Suppose we have an economy with a representative consumer with the following utility function: $$U=\sum_{t=0}^\infty\beta^t\frac{c_t^{1-\sigma}}{1-\sigma} $$ There is no uncertainty and the household can invest in a single risk-free asset bearing a fixed one-period rate of return $R > 1$. Its budget constraint is $$c_t+R^{-1}a_{t+1}\leq y_t+a_t$$ with $a_0$ given. Its borrowing constraint is $$a_{t+1}\geq \bar{a}_{t+1}$$ For simplicity, assume $R\beta = 1$. The stream of endowments $$y_t=\lambda_t$$ where $1 < \lambda < R$ so that $\sum_{t=0}^\infty \beta^ty_t$ holds. The borrowing constraint $\bar{a}_{t+1}$ is either a borrowing limit given by $\bar{a}_{t+1}=0$ in the case of the no-borrowing constraint or given by $\bar{a}_{t+1}=\tilde{a}_{t+1}$ in the case of the natural-borrowing constraint.

Now I guess the maximization problem of the household and the Euler equation would look something like this: $$\text{max}\ \sum_{t=0}^\infty\beta^t\frac{c_t^{1-\sigma}}{1-\sigma} $$ $$\text{s.t}\ c_t+R^{-1}a_{t+1}\leq y_t+a_t $$ $$a_{t+1}\geq \bar{a}_{t+1} $$

FOC w.r.t $\{c_t\}$ : $\beta^t c_t^{-\sigma}=\lambda_t$

FOC w.r.t $\{c_{t+1}\}$ : $\beta^{t+1} c_t^{-\sigma}=\lambda_{t+1}$

FOC w.r.t $\{a_{t+1}\}$ : $R^{-1}\lambda_t=\lambda_{t+1}+ \mu_t$ where $\mu_t$ is the multiplier on the borrowing constraint.

The EE is : $u'(c_t)=\underbrace{R\beta}_{=1}u'(c_{t+1})+\mu_t$ this is for the non-binding case but the situation when the borrowing limit is binding. $\rightarrow$ I guess we would have $c_{t+1}>c_t$, so consumption is a nondecreasing sequence.

My question is if we assume $a_0=0$ and the agent is exposed to the no-borrowing constraint. How can we prove that the economy will always be borrowing constrained? And if $\tilde{a}_{t+1}$ is equal to the natural debt limit (present discounted value of future endowment stream), why is now the case that the economy will never be borrowing constraint?


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