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Strategic risk taking when there is a public good to be provided privately
Journal of Public Economics 71 (1999) 403–414
Strategic risk taking when there is a public good to be provided privately Julio R. Robledo* ¨ offentliche ¨ Institut f ur Finanzen und Sozialpolitik, Fachbereich Wirtschaftswissenschaft, Freie ¨ Berlin, Boltzmannstraße 20, D-14195 Berlin, Germany Universitat 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
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a group of players who make voluntary contributions to a public good, it may be a strategic advantage for him to stay uninsured. He may elicit higher contributions from others and may be expected to make lower contributions to the public good if his income prospects are uncertain. This advantage can overturn his risk taking disadvantage. Consider the following situation where it can pay to face uncertainty. 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 faces uncertainty, i will be expected to reduce his contribution ‘precautionarily’. The other players will anticipate this behaviour and will increase their contributions accordingly. In the Nash equilibrium, the player with, say, uncertain income enjoys 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 and in general lower than the optimal Samuelson level. No player takes account of the positive effect of his contribution on the other players’ utility. This paper considers uncertain income and derives the full comparative statics for symmetric and asymmetric situations. There are two players and the second player’s income is more risky in a mean preserving spread sense than the income of the first player. The focus of this paper, however, is on the ‘dynamics’. We consider 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 perfect information. But closer inspection shows that these other motives are quite distinct from the strategic motive considered here. Full coverage can be suboptimal if the insurance market is incomplete in the sense that not all states of nature can be covered by an insurance contract. This case was considered by Doherty and Schlesinger (1983) for two simultaneous risks: an uninsurable risk and a risk for which fair insurance is offered. The individual may refuse to buy full insurance for the insurable risk because the overall riskiness of his portfolio is lower if he buys a different coverage when the two risks are not stochastically independent. Similarly, Doherty and Schlesinger (1990) consider contract nonperformance (e.g. because of insurer insolvency). Depending on the stochastic relationship between the different risks, there are situations where less than full insurance is bought at the optimum even with a fair premium. Basically, what drives these results is that purchase of ‘full coverage’ for the insurable risk would not really convert the buyer’s risky income distribution into a safe income with the same mean: other
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uninsurable risks combined with the insurable risk jointly determine the riskiness of his ‘income portfolio’. In the case considered in this paper, even if the player could convert his income portfolio into a safe income with the same mean, he would 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 that they will be bailed out in the case of a serious loss. They can elicit a transfer from others if a serious loss occurs. As in the case considered in this paper, taking a risk induces other individuals to do something that is beneficial for the player taking the risk. There are however some important differences in the approach in this paper. In the limited liability case the beneficial transfer occurs conditional on the loss. 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 altruistic motives, by some exogenous institutional limited liability constraints, or by bail out clauses. In the approach in this paper, all players are perfectly selfish in a narrow sense and there are no institutionally set limited liability constraints. Consider as an example several risk averse firms planning their lobbying expenditures. Each firm’s contribution is a private expenditure towards the public good ‘‘government lobbying’’. It seems fair to assume that the firms only care about the total amount of lobbying funds and not about the ‘warm glow’ of the firm’s own contribution. Think of a situation where firm i’s total profit is uncertain. Firm i reduces its public good commitment thereby inducing the other firms to increase their contributions to compensate for i’s behaviour. This amounts, in effect, to a transfer to firm i through the lobbying expenditures. If i’s risk premium is small enough, the positive effect of the subsidy might outweigh the cost of risk. Similarly, when a firm reduces its output in a Cournot oligopoly game, the price increase benefits all oligopolists. The output reduction is the firm’s private contribution to the public good ‘‘quantity reduction and price increase’’ (from the firms’ 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 decisions about the private provision of environmental public goods. There is a growing concern about the increase in pollution leading to global warming, the depletion of the ozone layer or the increasing deforestation which diminishes the earth’s biodiversity. If a country unilaterally reduces its polluting emissions, other countries will free-ride on this reduction and may even offset it by increasing their emissions. The reduction in polluting emissions of pollutants like CO 2 or CFCs is an international public good which is privately provided by individual countries.1
1
See Hoel (1991) and Sandler (1992) for analysis of environmental problems in a public good framework and Murdoch and Sandler (1997) for an empirical assessment of the theory.
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Now, if country i’s economic prospects become riskier, it will make a smaller reduction in its emissions. This induces the other countries to reduce their emissions further, since every country knows the economic risks faced by country i. This amounts to an income transfer to country i and induces i to further reduce its commitment. This paper proceeds as follows. Section 2 explains the model. To calculate the subgame perfect equilibrium, the private provision game under uncertain income has to be solved. This is done briefly in Section 3 for the case where both individuals face the same amount of risk (‘‘symmetric’’ uncertainty). In Sections 4, 5 and 6 the main result of the analysis, the welfare implications and a numerical example are presented. Section 7 concludes.
