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Rothschild-Stiglitz competitive insurance market under quasilinear preferences
Economics Letters North-Holland
27 (1988) 27-30
27
ROTHSCHILD-STIGLITZ COMPETITIVE UNDER QUASILINEAR PREFERENCES
INSURANCE
MARKET
Shou Yong SHI * Unwersity
Received Accepted
of
Toronto, Toronto, Ont., Crrnada MSS IA1
16 December 1987 19 January 1988
In a heterogeneous agents economy under quasilinear preferences, if preferences are unidirectional, the pooling equilibrium may exist. If so, it is Pareto optimal; if not, the separating equilibrium (which is Pareto optimal) may or may not exist depending on preferences. If the preferences are multidirectional, the pooling equilibrium always exists and is Pareto optimal.
1. Introduction The Rothschild-Stiglitz (1976) competitive insurance market model characterized the equilibrium in an uncertain economy where there are two types of agents: high-risk agents with number nH and 7r,_of accident, probability mu of no accident and low-risk agents with number n,_ and probability r,_ > ru. The economy contains one physical commodity and two states. Initial endowments for both types are M,= (wl, ~2) under two states. Consumptions for two states are I1 and 12. The budget constraints are identical to the insurance company’s zero profit conditions for each agents: high risk: ~~11 + (1 - au)12
= n,wl
low risk: rt.11 + (1 - 7r,)12 = n,wl
+ (1 - m,)w2, + (1 - r,)w2.
Rothschild and Stiglitz analyzed the equilibrium when two types of agents had the same utility functions except the probabilities in the expected utility [Campbell (1987)]. Extending the model to non-expected utility maximization, we use quasilinear or weighted linear mean (WLM) preferences described by Chew (1983). The pooling equilibrium and the separating equilibrium may exist and be Pareto optimal in specific cases for heterogeneous agents.
2. Equilibrium for expected utility maximization For expected utility maximization, the indifference curves are parallel lines defined on the c.d.f. space D(X). Following Shi (nd.), budget curves are concave to the lower utility points while the low-risk agents’ budget curves are less curvatured than the high-risk agents’ since ru < 7~~, as in * The author has had the benefit of helpful remains solely responsible for errors.
0165-1765/88/$3.50
conversations
8 1988, Elsevier Science Publishers
with Professor
D. Campbell
B.V. (North-Holland)
and S.H. Chew. As usual,
the author
2x
Fig. 1.
S. Y. Shi / Rothschrld-Stiglitz
Rothschild-Stiglitz
market
under expected
competitiw
mwrance
market
utility maximization
fig. 1. Since the budget curve is completely determined by the corresponding call the former by the latter. Rothschild and Stiglitz’s conclusions can be re-interpreted in this fashion.
probability
7~. we will
(2.1.) The pooling equilibrium does not exist if rru # 7~~. Since the marginal rates of substitution are different for two types of agents. their utility-maximizing points are differently curvatured on pooling budget curve rO. In fig. 1, 7rCois the pooling probability of no accident, 7~~ < r. c ri Since the 7~” budget curve is more curvatured than the 7~~ budget curve, under the pooling budget curve 7r0, the maximization point for low-risk people, N, is on the left of the high-risk people’s maximization point, M. This means that in state 1 (no accident), the low-risk people give less than what the high-risk people want. Therefore, the insurance company suffers a loss. (M, N) is not an equilibrium. (2.2.) One possible equilibrium can be (H, Q), i.e., offering the high-risk people the utility maximized bundle H under their own budget curve, and the low-risk people the maximum utility bundle Q which the high-risk people will not take the advantage of. The necessary condition for (H. Q) to be an equilibrium is that the low-risk people’s indifference curve through Q should be fully above the pooling budget curve. Suppose not. Then offering N on the pooling budget curve 7~~) for both agents can make both agents better off than (H, Q). Furthermore, we can make a positive profit for the insurance company by slightly lowering N to S. S dominates (H, Q). n Jn has to be sufficiently big to ensure the equilibrium (H, Q). (2.3.) The possible equilibrium in (2.2) may not be Pareto optimal if cross substitution is allowed and nH/n is sufficiently low. Since we can make a small move towards the pooling budget curve but make the low-risk people better off and the insurance company positively profited. If nH/n is sufficiently low, the insurance company can compensate the high-risk people’s loss by only a part of the profit. (H, Q) is dominated. close. This is different from (2.4.) Separating equilibrium may exist if rH and rL are sufficiently Rothschild-Stiglitz’s conclusions. For example, in fig. 1, if rH has at least the value of rrU shown in L, is below the high-risk people’s the diagram, the low-risk people’s utility-maximizing point, indifference curve which goes through their own maximization point M. None of the two types of agents will take the advantage of the other. (M, L) is a separating equilibrium. Drawn in the standard diagram, this possibility also exists. Intuitively, when the two types agents have very close probabilities of accident, the insurance company can eventually offer the separating equilibrium bundles. This was ruled out by Rothschild and Stiglitz by assuming no cross-substitution between agents.
