally refuse to buy fair insurance in this game.The insurance literature has identified some situations in which risk averse

individuals do not find it in their interest to buy full insurance under perfectinformation. But closer inspection shows that these other motives are quite distinctfrom the strategic motive considered here. Full coverage can be suboptimal if theinsurance market is incomplete in the sense that not all states of nature can becovered by an insurance contract. This case was considered by Doherty andSchlesinger (1983) for two simultaneous risks: an uninsurable risk and a risk forwhich fair insurance is offered. The individual may refuse to buy full insurance forthe insurable risk because the overall riskiness of his portfolio is lower if he buys adifferent coverage when the two risks are not stochastically independent. Similar-ly, Doherty and Schlesinger (1990) consider contract nonperformance (e.g.because of insurer insolvency). Depending on the stochastic relationship betweenthe different risks, there are situations where less than full insurance is bought atthe optimum even with a fair premium. Basically, what drives these results is thatpurchase of ‘full coverage’ for the insurable risk would not really convert thebuyer’s risky income distribution into a safe income with the same mean: other

J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414 405

uninsurable risks combined with the insurable risk jointly determine the riskinessof his ‘income portfolio’. In the case considered in this paper, even if the playercould convert his income portfolio into a safe income with the same mean, hewould prefer staying uninsured.

The papers most closely related to this paper are Sinn (1982) and Coate (1995).These papers argue that individuals may intentionally take risks if they know thatthey will be bailed out in the case of a serious loss. They can elicit a transfer fromothers if a serious loss occurs. As in the case considered in this paper, taking a riskinduces other individuals to do something that is beneficial for the player takingthe risk. There are however some important differences in the approach in thispaper. In the limited liability case the beneficial transfer occurs conditional on theloss. In the approach in this paper, the transfer occurs unconditionally. In addition,the transfers in the papers by Sinn and Coate are motivated by some altruisticmotives, by some exogenous institutional limited liability constraints, or by bailout clauses. In the approach in this paper, all players are perfectly selfish in anarrow sense and there are no institutionally set limited liability constraints.

Consider as an example several risk averse firms planning their lobbyingexpenditures. Each firm’s contribution is a private expenditure towards the publicgood ‘‘government lobbying’’. It seems fair to assume that the firms only careabout the total amount of lobbying funds and not about the ‘warm glow’ of thefirm’s own contribution. Think of a situation where firm i’s total profit isuncertain. Firm i reduces its public good commitment thereby inducing the otherfirms to increase their contributions to compensate for i’s behaviour. Thisamounts, in effect, to a transfer to firm i through the lobbying expenditures. If i’srisk premium is small enough, the positive effect of the subsidy might outweighthe cost of risk.

Similarly, when a firm reduces its output in a Cournot oligopoly game, the priceincrease benefits all oligopolists. The output reduction is the firm’s privatecontribution to the public good ‘‘quantity reduction and price increase’’ (from thefirms’ point of view). Under uncertainty, a firm may decrease its commitment, e.g.increase its production.

Uncertain income or uncertain initial endowment may also influence decisionsabout the private provision of environmental public goods. There is a growingconcern about the increase in pollution leading to global warming, the depletion ofthe ozone layer or the increasing deforestation which diminishes the earth’sbiodiversity. If a country unilaterally reduces its polluting emissions, othercountries will free-ride on this reduction and may even offset it by increasing theiremissions. The reduction in polluting emissions of pollutants like CO or CFCs is2

1an international public good which is privately provided by individual countries.

1See Hoel (1991) and Sandler (1992) for analysis of environmental problems in a public goodframework and Murdoch and Sandler (1997) for an empirical assessment of the theory.

406 J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414

Now, if country i’s economic prospects become riskier, it will make a smallerreduction in its emissions. This induces the other countries to reduce theiremissions further, since every country knows the economic risks faced by countryi. This amounts to an income transfer to country i and induces i to further reduceits commitment.

This paper proceeds as follows. Section 2 explains the model. To calculate thesubgame perfect equilibrium, the private provision game under uncertain incomehas to be solved. This is done briefly in Section 3 for the case where bothindividuals face the same amount of risk (‘‘symmetric’’ uncertainty). In Sections4, 5 and 6 the main result of the analysis, the welfare implications and a numericalexample are presented. Section 7 concludes.

