How to derive the input demand functions from a perfect substitutes production function

Question

I am struggling to derive the input demand functions from a production function with inputs that are perfect substitutes. The production function is as follows: $f(x_1,x_2) = (x_1+x_2)^\frac{1}{2}$

I think the solution would depend on the Marginal rate of technical substitution and the input price ratio. The Marginal rate of technical substitution in this case is $1$ and I am not sure what the input demand functions would be when the Isoquant's slope is the same as the Isocost given that the isoquant is also a straight line. I know that for $\frac{w_1}{w_2} > 1$ or $\frac{w_1}{w_"} < 1$ (where $w_1$ and $w_2$ are the input prices) there would be corner solutions but how do we approach the case where the slopes are the same especially when the isoquant is a straight line?

Answer 1

Let $w:= w_1 = w_2$

The optimization problem you'd solve is profit maximization:

$\max \Pi = P (x_1 + x_2)^\frac{1}{2} - w x_1 - w x_2$

The first order condition for the $i$-th factor is

$\frac{\partial \Pi}{\partial x_i} = \frac{P}{2} (x_1 + x_2)^{-\frac{1}{2}} - w = 0$

$\implies \frac{P}{2} (x_1 + x_2)^{-\frac{1}{2}} = w$

$\implies \frac{2}{P} (x_1 + x_2)^\frac{1}{2} = \frac{1}{w}$

$\implies (x_1 + x_2)^\frac{1}{2} = \frac{P}{2w}$

$\implies x_1^\star + x_2^\star = (\frac{P}{2w})^2$

Note in this case, both f.o.c. are the same, I did it with the index $i$ to save some algebra.

Note the isoquants are given by

$(x_1 + x_2)^\frac{1}{2} = \overline{Y} \implies x_1 + x_2 = (\overline{Y})^2$

This implies the imput demands can be any combination on the isoquant for $\overline{Y} = \frac{P}{2w}$, which is also an isocost curve because

$w x_1 + w x_2 = \overline{C} \implies x_1 + x_2 = \frac{\overline{C}}{w}$

This implies it is also the isocost for $\overline{C} = \frac{P^2}{4w}$

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Edit from discussion in the comments:

Here the input demand functions are not uniquely determined.

The condition I got is that both demands have to add up to $(\frac{P}{2w})^2$. Since the demands are non-negative, we can pick any $x_1^\star$ on the interval $x_1^\star \in [0,(\frac{P}{2w})^2]$.

Then, for the above equation to be satisfied, given an $x_1^\star$, we pick $x_2^\star = (\frac{P}{2w})^2 - x_1^\star$

With this, we can say that the input demands are given by

$x_1^\star \in [0,(\frac{P}{2w})^2]$

$x_2^\star = (\frac{P}{2w})^2 - x_1^\star$

The above describes both interior and corner optimal points, the corner ones being if we pick $x_1^\star = 0$ or $(\frac{P}{2w})^2$, while the interior ones are given by choices of $x_1^\star \in (0,(\frac{P}{2w})^2)$.