<p>Full task condition: A monopolistic firm’s costs are <span class="math-container">$TC=\begin{cases} 170+20q, \ q&gt;0 \\ 0 \ q=0 \end{cases}$</span>. There are two groups of consumers of equal size in the monopolist’s market: <span class="math-container">$Q_{d1} = 120-p$</span>, <span class="math-container">$Q_{d2} = 90 - p$</span>. (The groups are homogeneous) The state allows the monopoly to offer a menu of two-part tariffs. The monopoly can’t distinguish consumers from different groups. The state imposes a price ceiling (it also cannot distinguish between consumers). Plot the dependence of the monopolist’s output (Q) on the price ceiling (<span class="math-container">$\overline{P}$</span>)</p>
<p>I’ve tried realizing the output without the price ceiling, but didn’t have much luck with it.</p>
<p><span class="math-container">$$\begin{cases}
    \pi = t_1+t_2 - TC\rightarrow max \ to \ t_1, t_2 \\
CS_1\geq 0 \\
CS_2\geq 0 \\
CS_1(t_1)\geq CS_1(t_2) \\
CS_2(t_2)\geq CS_2(t_1) \\
\end{cases}$$</span></p>
<p><span class="math-container">$$t_i =\begin{cases}
    A_i+p_iQ_i, \ Q_i \geq 0 \\
    0, \ Q_i = 0
\end{cases}$$</span>
<span class="math-container">$$CS_1 = \frac{(120-p_1)^2}{2}-A_1$$</span>
<span class="math-container">$$CS_2 = \frac{(90-p_2)^2}{2}-A_2$$</span>
<span class="math-container">$$\pi = (A_1+p_1Q_1)+(A_2+p_2Q_2)-170-20(Q_1+Q_2)$$</span></p>
<p>Trying to find the demand equation with the tariff:
<span class="math-container">$$TC_{D_1} = p_1*Q_1+A_1=(120-Q_1)*Q_1$$</span>
<span class="math-container">$$p_1=120-Q_1-\frac{A_1}{Q_1}$$</span>
For the second group
<span class="math-container">$$p_2=90-Q_2-\frac{A_2}{Q_2}$$</span>
<span class="math-container">$$\pi =Q_1(120-Q_1-\frac{A_1}{Q_1})+A_1+Q_2(90-Q_2-\frac{A_2}{Q_2})+A_2-20(Q_1+Q_2)-170$$</span>
<span class="math-container">$$\pi = 100Q_1-Q_1^2+70Q_2-Q_2^2-170$$</span>
Maximizing the profit:
<span class="math-container">$$Q_1^* = 50$$</span>
<span class="math-container">$$Q_2^* = 35$$</span>
<span class="math-container">$$\pi = 3555$$</span>
<span class="math-container">$$p_1=120-Q_1-\frac{A_1}{Q_1} = 70-\frac{A_1}{50}$$</span>
<span class="math-container">$$p_2=90-Q_2-\frac{A_2}{Q_2} = 55-\frac{A_2}{35}$$</span>
<span class="math-container">$$CS_1= \frac{(120-70+\frac{A_1}{50})^2}{2}-A_1\geq 0\Longleftrightarrow x \in R$$</span></p>
<p><span class="math-container">$$CS_2 = \frac{(90-55+\frac{A_2}{35})^2}{2}-A_2 \geq 0 \Longleftrightarrow x \in R$$</span> \
<span class="math-container">$$\frac{(120-70+\frac{A_1}{50})^2}{2}-A_1 \geq \frac{(120-55+\frac{A_2}{35})^2}{2}-A_2$$</span>
<span class="math-container">$$\frac{2500^2+A_1^2}{5000} \geq \frac{2275^2+2100A_2+A_2^2}{2450}$$</span>
<span class="math-container">$$\frac{(90-55+\frac{A_2}{35})^2}{2}-A_2\geq \frac{(90-70+\frac{A_1}{50})^2}{2}-A_1$$</span>
<span class="math-container">$$\frac{1225^2+A_2^2}{2450} \geq \frac{1000^2-3000A_1+A_1^2}{5000}$$</span></p>
<p>This way the monopolist’s should set infinitely high <span class="math-container">$A_1$</span> and <span class="math-container">$A_2$</span> and this doesn’t seem like the right answer.</p>
<p>And after this I couldn’t solve it with the price ceiling for the same reasons.</p>
<p>PS I could have misunderstood the task so feel free to suggest an alternative interpretation.</p>

