Fixing first-order part of second-order approx


I want to know if I can solve for a second order approximation given a fixed first-order solution.

To be a little less vague, I’m thinking about a portfolio problem where the assets have the same mean return but different risk characteristics. So to first order, the assets are the same and so I cannot solve for holdings of each individually. However, I can solve for the first some linear combination of the two. To be more specific, imagine that assets 1 and 2 have the same mean return, but different state-dependent payoffs. Then I can solve a first-order approximation to the model where agent’s gross asset position A = A1 + A2 is defined. But when where A1 & A2 are distinct dynare fails because there are a continuum of first-order solutions (because the two are perfect substitutes in a first-order world). This is a problem because solving for the first-order approximation is a necessary step for solving the second-order approximation for the problem with the two assets, which is what I really want to do.

So here’s a method that I thought might work:


  1. Solve the first-order approximation to the model where only the gross position is defined.
  2. Create a first-order approximation to the full problem using some arbitrary rule satisfying A=A1+A2. For example, A1=A2=A/2
  3. Then ask dynare to solve for the second-order solution given this first-order approximation.

In the typical language that people use to describe these approximations, I want to define g_y and g_u myself, and then ask dynare to recover g_{yy}, g_{uu} and g_{\sigma\sigma}.

This seems like it should not be a novel problem, so perhaps someone has asked this before, although I couldn’t find it. And maybe the method won’t work - the lines of g_y and g_u defining the policy rules for A1 and A2 will be colinear, which might preclude solving the problem this way.