S. Y. Shi / Rothschild-Stiglitz
competitive
insurance
market
29
3. Equilibrium for weighted linear mean (WLM) preferences We first assume unidirectional curves share the common origin, fig. 2.
preferences [Shi (n.d.)]. For WLM preferences, all the indifference such as A for the low-risk people and B for the high-risk people in
(3.1.) The pooling equilibrium may exist. If it exists, it is Pareto optimal. In fig. 2, the high-risk people maximize their utility at P under the pooling budget curve r,,. Their indifference curve BP happens to extend through the low-risk people’s origin A. Then P is also the low-risk people’s utility maximization point under the pooling budget curve. P is a pooling equilibrium. For any bundle offered to the low-risk people along v,_ and above N, the high-risk people will pretend to be low-risk people so that the insurance company suffers a loss. The non-negative profit separating equilibrium must be (H, K ), where K is along 7~~ and below N. But (H, K) is obviously dominated by P. Along the pooling budget curve rO, any points are dominated by P.Therefore P is Pareto optimal. If pooling equilibrium does not exist, we have to distinguish whether or not the low risk people’s origin A is below the extension of BP. (3.2.) If A is below BP extension, then the pooling equilibrium does not exist, but the separating equilibrium may exist and be Pareto optimal. In fig. 3, the pooling equilibrium does not exist because under 7r0, the low-risk people’s maximization point P ’ is on the left of the high-risk people’s maximization point P. The insurance company suffers a loss. The separating equilibrium (H, Q) may exist and be Pareto optimal if the low-risk people’s maximizing point Q under 7~,_is below the high-risk people’s indifference curve BH which goes through their maximizing point H under rn.
Fig. 2.
Pooling
Fig. 3.
Separating
equilibrium
equilibrium
exists under WLM preferences.
exists while pooling
equilibrium
does not.
Fig. 4.
Standard
Fig. 5.
Pooling
Rothschild-Shglitz
equilibrium
equilibrium
exists but is not Pareto
always exists and is Pareto
optimal
under
optimal.
multidirectional
preferences.
(3.3.) If A is above the extension BP. the pooling equilibrium does not exist. Because in state 2, the low-risk people consume more than what the high-risk people give up. In fig. 4, P’ is on the right of P. The company suffers a loss. Separation equilibrium does not exist since the low-risk people’s maximization point Q is above BP, the high-risk people take the advantage of the low-risk people so that the company suffers a loss. However, there may be a standard Rothschild&Stiglitz equilibrium (H, K ). But this equilibrium is not Pareto optimal since AK has to intersect the pooling budget curve at least twice by the fact that A is above BP and K is below BP. (3.4.) If the preferences are multidirectional. the pooling equilibrium always exists and is Pareto optimal, as in fig. 5. This is because the pooling budget curve is always above the two types of agents’ budget curves rH and pi,_. relative to their original lines BW and A W, respectively. As a summary, for unidirectional preferences, the pooling equilibrium may exist. If so, it is Pareto optimal; if not. the separating equilibrium may or may not exist depending on the preferences. For multidirectional preferences, the pooling equilibrium always exists and is Pareto optimal.
References Campbell. D.. 19X7. Resources allocation mechanisms (Cambridge University Press. Cambridge) 82-95. Cheu. S.H.. 1983. A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox, Econometrica 51. no. 4. July. Rothschdd. M. and J. Stiglitz. 1976, Equilibrium in competitive inaurancc markets: An essay on the economics of imperfect information. Quarterly Journal of Economics 90. 629-650. Shi. S.Y., 1988, Rational expectation equilibrium under quasilinear preferences - A diagrammatic approach, Economics Letters, forthcoming.
