<p>I don't know what is in the answer pamphlet.</p>
<p>Your solution is fine.</p>
<p>One thing to be wary of is that you make the assumption that both types of consumers are served, <span class="math-container">$Q_1,Q_2&gt;0$</span>. You could still consider the case that the price is set so high (<span class="math-container">$p&gt;50$</span>) that type <span class="math-container">$1$</span>'s do not purchase any units. Seems like maximal profit under that condition occurs at <span class="math-container">$p=60$</span>, <span class="math-container">$Q_2 = 4$</span>, neting
<span class="math-container">$$
\Pi = (60 - 20) \cdot (0+4) - 90 = 70,
$$</span>
so your solution is still the global maximum.</p>
<p>Your S.O.C. only checks whether your solution is a local maximum; here the profit function was not globally concave, it had a kink, thus your local maximum is not necessarily global.</p>
<p><a href="https://i.sstatic.net/DAbH5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/DAbH5.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-07-23T16:15:43.273Z
Thank you so much! I just want to confirm my understanding is correct: The reason that there is a kink is because it is possible for the monopolist to set the price so high so that customers of type 1 do not consume at all. Is that what you mean?
Beerus
CC BY-SA 4.0
2023-07-23T19:26:14.263Z
Yes, that is what I mean.
Giskard
CC BY-SA 4.0
2023-07-23T07:46:28.737Z
2023-07-23T19:27:20.227Z
2023-07-23T19:27:20.227Z
Giskard
<p>Price discrimination implies that the monopolist is selling different units of output at different prices. In the above case, since there is no price discrimination the monopolist is selling different units of output at the <strong>same</strong> price.</p>

CC BY-SA 4.0
2023-07-23T07:19:41.490Z
2023-07-23T07:19:41.490Z
econman
<p>I am working on <em>Mathematics for Economists</em> by Simon and Blume Exercise 17.7. I know there is an Answers Pamphlet. However, the solution to this question does not make any sense to me. It seems that they put a solution to a wrong question there. But anyway, I tried that exercise myself, but I am not totally sure whether my answer is correct or not. I would really appreciate it if someone could help me check! Thanks in advance!</p>
<p>Here is the exercise:</p>
<blockquote>
<p>A monopolist producing a single output has two types of customers. If it produces <span class="math-container">$Q_1$</span> units for customers of type 1, then these customers are willing to pay a price of <span class="math-container">$50 - 5Q_1$</span> dollars per unit. If it produces <span class="math-container">$Q_2$</span> units for customers of type 2, then these customers are willing to pay a price of <span class="math-container">$100 - 10Q_2$</span> dollars per unit. The monopolist's cost of manufacturing <span class="math-container">$Q$</span> units of output is <span class="math-container">$90 + 20Q$</span> dollars. Compute the demand function for the market as a whole, without price discrimination. Compute the firm's profit maximizing output for this situation.</p>
</blockquote>
<p>This is how I did it:</p>
<blockquote>
<p>Without price discrimination, we have <span class="math-container">$p_1 = p_2$</span>. Then
<span class="math-container">\begin{equation}
50 - 5Q_1 = 100 - 10Q_2.
\end{equation}</span>
Plug in <span class="math-container">$p = 100 - 10Q_2$</span>, we have the market demand function:
<span class="math-container">\begin{equation}
p = -\frac{10}{3}Q + \frac{200}{3}.
\end{equation}</span>
The firm's profit function is
<span class="math-container">\begin{equation}
F(Q) = (-\frac{10}{3}Q + \frac{200}{3})Q - (90 + 20Q).
\end{equation}</span>
F.O.C.:
<span class="math-container">\begin{equation}
\frac{\partial F}{\partial Q} = -\frac{20}{3}Q + \frac{140}{3} = 0 \Rightarrow Q = 7.
\end{equation}</span>
S.O.C.:
<span class="math-container">\begin{equation}
\frac{\partial^2 F}{\partial Q^2} = -\frac{20}{3} &lt; 0.
\end{equation}</span>
Thus, <span class="math-container">$Q = 7$</span> is the profit maximizing production. The profit then is <span class="math-container">$F = \frac{220}{3}$</span>.</p>
</blockquote>
<p>Basically, I am not sure whether it is correct to just set <span class="math-container">$p_1 = p_2$</span> to get the market demand function. I cannot figure out how intuitively that will make sense.</p>