2. The model Consider a situation with two individuals. Each player has a continuous increasing strictly concave utility function U i (G, x i ), where G 5 g1 1 g2 , gi is i’s contribution to the public good and x i i’s private consumption (i 5 1, 2). As usual in the literature, we will assume that both goods are strictly normal. Under certainty, this ensures uniqueness of the Nash equilibrium.2 Each player i is ] endowed with income Mi , where Mi is a random variable with domain [M, ] M], M ] . 0. The budget constraint for each individual is given by x is 1 gi 5 m is , where gi #M ] where the index i stands for the individual and s[h1, . . . ,Sj for the outcome of the random variable Mi . We set both prices equal to 1. This normalization is standard in the literature and implies that the individuals make real commitments toward the purchase of the public good. We also assume that both players choose their expenditures before knowing what the realization of the random income will turn out to be. Thus, it is real private consumption which is random. The players decide in the first place on their public good commitment. This timing is crucial in our model. In the environmental goods case, Buchholz and Konrad (1995) argue that contributions to global environmental goods can be seen as a change of an investment path or of the technology. A further change of the technology (e.g. a change from nuclear power stations to gas powered stations) would imply high switching costs. Hence, the environmental decision is a long term commitment. One can interpret the right hand side of the budget constraint as 2
To be strict, under uncertainty the normality of the goods has to be suitably defined. If the mean of the distribution M increases and the reaction of the demand function of the public good dG / dM lies strictly between 0 and 1, then we can apply the result of Bergstrom et al. (1986) concerning existence and uniqueness of a Nash equilibrium.
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a lifetime income, which is uncertain by nature. The decisions about the contributions to the public good have to be taken in the present.3 What is more, countries have to pledge their contribution to the public good in real terms regardless of realized future income, e.g. by committing to the closure of air polluting coal fired power stations. The first-order conditions describing the Nash equilibrium are given by EU iG (G, M 2 g i ) 2 EU xi (G, M 2 g i ) 5 0, i 5 1, 2,
(1)
where the expectation is taken with respect to the random income and the indexes denote the first derivative with respect to the variable. Let F i (G, M)5EU Gi 2EU xi i denote 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 where U i 5U and income is an identically distributed random variable Mi with E(Mi )5m i 5m for all i: All individuals are identical, so g i 5g.
3. Symmetric income uncertainty Before analyzing the game under asymmetric uncertainty let us summarize the effect of increased symmetric uncertainty. We start from an equilibrium situation under uncertainty where g c is the optimal commitment for an income distribution M c . Consider now an exogenous increase in riskiness in the distribution in a mean preserving spread sense, such that g u becomes the new equilibrium commitment for distribution M u . This case was analyzed by Dardanoni (1988) for a two argument utility function and by Gradstein et al. (1992) for the private provision of a public good under price uncertainty. If both the private and the public good are normal and (U ixxx 2U iGxx ).0, i51, 2, then the equilibrium level of the public good under private provision decreases when income uncertainty increases: G u , G c. (U ixxx 2U iGxx ).0 is a sufficient condition for the first order conditions (FOC) to be concave. Concavity of the FOC drives the result, and can be interpreted with the help of the Arrow–Pratt measure of absolute risk aversion with respect to the random private good x,4 Uxx (G, m 2 gi ) Abs R x ( gi ) 5 2 ]]]]. Ux (G, m 2 gi )
(2)
If the Arrow–Pratt measure R Abs x ( gi ) is an increasing function of the commitment gi , holding all other g2i constant (e.g. a decreasing function of x), then the 3
This interpretation of the budget constraint in time terms was suggested by a referee. We use the extension of the one-dimensional Arrow–Pratt measure to the 2-dimensional case suggested by Sandmo (1969). 4
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condition (Uxxx 2UGxx ).0 is satisfied. For a separable utility function, the condition reduces to the prudence requirement (2Uxxx /Uxx ).0. In the onedimensional case, prudence is a necessary condition for a decreasing Arrow–Pratt measure of absolute risk aversion. In our model, the commitment to the public good plays the role of saving in a precautionary saving framework. The players increase (expected) private consumption by reducing the public good commitments in order to avoid very low levels of the random good x. Prudence describes how the individual changes his optimal, risk-averse behaviour under uncertainty in a way analogous to the way as risk 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 his marginal rate of substitution equals 1. On average, he will end with a higher private consumption than at the corresponding level of x i (G) under certainty.
4. Asymmetric income uncertainty We start from an equilibrium situation under uncertainty and analyze the effect of a one-sided increase in risk. Assume that, for exogenous reasons, player 2 faces a mean preserving spread in risk. We will show that, under normality and prudence assumptions, this behaviour may be rational if this player participates in the game of private provision of the public good at a later stage. Although the equilibrium level of G decreases due to the increased uncertainty, this reduction is not shared equally between the players. The commitment g˜ 2 decreases more than proportionally. Effectively, this is an income transfer from player 1 to player 2.5 Proposition 1. (Effect on G of asymmetric income uncertainty). Suppose both the private and the public good are normal, and both individuals contribute a positive i amount to the public good before and after the risk increase. If (U ixxx 2U Gxx ).0, i51, 2, then a one-sided increase in uncertainty in 1’ s income leads to a reduction of both 1’ s commitment and the equilibrium level of G. Player 2’ s commitment increases. Proof. Denote the equilibrium level for the common income distribution M with G ˜ where M ˜ and consider the new situation where the distribution of 2’s income is M, is more risky than M. Write the corresponding first-order conditions:
5
˜ M 2 g˜ ) 2 EU 1 (G, ˜ M 2 g˜ ) 5 0 FOC player 1, EU 1G (G, 1 x 1
(3)
˜ M ˜ 2 g˜ ) 2 EU 2 (G, ˜ M ˜ 2 g˜ ) 5 0 FOC player 2. EU 2G (G, 2 x 2
(4)
We denote the levels in the asymmetric case with the asymmetric symbol |.