27 (1988) 27-30
27
ROTHSCHILD-STIGLITZ COMPETITIVE UNDER QUASILINEAR PREFERENCES
INSURANCE
MARKET
Shou Yong SHI * Unwersity
Received Accepted
of
Toronto, Toronto, Ont., Crrnada MSS IA1
16 December 1987 19 January 1988
In a heterogeneous agents economy under quasilinear preferences, if preferences are unidirectional, the pooling equilibrium may exist. If so, it is Pareto optimal; if not, the separating equilibrium (which is Pareto optimal) may or may not exist depending on preferences. If the preferences are multidirectional, the pooling equilibrium always exists and is Pareto optimal.
1. Introduction The Rothschild-Stiglitz (1976) competitive insurance market model characterized the equilibrium in an uncertain economy where there are two types of agents: high-risk agents with number nH and 7r,_of accident, probability mu of no accident and low-risk agents with number n,_ and probability r,_ > ru. The economy contains one physical commodity and two states. Initial endowments for both types are M,= (wl, ~2) under two states. Consumptions for two states are I1 and 12. The budget constraints are identical to the insurance company’s zero profit conditions for each agents: high risk: ~~11 + (1 - au)12
= n,wl
low risk: rt.11 + (1 - 7r,)12 = n,wl
+ (1 - m,)w2, + (1 - r,)w2.
Rothschild and Stiglitz analyzed the equilibrium when two types of agents had the same utility functions except the probabilities in the expected utility [Campbell (1987)]. Extending the model to non-expected utility maximization, we use quasilinear or weighted linear mean (WLM) preferences described by Chew (1983). The pooling equilibrium and the separating equilibrium may exist and be Pareto optimal in specific cases for heterogeneous agents.
2. Equilibrium for expected utility maximization For expected utility maximization, the indifference curves are parallel lines defined on the c.d.f. space D(X). Following Shi (nd.), budget curves are concave to the lower utility points while the low-risk agents’ budget curves are less curvatured than the high-risk agents’ since ru < 7~~, as in * The author has had the benefit of helpful remains solely responsible for errors.
0165-1765/88/$3.50
conversations
8 1988, Elsevier Science Publishers
with Professor
D. Campbell
B.V. (North-Holland)
and S.H. Chew. As usual,
the author
2x
Fig. 1.
S. Y. Shi / Rothschrld-Stiglitz
Rothschild-Stiglitz
market
under expected
competitiw
mwrance
market
utility maximization
fig. 1. Since the budget curve is completely determined by the corresponding call the former by the latter. Rothschild and Stiglitz’s conclusions can be re-interpreted in this fashion.
probability
7~. we will
(2.1.) The pooling equilibrium does not exist if rru # 7~~. Since the marginal rates of substitution are different for two types of agents. their utility-maximizing points are differently curvatured on pooling budget curve rO. In fig. 1, 7rCois the pooling probability of no accident, 7~~ < r. c ri Since the 7~” budget curve is more curvatured than the 7~~ budget curve, under the pooling budget curve 7r0, the maximization point for low-risk people, N, is on the left of the high-risk people’s maximization point, M. This means that in state 1 (no accident), the low-risk people give less than what the high-risk people want. Therefore, the insurance company suffers a loss. (M, N) is not an equilibrium. (2.2.) One possible equilibrium can be (H, Q), i.e., offering the high-risk people the utility maximized bundle H under their own budget curve, and the low-risk people the maximum utility bundle Q which the high-risk people will not take the advantage of. The necessary condition for (H. Q) to be an equilibrium is that the low-risk people’s indifference curve through Q should be fully above the pooling budget curve. Suppose not. Then offering N on the pooling budget curve 7~~) for both agents can make both agents better off than (H, Q). Furthermore, we can make a positive profit for the insurance company by slightly lowering N to S. S dominates (H, Q). n Jn has to be sufficiently big to ensure the equilibrium (H, Q). (2.3.) The possible equilibrium in (2.2) may not be Pareto optimal if cross substitution is allowed and nH/n is sufficiently low. Since we can make a small move towards the pooling budget curve but make the low-risk people better off and the insurance company positively profited. If nH/n is sufficiently low, the insurance company can compensate the high-risk people’s loss by only a part of the profit. (H, Q) is dominated. close. This is different from (2.4.) Separating equilibrium may exist if rH and rL are sufficiently Rothschild-Stiglitz’s conclusions. For example, in fig. 1, if rH has at least the value of rrU shown in L, is below the high-risk people’s the diagram, the low-risk people’s utility-maximizing point, indifference curve which goes through their own maximization point M. None of the two types of agents will take the advantage of the other. (M, L) is a separating equilibrium. Drawn in the standard diagram, this possibility also exists. Intuitively, when the two types agents have very close probabilities of accident, the insurance company can eventually offer the separating equilibrium bundles. This was ruled out by Rothschild and Stiglitz by assuming no cross-substitution between agents.