2. The model

Consider a situation with two individuals. Each player has a continuousiincreasing strictly concave utility function U (G, x ), where G 5 g 1 g , g is i’si 1 2 i

contribution to the public good and x i’s private consumption (i 5 1, 2). As usuali

in the literature, we will assume that both goods are strictly normal. Under2certainty, this ensures uniqueness of the Nash equilibrium. Each player i is

]endowed with income M , where M is a random variable with domain [M, M],i i ]M . 0. The budget constraint for each individual is given by]

x 1 g 5 m , where g #Mis i is i ]

where the index i stands for the individual and s[h1, . . . ,Sj for the outcome of therandom variable M . We set both prices equal to 1. This normalization is standardi

in the literature and implies that the individuals make real commitments toward thepurchase of the public good. We also assume that both players choose theirexpenditures before knowing what the realization of the random income will turnout to be. Thus, it is real private consumption which is random. The players decidein the first place on their public good commitment.

This timing is crucial in our model. In the environmental goods case, Buchholzand Konrad (1995) argue that contributions to global environmental goods can beseen as a change of an investment path or of the technology. A further change ofthe technology (e.g. a change from nuclear power stations to gas powered stations)would imply high switching costs. Hence, the environmental decision is a longterm commitment. One can interpret the right hand side of the budget constraint as

2To be strict, under uncertainty the normality of the goods has to be suitably defined. If the mean ofthe distribution M increases and the reaction of the demand function of the public good dG /dM liesstrictly between 0 and 1, then we can apply the result of Bergstrom et al. (1986) concerning existenceand uniqueness of a Nash equilibrium.

J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414 407

a lifetime income, which is uncertain by nature. The decisions about the3contributions to the public good have to be taken in the present. What is more,

countries have to pledge their contribution to the public good in real termsregardless of realized future income, e.g. by committing to the closure of airpolluting coal fired power stations.

The first-order conditions describing the Nash equilibrium are given by

i i i iEU (G, M 2 g ) 2 EU (G, M 2 g ) 5 0, i 5 1, 2, (1)G x

where the expectation is taken with respect to the random income and the indexesi i idenote the first derivative with respect to the variable. Let F (G, M)5EU 2EUG x

idenote player i’s expected marginal utility with respect to his strategic variable g .Equations (1) collapse to a single equation if we consider the symmetric game

iwhere U 5U and income is an identically distributed random variable M withiiE(M )5m 5m for all i: All individuals are identical, so g 5g.i i

3. Symmetric income uncertainty

Before analyzing the game under asymmetric uncertainty let us summarize theeffect of increased symmetric uncertainty. We start from an equilibrium situation

cunder uncertainty where g is the optimal commitment for an income distributioncM . Consider now an exogenous increase in riskiness in the distribution in a mean

upreserving spread sense, such that g becomes the new equilibrium commitmentufor distribution M . This case was analyzed by Dardanoni (1988) for a two

argument utility function and by Gradstein et al. (1992) for the private provisionof a public good under price uncertainty. If both the private and the public good

i iare normal and (U 2U ).0, i51, 2, then the equilibrium level of the publicxxx Gxxugood under private provision decreases when income uncertainty increases: G ,

cG .i i(U 2U ).0 is a sufficient condition for the first order conditions (FOC) toxxx Gxx

be concave. Concavity of the FOC drives the result, and can be interpreted withthe help of the Arrow–Pratt measure of absolute risk aversion with respect to the

4random private good x,

U (G, m 2 g )xx iAbs ]]]]R ( g ) 5 2 . (2)x i U (G, m 2 g )x i

AbsIf the Arrow–Pratt measure R ( g ) is an increasing function of the commitmentx i

g , holding all other g constant (e.g. a decreasing function of x), then thei 2i

3This interpretation of the budget constraint in time terms was suggested by a referee.4We use the extension of the one-dimensional Arrow–Pratt measure to the 2-dimensional case

suggested by Sandmo (1969).

408 J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414

condition (U 2U ).0 is satisfied. For a separable utility function, thexxx Gxx

condition reduces to the prudence requirement (2U /U ).0. In the one-xxx xx

dimensional case, prudence is a necessary condition for a decreasing Arrow–Prattmeasure of absolute risk aversion.

