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The Inverse Optimal Problem: A Dynamic Programming Approach

Fwu-Ranq Chang

Abstract

The paper solves the stochastic inverse optimal problem. Dynamic programming is used to transform the original problem into a differential equation. Such an equation is well-defined (with probability one) if the production function is sufficiently concave at infinity. When the production function has a finite slope at the origin, we show that a solution to the aforementioned problem exists for a twice continuously differentiable, strictly increasing consumption function provided the savings function, starting from the origin, is steep initially and flat eventually. Three well-known consumption functions, linear (in the capital-labor ratio), Keynesian, and Cantabrigian, are also studied within the stochastic framework. A well-known result in discrete time models--that a logarithmic utility function and a Cobb-Douglas production function imply a Keynesian consumption function--does not carry through to the continuous time case.