# Plot the dependence of the monopolist’s output (Q) on the price ceiling ($\overline{P}$)

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.

• 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 eprints.stiperdharmawacana.ac.id/176. Nov 8, 2021 at 13:25
• @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 $CS_2(tariff_2)>CS_2(tariff_1)$ Nov 9, 2021 at 19:57
• 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). Nov 11, 2021 at 11:01
• @VARulle Oh I see, I’ll add that to the task thx Nov 11, 2021 at 12:01
• 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? Nov 12, 2021 at 13:14