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From the FOC from individual 1 we may calculate 1 dg˜ 1 E(U GG 2 U 1Gx ) ] 5 2 ]]]]]]] , 0, 1 dg˜ 2 E(U GG 2 2U 1Gx 1 U 1xx ) 1
1
1
1
1
where E(U GG 22U Gx 1U xx ),0 by the second-order condition and (U GG 2U Gx ), 0 by normality. Thus, we can subsume both Eqs. (3) and (4) under ˜ M ˜ ) 5 EU 2 (g˜ (g˜ ) 1 g˜ , M ˜ 2 g˜ ) 2 EU 2 (g˜ (g˜ ) 1 g˜ , M ˜ 2 g˜ ) F 2 (G, G 1 2 2 2 x 1 2 2 2 5
E
] M
M ]
˜ 2 g˜ ) 2 U (g˜ (g˜ ) 1 g˜ , M ˜ (U G (g˜ 1 (g˜ 2 ) 1 g˜ 2 , M 2 x 1 2 2 2
˜ u ) 5 0, 2 g˜ 2 )) dF(M,
2
(5)
and
] 1 where 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 continuously differentiable with (≠F / ≠M)5f(M, u ) and (≠F / ≠u )5Fu . An increase in u is called a mean preserving spread iff (Diamond and Stiglitz, 1974)
E
] M
E
M0
M ]
M ]
Fu (M, u ) dM 5 0 and ] Fu (M, u ) dM $ 0 ;M0 [ [M, ] M].
(6)
(7)
To calculate the effect of an increase in risk on the optimal commitment (dg˜ 2 / du ) we implicitly differentiate Eq. (5) following Dardanoni (1988): ≠F 2 ]] dg˜ 2 ≠u ] 5 2 ]] . du ≠F 2 ]] ≠g˜ 2 Since (≠F 2 / ≠g˜ 2 ),0, the sign of (dg˜ 2 / du ) is equal to the sign of (≠F 2 / ≠u ). We obtain
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(8) Thus, the commitment of the player 2, whose income uncertainty has increased, falls (g˜ 2 ,g2 ). By (dg˜ 1 / dg˜ 2 ),0, the other player’s commitment must increase: g˜ 1 .g1 . To show that the sum of commitments is actually lower when one agent faces more uncertainty, consider the first-order conditions of individual 1 under symmetric and asymmetric uncertainty:
,
(9)
(10) We know that the expected private consumption of player 1 falls: Ex˜ R1 ,Ex R1 5Ex R . ˜ Then, due to normality, given Eq. (9), the left hand would fall and Suppose G,G. the right hand would increase. This is a contradiction to Eq. (10), so G˜ ,G. Q.E.D. Corollary 1. Both goods are normal and (U ixxx 2U iGxx ).0, i51, 2. Starting from a noncorner equilibrium situation with given G and Ex 1R 5Ex 2R 5Ex R , an increase in uncertainty for player 2 leads to the following results: G˜ , G,
g˜ 2 , g , g˜ 1 , and Ex˜ R1 , Ex R , Ex˜ 2R .
(11)
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5. Welfare effects Under certainty, the private provision of a public good leads to a suboptimal equilibrium level: some joint increase in commitments benefits both players. The introduction 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 (Uxxx 2UGxx ).0), they will decrease their contributions under uncertainty and widen the gap between the resulting equilibrium level and the Samuelson level under certainty. In this case, welfare must decrease, too (Gradstein et al., 1992). Both effects (increased uncertainty and decreased provision of G) work in the same direction. If uncertainty increases the contributions, then welfare (under uncertainty and private provision) may increase or decrease when compared to the certainty benchmark. When uncertainty is asymmetric and affects only player 2, it constitutes a strategic advantage and therefore the other player is worse off. The welfare results in 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˜ 2 of player 2 increases player 1’s commitment. This amounts to an income transfer from 1 to 2: the expected private consumption of player 2 increases. Player 1 is definitely worse off. The first and the third effects both reduce his utility, while the second effect does not affect him directly. He consumes less of G and less of (expected) x. For player 2, the situation is ambiguous. The first and second effects reduce 2’s utility, while the income transfer increases it. Intuitively, if an individual is not very risk averse, the resulting income transfer may overcompensate his risk premium. To analyze formally the positive and the negative effect for player 2, it is useful to break up the difference in utility in two terms as Gradstein et al. (1992) do in their context: 2
˜ M ˜ 2 g˜ ) DU 2 5 EU 2 2 EU˜ 5 EU 2 (G, M 2 g) 2 EU 2 (G, 2
(12)
˜ M 2 g˜ )] 5 [EU 2 (G, M 2 g) 2 EU 2 (G, 2 2 ˜ 2 ˜ ˜ 1 [EU (G, M 2 g˜ 2 ) 2 EU (G,M 2 g˜ 2 )].