<microeconomics><optimization><monopoly><demand>
Question About Non-Discriminating Monopolist - Mathematics for Economists by Simon and Blume Chapter 17 Exercise 7
Beerus
# Question About Non-Discriminating Monopolist - Mathematics for Economists by Simon and Blume Chapter 17 Exercise 7

(Asked by: username_0 on 2023-07-23)

I am working on *Mathematics for Economists* by Simon and Blume Exercise 17.7. I know there is an Answers Pamphlet. However, the solution to this question does not make any sense to me. It seems that they put a solution to a wrong question there. But anyway, I tried that exercise myself, but I am not totally sure whether my answer is correct or not. I would really appreciate it if someone could help me check! Thanks in advance!

Here is the exercise:

> A monopolist producing a single output has two types of customers. If it produces $Q\_1$ units for customers of type 1, then these customers are willing to pay a price of $50 - 5Q\_1$ dollars per unit. If it produces $Q\_2$ units for customers of type 2, then these customers are willing to pay a price of $100 - 10Q\_2$ dollars per unit. The monopolist's cost of manufacturing $Q$ units of output is $90 + 20Q$ dollars. Compute the demand function for the market as a whole, without price discrimination. Compute the firm's profit maximizing output for this situation.

This is how I did it:

> Without price discrimination, we have $p\_1 = p\_2$. Then
> \begin{equation}
> 50 - 5Q\_1 = 100 - 10Q\_2.
> \end{equation}
> Plug in $p = 100 - 10Q\_2$, we have the market demand function:
> \begin{equation}
> p = -\frac{10}{3}Q + \frac{200}{3}.
> \end{equation}
> The firm's profit function is
> \begin{equation}
> F(Q) = (-\frac{10}{3}Q + \frac{200}{3})Q - (90 + 20Q).
> \end{equation}
> F.O.C.:
> \begin{equation}
> \frac{\partial F}{\partial Q} = -\frac{20}{3}Q + \frac{140}{3} = 0 \Rightarrow Q = 7.
> \end{equation}
> S.O.C.:
> \begin{equation}
> \frac{\partial^2 F}{\partial Q^2} = -\frac{20}{3} < 0.
> \end{equation}
> Thus, $Q = 7$ is the profit maximizing production. The profit then is $F = \frac{220}{3}$.

Basically, I am not sure whether it is correct to just set $p\_1 = p\_2$ to get the market demand function. I cannot figure out how intuitively that will make sense.

**Tags:** microeconomics, optimization, monopoly, demand


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## 2 answers

### Answer by username_1 on 2023-07-23. Score: 1 | Accepted answer

I don't know what is in the answer pamphlet.

Your solution is fine.

One thing to be wary of is that you make the assumption that both types of consumers are served, $Q\_1,Q\_2>0$. You could still consider the case that the price is set so high ($p>50$) that type $1$'s do not purchase any units. Seems like maximal profit under that condition occurs at $p=60$, $Q\_2 = 4$, neting
$$
\Pi = (60 - 20) \cdot (0+4) - 90 = 70,
$$
so your solution is still the global maximum.

Your S.O.C. only checks whether your solution is a local maximum; here the profit function was not globally concave, it had a kink, thus your local maximum is not necessarily global.

[![enter image description here](https://i.sstatic.net/DAbH5.png)](https://i.sstatic.net/DAbH5.png)

> **Comments:**
> **username_0**: Thank you so much! I just want to confirm my understanding is correct: The reason that there is a kink is because it is possible for the monopolist to set the price so high so that customers of type 1 do not consume at all. Is that what you mean?  
> **username_1**: Yes, that is what I mean.  

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### Answer by username_2 on 2023-07-23. Score: -1

Price discrimination implies that the monopolist is selling different units of output at different prices. In the above case, since there is no price discrimination the monopolist is selling different units of output at the **same** price.


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