The TYPICAL law of motion of capital

Question

Consider a typical macro firm set-up where the capital can be produced from output goods using a linear technology. It costs some X units of output good to make a unit of the capital, and the capital production is instantaneous. For simplicity, let the numeraire be the output good.

The typical law of motion of the capital is (with a geometric depreciation rate):

$k_t=k_{t-1}(1-\delta)+i_t$

The firm's technology meets the typical strictly concave, increasing, Inada, and TFP conditions: the amount of capital invested by a firm at time $t$ produces output $y_t$ at time $t+1$ where $A>0$ denotes the TFP.

$y_{t+1}=Af(k_t)$

$\textbf{My question}:$ How do you understand the $\textit{investment TIMING}$ in the law of motion of capital? I read it as the capital that is available today is the sum of what you have carried over from yesterday as depreciated capital plus how much you invest. But how do you interpret this rearrangement:

$i_t=k_t-k_{t-1}(1-\delta)$

So if I stand at $t$, then I have the following sources for capital, correct?:

(1) $(1-\delta)k_{t-1}$: This simply what I carried over as depreciated capital

(2) $k_t$: The capital I obatained from my linear production technology using the capital I used yesterday. (The remainder of unused capital is depreciated and carried over as in (1))

But why is (2)-(1), today's firm investment?

Answer 1

$k_t$ is your capital after investment, so if you subtract the capital you carried over from last period, $k_{t-1}(1-\delta)$, you obtain the amount that must have been invested in order to have $k_t$ of capital today.

Remember that $k$ is a stock variable, while $i$ is a flow variable. This is why the flow $i_t$ is the difference between the current and the former stock: $k_t-k_{t-1}(1-\delta)$.

Answer 2

According to how you've defined the firm's LOMOC, investment produces capital simultaneously. That is, $i_t = k_t - (1-\delta)k_{t-1}$ defines how much capital the firm created in period $t$. If you want a delayed investment function, you could simply set the LOMOC to $k_t = (1-\delta)k_{t-1} + i_{t-1}$. In this setting, firms invest output into capital in period $t-1$, then realize their investment in period $t$.