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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.