Hello, I am a bit confused about the timing and expectation operator in dynare and how it is related to model local variables,. Suppose I have the following utility function u\left(c_{t}\right)=\frac{\left(c_{t}-hc_{t-1}\right)^{1-\gamma}}{1-\gamma} and an asset (not risk free) with R_{t}^{p} gross return for time t for asset purchased at time t-1. Then Euler eqn \left(c_{t}-hc_{t-1}\right)^{-\gamma}-\beta h\left(c_{t+1}-hc_{t}\right)^{-\gamma}=\beta\mathbb{E}_{t}\left[R_{t+1}^{p}\left(\left(c_{t+1}-hc_{t}\right)^{-\gamma}-\beta h\left(c_{t+2}-hc_{t+1}\right)^{-\gamma}\right)^{-\gamma}\right]
Which of the following is correct in dynare if I am doing 2nd - 3rd order simlations as in Basu-Bundick
Case-1 (with model-local variables)
model_local_variable
uc mu_{t} (long_name=‘marginal utility at time t’)
end;
model
#uc=( (c-h*c(-1) -betta*h*(c(+1)-h*c) )^(-gamma);
uc=betta*uc(+1)*Rp(+1);
end;
Case-2 (with model-local variables)
model
#util=c(+1)-h*c;
#uc=( util(-1) -betta*h*util )^(-gamma);
#uctp1=(( util-betta*h*util(+1) )^(-gamma))^(-gamma);
uc=betta*uc(+1)*Rp(+1);
end;
Case-3 (without model-local variables)
model
uc=(c-h*c(-1))^(-gamma)-betta*h*(c(+1)-h*c)^(-gamma);
uc=betta*uc(+1)*Rp(+1);
end;
Thank you