# Optimal consumption for infinite number of periods and exogenous income

I have the following optimization problem:

$$\max_{\{c_t, s_{t+1}\}} \Pi_{t=0}^\infty c_t^{\beta^t}$$

$$\text{subject to } \space c_t + s_{t+1} = y_t + (1 + r) s_t \text{ and } s_0 = 0$$

How do I find the optimal consumption level for each time period $$t$$?

Here's what I have so far:

Let $$p_t = \frac{1}{(1 + r)^t}$$

Then, from the budget constraint $$p_{t+1} s_{t+1} - p_{t+1} s_t = p_{t+1}(y_t - c_t)$$.

If we sum over $$T$$ periods: $$(p_1 s_1 - p_0 0) + (p_2 s_2 - p_1 s_1) + ... (p_{T+1} s_{T+1} - p_T s_T) = \sum_{t=0}^\infty p_{t+1}(y_t - c_t)$$

If I use a natural logarithm transformation on the utility function and assume $$\lim_{T\to\infty} p_T s_T = 0$$ (no ponzi schemes allowed), I can rewrite the optimization problem as:

$$\max_{\{c_t\}} \sum_{t=0}^\infty \beta^t\ln(c_t)$$

$$\text{subject to} \sum_{t=0}^\infty p_{t+1} c_t = \sum_{t=0}^\infty p_{t+1} y_t$$

Hence, my lagrangian function should look like (unless I'm missing something):

$$\mathcal{L} = \sum \beta^t\ln(c_t) - \lambda(\sum_{t=0}^\infty p_{t+1} c_t - \sum_{t=0}^\infty p_{t+1} y_t) + \mu c_t$$

How do I get $$c_t^* \space \forall \space t$$?

From the Lagrangian $$\max _{\left\{c_{t}, s_{t+1}\right\}} \Pi_{t=0}^{\infty} c_{t}^{\beta^{t}} - \Pi_{t=0}^{\infty}\lambda_t(c_{t}+s_{t+1}-y_{t}-(1+r) s_{t}) \\ \text{s.t.} \ s_0 = 0, c_t > 0$$, you can get the Euler equation $$\frac{\beta_{t+1}c_{t+1}^{\beta_{t+1}-1}}{\beta_{t}c_{t}^{\beta_{t}-1}}= \frac{\beta_{t+1}\left[y_{t+1}+(1+r)s_{t+1}-s_{t+2}\right]^{\beta_{t+1}-1}}{\beta_{t}\left[y_{t}+(1+r)s_{t}-s_{t+1}\right]^{\beta_{t}-1}} = \frac{1}{1+r}$$, and then you can write $$s_{t}^{*} \forall t>1$$ as a function of $$s_1$$ with the initial condition $$s_0$$. By using the transversality condition as a terminal condition: $$\lim _{t \rightarrow \infty} \lambda_ts_{t+1}= \lim _{t \rightarrow \infty} {\beta_{t}\left[y_{t}+(1+r)s_{t}-s_{t+1}\right]^{\beta_{t}-1}}s_{t+1} =0$$, you can solve for $$\left\{s_{t}\right\}_{t=1}^{\infty}$$ forwards by guessing an initial value for $$s_1$$ and yielding the sequence that does not violate the TVC or the nonnegativity constraint on consumption. And once you get $$s_{t}^{*} \forall t$$ you get $$c_{t}^{*} \forall t$$.
• Sorry, I do not understand this answer. Are you saying $s_0 = 0$ implies that the agent doesn't save any money in $t=0$? That is, $c_0 + s_1 = c_0 = y_0$? Commented Sep 15, 2021 at 13:56