How to derive the standard euler equation from the Expanding Variety Model In the standard expanding variety model where the representative household faces the following CRRA utility function
$$
\int_{0}^{\infty} \exp(-\rho t) \frac{C(t)^{1-\theta}-1}{1-\theta}dt
$$
and final goods that is produced competitively with the following production
$$
Y(t) = \frac{1}{1-\beta} \left( \int_{0}^{N(t)}x(v,t)^{1-\beta}dv\right)L^{\beta}
$$
with the following resource constraint
$$
C(t) + X(t) + Z(t) \leq Y(t)
$$
where $X(t)$ is the total spending on machines, and $Z(t)$ is the total expenditure on research and development.
The innovation possibilities frontier takes the form
$$
\dot{N}(t) = \eta Z(t)
$$
The intermediary goods firm faces the following problem where
$$
r(t) V(v,t) - \dot{V}(v,t) = \pi(v,t)
$$
The thing I don't understand about the model is how the Euler Equation is derived. In almost everywhere I look the Euler Equation is the standard
$$
\frac{\dot{C}(t)}{C(t)} = \frac{1}{\theta}(r(t)-\rho)
$$
I don't understand how you would derive this equation.
The way you would go about solving this problem is as the ChinG said is by setting up the Hamiltonian. In this case this is:
$$\mathcal{H}:e^{-\rho t} \frac{C(t)^{1-\theta}-1}{1-\theta}+\mu(t)\left[Y(t)-C(t)-X(t)-Z(t)\right]$$
Taking the first order condition for this problem we get:
$$\frac{\partial \mathcal{H}}{\partial C(t)}:e^{-\rho t}C(t)^{-\theta}-\mu(t)=0 $$
or
$$e^{-\rho t}C(t)^{-\theta}=\mu(t)$$
Taking the natural logs of this equation and differentiating with respect to time we get:
$$\ln\left[\frac{\partial \mathcal{H}}{\partial C(t)}\right]:-\rho t -\theta \ln [C(t)]=\ln[\mu(t)]$$
$$\frac{\partial\ln\left[\mathcal{H}_{C(t)}\right]}{\partial t}:-\rho-\theta\frac{\dot{C}(t)}{C(t)}=-\frac{\dot{\mu}(t)}{\mu(t)}$$
Rearranging the above and noting that $\frac{\dot{\mu}(t)}{\mu(t)}=r(t)$
we get:
$$\frac{\dot{C}(t)}{C(t)}=\frac{1}{\theta}(r(t)-\rho)$$
Which is the Euler equation.
Hope this helps.