Interest/Rental Rate is today's or tomorrow's MPK?

Question

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.

Answer 1

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.