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If I have two demand functions

$P_1=100-10X_1$ and $P_2=50-10X_2$

and $X_1=40$ and $X_2=60$ with $MC=10$

We have two consumer groups: 1 and 2, where 1 has a higher willingness to pay, and we have one single good. $X_1=40$ refers to number of customers.

How might I derive the optimal uniform price and its aggregate demand function from this?

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Let the market price for the good be $P$.

It is easy to see that a consumer from group $1$'s demand for the good is $$ X_1 = \frac{100-P}{10} $$

Similarly, a consumer from group $2$'s demand for the good is $$ X_2 = \frac{50-P}{10} $$

Total demand $Q$ is then given by $$ Q = 40X_1 + 60X_2 = 700-10P $$

This means that the market inverse demand curve (i.e. aggregate demand) is $$ P(Q) = 70 - \frac{Q}{10}$$

Suppose a single monopolist were serving this market. The monopolist's profit is $$ \Pi(Q) =P(Q) \cdot Q -MC \cdot Q $$

If we take the first-order condition (i.e. take the first derivative of $\Pi$ and set it equal to $0$), we know that the optimal quantity sold satisfies $$ 70 - \frac{Q^*}{5} = 10 $$

This means that $Q^* = 300$. We can plug this back into the market demand function to find that $P^* = 40$, which is the optimal uniform price.

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  • $\begingroup$ Hmm.. Something is weird. Either the solution to my problem: $Q=700-10P$ and $P=40$ is wrong, or your calculations are wrong... $\endgroup$
    – Mataunited17
    Commented Nov 25, 2016 at 21:47
  • $\begingroup$ Ah, you have different numbers of customers. The solution above will change, but I think how to do that should be obvious given what I've done. I'll edit my post shortly. It'd be helpful if you could mark this as the answer to your question once I've done that! That way this question doesn't stay in the unanswered queue. $\endgroup$
    – Theoretical Economist
    Commented Nov 25, 2016 at 21:49
  • $\begingroup$ I was initially assuming there was one of each consumer. I've edited the answer to account for the fact that there are $40$ of type $1$ and $60$ of type $2$. $\endgroup$
    – Theoretical Economist
    Commented Nov 25, 2016 at 22:01
  • $\begingroup$ Did you assume that one customer of each type would demand X units of goods? $\endgroup$
    – Mataunited17
    Commented Nov 25, 2016 at 22:10
  • $\begingroup$ As I stated in the original version of the post, I assumed there was only one customer of each type (as opposed to $40$ of type $1$ and $60$ of type $2$). I've edited the post now, and my answer matches yours. $\endgroup$
    – Theoretical Economist
    Commented Nov 25, 2016 at 22:16

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