(13)
The first term reflects the expected utility difference between the levels (G, x 2 ) and ˜ x˜ ), while the second term measures the difference in utility due to the (G, 2 increase in uncertainty. The second term is positive since the individual is risk averse and strictly prefers less income uncertainty. The first term is ambiguous. If the increase in private consumption x˜ 2 .x 2 5x outweighs the decrease in G, the
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first term might be negative. An individual with a small enough risk premium and big enough income transfer might prefer the situation where he faces more uncertainty. Corollary 2. (Risk Loving Behaviour). Suppose both the private and the public good are normal and (Uxxx 2UGxx ).0. If the effect of increased private consumption 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 other player. In the light of these results, compulsory insurance constitutes a Pareto improvement. It has been shown that uncertainty is a strategic advantage. No player will surrender this advantage voluntarily. But if both agents must buy full insurance and are prudent, the public good level will increase and the agents will be at the Nash equilibrium under certainty. This welfare improvement may be unattainable in the international public goods case, because there is no supranational authority which can enforce a compulsory insurance scheme upon sovereign countries 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 that drives the result. We will present an extreme case where the original situation is characterized by income certainty for both players and player 2 faces an exogenous increase in risk. Let the players have the utility functions U 1 (G, x 1 )5x 1 1 ]34 ln(G) and U 2 (G, x 2 )5ln(x 2 )1ln(G) and fixed incomes m 1 5m 2 52.6 Under certainty and assuming an interior solution, the first-order condition for player 1 uniquely determines the overall level of provision of the public good: G*5 ]43 . This implies g1 5 ]43 2g2 and x 1 522g1 . For player 2 we obtain Ux 5(1 / x 2 )5(1 /G)5UG and, hence, (1 /(22g2 ))5(1 /( g1 1g2 )). The unique equilibrium 1 2 is given by g1 5g2 5 ]32 , x 1 5x 2 5 ]34 , U 5 ]34 1 ]34 ln( ]34 ), and U 5 2 ln( ]34 ). Now let the income of player 2 be a discrete random variable which is equal to 6 For utility U 1 , private consumption is normal but not strictly normal. However, uniqueness of the equilibrium is ensured here by the strict normality of both goods for the player 2. Also, U 1 is the limiting case U 1 5lima →0 (x 1 1 a ln x 1 1 ]43 ln(G)), with (x 1 1 a ln x 1 1 ]43 ln(G)) fulfilling strict normality for both the private and the public good, for all a .0. The quantitative result of the example carries over to sufficiently small positive a.
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] m ] 2 51.5 or m 2 52.5 with probability 0.5. Player 1’s optimal decision is unchanged. The risk averse individual 2 now has the first-order condition 1 1 1 EUx 5 ] 1 ] 5 ] 5 UG , 2x] 2 2x] 2 G˜
] 2g˜ and x 5m 2g˜ denote his private consumption x˜ when the where ]x 2 5m 2 2 2 2 ]2 ]2 high or the low value of the random variable is realized. Straightforward calculations lead to g˜ 1 5 ]65 , g˜ 2 5 ]21 , x˜ 1 5 ]67 and E(x˜ 2 )5 ]23 . The player with uncertain income free-rides on the public good contribution of player one. Since the first individual always fully compensates every reduction in 2’s commitment to the public good, the public good level is unchanged. As expected, the player facing increased uncertainty increases his utility at the expense of the risk free individual: 2 1 EU˜ 5 ]12 ln(2) 1 ln( ]43 ) . U 2 . Moreover, player 1’s expected utility is U˜ 5 ]76 1 1 4 4 ] ln( ] ) , U . 3 3
7. Conclusions We have identified a case where a risk averse agent rationally prefers a riskier situation. In such a setting, where the player’s risk aversion is small and her prudence big enough, she refuses to buy any insurance. Actually, the player has a rational incentive to engage in some risky activity. Uncertainty yields a strategic advantage. Participation in a voluntary contributions game has adverse commitment incentives. Konrad (1994) has shown that participants have an incentive to distort their intertemporal consumption decision. This paper shows that participation in a voluntary contributions game also distorts risk-rating incentives. When a public good is provided privately, the agent facing higher uncertainty elicits an income transfer from the other players, who need not have altruistic preferences. This contrasts with most literature where individuals rely on altruistic individuals to bail them out when they suffer a loss. In our present case, it is in the other players’ interest to make the income transfer because this increases the provision level of the public good. This strategic advantage of risk when facing altruistic individuals provides additional justification for some compulsory insurance schemes observed in many countries. Insurance eliminates one distortion and leads to the still suboptimal Nash equilibrium of private provision of a public good. If the agents are sovereign countries, welfare improving compulsory insurance may not be an option. A possible solution may involve the countries with more stable incomes assuming the role of insurers. This would still mean a kind of income transfer, since these countries would, in fact, be assuming the risk premium of countries facing higher risks.
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Acknowledgements I thank Anette Boom, Kai A. Konrad, Roland Strausz and two anonymous referees for helpful comments. The usual caveat applies.