In our model, the commitment to the public good plays the role of saving in aprecautionary saving framework. The players increase (expected) private con-sumption by reducing the public good commitments in order to avoid very lowlevels of the random good x. Prudence describes how the individual changes hisoptimal, risk-averse behaviour under uncertainty in a way analogous to the way asrisk aversion measures how much the individual dislikes uncertainty (see Kimball,1990). A prudent player is most likely not to be at an ex post optimum where hismarginal rate of substitution equals 1. On average, he will end with a higherprivate consumption than at the corresponding level of x (G) under certainty.i

4. Asymmetric income uncertainty

We start from an equilibrium situation under uncertainty and analyze the effectof a one-sided increase in risk. Assume that, for exogenous reasons, player 2 facesa mean preserving spread in risk. We will show that, under normality and prudenceassumptions, this behaviour may be rational if this player participates in the gameof private provision of the public good at a later stage. Although the equilibriumlevel of G decreases due to the increased uncertainty, this reduction is not shared

˜equally between the players. The commitment g decreases more than pro-25portionally. Effectively, this is an income transfer from player 1 to player 2.

Proposition 1. (Effect on G of asymmetric income uncertainty). Suppose both theprivate and the public good are normal, and both individuals contribute a positive

i iamount to the public good before and after the risk increase. If (U 2U ).0,xxx Gxx

i51, 2, then a one-sided increase in uncertainty in 1’s income leads to areduction of both 1’s commitment and the equilibrium level of G. Player 2’scommitment increases.

Proof. Denote the equilibrium level for the common income distribution M with G˜ ˜and consider the new situation where the distribution of 2’s income is M, where M

is more risky than M. Write the corresponding first-order conditions:

1 1˜ ˜˜ ˜EU (G, M 2 g ) 2 EU (G, M 2 g ) 5 0 FOC player 1, (3)G 1 x 1

2 2˜ ˜ ˜ ˜˜ ˜EU (G, M 2 g ) 2 EU (G, M 2 g ) 5 0 FOC player 2. (4)G 2 x 2

5We denote the levels in the asymmetric case with the asymmetric symbol |.

J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414 409

From the FOC from individual 1 we may calculate

1 1˜dg E(U 2 U )1 GG Gx] ]]]]]]]5 2 , 0,1 1 1˜dg E(U 2 2U 1 U )2 GG Gx xx

1 1 1 1 1where E(U 22U 1U ),0 by the second-order condition and (U 2U ),GG Gx xx GG Gx

0 by normality. Thus, we can subsume both Eqs. (3) and (4) under

2 2 2˜ ˜ ˜ ˜˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜F (G, M ) 5 EU (g (g ) 1 g , M 2 g ) 2 EU (g (g ) 1 g , M 2 g )G 1 2 2 2 x 1 2 2 2

]M

2 2˜ ˜˜ ˜ ˜ ˜ ˜ ˜ ˜5E (U (g (g ) 1 g , M 2 g ) 2 U (g (g ) 1 g , MG 1 2 2 2 x 1 2 2M]

˜˜2 g )) dF(M, u ) 5 0, (5)2

and

] 1where F(M, u ) is the distribution of M defined on the support [M, M] and u [R]

is a riskiness parameter. The distribution function F is twice continuouslydifferentiable with (≠F /≠M)5f(M, u ) and (≠F /≠u )5F . An increase in u is calledu

a mean preserving spread iff (Diamond and Stiglitz, 1974)

]ME F (M, u ) dM 5 0 and (6)u

M]

M0 ]E F (M, u ) dM $ 0 ;M [ [M, M]. (7)u 0 ]M]

˜To calculate the effect of an increase in risk on the optimal commitment (dg /du )2

we implicitly differentiate Eq. (5) following Dardanoni (1988):

2≠F]]˜dg2 ≠u

] ]]5 2 .2du ≠F]]

˜≠g2

2 2˜ ˜Since (≠F /≠g ),0, the sign of (dg /du ) is equal to the sign of (≠F /≠u ). We2 2

obtain

410 J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414

(8)

Thus, the commitment of the player 2, whose income uncertainty has increased,˜ ˜ ˜falls (g ,g ). By (dg /dg ),0, the other player’s commitment must increase:2 2 1 2

g̃ .g . To show that the sum of commitments is actually lower when one agent1 1

faces more uncertainty, consider the first-order conditions of individual 1 undersymmetric and asymmetric uncertainty:

