<p>Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant.  If you change the exponent 1-alpha to beta where alpha+beta &lt; 1, there will be decreasing returns to scale (but still homotheticity) and you will get increasing marginal cost.</p>

CC BY-SA 4.0
2021-01-03T05:57:14.483Z
Another important result of constant returns to scale (and hence, constant marginal cost) is that it implies there are zero profits in a perfectly competitive economy
Brennan
CC BY-SA 4.0
2021-01-03T01:47:27.697Z
2021-01-03T01:47:27.697Z
Mark Machina
<p>A company has a production function:
<span class="math-container">$$y=x_1^{\alpha}x_2^{1-\alpha}$$</span></p>
<p>where <span class="math-container">$0&lt;\alpha&lt;1$</span>. Factor input 1 costs <span class="math-container">$w_1&gt; 0$</span> and factor input 2 costs <span class="math-container">$w_2&gt; 0$</span>. The company wants to minimize its production costs when it produces y&gt; 0 units of the output.</p>
<p>I have in a problem found that MC:
<span class="math-container">$$MC=\frac{\partial c}{\partial y}=w_1((\frac{w_2\alpha}{w_1(1-\alpha)})^{1-\alpha})+w_2((\frac{w_1(1-\alpha)}{w_2 \alpha})^\alpha)$$</span></p>
<p>We see that marginal costs only depending on <span class="math-container">$w_1$</span> and <span class="math-container">$w_2$</span> and thus not on y. So we have that the marginal costs are constant. But why is that the case?
What is the economic interpretation?
I think maybe I should use returns to scale?</p>

<microeconomics><production-function><cobb-douglas><cost-functions>
CobbDouglas: Constant marginal costs and constant returns to scale
Lifeni
# CobbDouglas: Constant marginal costs and constant returns to scale

(Asked by: username_0 on 2021-01-03)

A company has a production function:
$$y=x\_1^{\alpha}x\_2^{1-\alpha}$$

where $0<\alpha<1$. Factor input 1 costs $w\_1> 0$ and factor input 2 costs $w\_2> 0$. The company wants to minimize its production costs when it produces y> 0 units of the output.

I have in a problem found that MC:
$$MC=\frac{\partial c}{\partial y}=w\_1((\frac{w\_2\alpha}{w\_1(1-\alpha)})^{1-\alpha})+w\_2((\frac{w\_1(1-\alpha)}{w\_2 \alpha})^\alpha)$$

We see that marginal costs only depending on $w\_1$ and $w\_2$ and thus not on y. So we have that the marginal costs are constant. But why is that the case?
What is the economic interpretation?
I think maybe I should use returns to scale?

**Tags:** microeconomics, production-function, cobb-douglas, cost-functions


---

## 1 answers

### Answer by username_1 on 2021-01-03. Score: 8

Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant. If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will get increasing marginal cost.

> **Comments:**
> **username_2**: Another important result of constant returns to scale (and hence, constant marginal cost) is that it implies there are zero profits in a perfectly competitive economy  

---

