A Weaker Condition for Higher-Order Differentiability of Policy Functions in Recursive Models

This paper analyzes the differentiability of the value and policy functions in recursive

optimization models. We start with a well-known result: if the one-period utility function

is strongly concave and twice continuously differentiable (C

2

), then the value function is

also C

2 and the policy function is C

1

. This regularity is crucial for both the qualitative

analysis and computation of optimal solutions.

However, some counterexamples demonstrate that simply increasing the differentiability

of the utility function to C

k

for k ≥ 3 isn’t enough to guarantee that the policy function

is C

2

. A common assumption that ensures that the value function has the same degree of

differentiability as the utility function is the “dominant diagonal block” property, but this

assumption is hard to check in practice.

We present a weaker condition based on the eigenvalues of the policy function’s Jacobian

matrix and the discount factor. Our findings reveal that the order of differentiability of

the policy function is limited by the size of these eigenvalues. Consequently, even when the

utility function is differentiable to an arbitrary order, the policy function’s differentiability

is restricted by our condition. Besides some counterexamples, we discuss various economic

models where the present results may apply. Joint paper with Manuel S. Santos.