<p>MC = 2Q1 + 2Q2 for both demand functions. Set MR1 = 2Q1 + 2Q2, and MR2 = 2Q1 + 2Q1 and then you have a system of 2 unknowns and 2 equations. The solution ends up being Q1 = 15, P1 = 72.5, Q2 = 17.5, P2 = 82.5. In this case Profits under PD > Profits Under single price. See you in 468 tomorrow. </p>

CC BY-SA 3.0
2017-10-03T02:40:56.327Z
2017-10-03T02:40:56.327Z
Alex Toy
<p>I think if the government prohibits price discrimination, then the sum of MR in each market should equal to the MC: <span class="math-container">$$2(Q_1+Q_2) = 180 - Q_1 - 2Q_2$$</span> Also the price in each market should be the same: <span class="math-container">$P_1 = P_2$</span>. Therefore: 
<span class="math-container">$$ 80-Q_1/2 = 100-Q_2$$</span>. Aligning these two equations we get <span class="math-container">$$Q_1 = 20; Q_2 = 30; P_1=P_2 = 70.$$</span> Hope this helps!</p>

CC BY-SA 4.0
2018-12-18T18:17:58.377Z
This answer is simply **wrong**. Where did you learn "if the government prohibits price discrimination, then the sum of MR in each market should equal to the MC"?
Herr K.
CC BY-SA 4.0
2018-12-17T15:27:13.390Z
2018-12-18T12:21:33.387Z
2018-12-18T12:21:33.387Z
Becky
<blockquote> A price discriminating monopolist sells in two markets. Inverse demand in market 1 is given by: $$P_1(Q_1) = 80 - (1/2)Q_1$$ and inverse demand in market 2 is given by: $$P_2(Q_2) = 100 - Q_2$$ The monopolist's cost function is $C(Q) = (Q_1 + Q_2)^2$<br>
a. Formulate the monopoly's function as a function of $Q_1$ and $Q_2$. <br>
b. Calculate the monopoly's profit maximizing quantity sold in markets 1 and 2 and the corresponding prices.<br><br>
Now suppose the government forbids price discrimination, so the monopolist can only set a single price for the two markets.<br><br>
c. Compute the monopoly price and quantity.<br>
d. How much is sold in market 1 and market 2, respectively?<br>
e. Was the government intervention beneficial for social welfare or not? </blockquote>

<p>a. This one is straightforward. $$= Q_1(80 - Q_1/2) + Q_2(100 - Q_2) - (Q_1 + Q_2)^2$$
b. This one I believed to be straightforward - set MR = MC. Thus, $$MR_1 = 80 - Q_1$$ $$MR_2 = 100 - 2Q_2$$ $$MC_1 = 2Q_1$$ $$MC_2 = 2Q_2$$ which gives $$(Q_1, Q_2, P_1, P_2) = (\frac{80}{3}, 25, \frac{200}{3}, 75)$$
c. This is where I start to get a bit messed up. Setting $P_1 = P_2$ gives a total demand function $$Q_1 + Q_2 = Q = 260 - 3P$$ I then solve for price, and get $$MR = \frac{260}{3} - \frac{2Q}{3}$$ and $$MC = 2Q$$ Thus, I'm left with $$(P,Q) = (75.8333, 32.5)$$</p>

<p>However, I feel as though I must have made a mistake. The profit when the monopolist is able to price discriminate actually ends up being lower than the profit when limited to one price. This surely can't be the case - where is my thinking going wrong here? Is my methodology off?</p>

CC BY-SA 3.0
2017-10-02T01:58:41.363Z
Wouldn't that make sense that the monopolist's profit is higher when he can charge different customers the maximum price for that customer?
CWill
CC BY-SA 3.0
2017-10-02T02:06:36.730Z
@CWill that certainly does make sense - I actually mistyped, because it in fact turns out that the profit is *lower* when he's able to price discriminate. Thus, I'm afraid my methods might be incorrect.
mizichael
CC BY-SA 3.0
2017-10-02T06:43:30.700Z
Why do you say $MC_i=2Q_i$?
Ubiquitous
CC BY-SA 3.0
2017-10-02T13:53:35.280Z
@Ubiquitous $$C(Q_1, Q_2) = (Q_1 + Q_2)^2$$ $$Q_1 + Q_2 = Q_i$$ $$C(Q_i) = (Q_i)^2$$ $$MC_i = 2Q_i$$ Or do you mean for $i \in 1,2$? This I was unsure about.
mizichael
CC BY-SA 3.0
2017-10-02T14:41:40.477Z
I meant $i\in1,2$. Marginal cost for good $1$ is calculated as $\partial C/\partial Q_1=2(Q_1+Q_2)$. So it is only true that $MC_1=2Q_1$ if $Q_2=0$. But you are looking for a solution in which $Q_1$ and $Q_2$ are both different from zero, so you have used the wrong marginal cost function.
Ubiquitous
CC BY-SA 3.0
2017-10-02T21:33:43.710Z
@Ubiquitous I see - since $MC_1 = MC_2 = 2(Q_1 + Q_2)$, would the profit maximizing condition be $MR_1 = MR_2$? So that we get $Q_1 = 2Q_2 - 20$? Or is this the wrong approach?
mizichael
<general-equilibrium><monopoly><price-discrimination>
Price discriminating monopolist question
mizichael
# Price discriminating monopolist question

