# Welfare cost of business cycle stochastic simulation

I am doing research on the welfare cost of business cycle using two-agent DSGE model.

I compare lifetime welfare of fluctuating economy(where aggregate shock prevails) and the lifetime welfare of non-fluctuating economy(consuming steady state level consumption forever). Since I am using unconditonal welfare cost measure, I specify `V = U + beta*V(+1)` in the mod file, simulate economy for 100,000 periods(with order=2), and take simulated mean(oo_.mean) as a lifetime welfare for fluctuating economy. For the steady state level of consumption, I solved it manually by hands and calculate welfare analytically. Lastly I compare the two welfare estimates(Since I specified log consumption utility function, there exists closed form solution for welfare cost.).

Question is that

1. Conceptually, am I doing right for calcuating welfare cost of business cycle using second order approximation?
2. I specified shock as `A = (1-rhoA)*As + rhoA*A(-1) + sA*eA` for TFP shock and found out that welfare cost and simulated means are significantly different when I use above shock process and `A = (1-rhoA)*As + rhoA*A(-1) - sA*eA`. This is somewhat counter-intuitive since if eA are drawn from symmetric normal distribution + or - sign in front of eA should not cause large difference.( I checked this phenomenon by changing seeds of simulation and cound observe same pattern for all cases.) . I wonder if there exist right way to specify shock process, simulation period, approximation order in calculating welfare cost( should I use `A = (1-rhoA)*As + rhoA*A(-1) + sA*eA` only for stochastic simulation?).