References Bergstrom, T., Blume, L., Varian, H., 1986. On the private provision of public goods. Journal of Public Economics 29, 25–49. Buchholz, W., Konrad, K.A., 1995. Strategic transfers and private provision of public goods. Journal of Public Economics 57, 489–505. Coate, S., 1995. Altruism, the Samaritan’s dilemma, and government transfer policy. American Economic Review 85, 46–57. Dardanoni, V., 1988. Optimal choices under uncertainty: The case of two-argument utility functions. Economic Journal 98, 429–450. Diamond, P., Stiglitz, J.E., 1974. Increases in risk and risk aversion. Journal of Economic Theory 8, 337–360. Doherty, N., Schlesinger, H., 1983. Optimal insurance in incomplete markets. Journal of Political Economy 91, 1045–1054. Doherty, N., Schlesinger, H., 1990. Rational insurance purchasing: consideration of contract nonperformance. The Quarterly Journal of Economics 105, 243–253. Gradstein, M., Nitzan, S., Slutsky, S., 1992. The effect of uncertainty on interactive economic behaviour. Economic Journal 102, 554–561. Hoel, M., 1991. Global environmental problems: the effects of unilateral actions taken by one country. Journal of Environmental Economics and Management 20, 55–70. Kimball, M.S., 1990. Precautionary saving in the small and in the large. Econometrica 58, 53–73. Konrad, K.A., 1994. The strategic advantage of being poor: private and public provision of public goods. Economica 61, 79–92. Murdoch, J.C., Sandler, T., 1997. The voluntary provision of a pure public good: the case of reduced CFC emissions and the Montreal protocol. Journal of Public Economics 63, 331–349. Sandler, T., 1992. Collective Action: Theory and Applications. University of Michigan Press, Ann Arbor. Sandmo, A., 1969. Capital risk, consumption and portfolio choice. Econometrica 37, 586–599. Sinn, H.-W., 1982. Kinked utility and the demand for human wealth and liability insurance. European Economic Review 17, 149–162.
Strategic risk taking when there is a public good to be provided privately Julio R. Robledo* ¨ offentliche ¨ Institut f ur Finanzen und Sozialpolitik, Fachbereich Wirtschaftswissenschaft, Freie ¨ Berlin, Boltzmannstraße 20, D-14195 Berlin, Germany Universitat 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
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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 a strategic advantage for him to stay uninsured. He may elicit higher contributions from others and may be expected to make lower contributions to the public good if his income prospects are uncertain. This advantage can overturn his risk taking disadvantage. Consider the following situation where it can pay to face uncertainty. 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 faces uncertainty, i will be expected to reduce his contribution ‘precautionarily’. The other players will anticipate this behaviour and will increase their contributions accordingly. In the Nash equilibrium, the player with, say, uncertain income enjoys 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 and in general lower than the optimal Samuelson level. No player takes account of the positive effect of his contribution on the other players’ utility. This paper considers uncertain income and derives the full comparative statics for symmetric and asymmetric situations. There are two players and the second player’s income is more risky in a mean preserving spread sense than the income of the first player. The focus of this paper, however, is on the ‘dynamics’. We consider 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 perfect information. But closer inspection shows that these other motives are quite distinct from the strategic motive considered here. Full coverage can be suboptimal if the insurance market is incomplete in the sense that not all states of nature can be covered by an insurance contract. This case was considered by Doherty and Schlesinger (1983) for two simultaneous risks: an uninsurable risk and a risk for which fair insurance is offered. The individual may refuse to buy full insurance for the insurable risk because the overall riskiness of his portfolio is lower if he buys a different coverage when the two risks are not stochastically independent. Similarly, Doherty and Schlesinger (1990) consider contract nonperformance (e.g. because of insurer insolvency). Depending on the stochastic relationship between the different risks, there are situations where less than full insurance is bought at the optimum even with a fair premium. Basically, what drives these results is that purchase of ‘full coverage’ for the insurable risk would not really convert the buyer’s risky income distribution into a safe income with the same mean: other
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uninsurable risks combined with the insurable risk jointly determine the riskiness of his ‘income portfolio’. In the case considered in this paper, even if the player could convert his income portfolio into a safe income with the same mean, he would 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 that they will be bailed out in the case of a serious loss. They can elicit a transfer from others if a serious loss occurs. As in the case considered in this paper, taking a risk induces other individuals to do something that is beneficial for the player taking the risk. There are however some important differences in the approach in this paper. In the limited liability case the beneficial transfer occurs conditional on the loss. 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 altruistic motives, by some exogenous institutional limited liability constraints, or by bail out clauses. In the approach in this paper, all players are perfectly selfish in a narrow sense and there are no institutionally set limited liability constraints. Consider as an example several risk averse firms planning their lobbying expenditures. Each firm’s contribution is a private expenditure towards the public good ‘‘government lobbying’’. It seems fair to assume that the firms only care about the total amount of lobbying funds and not about the ‘warm glow’ of the firm’s own contribution. Think of a situation where firm i’s total profit is uncertain. Firm i reduces its public good commitment thereby inducing the other firms to increase their contributions to compensate for i’s behaviour. This amounts, in effect, to a transfer to firm i through the lobbying expenditures. If i’s risk premium is small enough, the positive effect of the subsidy might outweigh the cost of risk. Similarly, when a firm reduces its output in a Cournot oligopoly game, the price increase benefits all oligopolists. The output reduction is the firm’s private contribution to the public good ‘‘quantity reduction and price increase’’ (from the firms’ 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 decisions about the private provision of environmental public goods. There is a growing concern about the increase in pollution leading to global warming, the depletion of the ozone layer or the increasing deforestation which diminishes the earth’s biodiversity. If a country unilaterally reduces its polluting emissions, other countries will free-ride on this reduction and may even offset it by increasing their emissions. The reduction in polluting emissions of pollutants like CO 2 or CFCs is an international public good which is privately provided by individual countries.1
1
See Hoel (1991) and Sandler (1992) for analysis of environmental problems in a public good framework and Murdoch and Sandler (1997) for an empirical assessment of the theory.
