# Natural borrowing/debt limit and other borrowing constraints

When confronted with the simple household consumption maximization problem under uncertainty (and with Arrow security sequential trading)

$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\sum_{t=0}^{\infty}\sum_{s^t\in S^t}\beta^tprob(s^t) u(c_t(s^t))$$

$$\text{s.t.}\,\,\,c_t(s^t)+\sum_{s_{t+1}|s^t}Q_t(s^t,s_{t+1})a_{t+1}(s^t,s_{t+1})=y_t(s^t)+a_t(s^t)$$ $$-a_{t+1}(s^t,s_{t+1})\le A_{t+1}(s^t,s_{t+1})$$

$$a_0(s^t)=0$$

we call $A_{t+1}(s^t,s_{t+1})$ the natural debt/borrowing limit and define it as the loosest possible limit. But what exactly does this mean? And what form does it take?

Moreover, in the FOCs for $a_{t+1}(s^t,s_{t+1})$ we cancel the Lagrangian multiplier of the second constraint, it doesn't bind because of the Inada conditions on the function of utility. Is the intuition here that even when you borrow an infinitesimal amount $\epsilon$, if it is repeated enough times you'd reach a point where the debt stock is so high that consumption needs to be zero (not negative because above we implicitly assume $c_t\ge0$ right?) to repay it, which would imply an infinite (?) loss of utility, and thus you shouldn't borrow at all?

Finally, what is the relation with lower asset bound, no-Ponzi scheme and no-default constraint? How do their implications differ?