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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?

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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.

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  • $\begingroup$ 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. $k(t+1) = \frac{s(t)}{1+n}$. $\endgroup$ – Chris Cheung Apr 11 '17 at 3:24
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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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