Patching utility functions

Some utility functions have singularity u(c) --> -infty as c-->0. CRRA utility, which is a good example, is one of the most popular specifications that are de facto standard in today’s macroeconomic modeling. However, the singularity u(c) --> -infty as c approaching zero brings annoying numerical challenges in practice. Even though in mathematics, such singularity ensures the optimal c >0 strictly, the numerical solvers, however, do not understand this implication and may try evaluating the Lagrangian/Hamiltonian at c >= 0. This especially happens when the budget constraint leads to an irregular admissible space of c and one is using interior point solvers.

A traditional numerical trick is to set a small tolerace d>0:

$$ u(c) \approx \frac{1}{1-\gamma}(c+d)^{1-\gamma} $$

such that some small negative c would not crash the whole program. Of course, the cost of such shifting operation is a little degree of risk aversion twist.

However, this trick is not good enough: how large should the d be to handle all the millions of grid points in each iteration? If d is too large, can we still claim that this is a CRRA utility?

Thus, a natural idea is to find a way to “extend” the CRRA utility function to the whole real line, while keeping the true CRRA utility function form without d in the regions where the optimal solution locate.

To be accurate: by setting a small enough positive threshold m>0, for any c<m, we replace the true CRRA utility function with a smooth extrapolation. The patched CRRA utility function is a piecewise function:

$$ \hat{u}(c) = \begin{cases} \frac{1}{1-\gamma}c^{1-\gamma} , c \geq m \ P(c|m) , c < m \end{cases} $$

The extrapolation part P is strictly monotonic and smooth enough. We call this operation as “patching”, i.e. cut off the undesirable part of the true function at some point then “patch” the removed part with a patching function P. Intuitively, such patching does not affect the final solution when m is small enough regarding the final optimal consumption level. It only provides a large enough evaluable region for the numerical solvers to explore.

In the following sections, we discuss some widely used utility function forms and their C^2 smooth patching.

One dimensional CRRA

OK, let’s do some math. Suppose we want a C^2 patch function P for the one-dimensional CRRA utility u(c). Let’s consider the following polynomial up to the 2nd order:

$$ P(c|m) := \beta_0 + \beta_1 c + \beta_2 c^2 $$

The C^2 smoothness requries conditions at the boundary c =m:

$$ P(m|m) = \beta_0 + \beta_1 m + \beta_2 m^2 = \frac{1}{1-\gamma}m^{1-\gamma} $$

$$ P_m (m|m)=\beta_1 + 2\beta_2 m = m^{-\gamma} $$

$$ P_{mm}(m|m)=2\beta_2 = -\gamma m^{-\gamma-1} $$

Recursively solving the coefficients gives:

$$ \beta_2 = -\frac{\gamma}{2} m^{-\gamma-1} $$

$$ \beta_1 = (1-m \gamma) m^{-\gamma} $$

$$ \beta_0 = m^{1-\gamma} ( \frac{1}{1-\gamma} - 1 + m\gamma + \frac{\gamma}{2} ) $$

The separable CRRA utility used in Jones, C., Midrigan, V. & Philippon, T. (2022, Econometrica) (“Household Leverage and the Recession.”) follows the same structure because of the additivity.

(more on the way…)