<p>Let <span class="math-container">$w:= w_1 = w_2$</span></p>
<p>The optimization problem you'd solve is profit maximization:</p>
<p><span class="math-container">$\max \Pi = P (x_1 + x_2)^\frac{1}{2} - w x_1 - w x_2$</span></p>
<p>The first order condition for the <span class="math-container">$i$</span>-th factor is</p>
<p><span class="math-container">$\frac{\partial \Pi}{\partial x_i} = \frac{P}{2} (x_1 + x_2)^{-\frac{1}{2}} - w = 0$</span></p>
<p><span class="math-container">$\implies \frac{P}{2} (x_1 + x_2)^{-\frac{1}{2}} = w$</span></p>
<p><span class="math-container">$\implies \frac{2}{P} (x_1 + x_2)^\frac{1}{2} = \frac{1}{w}$</span></p>
<p><span class="math-container">$\implies (x_1 + x_2)^\frac{1}{2} = \frac{P}{2w}$</span></p>
<p><span class="math-container">$\implies x_1^\star + x_2^\star = (\frac{P}{2w})^2$</span></p>
<p>Note in this case, both f.o.c. are the same, I did it with the index <span class="math-container">$i$</span> to save some algebra.</p>
<p>Note the isoquants are given by</p>
<p><span class="math-container">$(x_1 + x_2)^\frac{1}{2} = \overline{Y} \implies x_1 + x_2 = (\overline{Y})^2$</span></p>
<p>This implies the imput demands can be any combination on the isoquant for <span class="math-container">$\overline{Y} = \frac{P}{2w}$</span>, which is also an isocost curve because</p>
<p><span class="math-container">$w x_1 + w x_2 = \overline{C} \implies x_1 + x_2 = \frac{\overline{C}}{w}$</span></p>
<p>This implies it is also the isocost for <span class="math-container">$\overline{C} = \frac{P^2}{4w}$</span></p>
<hr />
<p>Edit from discussion in the comments:</p>
<p>Here the input demand functions are not uniquely determined.</p>
<p>The condition I got is that both demands have to add up to <span class="math-container">$(\frac{P}{2w})^2$</span>. Since the demands are non-negative, we can pick any <span class="math-container">$x_1^\star$</span> on the interval <span class="math-container">$x_1^\star \in [0,(\frac{P}{2w})^2]$</span>.</p>
<p>Then, for the above equation to be satisfied, given an <span class="math-container">$x_1^\star$</span>, we pick <span class="math-container">$x_2^\star = (\frac{P}{2w})^2 - x_1^\star$</span></p>
<p>With this, we can say that the input demands are given by</p>
<p><span class="math-container">$x_1^\star \in [0,(\frac{P}{2w})^2]$</span></p>
<p><span class="math-container">$x_2^\star = (\frac{P}{2w})^2 - x_1^\star$</span></p>
<p>The above describes both interior and corner optimal points, the corner ones being if we pick <span class="math-container">$x_1^\star = 0$</span> or <span class="math-container">$(\frac{P}{2w})^2$</span>, while the interior ones are given by choices of <span class="math-container">$x_1^\star \in (0,(\frac{P}{2w})^2)$</span>.</p>

CC BY-SA 4.0
2023-05-29T20:19:59.793Z
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?
Debbie
CC BY-SA 4.0
2023-05-29T20:24:34.543Z
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).
Nicolas Torres
CC BY-SA 4.0
2023-05-29T20:26:08.900Z
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?
Debbie
CC BY-SA 4.0
2023-05-29T20:37:27.967Z
I just edited my answer to try to make things clearer for you, check it out!
Nicolas Torres
CC BY-SA 4.0
2023-05-29T20:12:17.803Z
2023-05-29T21:09:20.320Z
2023-05-29T21:09:20.320Z
Nicolas Torres
<p>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:
<span class="math-container">$f(x_1,x_2) = (x_1+x_2)^\frac{1}{2}$</span></p>
<p>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 <span class="math-container">$1$</span> 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 <span class="math-container">$\frac{w_1}{w_2} &gt; 1$</span> or <span class="math-container">$\frac{w_1}{w_"} &lt; 1$</span> (where <span class="math-container">$w_1$</span> and <span class="math-container">$w_2$</span> 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?</p>

<microeconomics><production-function><profit-maximization><producer-theory>
How to derive the input demand functions from a perfect substitutes production function
Debbie
# How to derive the input demand functions from a perfect substitutes production function

(Asked by: username_0 on 2023-05-29)

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?

**Tags:** microeconomics, production-function, profit-maximization, producer-theory

---

## 1 answers

### Answer by username_1 on 2023-05-29. Score: 1 | Accepted answer

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 $x\_1^\star \in (0,(\frac{P}{2w})^2)$.

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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).  
> **username_0**: 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?  
> **username_1**: I just edited my answer to try to make things clearer for you, check it out!  

---

