A general and intuitive envelope theorem

Andrew Clausen, Carlo Strub

Research output: Working paperDiscussion paper


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.
Original languageEnglish
PublisherEdinburgh School of Economics Discussion Paper Series
Number of pages35
Publication statusPublished - 31 Dec 2013

Publication series

NameESE Discussion Papers


  • First-order conditions
  • Discrete choice
  • Unsecured credit
  • Adjustment costs
  • Informal
  • Insurance arrangements


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