<p>Let $y_1$ and $y_2$ denote the output of firms $1$ and $2$. I suspect by price function you mean the inverse demand function. So if
$$
y_1 + y_2 = Y = D(p)
$$
then
$$
D^{-1}(Y) = p(Y) = p(y_1 + y_2).
$$
Let us introduce the notation
$$
MR_i(y_1,y_2) = \frac{\partial p(y_1 + y_2)}{\partial y_i}.
$$
Generally
$$
MR_i(y_1,y_2) \neq MC_i(y_i),
$$
that is the functions are not the same. But if the outputs $y_1^*,y_2^*$ constitute an equilibrium, the functions take the same value.
$$
\forall i: \ MR_i(y_1^*,y_2^*) = MC_i(y_i^*)
$$</p>

<p>Another example:
<br> The function $x^2$ is not the same as the constant function 4. But if $x=2$ they take the same value.</p>

CC BY-SA 3.0
2015-12-08T07:31:55.690Z
2015-12-08T07:31:55.690Z
Giskard
<p>This example is in a oligopoly market with two firms. The demand curve is given and also two firms' MC is given. How would one calculate price function in this scenario?</p>

<p>I found the slope using the demand curve and then found the y intercept to the get the price function. However, I also know that MC is the derivative of the price function. If I take the derivative of the price function I got, it's not the same as the MC.. Any thoughts on this?</p>

CC BY-SA 3.0
2015-12-08T02:52:18.093Z
Could you include the specific demand curve and marginal costs?
cpage
CC BY-SA 3.0
2015-12-08T05:03:01.667Z
It also matters whether the firms are competing in quantity or price.
Herr K.
<price-theory>
How to calculate price function from demand curve when marginal cost is given?
anusha
# How to calculate price function from demand curve when marginal cost is given?

(Asked by: username_0 on 2015-12-08)

This example is in a oligopoly market with two firms. The demand curve is given and also two firms' MC is given. How would one calculate price function in this scenario?

I found the slope using the demand curve and then found the y intercept to the get the price function. However, I also know that MC is the derivative of the price function. If I take the derivative of the price function I got, it's not the same as the MC.. Any thoughts on this?

**Tags:** price-theory
> **Comments:**
> **username_1**: Could you include the specific demand curve and marginal costs?  
> **username_2**: It also matters whether the firms are competing in quantity or price.  

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## 1 answers

### Answer by username_3 on 2015-12-08. Score: 1

Let $y\_1$ and $y\_2$ denote the output of firms $1$ and $2$. I suspect by price function you mean the inverse demand function. So if
$$
y\_1 + y\_2 = Y = D(p)
$$
then
$$
D^{-1}(Y) = p(Y) = p(y\_1 + y\_2).
$$
Let us introduce the notation
$$
MR\_i(y\_1,y\_2) = \frac{\partial p(y\_1 + y\_2)}{\partial y\_i}.
$$
Generally
$$
MR\_i(y\_1,y\_2) \neq MC\_i(y\_i),
$$
that is the functions are not the same. But if the outputs $y\_1^\*,y\_2^\*$ constitute an equilibrium, the functions take the same value.
$$
\forall i: \ MR\_i(y\_1^\*,y\_2^\*) = MC\_i(y\_i^\*)
$$

Another example:
  
 The function $x^2$ is not the same as the constant function 4. But if $x=2$ they take the same value.


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