In Mankiw's Macreoconomics, how come capital is assumed to be fixed while investment is assumed to be variable?

Question

I'm looking at Greg Mankiw's Macroeconomics (7th edition), at the model presented in chapter 3:

$Y = C + I + G,$

where $Y$ is total output, $C = C(Y-T)$ is consumption, $T$ are net taxes (fixed by policy), $I = I(r)$ is investment, $r$ is the interest rate, and $G$ are government purchases (fixed by policy). The model further makes the assumption that

$Y = F(K,L),$

where $K,L$ are the factors of production, capital and labor, and are assumed to be fixed, so that $Y$ is therefore fixed as well. Using the fact that $Y,T,G$ are fixed, it follows that $C$ is also fixed and thus saving,

$S = Y-C-G,$

is fixed. But the model entails that $S = I(r)$ and therefore any movement in saving translates into a movement in investment (and thus the interest rate). How is it possible that investment is considered variable, if investment is the purchase of new capital goods and capital is taken as fixed? If investment increases, then so should capital.

Answer 1

It seems that Mankiw has implicitly assumed that all the investment is used to replace the depreciated capital. See the second sentence in the Investment section of Chapter 3-3:"Firms buy investment goods to add to their stock of capital and to replace existing capital as it wears out. "

Usually, the law of motion of capital is written as \begin{equation} K_{t+1}=I_{t+1}+(1-\delta)K_{t} \end{equation} where $\delta$ is the depreciation rate of capital. By assuming capital to remain at a fixed level $\overline{K}$, I believe that Mankiw is trying to only discuss the steady state in which capital is remained at a constant level, while avoiding introducing any more details. Since this is just the beginning part of textbook, everything is made as simple as possible. There is even no time subscripts for variables!