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Clarke Differentials and the Envelope Theorem in Dynamic Programming

Yuhki Hosoya

TL;DR

This paper extends the envelope theorem to deterministic dynamic programming without requiring differentiability, convexity, or boundedness, by employing the Clarke differential. It develops a self-contained framework of locally Lipschitz functions and Clarke calculus, culminating in a main result that bounds ∂°ĤV(x̄) by the x-derivative of the stage function w via a suitable optimizer ȳ in the policy correspondences. The contribution lies in proving the envelope property under very weak assumptions, enabling application to standard economic models such as CRRA utilities and Cobb–Douglas or AK technologies. The work broadens the scope of envelope-type results to non-smooth settings, with practical implications for sensitivity analysis in deterministic DP models and reduced-form economic applications.

Abstract

In this paper, we consider a deterministic dynamic programming model, and derive the envelope theorem using the Clarke differential. Compared with previous research, we do not require differentiability, convexity, or boundedness.

Clarke Differentials and the Envelope Theorem in Dynamic Programming

TL;DR

This paper extends the envelope theorem to deterministic dynamic programming without requiring differentiability, convexity, or boundedness, by employing the Clarke differential. It develops a self-contained framework of locally Lipschitz functions and Clarke calculus, culminating in a main result that bounds ∂°ĤV(x̄) by the x-derivative of the stage function w via a suitable optimizer ȳ in the policy correspondences. The contribution lies in proving the envelope property under very weak assumptions, enabling application to standard economic models such as CRRA utilities and Cobb–Douglas or AK technologies. The work broadens the scope of envelope-type results to non-smooth settings, with practical implications for sensitivity analysis in deterministic DP models and reduced-form economic applications.

Abstract

In this paper, we consider a deterministic dynamic programming model, and derive the envelope theorem using the Clarke differential. Compared with previous research, we do not require differentiability, convexity, or boundedness.

Paper Structure

This paper contains 19 sections, 14 theorems, 101 equations, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and Preliminary Results
  3. Locally Lipschitz Functions
  4. Clarke Derivatives
  5. Strong Differentiability
  6. Main Result
  7. The Model
  8. The Result
  9. Application to Economic Models
  10. Conclusion
  11. Proofs of Several Results
  12. Proof of Proposition \ref{['prop1']}
  13. Proof of Proposition \ref{['prop2']}
  14. Proof of Proposition \ref{['prop3']}
  15. Proof of Proposition \ref{['prop4']}
  16. ...and 4 more sections

Key Result

Proposition 1

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Theorem 1
  • Theorem 2
  • ...and 4 more