Risk aversion and IES in EZ preference

General DSGE Modeling
1

Hello everyone! I have recently been reproducing Gourio, F. (2012). “Disaster Risk and Business Cycles.” American Economic Review, 102, 2734–2766. https://doi.org/10.1257/aer.102.6.2734. However, I encountered problems calculating risk aversion and IES under the EZ preference.
In Gourio (2012), the representative consumer has recursive preferences (Epstein and Zin 1989):

V_t=\left(U_t^{1-\psi}+\beta E_t\left(V_{t+1}^{1-\gamma}\right)^{\frac{1-\psi}{1-\gamma}}\right)^{\frac{1}{1-\psi}}

where the utility index U_t depends on consumption C_t as well as hours worked N_t, and takes the following standard Cobb-Douglas form, consistent with balanced growth:

U_t=u\left(C_t, N_t\right)=C_t\left(1-N_t\right)^v

For this specification, \gamma is the risk aversion coefficient and the parameter \psi is inversely related to elasticity of substitution (IES). Specifically, the IES is 1 / \hat{\psi}, where \hat{\psi}=1-(1+v)(1-\psi), and it is larger than unity if and only if \psi <1 .
I successfully calculated the discount factor by referring to
EZPreferences.pdf (124.7 KB)
, but when I computed IES according to the formula, I found that IES is exactly \frac{1}{\psi}, which is the erroneous result emphasized by Gourio (2012). Moreover, I am unsure how to calculate risk aversion; in fact, the risk aversion I calculated seems to have no relation to \gamma at all.

I hope someone can point me in the right direction! Thank you in advance!

2

Did you find a solution yet? If not, please post your current computations. It’s easier to spot issues than start from scratch.