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Determine the linear production function in the form $Q=aK+bL$ if it takes $3.5$ labor ($L$) hours to produce what a machine ($K$) can do in one hour?

I was given the answer is: $Q=3.5 \times K+L$

But I don't understand why the function is not $Q=K+3.5 \times L$

Amaan M
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1 Answers1

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Great question, it trips up intro econ students all the time. Think about it in terms of marginal products, and it's a lot easier.

Given $Q = aK + bL$, the marginal amount that $L$ can produce per unit of labor is the marginal product of labor, $\text{MPL}$, which is

$$ \text{MPL} = \frac{\partial Q}{\partial L} = b . $$

Similarly, the marginal amount that capital can produce per unit of capital is the marginal product of capital, $\text{MPK}$, which is

$$ \text{MPK} = \frac{\partial Q}{\partial K} = a .$$

Then, if capital can produce $3.5$ times what labor can produce in one hour, we have

$$ \text{MPK} = 3.5 \text{MPL} $$ $$ \Rightarrow a = 3.5 b . $$

Finally, substitute that back into $Q$, and you have

$$ Q = 3.5bK + bL . $$

The answer you provided is the case when $b = 1$, but it holds for any value of $b$.

Amaan M
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