3. Consider a representative consumer who lives for two periods ($t = 0,1$), and does not leave any bequests. The consumer chooses consumption in each period, $C_0$ and $C_1$, to maximize lifetime utility, taking income in each period $M_0$ and $M_1$, the real interest rate $r$, and the time preference factor $\beta \in (0,1)$ as given. The utility function is:

$$U(C_0, C_1) = \ln C_0 + \beta \ln(C_1)$$

The consumer can borrow or lend at the real interest rate $r$ and therefore faces the following intertemporal budget constraint:

$$C_0 + \frac{C_1}{1+r} = M_0 + \frac{M_1}{1+r}$$

Please answer the following questions:

(a) Derive the Euler equation and discuss the economic intuition behind it. (5%)