A General and Intuitive Envelope Theorem

We present an envelope theorem for establishing first-order conditions in decision problems involving continuous and discrete choices. Our theorem accommodates general dynamic programming problems, even with unbounded marginal utilities. And, unlike classical envelope theorems that focus only on differentiating value functions, we accommodate other endogenous functions such as default probabilities and interest rates. Our main technical ingredient is how we establish the differentiability of a function at a point: we sandwich the function between two differentiable functions from above and below. Our theory is widely applicable. In unsecured credit models, neither interest rates nor continuation values are globally differentiable. Nevertheless, we establish an Euler equation involving marginal prices and values. In adjustment cost models, we show that first-order conditions apply universally, even if optimal policies are not (S,s). Finally, we incorporate indivisible choices into a classic dynamic insurance analysis.

Harmful competition in the insurance markets

There is a general presumption that competition is a good thing. In this paper we show that competition in the insurance markets can be bad and that adverse selection is in general worse under competition than under monopoly. The reason is that monopoly can exploit its market power to relax incentive constraints by cross-subsidization between different risk types. Cream-skimming behavior, on the contrary, prevents competitive firms from using implicit transfers. In effect monopoly is shown to provide better coverage to those buying insurance but at the cost of limiting participation to insurance. Performing simulation for di erent distributions of risk, we find that monopoly in general performs (much) better than competition in terms of the realization of the gains from trade across all traders in equilibrium. However, most of the surplus is retained by the firm and, as a result, most individuals prefer competitive markets notwithstanding their performance is generally poorer than monopoly.