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A General and Intuitive Envelope Theorem

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Abstract

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.

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  • Andrew Clausen & Carlo Strub, 2014. "A General and Intuitive Envelope Theorem," ESE Discussion Papers 248, Edinburgh School of Economics, University of Edinburgh.
  • Handle: RePEc:edn:esedps:248
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    Keywords

    first-order conditions; discrete choice; unsecured credit; adjustment costs; informal insurance arrangements;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • E20 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - General (includes Measurement and Data)

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