<p>You are asked to find the price (not quantity) that the monopoly would choose to maximize its profit.</p>
<ul>
<li>If it chooses a price <span class="math-container">$p&gt;10$</span>, then its profit is zero</li>
<li>If it chooses a price <span class="math-container">$p=10$</span>, then its profit is <span class="math-container">$2Q^d(10)$</span></li>
<li>If it chooses a price <span class="math-container">$p&lt;10$</span>, then its profit is <span class="math-container">$25(p-8)$</span></li>
</ul>
<p>Profit is strictly increasing in <span class="math-container">$p$</span> for <span class="math-container">$p&lt;10$</span>, so there can only be a solution to the profit-maximization problem if we assume at least one of the following:</p>
<ol>
<li><span class="math-container">$Q^d(10)=25$</span> (i.e. all buyers buy if indifferent)</li>
<li>There is some maximum allowable value of <span class="math-container">$p$</span> on the interval <span class="math-container">$[0,10)$</span> (e.g. $9.99 - because only  one cent increments are allowed)</li>
</ol>
<p>Neither assumption is unreasonable. In particular, assumption 1 is not unreasonable as the monopolist would get almost the same profit by pricing just a little below <span class="math-container">$10$</span> and making the consumers strictly prefer buying to not.</p>

CC BY-SA 4.0
2023-11-02T10:22:21.760Z
In second assumption there no global maximser in [0,10), so solution won't exist in second assumption .I'm talking about the first assumption. If we consider 1st assumption the demand curve would be discontinuous (i.e when price reaches 10 Q=25, there won't be horizontal line between 0 and 25 ).But as per the definition of reservation price what is correct, at reservation price the consumer would surely buy or will be indifferent?
hr08
CC BY-SA 4.0
2023-11-02T10:44:58.087Z
The second assumption ensures there is a maximum: either profit will be maximized at the maximum value of $p$ in $[0,10)$ or at $p=10$. The usual assumption is 1, i.e. that buyers will purchase at a price equal to their reservation price. If we don't assume 1 or 2, then there is no profit maximizing price. Note that if we think of this as a game with a continuum of consumers and one firm, then assumption 1 just amounts to selecting the only Nash equilibrium of that game.
smcc
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2023-11-02T10:51:52.207Z
Oh right I didn't notice the discrete increment example you mentioned.
hr08
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2023-11-02T09:06:59.777Z
2023-11-02T11:51:06.390Z
2023-11-02T11:51:06.390Z
smcc
<p>The monopolist's profit is revenue minus cost where revenue is <span class="math-container">$ 10\cdot q $</span> and cost is <span class="math-container">$FC +8\cdot q$</span> . So it looks like it would maximize <span class="math-container">$q$</span> at 25 (unless FC is so large that profit is negative; but this problem might assume FC=0).</p>

CC BY-SA 4.0
2023-10-30T14:24:14.343Z
Yes FC IS zero, I have edited it. But I'm mainly confused about the demand at p=10 which can be anything [0,25], so the supply would not be a single value but a range [0,25]
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CC BY-SA 4.0
2023-10-30T14:31:27.687Z
Normally a monopolist sets marginal revenue equal to marginal cost. But that doesn't work here because both are constant (MR=10 and MC=8). So the monopolist will produce as much as he can sell.
Daniel
CC BY-SA 4.0
2023-10-30T14:38:00.430Z
As the demand curve here is plotted from reservation price so what would be the equilibrium quantity since at p=10 consumers are indifferent between buying and not buying the product, so the quantity demanded would not be exactly 25.
hr08
CC BY-SA 4.0
2023-10-30T14:17:46.217Z
2023-10-30T14:17:46.217Z
Daniel
<p><strong>Q1:</strong> There are <span class="math-container">$25$</span> consumers with each demanding 1 unit and each consumers' valuation of the product is <span class="math-container">$10$</span>. How would the demand curve look like.<br />
<strong>Q2:</strong> Now suppose there's a monopolist with <span class="math-container">$MC=8$</span> and <span class="math-container">$FC=0$</span> what price would he set to maximize its profit.</p>
<p><strong>Ans 1:</strong>
Considering valuation and reservation price same the aggregate demand should look like this:</p>
<p><a href="https://i.sstatic.net/Q80cBm.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Q80cBm.jpg" alt="enter image description here" /></a></p>
<p><strong>Ans 2:</strong> Now im stuck here.<br />
I need to understand the difference between the demand curve drawn from solution to utility maximization problem and demand curve drawn form reservation price or are they same?<br />
Because the demand from Util max is what the consumer will surely buy, but in case of reservation price he is indifferent between buying and not buying (as mentioned in Hal Varian) i.e <span class="math-container">$q^d=\{0,1\}$</span>.<br />
Here we are plotting demand curve from reservation price so, how does the seller decide what qty to supply at the reservation price since it is not sure that the consumers will buy all of it?</p>