CC BY-SA 4.0
2021-11-08T13:25:21.813Z
The monopolist cannot distinguish consumers from different groups, so he can set only one subscription fee and one price. See page 415 of the textbook available at http://eprints.stiperdharmawacana.ac.id/176/.
VARulle
CC BY-SA 4.0
2021-11-09T19:57:07.843Z
@VARulle hmm (the book you’ve added doesn’t have this page but managed to find it myself) Still I don’t understand why can’t our firm set two different tariffs? Yes it cannot distinguish consumers but they can do it themselves. The tariff should be most appealing to the group it is made for (I hope this makes sense) i.e. $CS_1(tariff_1)>CS_1(tariff_2)$ and $CS_2(tariff_2)>CS_2(tariff_1)$
Foorgy Infifcio
CC BY-SA 4.0
2021-11-11T11:01:21.110Z
Yeah, sorry, wrong link. But "The state allows the monopoly to introduce a subscription fee" seems to suggest an interpretation á la Pindyck-Rubinfeld: The monopoly may offer a (single) two-part tariff. If you meant to say that the monopoly is allowed to offer a menu of two-part tariffs (or, even more general, a menu of contracts), then you should say so explicitly in the question. Additionally it should also be clarified whether or not you assume that each group consists of identical consumers (and can thus be treated as a single representative consumer).
VARulle
CC BY-SA 4.0
2021-11-11T12:01:23.410Z
@VARulle Oh I see, I’ll add that to the task thx
Foorgy Infifcio
CC BY-SA 4.0
2021-11-12T13:14:17.387Z
I don't understand the equation following "Trying to find the demand equation with the tariff". And what does $TC_{D_1}$ stand for? Total costs? Why?
VARulle
<microeconomics><monopoly><price-discrimination>
Plot the dependence of the monopolist’s output (Q) on the price ceiling ($\overline{P}$)
Foorgy Infifcio
# Plot the dependence of the monopolist’s output (Q) on the price ceiling ($\overline{P}$)

(Asked by: username_0 on 2021-11-06)

Full task condition: A monopolistic firm’s costs are $TC=\begin{cases} 170+20q, \ q>0 \\ 0 \ q=0 \end{cases}$. There are two groups of consumers of equal size in the monopolist’s market: $Q\_{d1} = 120-p$, $Q\_{d2} = 90 - p$. (The groups are homogeneous) The state allows the monopoly to offer a menu of two-part tariffs. The monopoly can’t distinguish consumers from different groups. The state imposes a price ceiling (it also cannot distinguish between consumers). Plot the dependence of the monopolist’s output (Q) on the price ceiling ($\overline{P}$)

I’ve tried realizing the output without the price ceiling, but didn’t have much luck with it.

$$\begin{cases}
\pi = t\_1+t\_2 - TC\rightarrow max \ to \ t\_1, t\_2 \\
CS\_1\geq 0 \\
CS\_2\geq 0 \\
CS\_1(t\_1)\geq CS\_1(t\_2) \\
CS\_2(t\_2)\geq CS\_2(t\_1) \\
\end{cases}$$

$$t\_i =\begin{cases}
A\_i+p\_iQ\_i, \ Q\_i \geq 0 \\
0, \ Q\_i = 0
\end{cases}$$
$$CS\_1 = \frac{(120-p\_1)^2}{2}-A\_1$$
$$CS\_2 = \frac{(90-p\_2)^2}{2}-A\_2$$
$$\pi = (A\_1+p\_1Q\_1)+(A\_2+p\_2Q\_2)-170-20(Q\_1+Q\_2)$$

