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, 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.

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.