<p>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.</p>

<p>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 <em>tomorrow,</em> which is of course equal to <em>tomorrow's</em> 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!</p>

<p><strong>Here's why</strong> $r_t = \text{rental}_{t+1}$ <strong>(summarized from text):</strong> If $r_t &lt; \text{rental}_{t+1}$, 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 $r_t &gt; \text{rental}_{t+1}$, then nobody will want to hold capital until the next term, instead attempting to lend all of their funds out.</p>

CC BY-SA 3.0
2017-05-18T21:40:17.413Z
2017-05-18T21:49:48.783Z
2017-05-18T21:49:48.783Z
economicist
<p>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.</p>

<p>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 <em>today's</em> 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 <em>tomorrow's</em> marginal product of capital, the problem is solved with just a couple steps of algebra. But you can't just do that, can you?</p>

<p>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.</p>

CC BY-SA 3.0
2017-05-18T17:42:37.693Z
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?
Giskard
CC BY-SA 3.0
2017-05-18T21:35:33.590Z
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 $r_t = \text{rental}_{t+1}$
economicist
<interest-rate><olg>
Interest/Rental Rate is today's or tomorrow's MPK?
economicist
# Interest/Rental Rate is today's or tomorrow's MPK?

(Asked by: username_0 on 2017-05-18)

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.

**Tags:** interest-rate, olg
> **Comments:**
> **username_1**: 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?  
> **username_0**: 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 $r_t = \text{rental}_{t+1}$  

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## 1 answers

### Answer by username_0 on 2017-05-18. Score: 1 | Accepted answer

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.


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