# Labor capacity utilization shock attached to supply of working hours

Dear Johannes,
In my DSGE model:
Utility: U=log(C)+log(1-VL)
where U is utility, C is consumption, L is working hours (maximum of working hours is 1), V is labor utilization shock, in this setting 1-V
L should be smaller than 1.
Labor utilization shock follows an autoregressive process:
V=Vbar+Rho* (V(-1)-Vbar)+Epsilon
where V is labor utilization shock, Vbar is steady state of labor utilization shock, Rho is autoregressive parameter, Epsilon is labor utilization innovation and follows N(0, SigmaV^2)
SigmaV is the standard deviation of labor utilization innovation.
In this setting, because L represents working hours supply, L is defined to be smaller than 1 (only a proportion of hours will be devoted to work),
However, after I attach labor utilization shock V to working hours supply L,
will VL still be smaller than 1 or is there a possibility that VL may be greater than 1?

Thank you very much and look forward to hearing from you.
Best regards,
Jesse

Conceptually, VL will never reach 1 due to the asymptote in log-utility. The household would receive infinite disutility.
The problem is the solution technique. A polynomial approximation (perturbation) can always move beyond any bound.

Thank you for your answer. Could the solution method embedded in Dynare handle this? Will Dynare ensure VL never reach above 1?
Thank you again and look forward to hearing from you.

As I wrote above, this cannot be guaranteed with `stoch_simul` doe to the use of polynomial approximations. But often, this is not actually a problem in simulations.

Dear Johannes,
Thank you for your helpful and inspirational guidance. If I set the size of labor intensity shock V to 2 instead of infinity, will this problem be alleviated? If prior standard deviation of labor intensity shock is small, the chance of 1-VL being negative is very small and can be neglected?
Thank you again and look forward to hearing from you.
Best regards,
Jesse

Having smaller shocks indeed alleviates the problem. Usually, you can ex-post check whether violations occurred.

Dear Johannes,
Thank you very much! Could I know how to ex-post check whether violations occur?
Thank you very much and look forward to hearing from you.
Best regards,
Jesse

Simply simulate the model for a couple of hundred `periods`