, (9)

(10)

R R R˜We know that the expected private consumption of player 1 falls: Ex ,Ex 5Ex .1 1˜Suppose G,G. Then, due to normality, given Eq. (9), the left hand would fall and

˜the right hand would increase. This is a contradiction to Eq. (10), so G ,G.Q.E.D.

i iCorollary 1. Both goods are normal and (U 2U ).0, i51, 2. Starting fromxxx GxxR R Ra noncorner equilibrium situation with given G and Ex 5Ex 5Ex , an increase1 2

in uncertainty for player 2 leads to the following results:

R R R˜ ˜ ˜ ˜ ˜G , G, g , g , g , and Ex , Ex , Ex . (11)2 1 1 2

J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414 411

5. Welfare effects

Under certainty, the private provision of a public good leads to a suboptimalequilibrium level: some joint increase in commitments benefits both players. Theintroduction of uncertainty leads to a different Samuelson solution.

If the agents are prudent towards the consumption of the random private good(in the more general case, if (U 2U ).0), they will decrease their contribu-xxx Gxx

tions under uncertainty and widen the gap between the resulting equilibrium leveland the Samuelson level under certainty. In this case, welfare must decrease, too(Gradstein et al., 1992). Both effects (increased uncertainty and decreasedprovision of G) work in the same direction. If uncertainty increases thecontributions, then welfare (under uncertainty and private provision) may increaseor decrease when compared to the certainty benchmark.

When uncertainty is asymmetric and affects only player 2, it constitutes astrategic advantage and therefore the other player is worse off. The welfare resultsin the 2-player-game depend on the following effects:

˜1. The amount of public good provision decreases, G ,G.2. Individual 2 faces more uncertainty regarding his private consumption.

˜3. The anticipated decrease in the public good commitment g of player 22

increases player 1’s commitment. This amounts to an income transfer from 1 to2: the expected private consumption of player 2 increases.

Player 1 is definitely worse off. The first and the third effects both reduce hisutility, while the second effect does not affect him directly. He consumes less of Gand less of (expected) x. For player 2, the situation is ambiguous. The first andsecond effects reduce 2’s utility, while the income transfer increases it. Intuitively,if an individual is not very risk averse, the resulting income transfer mayovercompensate his risk premium. To analyze formally the positive and thenegative effect for player 2, it is useful to break up the difference in utility in twoterms as Gradstein et al. (1992) do in their context:

22 2 2 2˜ ˜ ˜ ˜DU 5 EU 2 EU 5 EU (G, M 2 g) 2 EU (G, M 2 g ) (12)2

2 2 ˜ ˜5 [EU (G, M 2 g) 2 EU (G, M 2 g )]2

2 2˜ ˜ ˜˜ ˜1 [EU (G, M 2 g ) 2 EU (G,M 2 g )]. (13)2 2

The first term reflects the expected utility difference between the levels (G, x ) and2˜ ˜(G, x ), while the second term measures the difference in utility due to the2

increase in uncertainty. The second term is positive since the individual is riskaverse and strictly prefers less income uncertainty. The first term is ambiguous. If

˜the increase in private consumption x .x 5x outweighs the decrease in G, the2 2

412 J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414

first term might be negative. An individual with a small enough risk premium andbig enough income transfer might prefer the situation where he faces moreuncertainty.

Corollary 2. (Risk Loving Behaviour). Suppose both the private and the publicgood are normal and (U 2U ).0. If the effect of increased private consump-xxx Gxx

tion outweighs the effects of the reduction in G and of the increased risk premium,then the player will not buy any fair insurance for the additional risk or,alternatively, will take a risk in order to elicit the income transfer from the otherplayer.

In the light of these results, compulsory insurance constitutes a Paretoimprovement. It has been shown that uncertainty is a strategic advantage. Noplayer will surrender this advantage voluntarily. But if both agents must buy fullinsurance and are prudent, the public good level will increase and the agents willbe at the Nash equilibrium under certainty. This welfare improvement may beunattainable in the international public goods case, because there is no suprana-tional authority which can enforce a compulsory insurance scheme upon sovereigncountries and maybe even no insurer willing to insure the risk.

