In Essays in the theory of risk-bearing

Arrow, “Essays in the Theory of Risk Bearing,” Amsterdam, North Holland, 1971

Arrow (1965) Aspects of the Theory of Risk-Bearing

thus the indifference curves are tangent to each other and to the equilibrium priceline (shown in Figure F with slope -p1/p2). That an Arrow-Debreuequilibrium is an optimal risk-bearing allocation is clear enough from the tangencies andstandard convexity proofs will confirm this more generally. Notice that weassume agents have the same subjective probabilities, i.e. 1A/ 2A = 1B/ 2B, then the tangency condition on thecontract curve reduces simply to uA (x1)/uA (x2) = uB (x1)/uB (x2), or cross-multiplying:

K.J. Arrow (1953)

Essays on the Theory of Risk Bearing

Of course, this particular example assumes that only one agent faces randomnessdirectly. What if agents face individual risks of some sort? In this case, weneed to specify "social states" as distributions of specific agents overdifferent individual states, (e.g. "I am ill, you are ill, he is healthy, she ishealthy" would be one social state, while "I am ill, you are healthy, he ishealthy, she is healthy" would be another social state). However. with S nowredefined this way, we can resurrect the fundamental theorem of risk-bearing and, by thesame law of large numbers argument, achieve the same result, i.e. that pis = spi, where pi is the"sure" price and s the probability ofsocial state s S.

Essays in the Theory of Risk Bearing

spi pis
so the state-contingent price for good i in state s, pis, is equal to theproduct of the price of that good and the probability of that state. Thus,state-contingent prices are directly proportional to the probability of the social statein which they occur. This means that if we know the price, pi, andthe probabilities, then we can easily determine what the state-contingent price is.The implications of this can be understood as follows: in principle, in theArrow-Debreu economy, when we have n goods and S states, we need nS markets in order todetermined nS prices. This is particularly troublesome when considering individual risksas it seems to imply that there are separate markets and prices for "delivery of eggswhen I am ill" and "delivery of bread when he is ill", etc. which are fartoo particular. However, what 'stheorem implies is that when there is a large number of agents, then the price of"eggs when I am ill" becomes merely the sure price of eggs (determined in thesure market, pi, which delivers in states) multiplied by theprobability of "I am ill". Consequently, instead of nS markets, we merely need nsure markets and S insurance markets (or Arrow securities, if we wish - see our ) which deliver individualpurchasing power in the case of particular states. Thus the number of necessary marketsgets reduced considerably when the number of agents is large. Of course, this particular example assumes that only one agent faces randomnessdirectly. What if agents face individual risks of some sort? In this case, weneed to specify "social states" as distributions of specific agents overdifferent individual states, (e.g. "I am ill, you are ill, he is healthy, she ishealthy" would be one social state, while "I am ill, you are healthy, he ishealthy, she is healthy" would be another social state). However. with S nowredefined this way, we can resurrect the fundamental theorem of risk-bearing and, by thesame law of large numbers argument, achieve the same result, i.e. that pis = spi, where pi is the"sure" price and s the probability ofsocial state s S.

Arrow k 1971 essays in the theory of risk bearing