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Now, if country i’s economic prospects become riskier, it will make a smaller reduction in its emissions. This induces the other countries to reduce their emissions further, since every country knows the economic risks faced by country i. This amounts to an income transfer to country i and induces i to further reduce its commitment. This paper proceeds as follows. Section 2 explains the model. To calculate the subgame perfect equilibrium, the private provision game under uncertain income has to be solved. This is done briefly in Section 3 for the case where both individuals face the same amount of risk (‘‘symmetric’’ uncertainty). In Sections 4, 5 and 6 the main result of the analysis, the welfare implications and a numerical example are presented. Section 7 concludes.
2. The model Consider a situation with two individuals. Each player has a continuous increasing strictly concave utility function U i (G, x i ), where G 5 g1 1 g2 , gi is i’s contribution to the public good and x i i’s private consumption (i 5 1, 2). As usual in the literature, we will assume that both goods are strictly normal. Under certainty, this ensures uniqueness of the Nash equilibrium.2 Each player i is ] endowed with income Mi , where Mi is a random variable with domain [M, ] M], M ] . 0. The budget constraint for each individual is given by x is 1 gi 5 m is , where gi #M ] where the index i stands for the individual and s[h1, . . . ,Sj for the outcome of the random variable Mi . We set both prices equal to 1. This normalization is standard in the literature and implies that the individuals make real commitments toward the purchase of the public good. We also assume that both players choose their expenditures before knowing what the realization of the random income will turn out to be. Thus, it is real private consumption which is random. The players decide in the first place on their public good commitment. This timing is crucial in our model. In the environmental goods case, Buchholz and Konrad (1995) argue that contributions to global environmental goods can be seen as a change of an investment path or of the technology. A further change of the technology (e.g. a change from nuclear power stations to gas powered stations) would imply high switching costs. Hence, the environmental decision is a long term commitment. One can interpret the right hand side of the budget constraint as 2
To be strict, under uncertainty the normality of the goods has to be suitably defined. If the mean of the distribution M increases and the reaction of the demand function of the public good dG / dM lies strictly between 0 and 1, then we can apply the result of Bergstrom et al. (1986) concerning existence and uniqueness of a Nash equilibrium.
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a lifetime income, which is uncertain by nature. The decisions about the contributions to the public good have to be taken in the present.3 What is more, countries have to pledge their contribution to the public good in real terms regardless of realized future income, e.g. by committing to the closure of air polluting coal fired power stations. The first-order conditions describing the Nash equilibrium are given by EU iG (G, M 2 g i ) 2 EU xi (G, M 2 g i ) 5 0, i 5 1, 2,
(1)
where the expectation is taken with respect to the random income and the indexes denote the first derivative with respect to the variable. Let F i (G, M)5EU Gi 2EU xi i denote 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 where U i 5U and income is an identically distributed random variable Mi with E(Mi )5m i 5m for all i: All individuals are identical, so g i 5g.
3. Symmetric income uncertainty Before analyzing the game under asymmetric uncertainty let us summarize the effect of increased symmetric uncertainty. We start from an equilibrium situation under uncertainty where g c is the optimal commitment for an income distribution M c . Consider now an exogenous increase in riskiness in the distribution in a mean preserving spread sense, such that g u becomes the new equilibrium commitment for distribution M u . This case was analyzed by Dardanoni (1988) for a two argument utility function and by Gradstein et al. (1992) for the private provision of a public good under price uncertainty. If both the private and the public good are normal and (U ixxx 2U iGxx ).0, i51, 2, then the equilibrium level of the public good under private provision decreases when income uncertainty increases: G u , G c. (U ixxx 2U iGxx ).0 is a sufficient condition for the first order conditions (FOC) to be concave. Concavity of the FOC drives the result, and can be interpreted with the help of the Arrow–Pratt measure of absolute risk aversion with respect to the random private good x,4 Uxx (G, m 2 gi ) Abs R x ( gi ) 5 2 ]]]]. Ux (G, m 2 gi )
(2)
If the Arrow–Pratt measure R Abs x ( gi ) is an increasing function of the commitment gi , holding all other g2i constant (e.g. a decreasing function of x), then the 3
This interpretation of the budget constraint in time terms was suggested by a referee. We use the extension of the one-dimensional Arrow–Pratt measure to the 2-dimensional case suggested by Sandmo (1969). 4
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condition (Uxxx 2UGxx ).0 is satisfied. For a separable utility function, the condition reduces to the prudence requirement (2Uxxx /Uxx ).0. In the onedimensional case, prudence is a necessary condition for a decreasing Arrow–Pratt measure of absolute risk aversion. In our model, the commitment to the public good plays the role of saving in a precautionary saving framework. The players increase (expected) private consumption by reducing the public good commitments in order to avoid very low levels of the random good x. Prudence describes how the individual changes his optimal, risk-averse behaviour under uncertainty in a way analogous to the way as risk 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 his marginal rate of substitution equals 1. On average, he will end with a higher private consumption than at the corresponding level of x i (G) under certainty.
