# Relationship between interest rate and rental price of capital

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?

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.

• 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 Apr 11 '17 at 3:24

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.