# Second order approximation welfare larger than the steady state welfare

I am writing a model to find the optimal policy rule coefficients to maximize social welfare. However, upon simulating the model a few times I found my second-order approximation of welfare with stochastic shocks is always larger than the steady-state welfare. While there is a legacy post mentioning that such a problem can happen when shocks are modeled as log-normal processes, which cause the average technology to be higher due to Jensen’s inequality effect. Is there any way to correctly measure the business cycle cost so I can compare the welfare between different assumptions on my policy rule coefficients?
I am assuming that maybe by writing to mod files, one with shock processes where the mean is not adjusted from the volatility, and another with the mean shifting by the amount implied by the level for volatility(hence the two models have different steady state). By comparing the steady-state of the first model and the second-order simulation of the second model, it may give my the correct business cycle cost?

Try modeling the shocks without the log, e.g. https://github.com/JohannesPfeifer/DSGE_mod/blob/master/Basu_Bundick_2017/Basu_Bundick_2017.mod

Thank you for your reply! Just a quick follow up question on modeling shocks without the log. If I model TFP (variable A) as A=(1-rho)Ass+rhoA(-1)+eps_A, where eps_A is the TFP shock, would it cause problems in my second order simulation as there’s the possibility that the TFP can be negative? My understanding of people usually modeling shocks with log is to prevent the TFP becoming negative, and also makes the interpretation of IRFs easier as it measured in percentage changes rather than in levels.

See e.g.

The same applies for any polynomial approximation. Thus, whether you use the log or not does not really matter. At first order the two specifications are even equivalent.

Regarding percentage deviations: if your mean `Ass` is 1, then you are getting percentage deviations as the log would imply a Jacobian transformation with `1/1=1`. Put differently, if `A` is above the mean of 1 by 0.01 units, this is also 1 percent.