<p>Both are economically sound. The notation is just a question of convention. The reason behind the ambiguity is that capital is a stock and investment is a flow variable. You are looking at capital in two different instants. Investment happens during the time between the two instants and its index is either the starting or the ending instant. </p>

CC BY-SA 3.0
2015-06-05T05:38:19.843Z
2015-06-08T21:05:51.643Z
2015-06-08T21:05:51.643Z
Giskard
<p>This is a bit more complicated than it looks.
The two are different but equivalent notational conventions, <strong>only if it is clear</strong> that they incorporate the same assumption of essence (that is rarely stated explicitly nowadays), i.e. that it takes only one period for investment to become part of capital and so productive.</p>

<p>Under this assumption
the notational difference comes from what is the meaning that we assign to the index for the <em><strong>state</strong></em> variable, the capital.</p>

<p>Some models designate $K_t$ to mean "Capital stock <strong>at the beginning</strong> of period $t$. In such a case, for example the production function for period $t$ should include $K_t$.  As for the law of motion of capital, since investment happens <em>during</em> period $t$, and augments the capital that was there at the beginning of the period, here the law of motion is usually written, for clarity,</p>

<p>$$K_{t+1} = (1-\delta) K_t + I_t$$
and lagging it, we obtain $K_t = (1-\delta)K_{t-1}+I_{t-1}$</p>

<p>Others designate $K_t$ to mean "Capital stock <strong>at the end</strong> of period $t$. With the same logic as before, the production function for period $t$ should include $K_{t-1}$ and the law of motion of capital should be written</p>

<p>$$K_{t} = (1-\delta)K_{t-1} + I_t$$</p>

<p><strong>But assume that I want to create a model where investment takes two periods to become part of capital (i.e. productive), and I also want to use the second notational convention of the above.</strong> I will have to write the law of motion of capital as
$$K_{t} = (1-\delta)K_{t-1} + I_{t-1}$$
which looks exactly like "capital at the beginning of the period plus investment productive in one period", <em>but</em> on the (totally) contrary, it is here meant to be translated "capital stock at the end of period $t$ (and hence available for production in period $t+1$), that is <em>productively</em> augmented by investment made during period $t-1$".</p>

<p>It would perhaps be better then to use the first notational convention and write</p>

<p>$$K_{t+1} = (1-\delta) K_t + I_{t-1}$$</p>

<p>But in general  one has to carefully read the assumptions and the notational conventions of the model. </p>

CC BY-SA 3.0
2015-06-05T13:09:45.677Z
2015-06-05T13:09:45.677Z
Alecos Papadopoulos
<p>In a lot of models with capital, I find different variants of capital accumulation formula as follows:
$K_t = (1-\delta)K_{t-1} +I_t$ or </p>

<p>$K_t = (1-\delta)K_{t-1}+I_{t-1}$</p>

<p>Which one is more economics-sound formula? </p>

<macroeconomics><investment>
Which capital accumulation is right? $K_t = (1-\delta)K_{t-1} +I_t$ or $K_t = (1-\delta)K_{t-1}+I_{t-1}$?
capkital
# Which capital accumulation is right? $K_t = (1-\delta)K_{t-1} +I_t$ or $K_t = (1-\delta)K_{t-1}+I_{t-1}$?

(Asked by: username_0 on 2015-06-05)

In a lot of models with capital, I find different variants of capital accumulation formula as follows:
$K\_t = (1-\delta)K\_{t-1} +I\_t$ or

$K\_t = (1-\delta)K\_{t-1}+I\_{t-1}$

Which one is more economics-sound formula?

**Tags:** macroeconomics, investment


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## 2 answers

### Answer by username_1 on 2015-06-05. Score: 9

Both are economically sound. The notation is just a question of convention. The reason behind the ambiguity is that capital is a stock and investment is a flow variable. You are looking at capital in two different instants. Investment happens during the time between the two instants and its index is either the starting or the ending instant.


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### Answer by username_2 on 2015-06-05. Score: 8

This is a bit more complicated than it looks.
The two are different but equivalent notational conventions, **only if it is clear** that they incorporate the same assumption of essence (that is rarely stated explicitly nowadays), i.e. that it takes only one period for investment to become part of capital and so productive.

Under this assumption
the notational difference comes from what is the meaning that we assign to the index for the ***state*** variable, the capital.

Some models designate $K\_t$ to mean "Capital stock **at the beginning** of period $t$. In such a case, for example the production function for period $t$ should include $K\_t$. As for the law of motion of capital, since investment happens *during* period $t$, and augments the capital that was there at the beginning of the period, here the law of motion is usually written, for clarity,

$$K\_{t+1} = (1-\delta) K\_t + I\_t$$
and lagging it, we obtain $K\_t = (1-\delta)K\_{t-1}+I\_{t-1}$

Others designate $K\_t$ to mean "Capital stock **at the end** of period $t$. With the same logic as before, the production function for period $t$ should include $K\_{t-1}$ and the law of motion of capital should be written

$$K\_{t} = (1-\delta)K\_{t-1} + I\_t$$

**But assume that I want to create a model where investment takes two periods to become part of capital (i.e. productive), and I also want to use the second notational convention of the above.** I will have to write the law of motion of capital as
$$K\_{t} = (1-\delta)K\_{t-1} + I\_{t-1}$$
which looks exactly like "capital at the beginning of the period plus investment productive in one period", *but* on the (totally) contrary, it is here meant to be translated "capital stock at the end of period $t$ (and hence available for production in period $t+1$), that is *productively* augmented by investment made during period $t-1$".

It would perhaps be better then to use the first notational convention and write

$$K\_{t+1} = (1-\delta) K\_t + I\_{t-1}$$

But in general one has to carefully read the assumptions and the notational conventions of the model.


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