4. Asymmetric income uncertainty We start from an equilibrium situation under uncertainty and analyze the effect of a one-sided increase in risk. Assume that, for exogenous reasons, player 2 faces a mean preserving spread in risk. We will show that, under normality and prudence assumptions, this behaviour may be rational if this player participates in the game of private provision of the public good at a later stage. Although the equilibrium level of G decreases due to the increased uncertainty, this reduction is not shared equally between the players. The commitment g˜ 2 decreases more than proportionally. Effectively, this is an income transfer from player 1 to player 2.5 Proposition 1. (Effect on G of asymmetric income uncertainty). Suppose both the private and the public good are normal, and both individuals contribute a positive i amount to the public good before and after the risk increase. If (U ixxx 2U Gxx ).0, i51, 2, then a one-sided increase in uncertainty in 1’ s income leads to a reduction of both 1’ s commitment and the equilibrium level of G. Player 2’ s commitment increases. Proof. Denote the equilibrium level for the common income distribution M with G ˜ where M ˜ and consider the new situation where the distribution of 2’s income is M, is more risky than M. Write the corresponding first-order conditions:
5
˜ M 2 g˜ ) 2 EU 1 (G, ˜ M 2 g˜ ) 5 0 FOC player 1, EU 1G (G, 1 x 1
(3)
˜ M ˜ 2 g˜ ) 2 EU 2 (G, ˜ M ˜ 2 g˜ ) 5 0 FOC player 2. EU 2G (G, 2 x 2
(4)
We denote the levels in the asymmetric case with the asymmetric symbol |.
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From the FOC from individual 1 we may calculate 1 dg˜ 1 E(U GG 2 U 1Gx ) ] 5 2 ]]]]]]] , 0, 1 dg˜ 2 E(U GG 2 2U 1Gx 1 U 1xx ) 1
1
1
1
1
where E(U GG 22U Gx 1U xx ),0 by the second-order condition and (U GG 2U Gx ), 0 by normality. Thus, we can subsume both Eqs. (3) and (4) under ˜ M ˜ ) 5 EU 2 (g˜ (g˜ ) 1 g˜ , M ˜ 2 g˜ ) 2 EU 2 (g˜ (g˜ ) 1 g˜ , M ˜ 2 g˜ ) F 2 (G, G 1 2 2 2 x 1 2 2 2 5
E
] M
M ]
˜ 2 g˜ ) 2 U (g˜ (g˜ ) 1 g˜ , M ˜ (U G (g˜ 1 (g˜ 2 ) 1 g˜ 2 , M 2 x 1 2 2 2
˜ u ) 5 0, 2 g˜ 2 )) dF(M,
2
(5)
and
] 1 where 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 continuously differentiable with (≠F / ≠M)5f(M, u ) and (≠F / ≠u )5Fu . An increase in u is called a mean preserving spread iff (Diamond and Stiglitz, 1974)
E
] M
E
M0
M ]
M ]
Fu (M, u ) dM 5 0 and ] Fu (M, u ) dM $ 0 ;M0 [ [M, ] M].
(6)
(7)
To calculate the effect of an increase in risk on the optimal commitment (dg˜ 2 / du ) we implicitly differentiate Eq. (5) following Dardanoni (1988): ≠F 2 ]] dg˜ 2 ≠u ] 5 2 ]] . du ≠F 2 ]] ≠g˜ 2 Since (≠F 2 / ≠g˜ 2 ),0, the sign of (dg˜ 2 / du ) is equal to the sign of (≠F 2 / ≠u ). We obtain
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(8) Thus, the commitment of the player 2, whose income uncertainty has increased, falls (g˜ 2 ,g2 ). By (dg˜ 1 / dg˜ 2 ),0, the other player’s commitment must increase: g˜ 1 .g1 . To show that the sum of commitments is actually lower when one agent faces more uncertainty, consider the first-order conditions of individual 1 under symmetric and asymmetric uncertainty:
,
(9)
(10) We know that the expected private consumption of player 1 falls: Ex˜ R1 ,Ex R1 5Ex R . ˜ Then, due to normality, given Eq. (9), the left hand would fall and Suppose G,G. the right hand would increase. This is a contradiction to Eq. (10), so G˜ ,G. Q.E.D. Corollary 1. Both goods are normal and (U ixxx 2U iGxx ).0, i51, 2. Starting from a noncorner equilibrium situation with given G and Ex 1R 5Ex 2R 5Ex R , an increase in uncertainty for player 2 leads to the following results: G˜ , G,
g˜ 2 , g , g˜ 1 , and Ex˜ R1 , Ex R , Ex˜ 2R .
(11)
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5. Welfare effects Under certainty, the private provision of a public good leads to a suboptimal equilibrium level: some joint increase in commitments benefits both players. The introduction 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 (Uxxx 2UGxx ).0), they will decrease their contributions under uncertainty and widen the gap between the resulting equilibrium level and the Samuelson level under certainty. In this case, welfare must decrease, too (Gradstein et al., 1992). Both effects (increased uncertainty and decreased provision of G) work in the same direction. If uncertainty increases the contributions, then welfare (under uncertainty and private provision) may increase or decrease when compared to the certainty benchmark. When uncertainty is asymmetric and affects only player 2, it constitutes a strategic advantage and therefore the other player is worse off. The welfare results in 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˜ 2 of player 2 increases player 1’s commitment. This amounts to an income transfer from 1 to 2: the expected private consumption of player 2 increases. Player 1 is definitely worse off. The first and the third effects both reduce his utility, while the second effect does not affect him directly. He consumes less of G and less of (expected) x. For player 2, the situation is ambiguous. The first and second effects reduce 2’s utility, while the income transfer increases it. Intuitively, if an individual is not very risk averse, the resulting income transfer may overcompensate his risk premium. To analyze formally the positive and the negative effect for player 2, it is useful to break up the difference in utility in two terms as Gradstein et al. (1992) do in their context: 2
˜ M ˜ 2 g˜ ) DU 2 5 EU 2 2 EU˜ 5 EU 2 (G, M 2 g) 2 EU 2 (G, 2
(12)
˜ M 2 g˜ )] 5 [EU 2 (G, M 2 g) 2 EU 2 (G, 2 2 ˜ 2 ˜ ˜ 1 [EU (G, M 2 g˜ 2 ) 2 EU (G,M 2 g˜ 2 )].