S. Y. Shi / Rothschild-Stiglitz
competitive
insurance
market
29
3. Equilibrium for weighted linear mean (WLM) preferences We first assume unidirectional curves share the common origin, fig. 2.
preferences [Shi (n.d.)]. For WLM preferences, all the indifference such as A for the low-risk people and B for the high-risk people in
(3.1.) The pooling equilibrium may exist. If it exists, it is Pareto optimal. In fig. 2, the high-risk people maximize their utility at P under the pooling budget curve r,,. Their indifference curve BP happens to extend through the low-risk people’s origin A. Then P is also the low-risk people’s utility maximization point under the pooling budget curve. P is a pooling equilibrium. For any bundle offered to the low-risk people along v,_ and above N, the high-risk people will pretend to be low-risk people so that the insurance company suffers a loss. The non-negative profit separating equilibrium must be (H, K ), where K is along 7~~ and below N. But (H, K) is obviously dominated by P. Along the pooling budget curve rO, any points are dominated by P.Therefore P is Pareto optimal. If pooling equilibrium does not exist, we have to distinguish whether or not the low risk people’s origin A is below the extension of BP. (3.2.) If A is below BP extension, then the pooling equilibrium does not exist, but the separating equilibrium may exist and be Pareto optimal. In fig. 3, the pooling equilibrium does not exist because under 7r0, the low-risk people’s maximization point P ’ is on the left of the high-risk people’s maximization point P. The insurance company suffers a loss. The separating equilibrium (H, Q) may exist and be Pareto optimal if the low-risk people’s maximizing point Q under 7~,_is below the high-risk people’s indifference curve BH which goes through their maximizing point H under rn.
Fig. 2.
Pooling
Fig. 3.
Separating
equilibrium
equilibrium
exists under WLM preferences.
exists while pooling
equilibrium
does not.
Fig. 4.
Standard
Fig. 5.
Pooling
Rothschild-Shglitz
equilibrium
equilibrium
exists but is not Pareto
always exists and is Pareto
optimal
under
optimal.
multidirectional
preferences.
(3.3.) If A is above the extension BP. the pooling equilibrium does not exist. Because in state 2, the low-risk people consume more than what the high-risk people give up. In fig. 4, P’ is on the right of P. The company suffers a loss. Separation equilibrium does not exist since the low-risk people’s maximization point Q is above BP, the high-risk people take the advantage of the low-risk people so that the company suffers a loss. However, there may be a standard Rothschild&Stiglitz equilibrium (H, K ). But this equilibrium is not Pareto optimal since AK has to intersect the pooling budget curve at least twice by the fact that A is above BP and K is below BP. (3.4.) If the preferences are multidirectional. the pooling equilibrium always exists and is Pareto optimal, as in fig. 5. This is because the pooling budget curve is always above the two types of agents’ budget curves rH and pi,_. relative to their original lines BW and A W, respectively. As a summary, for unidirectional preferences, the pooling equilibrium may exist. If so, it is Pareto optimal; if not. the separating equilibrium may or may not exist depending on the preferences. For multidirectional preferences, the pooling equilibrium always exists and is Pareto optimal.
References Campbell. D.. 19X7. Resources allocation mechanisms (Cambridge University Press. Cambridge) 82-95. Cheu. S.H.. 1983. A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox, Econometrica 51. no. 4. July. Rothschdd. M. and J. Stiglitz. 1976, Equilibrium in competitive inaurancc markets: An essay on the economics of imperfect information. Quarterly Journal of Economics 90. 629-650. Shi. S.Y., 1988, Rational expectation equilibrium under quasilinear preferences - A diagrammatic approach, Economics Letters, forthcoming.