<p>For a production function  <span class="math-container">$Y=F(K, L)$</span> of two inputs <span class="math-container">$K$</span> and <span class="math-container">$L$</span>, I think there can be two interpretations</p>
<p><strong>One interpretation could be the production function is characterized by fixed capital, usually assumed for the short run. This happens because we cannot increase or decrease the amount of land, manufacturing buildings, and machinery in a short production.</strong></p>
<p>With a fixed capital of <span class="math-container">$\hat{K}$</span>, the production function becomes
<span class="math-container">$$ Y=F(\hat{K},L) $$</span>
<span class="math-container">$$ Y=F(L) $$</span></p>
<p>This equation simply indicates that since capital is fixed, the production output depends only on the labor employed</p>
<p><strong>Another interpretation could be the production function is characterized by constant returns to scale, usually assumed for the long run. This happens when an increase in inputs (capital and labor) causes the same proportional increase in output.</strong></p>
<p>Constant returns to scale imply that if all factors of production are multiplied by any nonnegative real number <span class="math-container">$\mu$</span> then the scale of
production is also multiplied by μ.</p>
<p><span class="math-container">$$ \mu Y=F(\mu K, \mu L ) $$</span>
for any <span class="math-container">$\mu ≥ 0$</span></p>
<p>Because of the assumption of constant returns to scale, we can multiply all factors by <span class="math-container">$\dfrac{1}{K}$</span>, and the production function can be written as
<span class="math-container">$$\dfrac{Y}{K}=F( \frac{K}{K},\frac{L}{K} ) $$</span>
<span class="math-container">$$ y=F( 1, l ) $$</span>
<span class="math-container">$$ y=F( l ) $$</span></p>
<p>where <span class="math-container">$y = \dfrac{Y}{K}$</span> is output per capital, and <span class="math-container">$l =\dfrac{L}{K}$</span> is labor per capital.</p>

CC BY-SA 4.0
2022-06-07T07:29:52.043Z
2022-06-07T07:35:00.137Z
2022-06-07T07:35:00.137Z
Marble
<p>This is kind of a strange question, in the best kind of way.</p>
<p>The thing about capital in the first case is that it... <em>isn't.</em></p>
<p>What I mean is that a fundamental defining characteristic of capital is that it's a factor of production.  If this stuff isn't a factor of production, then whatever it is, it isn't capital in any meaningful sense.</p>

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2022-05-26T01:58:50.307Z
2022-05-26T01:58:50.307Z
Roger
<p>We just assume that the production does not depend on capital inputs.</p>
<p>This is consistent both with capital being fixed or flexible.</p>
<p>For example <span class="math-container">$F(L)$</span> could be <span class="math-container">$F(L,K)=L^{0.5}=F(L)$</span>. In such case the production function depends only on labor. In that case even when capital is not fixed increase in capital won’t lead to increase in output.</p>
<p>However, it is also possible that it is a production function for output at fixed level of capital <span class="math-container">$F(L,\bar{K})=F(L)$</span>. That is <strong>one</strong> of the possibilities.</p>
<p>Unless author mentions this there is no way of knowing which of the above two interpretations is correct</p>

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2022-05-25T10:17:52.713Z
2022-05-25T18:31:21.183Z
2022-05-25T18:31:21.183Z
1muflon1
<p>One way to think about it is the following:</p>
<p>Given a production function <span class="math-container">$F(L, K)$</span> of two inputs <span class="math-container">$L$</span> and <span class="math-container">$K$</span>, if <span class="math-container">$K$</span> is fixed at <span class="math-container">$\overline{K}$</span> in the short run, then we can define the short run production function in this way:</p>
<p><span class="math-container">$F_s(L) := F(L, \overline{K})$</span>.</p>
<p>Here <span class="math-container">$F_s(L)$</span> is the short run production function for <span class="math-container">$\overline{K}$</span> units of capital. This function will change if <span class="math-container">$\overline{K}$</span> changes.</p>

CC BY-SA 4.0
2022-05-25T11:45:34.077Z
2022-05-25T11:45:34.077Z
Amit
<p>We have two production functions, one <span class="math-container">$F(L)$</span> with labor as only factor of production, the other also includes capital <span class="math-container">$F(L,K)$</span>.</p>
<p>What do we assume about capital in the first case when we omit it from the production function? Is it fixed or completely flexible?</p>

<macroeconomics><production-function><theory>
What does omitting capital from production function assume about capital?
Papayapap
# What does omitting capital from production function assume about capital?

(Asked by: username_0 on 2022-05-25)

We have two production functions, one $F(L)$ with labor as only factor of production, the other also includes capital $F(L,K)$.

What do we assume about capital in the first case when we omit it from the production function? Is it fixed or completely flexible?

**Tags:** macroeconomics, production-function, theory


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## 4 answers

### Answer by username_4 on 2022-05-25. Score: 5 | Accepted answer

One way to think about it is the following:

Given a production function $F(L, K)$ of two inputs $L$ and $K$, if $K$ is fixed at $\overline{K}$ in the short run, then we can define the short run production function in this way:

$F\_s(L) := F(L, \overline{K})$.

Here $F\_s(L)$ is the short run production function for $\overline{K}$ units of capital. This function will change if $\overline{K}$ changes.


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### Answer by username_1 on 2022-06-07. Score: 1

For a production function $Y=F(K, L)$ of two inputs $K$ and $L$, I think there can be two interpretations

**One interpretation could be the production function is characterized by fixed capital, usually assumed for the short run. This happens because we cannot increase or decrease the amount of land, manufacturing buildings, and machinery in a short production.**

With a fixed capital of $\hat{K}$, the production function becomes
$$ Y=F(\hat{K},L) $$
$$ Y=F(L) $$

This equation simply indicates that since capital is fixed, the production output depends only on the labor employed

**Another interpretation could be the production function is characterized by constant returns to scale, usually assumed for the long run. This happens when an increase in inputs (capital and labor) causes the same proportional increase in output.**

Constant returns to scale imply that if all factors of production are multiplied by any nonnegative real number $\mu$ then the scale of
production is also multiplied by μ.

$$ \mu Y=F(\mu K, \mu L ) $$
for any $\mu ≥ 0$

Because of the assumption of constant returns to scale, we can multiply all factors by $\dfrac{1}{K}$, and the production function can be written as
$$\dfrac{Y}{K}=F( \frac{K}{K},\frac{L}{K} ) $$
$$ y=F( 1, l ) $$
$$ y=F( l ) $$

where $y = \dfrac{Y}{K}$ is output per capital, and $l =\dfrac{L}{K}$ is labor per capital.


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### Answer by username_2 on 2022-05-26. Score: 0

This is kind of a strange question, in the best kind of way.

The thing about capital in the first case is that it... *isn't.*

What I mean is that a fundamental defining characteristic of capital is that it's a factor of production. If this stuff isn't a factor of production, then whatever it is, it isn't capital in any meaningful sense.


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### Answer by username_3 on 2022-05-25. Score: 0

We just assume that the production does not depend on capital inputs.

This is consistent both with capital being fixed or flexible.

For example $F(L)$ could be $F(L,K)=L^{0.5}=F(L)$. In such case the production function depends only on labor. In that case even when capital is not fixed increase in capital won’t lead to increase in output.

However, it is also possible that it is a production function for output at fixed level of capital $F(L,\bar{K})=F(L)$. That is **one** of the possibilities.

Unless author mentions this there is no way of knowing which of the above two interpretations is correct


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