<p>The equation is exactly the same,</p>

<p>$$r(t) = R(t) - \delta$$</p>

<p>Now, set $\delta =1$ per assumptions, to obtain</p>

<p>$$r(t) = R(t) - 1 \implies 1+r(t) = R(t)$$</p>

<p>which reasonably continues to say that the accepted rental rate of capital $R(t)$, covers the depreciation rate plus $r(t)$ which is also the <em>net</em> return to capital.</p>

CC BY-SA 3.0
2017-04-11T03:24:08.997Z
Oh... I wrongly associate the overlapping generation model with no depreciation of capital - $\delta = 0$, but it should be full depreciation of capital - $\delta = 1$ as capital will then evolve according to $K(t+1) = (1-\delta)K(t) + s(t)L(t) = s(t)L(t)$ i.e. $k(t+1) = \frac{s(t)}{1+n}$.
Chris Cheung
CC BY-SA 3.0
2017-04-10T20:17:37.680Z
2017-04-10T20:17:37.680Z
Alecos Papadopoulos
<p>As interest rates go up, mortgage affordability goes down and therefore there is an increased demand in rents (offset by a lower demand in buying homes). Therefore rental rates have a higher relationship to interest rate than 1:1. </p>

CC BY-SA 4.0
2018-11-05T18:40:05.713Z
2018-11-06T02:47:22.080Z
2018-11-06T02:47:22.080Z
paolo ruggieri
<p>In Daron Acemoglu's Introduction to Modern Economic Growth (2009), P.32, it is stated that given assumption of exponential depreciation at the rate $\delta$ and normalization of price of final good to be 1, $r(t) = R(t) - \delta$ where $r(t)$ is interest rate, $R(t)$ is rental price of capital and $\delta$ is depreciation rate. The reason, which I agree, is that:</p>

<blockquote>
  <p>A unit of final good can be consumed now or used as capital and rented to firms. In the latter case, a household receives $R(t)$ units of good in the next period as rental price for its savings, but loses $\delta$ units of its capital holdings, since $\delta$ fraction of capital depreciates over time. Thus the household has given up one unit of commodity dated t-1 and receives $1 + r(t) = R(t) + 1-\delta$ units of commodity dated t.</p>
</blockquote>

<p>But, on P.330, when talking about overlapping generations model where $\delta = 0$, it is stated in equation (9.3): $1 + r(t) = R(t)$. Shouldn't it be $r(t) = R(t)$? I cannot find this is a typo like from <a href="http://www.econ.ku.dk/okocg/VV/VV-2015/Lectures%20and%20lecture%20notes/Errata-to-Acemoglu-book-VV-2015.pdf" rel="nofollow noreferrer">http://www.econ.ku.dk/okocg/VV/VV-2015/Lectures%20and%20lecture%20notes/Errata-to-Acemoglu-book-VV-2015.pdf</a>. Do I miss something? </p>

<macroeconomics><economic-growth>
Relationship between interest rate and rental price of capital
Chris Cheung
# Relationship between interest rate and rental price of capital

(Asked by: username_0 on 2017-04-10)

In Daron Acemoglu's Introduction to Modern Economic Growth (2009), P.32, it is stated that given assumption of exponential depreciation at the rate $\delta$ and normalization of price of final good to be 1, $r(t) = R(t) - \delta$ where $r(t)$ is interest rate, $R(t)$ is rental price of capital and $\delta$ is depreciation rate. The reason, which I agree, is that:

> A unit of final good can be consumed now or used as capital and rented to firms. In the latter case, a household receives $R(t)$ units of good in the next period as rental price for its savings, but loses $\delta$ units of its capital holdings, since $\delta$ fraction of capital depreciates over time. Thus the household has given up one unit of commodity dated t-1 and receives $1 + r(t) = R(t) + 1-\delta$ units of commodity dated t.

But, on P.330, when talking about overlapping generations model where $\delta = 0$, it is stated in equation (9.3): $1 + r(t) = R(t)$. Shouldn't it be $r(t) = R(t)$? I cannot find this is a typo like from <http://www.econ.ku.dk/okocg/VV/VV-2015/Lectures%20and%20lecture%20notes/Errata-to-Acemoglu-book-VV-2015.pdf>. Do I miss something?

**Tags:** macroeconomics, economic-growth


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## 2 answers

### Answer by username_1 on 2017-04-10. Score: 1 | Accepted answer

The equation is exactly the same,

$$r(t) = R(t) - \delta$$

Now, set $\delta =1$ per assumptions, to obtain

$$r(t) = R(t) - 1 \implies 1+r(t) = R(t)$$

which reasonably continues to say that the accepted rental rate of capital $R(t)$, covers the depreciation rate plus $r(t)$ which is also the *net* return to capital.

> **Comments:**
> **username_0**: Oh... I wrongly associate the overlapping generation model with no depreciation of capital - $\delta = 0$, but it should be full depreciation of capital - $\delta = 1$ as capital will then evolve according to $K(t+1) = (1-\delta)K(t) + s(t)L(t) = s(t)L(t)$ i.e. $k(t+1) = \frac{s(t)}{1+n}$.  

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### Answer by username_2 on 2018-11-05. Score: 0

As interest rates go up, mortgage affordability goes down and therefore there is an increased demand in rents (offset by a lower demand in buying homes). Therefore rental rates have a higher relationship to interest rate than 1:1.


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