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Below is the sketch of the solution. 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+...
First, the fact that $MRS=\frac{1}{4}$ does not tell you by itself that the consumer will only buy $x_2$. We need to go back to the 2nd Gossen's law: $\frac{Um{x_1}}{p_1}=\frac{Um{x_2}}{p_2}$ (this is from where the $MRS$ comes by the way) which is not the case since $\frac{1}{3}<\frac{4}{8}=\frac{1}{2}$. So you are right the consumer in the first place ...