(3) Suppose that β(1 + r) = 1 still holds. Now consider a small but positive value for σ. Under what condition on u do we get positive optimal savings a_2 > 0? (6%)

4. Consider the following two-period problem. The representative household utility function is $$\log(c1) + \beta \log(c2),$$ where $\beta \in (0,1)$ is the discount factor, $c_t$ is consumption in period $t, \forall t = 1, 2$. The budget constraints are $$c1 + a2 = y_1$$ and $$c2 = y2 + a_2(1 + r),$$ where $r$ is the exogenous net real interest rate, $a2$ is net savings at the end of period 1, and $yt$ is a deterministic endowment at period $t$, $\forall t = 1, 2$. The household faces a financial constraint: $$a_2 \ge -\ell,$$ where $\ell \ge 0$ represents the exogenous credit limit. (1) Define $\lambda$ as the Lagrangian multiplier of the financial constraint. Formulate the Lagrangian equation and derive the intertemporal Euler equation. Express the condition in terms of $c1, c2$ and $\lambda$. Please ensure that your representation always implies $\lambda \ge 0$. (8%) (Hint: You can define other Lagrangian multipliers if necessary)

(3) Suppose that $\beta(1 + r) = 1$ still holds. Now consider a small but positive value for $\sigma$. Under what condition on u do we get positive optimal savings $a_2 > 0$? (6%)

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