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

CC BY-SA 4.0
2019-06-04T16:37:45.783Z
2019-06-04T16:37:45.783Z
NBm424
<p><span class="math-container">$k_t$</span> is your capital after investment, so if you subtract the capital you carried over from last period, <span class="math-container">$k_{t-1}(1-\delta)$</span>, you obtain the amount that must have been invested in order to have <span class="math-container">$k_t$</span> of capital today. </p>

<p>Remember that <span class="math-container">$k$</span> is a stock variable, while <span class="math-container">$i$</span> is a flow variable. This is why the flow <span class="math-container">$i_t$</span> is the difference between the current and the former stock: <span class="math-container">$k_t-k_{t-1}(1-\delta)$</span>.</p>

CC BY-SA 4.0
2019-06-04T06:52:16.227Z
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
CC BY-SA 4.0
2019-06-04T07:23:49.250Z
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
CC BY-SA 4.0
2019-06-04T07:57:26.717Z
This question is really different from the one in your OP.
Giskard
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2019-06-04T16:53:21.097Z
@Giskard the recursive dynamics, and cake eating, timing of investment are exactly what I asked.
Frank Swanton
CC BY-SA 4.0
2019-06-04T18:21:54.327Z
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
CC BY-SA 4.0
2019-06-07T00:51:46.513Z
@Regio Hi Regio, just a quick related question. So in the set-up of the instantaneous capital production is assumed, does the investment occur like this? $\rightarrow$ When you stand today, you have the undepreciated capital carried over from yesterday, so you use this as input to the production and produce output. With this output, you produce today's capital, $k_t$. Hence, the available investment is this $k_t$ minus $k_{t-1}(1-\delta)$. Is this correct?
Frank Swanton
CC BY-SA 4.0
2019-06-07T02:29:04.943Z
I don't think so. I mean it really depends on how the person setting up the model chooses the timing. However, most likely, the timing is as follows. Investment, $i_t$ is chosen before production (perhaps from savings or borrowing money, etc). This creates $k_t$ along with the un-depreciated capital from yesterday. Finally, production occurs. I find this setup odd, I don't think there is a universal way to do the timing, but I feel that I've seen more often the assumption $k_{t+1}=(1-\delta)k_t+i_t$. I'm not sure why you think the one you presented is THE typical law of motion.
Regio
CC BY-SA 4.0
2021-09-20T09:50:20.250Z
@Regio Hope things are well with you. Just a quick clarification on the investment. The investment is basically the amount of output you put aside from consumption to put into the following period (or contemporaneous) production, correct? I just want to clarify the term "investment" in that it simply represents the amount put aside not consuming, right? It has nothing to do with the actual production technology $f(.)$?
Frank Swanton
CC BY-SA 4.0
2021-09-22T12:46:26.087Z
That’s correct, we typically think that investment is the part of the income that did not go to consumption (either domestic or foreign), and did not go to pay taxes. It is set aside for creating more capital, or in general for improving the production next period.
Regio
CC BY-SA 4.0
2019-06-04T06:16:20.807Z
2019-06-04T06:16:20.807Z
Regio
<p>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.</p>

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

<p><span class="math-container">$k_t=k_{t-1}(1-\delta)+i_t$</span></p>

<p>The firm's technology meets the typical strictly concave, increasing, Inada, and TFP conditions: the amount of capital invested by a firm at time <span class="math-container">$t$</span> produces output <span class="math-container">$y_t$</span> at time <span class="math-container">$t+1$</span> where <span class="math-container">$A&gt;0$</span> denotes the TFP.</p>

<p><span class="math-container">$y_{t+1}=Af(k_t)$</span></p>

<p><span class="math-container">$\textbf{My question}:$</span>
How do you understand the <span class="math-container">$\textit{investment TIMING}$</span> 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:</p>

<p><span class="math-container">$i_t=k_t-k_{t-1}(1-\delta)$</span></p>

<p>So if I stand at <span class="math-container">$t$</span>, then I have the following sources for capital, correct?:</p>

<p>(1) <span class="math-container">$(1-\delta)k_{t-1}$</span>: This simply what I carried over as depreciated capital</p>

<p>(2) <span class="math-container">$k_t$</span>: 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))</p>

<p>But why is (2)-(1), today's firm investment?</p>

<macroeconomics><capital>
The TYPICAL law of motion of capital
Frank Swanton
# The TYPICAL law of motion of capital

(Asked by: username_0 on 2019-06-04)

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?

**Tags:** macroeconomics, capital

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## 2 answers

### Answer by username_2 on 2019-06-04. Score: 3 | Accepted answer

$k\_t$ 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: $k\_t-k\_{t-1}(1-\delta)$.

> **Comments:**
> **username_0**: 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)?  
> **username_0**: 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.  
> **username_3**: This question is really different from the one in your OP.  
> **username_0**: @Giskard the recursive dynamics, and cake eating, timing of investment are exactly what I asked.  
> **username_2**: 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.  
> **username_0**: @Regio Hi Regio, just a quick related question. So in the set-up of the instantaneous capital production is assumed, does the investment occur like this? $\rightarrow$ When you stand today, you have the undepreciated capital carried over from yesterday, so you use this as input to the production and produce output. With this output, you produce today's capital, $k_t$. Hence, the available investment is this $k_t$ minus $k_{t-1}(1-\delta)$. Is this correct?  
> **username_2**: I don't think so. I mean it really depends on how the person setting up the model chooses the timing. However, most likely, the timing is as follows. Investment, $i_t$ is chosen before production (perhaps from savings or borrowing money, etc). This creates $k_t$ along with the un-depreciated capital from yesterday. Finally, production occurs. I find this setup odd, I don't think there is a universal way to do the timing, but I feel that I've seen more often the assumption $k_{t+1}=(1-\delta)k_t+i_t$. I'm not sure why you think the one you presented is THE typical law of motion.  
> **username_0**: @Regio Hope things are well with you. Just a quick clarification on the investment. The investment is basically the amount of output you put aside from consumption to put into the following period (or contemporaneous) production, correct? I just want to clarify the term "investment" in that it simply represents the amount put aside not consuming, right? It has nothing to do with the actual production technology $f(.)$?  
> **username_2**: That’s correct, we typically think that investment is the part of the income that did not go to consumption (either domestic or foreign), and did not go to pay taxes. It is set aside for creating more capital, or in general for improving the production next period.  

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### Answer by username_1 on 2019-06-04. Score: 1

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 $t$.


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