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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?

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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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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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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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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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