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

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?

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 $$\overline{C} = \frac{P^2}{4w}$$

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

• Can there be interior solutions since you're saying that the input demands can be any combination on the isoquant? I am thinking that because both of them are straight lines then there would only be a corner solution where either only x1 is used or only x2 is used in the case where their input prices are the same. Am I wrong? May 29 at 20:19
• Yes, all points on the isoquant (interior and corner) are optimal because if two input combinations yield the same production (same isoquant) and have the same cost (same isocost), then the $\text{Profit } = { Price } \times { Production } - { Cost}$ will be the same for both input combinations. Here both straight lines overlap (are the same). May 29 at 20:24
• Thanks for the clarification. Can you spell out what the input demand functions would be if it's an interior solution where the input prices are equal? May 29 at 20:26
• I just edited my answer to try to make things clearer for you, check it out! May 29 at 20:37
• This helps! Thanks! May 29 at 20:55