Trying to find the demand equation with the tariff:
$$TC\_{D\_1} = p\_1\*Q\_1+A\_1=(120-Q\_1)\*Q\_1$$
$$p\_1=120-Q\_1-\frac{A\_1}{Q\_1}$$
For the second group
$$p\_2=90-Q\_2-\frac{A\_2}{Q\_2}$$
$$\pi =Q\_1(120-Q\_1-\frac{A\_1}{Q\_1})+A\_1+Q\_2(90-Q\_2-\frac{A\_2}{Q\_2})+A\_2-20(Q\_1+Q\_2)-170$$
$$\pi = 100Q\_1-Q\_1^2+70Q\_2-Q\_2^2-170$$
Maximizing the profit:
$$Q\_1^\* = 50$$
$$Q\_2^\* = 35$$
$$\pi = 3555$$
$$p\_1=120-Q\_1-\frac{A\_1}{Q\_1} = 70-\frac{A\_1}{50}$$
$$p\_2=90-Q\_2-\frac{A\_2}{Q\_2} = 55-\frac{A\_2}{35}$$
$$CS\_1= \frac{(120-70+\frac{A\_1}{50})^2}{2}-A\_1\geq 0\Longleftrightarrow x \in R$$

$$CS\_2 = \frac{(90-55+\frac{A\_2}{35})^2}{2}-A\_2 \geq 0 \Longleftrightarrow x \in R$$ \
$$\frac{(120-70+\frac{A\_1}{50})^2}{2}-A\_1 \geq \frac{(120-55+\frac{A\_2}{35})^2}{2}-A\_2$$
$$\frac{2500^2+A\_1^2}{5000} \geq \frac{2275^2+2100A\_2+A\_2^2}{2450}$$
$$\frac{(90-55+\frac{A\_2}{35})^2}{2}-A\_2\geq \frac{(90-70+\frac{A\_1}{50})^2}{2}-A\_1$$
$$\frac{1225^2+A\_2^2}{2450} \geq \frac{1000^2-3000A\_1+A\_1^2}{5000}$$

This way the monopolist’s should set infinitely high $A\_1$ and $A\_2$ and this doesn’t seem like the right answer.

And after this I couldn’t solve it with the price ceiling for the same reasons.

PS I could have misunderstood the task so feel free to suggest an alternative interpretation.

**Tags:** microeconomics, monopoly, price-discrimination

> **Comments:**
> **username_1**: The monopolist cannot distinguish consumers from different groups, so he can set only one subscription fee and one price. See page 415 of the textbook available at http://eprints.stiperdharmawacana.ac.id/176/.  
> **username_0**: @username_1 hmm (the book you’ve added doesn’t have this page but managed to find it myself) Still I don’t understand why can’t our firm set two different tariffs? Yes it cannot distinguish consumers but they can do it themselves. The tariff should be most appealing to the group it is made for (I hope this makes sense) i.e. $CS_1(tariff_1)>CS_1(tariff_2)$ and $CS_2(tariff_2)>CS_2(tariff_1)$  
> **username_1**: Yeah, sorry, wrong link. But "The state allows the monopoly to introduce a subscription fee" seems to suggest an interpretation á la Pindyck-Rubinfeld: The monopoly may offer a (single) two-part tariff. If you meant to say that the monopoly is allowed to offer a menu of two-part tariffs (or, even more general, a menu of contracts), then you should say so explicitly in the question. Additionally it should also be clarified whether or not you assume that each group consists of identical consumers (and can thus be treated as a single representative consumer).  
> **username_0**: @username_1 Oh I see, I’ll add that to the task thx  
> **username_1**: I don't understand the equation following "Trying to find the demand equation with the tariff". And what does $TC_{D_1}$ stand for? Total costs? Why?  

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## 0 answers

