Economics of Contracts and Information
Professor: Dezsö Szalay
University of Bonn
2014
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Economics of Contracts and Information
Professor: Dezsö Szalay
University of Bonn
2014
Equilibrium in Competitive Insurance Markets (Rothschild and
Stiglitz (1976) QJE)
Consider a competitive version of Stiglitz’s (1977) two type model.
Results
Competitive Equilibria may not exist.
If an equilibrium exists it involves nonlinear pricing.
If a competitive equilibrium exists, it is a separating equilibrium.
Competitive Equilibria are pareto ine¢ cient. High risk individuals
exert a negative externality on low risk individuals. However, high risk
individuals do not bene…t from this.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Basic Model
W
d. Wealth in case of accident absent insurance.
W1 = W
α 1 , W2 = W
d + α2 is wealth in case of accident.
Let α = (α1 , α2 ) denote a contract.
Demand for Insurance
Individual’s utility:
V̂ (p, W1 , W2 ) = (1
p ) U (W1 ) + pU (W2 )
A contract α is worth
V (p, α) = V̂ (p, W
α1 , W
d + α2 )
Individuals select from all contracts the contract which maximizes
V (p, α) . The individual buys a contract only if V (p, α) V (p, 0) .
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Supply of Insurance contracts
Pro…ts of insurance company per customer
π (p, α) = (1
p ) α1
pα2 = α1
p ( α1 + α2 )
There is free entry and …rms are very wealthy.
De…nition of Equilibrium
Equilibrium in a competitive insurance market is a set of contracts such
that, when customers choose contracts to maximize expected utility, i) no
contract in the equilibrium set makes negative pro…ts; and ii) there is no
contract outside the equilibrium set that, if o¤ered, will make a
nonnegative pro…t.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Equilibrium with identical consumers
Free entry and perfect competition
π (p, α) = (1
p ) α1
pα2 = 0
The equilibrium is e¢ cient, and involves perfect insurance of the risk
averse customers by the risk neutral …rms.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Graph: Equilibrium with one type of customers
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Imperfect Information: Equilibrium with two Classes of Consumers
pH and pL denote the probabilities of accident for the two classes of
consumers. λ is the fraction of high risks in the market.
p = λpH + (1
λ) pL
One can imagine two kinds of equilibria:
Pooling equilibria: both customers buy the same contract.
Separating equilibria: customers buy di¤erent contracts.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Nonexistence of a pooling equilibrium
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Existence of separating equilibrium:
If there is an equilibrium, then it must be separating.
The set αH , αL is the only possible equilibrium for a market with low and
high risk individuals. (The restrictions are satis…ed by exactly one pair of
contracts.)
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
However, αH , αL need not be an equilibrium. In particular it is no
equilibrium if it can be upset by an o¤er γ. γ makes a pro…t if the fraction
of high risk individuals in the market is small.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
More generally, equilibrium does not exist if the costs to the low risk
consumers of pooling (cross subsidization) are low. This is the case if
there are few high risk individuals or if the probabilities of accidents are
not too di¤erent.
Welfare Economics
High risk individuals exert a negative externality on low risk individuals.
But they do not gain in the process. Therefore everybody could be made
better o¤ (the high risks being kept indi¤erent) if the high risks revealed
their type. Thus, the equilibrium is pareto ine¢ cient.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Alternative Equilibrium Concept
A Wilson Equilibrium is a set of contracts such that, when customers
choose among them so as to maximize pro…ts a) all contracts make
nonnegative pro…ts and b) there does not exist a new contract (or set of
contracts) which, if o¤ered, makes positive pro…ts even when all contracts
that lose money as a result of this entry are withdrawn.
If the separating equilibrium exists, then it is also a Wilson equilibrium. If
the separating equilibrium does not exist, then the Wilson equilibrium is
the pooling equilibrium which maximizes the utility of the low risks.
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1
Relation to recent literature
Substantial research has been carried through since Rothschild and
Stiglitz (1976)
Surveyed by Stole (2001) ”Price Discrimination in competitive
environments” (See Lars Stole’s webpage at Chicago)
Rothschild Stiglitz is a case of exclusive common agency.
Exclusive common agency can be analyzed as non-linear pricing with
type dependent outside options.
Nonexclusive common agency has been analyzed by Martimort and
Stole (2003) (See Lars Stole’s webpage at Chicago)
Professor: Dezsö Szalay University of Bonn () Economics of Contracts and Information
2014
1/1