# Euler Equation - small open economy

How would I find the Euler equation for: $$U=\sqrt{C_1}+\beta{\sqrt{C_2}}$$, where $$\beta=1/(1+r)$$

• try to be more specific in your question instead of just asking for the solution and try to provide your own attempt to the problem as well May 30, 2023 at 11:50

Consider a standard utility maximization problem of the form: \begin{align} \max_{\{C_t\}_{t=1}^{\infty}} \quad & \sum_{t=1}^{\infty}\beta ^tu(C_t) \\ \textrm{s.t.} \quad & \sum_{t=1}^{\infty}\frac{C_t}{(1+r)^{t-1}}\leq w \end{align} where, $$\beta \in (0,1)$$ , $$u$$ is strictly increasing and convex i.e., $$u'(.)>0, \; u''(.)<0$$ , and $$w$$ denotes the lifetime wealth
Note that the constraint will bind at optimum given our assumption about $$u(.)$$, therefore we can use the Lagrangian Method to solve the above. The Lagrangian function is given by: $$\mathcal{L}=\sum_{t=1}^{\infty}\beta ^tu(C_t)-\lambda\left (\sum_{t=1}^{\infty}\frac{C_t}{(1+r)^{t-1}} - w \right)$$
The FOC tangency condition for some arbitrary $$C_t, C_{t+1}$$ are: \begin{align} & \frac{\partial \mathcal{L}}{\partial C_t} = \beta^tu'(C_t)-\lambda \frac{1}{(1+r)^{t-1}} \overset{set}=0 \tag1\\ & \frac{\partial \mathcal{L}}{\partial C_{t+1}}=\beta^{t+1}u'(C_{t+1})-\lambda \frac{1}{(1+r)^{t}} \overset{set}=0 \tag2 \end{align}
From $$(1)$$ and $$(2)$$: $$\frac{\beta^{t}u'(C_{t})}{\beta^{t+1}u'(C_{t+1})}=\frac{(1+r)^t}{(1+r)^{t-1}}$$
The above gives us the Euler Equation as: $$\boxed{u'(C_t)=\beta(1+r) u'(C_{t+1})} \quad \forall \; t\geq 1$$