# Small Open Economy Without Government

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?