2
$\begingroup$

I recently posted and have since deleted a question regarding Exercise 2.1 in McCandless's "The ABCs of RBCs." The exercise involves deriving a first-order difference equation describing aggregate capital motion in a overlapping two-period generations model, and I could not solve it by using the definitions of $w_t$, $w_{t+1}$, and $r_t$ that I am accustomed to. It turns out that the exercise is easy to solve if you use the relationship $r_t = F_K(K(t+1), H(t+1))$ where $K$ and $H$ are aggregate capital and labor, respectively.

The reason I had so much difficulty with this exercise is that all of the economic theory I know dictates that $r_t = F_K(K(t), H(t))$, i.e. that today's interest/rental rate should equal today's marginal product of capital. Even this particular textbook says so a couple pages before the exercise in question. However, using that relationship yields an equation with no analytic solution for $K(t+1)$. If you use $r_t = F_K(K(t+1), H(t+1))$, i.e. that today's interest/rental rate is equal to tomorrow's marginal product of capital, the problem is solved with just a couple steps of algebra. But you can't just do that, can you?

If you need more context, I'm sorry that I deleted my previous question, but I had effectively shown 90% of the solution to the relevant exercise and I know that this site is not meant for giving homework solutions away. I hope someone can enlighten me on this particular stumbling block I've encountered, however.

$\endgroup$
2
  • 1
    $\begingroup$ When you are talking about today's interest rate, what exactly do you mean? Is it due today for capital rented yesterday, or is it due tomorrow for capital rented today? $\endgroup$ – Giskard May 18 '17 at 17:42
  • $\begingroup$ By today's interest rate, I mean the interest rate paid tomorrow for loans taken out today. My problem appears to have arisen from me setting that rate equal to today's rental rate, i.e. I have set $r_t = \text{rental}_t$. Instead, I should have $r_t = \text{rental}_{t+1}$ $\endgroup$ – economicist May 18 '17 at 21:35
1
$\begingroup$

I found the problem, and now I feel dumb but I suppose I should leave the question up so people can know the potential pitfall and prepare for it better than I did.

The problem arose from me misreading (and clearly misunderstanding) the equilibrium conditions for the problem, where it states that the interest rate $r_t$ on loans issued today and paid back tomorrow is equal to $\text{rental}_{t+1}$, the rental rate on capital used tomorrow, which is of course equal to tomorrow's marginal product of capital. I had my problem set up with $r_t = \text{rental}_{t+1}$, which is incorrect. This means that yes, we should have $r_t = \text{rental}_{t+1} = F_K(K(t+1), H(t+1))$, which makes the exercise a simple algebra problem. I just had my subscripts all messed up!

Here's why $r_t = \text{rental}_{t+1}$ (summarized from text): If $r_t < \text{rental}_{t+1}$, then under perfect foresight (an assumption of the basic model in the given chapter), people will attempt to borrow infinitely to capture the profit of $\text{rental}_{t+1} - r_t$, and this condition cannot hold in equilibrium. On the other hand, if $r_t > \text{rental}_{t+1}$, then nobody will want to hold capital until the next term, instead attempting to lend all of their funds out.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.