(Asked by: username_0 on 2017-10-02)

> A price discriminating monopolist sells in two markets. Inverse demand in market 1 is given by: $$P\_1(Q\_1) = 80 - (1/2)Q\_1$$ and inverse demand in market 2 is given by: $$P\_2(Q\_2) = 100 - Q\_2$$ The monopolist's cost function is $C(Q) = (Q\_1 + Q\_2)^2$  
> a. Formulate the monopoly's function as a function of $Q\_1$ and $Q\_2$.   
> b. Calculate the monopoly's profit maximizing quantity sold in markets 1 and 2 and the corresponding prices.  
>   
> Now suppose the government forbids price discrimination, so the monopolist can only set a single price for the two markets.  
>   
> c. Compute the monopoly price and quantity.  
> d. How much is sold in market 1 and market 2, respectively?  
> e. Was the government intervention beneficial for social welfare or not?

a. This one is straightforward. $$= Q\_1(80 - Q\_1/2) + Q\_2(100 - Q\_2) - (Q\_1 + Q\_2)^2$$
b. This one I believed to be straightforward - set MR = MC. Thus, $$MR\_1 = 80 - Q\_1$$ $$MR\_2 = 100 - 2Q\_2$$ $$MC\_1 = 2Q\_1$$ $$MC\_2 = 2Q\_2$$ which gives $$(Q\_1, Q\_2, P\_1, P\_2) = (\frac{80}{3}, 25, \frac{200}{3}, 75)$$
c. This is where I start to get a bit messed up. Setting $P\_1 = P\_2$ gives a total demand function $$Q\_1 + Q\_2 = Q = 260 - 3P$$ I then solve for price, and get $$MR = \frac{260}{3} - \frac{2Q}{3}$$ and $$MC = 2Q$$ Thus, I'm left with $$(P,Q) = (75.8333, 32.5)$$

However, I feel as though I must have made a mistake. The profit when the monopolist is able to price discriminate actually ends up being lower than the profit when limited to one price. This surely can't be the case - where is my thinking going wrong here? Is my methodology off?

**Tags:** general-equilibrium, monopoly, price-discrimination

> **Comments:**
> **username_1**: Wouldn't that make sense that the monopolist's profit is higher when he can charge different customers the maximum price for that customer?  
> **username_0**: @username_1 that certainly does make sense - I actually mistyped, because it in fact turns out that the profit is *lower* when he's able to price discriminate. Thus, I'm afraid my methods might be incorrect.  
> **username_2**: Why do you say $MC_i=2Q_i$?  
> **username_0**: @username_2 $$C(Q_1, Q_2) = (Q_1 + Q_2)^2$$ $$Q_1 + Q_2 = Q_i$$ $$C(Q_i) = (Q_i)^2$$ $$MC_i = 2Q_i$$ Or do you mean for $i \in 1,2$? This I was unsure about.  
> **username_2**: I meant $i\in1,2$. Marginal cost for good $1$ is calculated as $\partial C/\partial Q_1=2(Q_1+Q_2)$. So it is only true that $MC_1=2Q_1$ if $Q_2=0$. But you are looking for a solution in which $Q_1$ and $Q_2$ are both different from zero, so you have used the wrong marginal cost function.  
> **username_0**: @username_2 I see - since $MC_1 = MC_2 = 2(Q_1 + Q_2)$, would the profit maximizing condition be $MR_1 = MR_2$? So that we get $Q_1 = 2Q_2 - 20$? Or is this the wrong approach?  

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## 2 answers

### Answer by username_3 on 2017-10-03. Score: 2 | Accepted answer

MC = 2Q1 + 2Q2 for both demand functions. Set MR1 = 2Q1 + 2Q2, and MR2 = 2Q1 + 2Q1 and then you have a system of 2 unknowns and 2 equations. The solution ends up being Q1 = 15, P1 = 72.5, Q2 = 17.5, P2 = 82.5. In this case Profits under PD > Profits Under single price. See you in 468 tomorrow.


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### Answer by username_4 on 2018-12-17. Score: -1

I think if the government prohibits price discrimination, then the sum of MR in each market should equal to the MC: $$2(Q\_1+Q\_2) = 180 - Q\_1 - 2Q\_2$$ Also the price in each market should be the same: $P\_1 = P\_2$. Therefore:
$$ 80-Q\_1/2 = 100-Q\_2$$. Aligning these two equations we get $$Q\_1 = 20; Q\_2 = 30; P\_1=P\_2 = 70.$$ Hope this helps!

> **Comments:**
> **username_5**: This answer is simply **wrong**. Where did you learn "if the government prohibits price discrimination, then the sum of MR in each market should equal to the MC"?  

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