(13)
The first term reflects the expected utility difference between the levels (G, x 2 ) and ˜ x˜ ), while the second term measures the difference in utility due to the (G, 2 increase in uncertainty. The second term is positive since the individual is risk averse and strictly prefers less income uncertainty. The first term is ambiguous. If the increase in private consumption x˜ 2 .x 2 5x outweighs the decrease in G, the
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first term might be negative. An individual with a small enough risk premium and big enough income transfer might prefer the situation where he faces more uncertainty. Corollary 2. (Risk Loving Behaviour). Suppose both the private and the public good are normal and (Uxxx 2UGxx ).0. If the effect of increased private consumption 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 other player. In the light of these results, compulsory insurance constitutes a Pareto improvement. It has been shown that uncertainty is a strategic advantage. No player will surrender this advantage voluntarily. But if both agents must buy full insurance and are prudent, the public good level will increase and the agents will be at the Nash equilibrium under certainty. This welfare improvement may be unattainable in the international public goods case, because there is no supranational authority which can enforce a compulsory insurance scheme upon sovereign countries 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 that drives the result. We will present an extreme case where the original situation is characterized by income certainty for both players and player 2 faces an exogenous increase in risk. Let the players have the utility functions U 1 (G, x 1 )5x 1 1 ]34 ln(G) and U 2 (G, x 2 )5ln(x 2 )1ln(G) and fixed incomes m 1 5m 2 52.6 Under certainty and assuming an interior solution, the first-order condition for player 1 uniquely determines the overall level of provision of the public good: G*5 ]43 . This implies g1 5 ]43 2g2 and x 1 522g1 . For player 2 we obtain Ux 5(1 / x 2 )5(1 /G)5UG and, hence, (1 /(22g2 ))5(1 /( g1 1g2 )). The unique equilibrium 1 2 is given by g1 5g2 5 ]32 , x 1 5x 2 5 ]34 , U 5 ]34 1 ]34 ln( ]34 ), and U 5 2 ln( ]34 ). Now let the income of player 2 be a discrete random variable which is equal to 6 For utility U 1 , private consumption is normal but not strictly normal. However, uniqueness of the equilibrium is ensured here by the strict normality of both goods for the player 2. Also, U 1 is the limiting case U 1 5lima →0 (x 1 1 a ln x 1 1 ]43 ln(G)), with (x 1 1 a ln x 1 1 ]43 ln(G)) fulfilling strict normality for both the private and the public good, for all a .0. The quantitative result of the example carries over to sufficiently small positive a.
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] m ] 2 51.5 or m 2 52.5 with probability 0.5. Player 1’s optimal decision is unchanged. The risk averse individual 2 now has the first-order condition 1 1 1 EUx 5 ] 1 ] 5 ] 5 UG , 2x] 2 2x] 2 G˜
] 2g˜ and x 5m 2g˜ denote his private consumption x˜ when the where ]x 2 5m 2 2 2 2 ]2 ]2 high or the low value of the random variable is realized. Straightforward calculations lead to g˜ 1 5 ]65 , g˜ 2 5 ]21 , x˜ 1 5 ]67 and E(x˜ 2 )5 ]23 . The player with uncertain income free-rides on the public good contribution of player one. Since the first individual always fully compensates every reduction in 2’s commitment to the public good, the public good level is unchanged. As expected, the player facing increased uncertainty increases his utility at the expense of the risk free individual: 2 1 EU˜ 5 ]12 ln(2) 1 ln( ]43 ) . U 2 . Moreover, player 1’s expected utility is U˜ 5 ]76 1 1 4 4 ] ln( ] ) , U . 3 3
7. Conclusions We have identified a case where a risk averse agent rationally prefers a riskier situation. In such a setting, where the player’s risk aversion is small and her prudence big enough, she refuses to buy any insurance. Actually, the player has a rational incentive to engage in some risky activity. Uncertainty yields a strategic advantage. Participation in a voluntary contributions game has adverse commitment incentives. Konrad (1994) has shown that participants have an incentive to distort their intertemporal consumption decision. This paper shows that participation in a voluntary contributions game also distorts risk-rating incentives. When a public good is provided privately, the agent facing higher uncertainty elicits an income transfer from the other players, who need not have altruistic preferences. This contrasts with most literature where individuals rely on altruistic individuals to bail them out when they suffer a loss. In our present case, it is in the other players’ interest to make the income transfer because this increases the provision level of the public good. This strategic advantage of risk when facing altruistic individuals provides additional justification for some compulsory insurance schemes observed in many countries. Insurance eliminates one distortion and leads to the still suboptimal Nash equilibrium of private provision of a public good. If the agents are sovereign countries, welfare improving compulsory insurance may not be an option. A possible solution may involve the countries with more stable incomes assuming the role of insurers. This would still mean a kind of income transfer, since these countries would, in fact, be assuming the risk premium of countries facing higher risks.
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Acknowledgements I thank Anette Boom, Kai A. Konrad, Roland Strausz and two anonymous referees for helpful comments. The usual caveat applies.
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