Dear Johannes,
In my DSGE model, household utility function is U_t=ln(C_t)+\rho*ln(1-H_t)
where U_t is utility, C_t is consumption, H_t is hours worked, (1-H_t) is leisure, \rho is leisure weight.
Budget constraint C_t*P_t=H_t*W_t
I need to derive marginal rate of substitution (MRS_t) of leisure (1-H_t) for consumption C_t,
MRSt =(change in (1-Ht))/(change in Ct)=(partial derivative of Ut to Ct)/(partial derivative of Ut to (1-Ht))=(partial derivative of Ut to Ct)/[(partial derivative of Ut to Ht)*(partial derivative of Ht to (1-Ht))]=(1/C_t)/[-\rho/(1-H_t)* (-1)]=marginal utility of consumption/marginal disutility of labor service.
Is this correct?
Thank you very much and look forward to hearing from you.
Best regards,
Jesse
Yes, that looks correct.