<consumer-theory>
Reservation price and demand curve
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# Reservation price and demand curve

(Asked by: username_0 on 2023-10-30)

**Q1:** There are $25$ consumers with each demanding 1 unit and each consumers' valuation of the product is $10$. How would the demand curve look like.  
**Q2:** Now suppose there's a monopolist with $MC=8$ and $FC=0$ what price would he set to maximize its profit.

**Ans 1:**
Considering valuation and reservation price same the aggregate demand should look like this:

[![enter image description here](https://i.sstatic.net/Q80cBm.jpg)](https://i.sstatic.net/Q80cBm.jpg)

**Ans 2:** Now im stuck here.  
I need to understand the difference between the demand curve drawn from solution to utility maximization problem and demand curve drawn form reservation price or are they same?  
Because the demand from Util max is what the consumer will surely buy, but in case of reservation price he is indifferent between buying and not buying (as mentioned in Hal Varian) i.e $q^d=\{0,1\}$.  
Here we are plotting demand curve from reservation price so, how does the seller decide what qty to supply at the reservation price since it is not sure that the consumers will buy all of it?

**Tags:** consumer-theory

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## 2 answers

### Answer by username_1 on 2023-11-02. Score: 1 | Accepted answer

You are asked to find the price (not quantity) that the monopoly would choose to maximize its profit.

* If it chooses a price $p>10$, then its profit is zero
* If it chooses a price $p=10$, then its profit is $2Q^d(10)$
* If it chooses a price $p<10$, then its profit is $25(p-8)$

Profit is strictly increasing in $p$ for $p<10$, so there can only be a solution to the profit-maximization problem if we assume at least one of the following:

1. $Q^d(10)=25$ (i.e. all buyers buy if indifferent)
2. There is some maximum allowable value of $p$ on the interval $[0,10)$ (e.g. $9.99 - because only one cent increments are allowed)

Neither assumption is unreasonable. In particular, assumption 1 is not unreasonable as the monopolist would get almost the same profit by pricing just a little below $10$ and making the consumers strictly prefer buying to not.

> **Comments:**
> **username_0**: In second assumption there no global maximser in [0,10), so solution won't exist in second assumption .I'm talking about the first assumption. If we consider 1st assumption the demand curve would be discontinuous (i.e when price reaches 10 Q=25, there won't be horizontal line between 0 and 25 ).But as per the definition of reservation price what is correct, at reservation price the consumer would surely buy or will be indifferent?  
> **username_1**: The second assumption ensures there is a maximum: either profit will be maximized at the maximum value of $p$ in $[0,10)$ or at $p=10$. The usual assumption is 1, i.e. that buyers will purchase at a price equal to their reservation price. If we don't assume 1 or 2, then there is no profit maximizing price. Note that if we think of this as a game with a continuum of consumers and one firm, then assumption 1 just amounts to selecting the only Nash equilibrium of that game.  
> **username_0**: Oh right I didn't notice the discrete increment example you mentioned.  

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### Answer by username_2 on 2023-10-30. Score: 1

The monopolist's profit is revenue minus cost where revenue is $ 10\cdot q $ and cost is $FC +8\cdot q$ . So it looks like it would maximize $q$ at 25 (unless FC is so large that profit is negative; but this problem might assume FC=0).

> **Comments:**
> **username_0**: Yes FC IS zero, I have edited it. But I'm mainly confused about the demand at p=10 which can be anything [0,25], so the supply would not be a single value but a range [0,25]  
> **username_2**: Normally a monopolist sets marginal revenue equal to marginal cost. But that doesn't work here because both are constant (MR=10 and MC=8). So the monopolist will produce as much as he can sell.  
> **username_0**: As the demand curve here is plotted from reservation price so what would be the equilibrium quantity since at p=10 consumers are indifferent between buying and not buying the product, so the quantity demanded would not be exactly 25.  

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