# 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, 2023 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, 2023 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, 2023 at 20:26
• I just edited my answer to try to make things clearer for you, check it out! May 29, 2023 at 20:37
• This helps! Thanks! May 29, 2023 at 20:55