6. A numerical example: strategic demand for risk

Consider the following numerical example to better understand the intuition thatdrives the result. We will present an extreme case where the original situation ischaracterized by income certainty for both players and player 2 faces an

1exogenous increase in risk. Let the players have the utility functions U (G,2 64

]x )5x 1 ln(G) and U (G, x )5ln(x )1ln(G) and fixed incomes m 5m 52.1 1 2 2 1 23

Under certainty and assuming an interior solution, the first-order condition forplayer 1 uniquely determines the overall level of provision of the public good:

4 4] ]G*5 . This implies g 5 2g and x 522g . For player 2 we obtain U 5(1 /1 2 1 1 x3 3

x )5(1 /G)5U and, hence, (1 /(22g ))5(1 /( g 1g )). The unique equilibrium2 G 2 1 21 22 4 4 4 4 4

] ] ] ] ] ]is given by g 5g 5 , x 5x 5 , U 5 1 ln( ), and U 5 2 ln( ).1 2 1 23 3 3 3 3 3

Now let the income of player 2 be a discrete random variable which is equal to

6 1For utility U , private consumption is normal but not strictly normal. However, uniqueness of the1equilibrium is ensured here by the strict normality of both goods for the player 2. Also, U is the

1 4 4] ]limiting case U 5lim (x 1a ln x 1 ln(G)), with (x 1a ln x 1 ln(G)) fulfilling strict normalitya →0 1 1 1 13 3

for both the private and the public good, for all a .0. The quantitative result of the example carriesover to sufficiently small positive a.

J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414 413

]m 51.5 or m 52.5 with probability 0.5. Player 1’s optimal decision is un-2 2]changed. The risk averse individual 2 now has the first-order condition

1 1 1] ] ]EU 5 1 5 5 U ,]x G2x 2x ˜2 2 G]

] ] ˜ ˜ ˜where x 5m 2g and x 5m 2g denote his private consumption x when the2 2 2 2 2 2 2] ]high or the low value of the random variable is realized. Straightforward

5 1 7 3˜ ˜ ˜ ˜] ] ] ]calculations lead to g 5 , g 5 , x 5 and E(x )5 . The player with uncertain1 2 1 26 2 6 2

income free-rides on the public good contribution of player one. Since the firstindividual always fully compensates every reduction in 2’s commitment to thepublic good, the public good level is unchanged. As expected, the player facingincreased uncertainty increases his utility at the expense of the risk free individual:

2 121 4 7˜ ˜] ] ]EU 5 ln(2) 1 ln( ) . U . Moreover, player 1’s expected utility is U 5 12 3 614 4

] ]ln( ) , U .3 3

7. Conclusions

We have identified a case where a risk averse agent rationally prefers a riskiersituation. In such a setting, where the player’s risk aversion is small and herprudence big enough, she refuses to buy any insurance. Actually, the player has arational incentive to engage in some risky activity. Uncertainty yields a strategicadvantage. Participation in a voluntary contributions game has adverse commit-ment incentives. Konrad (1994) has shown that participants have an incentive todistort their intertemporal consumption decision. This paper shows that participa-tion in a voluntary contributions game also distorts risk-rating incentives.

When a public good is provided privately, the agent facing higher uncertaintyelicits an income transfer from the other players, who need not have altruisticpreferences. This contrasts with most literature where individuals rely on altruisticindividuals to bail them out when they suffer a loss. In our present case, it is in theother players’ interest to make the income transfer because this increases theprovision level of the public good.

This strategic advantage of risk when facing altruistic individuals providesadditional justification for some compulsory insurance schemes observed in manycountries. Insurance eliminates one distortion and leads to the still suboptimalNash equilibrium of private provision of a public good.

If the agents are sovereign countries, welfare improving compulsory insurancemay not be an option. A possible solution may involve the countries with morestable incomes assuming the role of insurers. This would still mean a kind ofincome transfer, since these countries would, in fact, be assuming the riskpremium of countries facing higher risks.

414 J.R. Robledo / Journal of Public Economics 71 (1999) 403 –414

Acknowledgements

I thank Anette Boom, Kai A. Konrad, Roland Strausz and two anonymousreferees for helpful comments. The usual caveat applies.

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