# The TYPICAL law of motion of capital

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?

## 2 Answers

$$k_t$$ 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: $$k_t-k_{t-1}(1-\delta)$$.

• Hi Regio, just to clarify. When you mean $i_t$ is the amount that must have been invested in order to obtain $k_t$, so what is the timing? When does today's investment happen today? For example, consider the capital to be the cake from the usual cake-eating problem. So today, I have the cake that is left-over after the rotten part available today, which is the term with $\delta$. In this case, how does the investment form? Do we put the remaining non-rotten cake part into the baking machine (i.e. linear technology)? – Frank Swanton Jun 4 '19 at 6:52
• I guess what I am primarily confused is that you invest today, but doesn't the firm reap the benefit of this investment tomorrow's tomorrow capital stock? If you plant the apple tree from the seeds available today, the farmer would reap the benefit of the fruition tomorrow, no? I am confused with the timing of the events, which dictate the law of motion. – Frank Swanton Jun 4 '19 at 7:23
• This question is really different from the one in your OP. – Giskard Jun 4 '19 at 7:57
• @Giskard the recursive dynamics, and cake eating, timing of investment are exactly what I asked. – Frank Swanton Jun 4 '19 at 16:53
• You have it in your setup "capital production is instantaneous", so your equations assume that investment becomes capital instantaneously. An alternative way to write the law of motion would be $k_{t+1}=(1-\delta)k_t +i_t$. This will assume that capital takes one period to be produced, so that you invest today and tomorrow you have capital (as your intuition is probably thinking of how things evolve). However your current setting is more akin to a situation where you buy a new plant and it can almost immediately be used for production in thee same period. – Regio Jun 4 '19 at 18:21

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 $$t$$.

• Thanks for the response! – Frank Swanton Jun 5 '19 at 0:20