<p>A monopolist prices by setting marginal revenue equal to marginal cost. The marginal revenue depends on demand.</p>
<p>If you have a fixed supply in the sense that the quantity offered is always some <span class="math-container">$q\in \mathbb R$</span> independent of the price, then you define a market equilibrium such that the demanded quantity at the equilibrium price must be equal to <span class="math-container">$q$</span>. As a dictator you may also impose that the entire quantity is traded at price zero and then somehow ration the excess demand, but then --some might argue that-- a secondary market would form such that the consumers with the highest willingness-to-pay end up with the good anyway.</p>
<p>If you have a monopolist who can produce up to <span class="math-container">$q$</span> units at cost zero (put differently, this monopolist already has <span class="math-container">$q$</span> goods and production is impossible), this monopolist would either trade the entire capacity if at <span class="math-container">$q$</span> marginal revenue &gt; 0 OR trade up to <span class="math-container">$q'&lt;q$</span> units where marginal revenue is zero at quantity <span class="math-container">$q'$</span>.</p>

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2020-11-17T14:16:13.657Z
The human monopolist obtains a "monopoly rent". If the monopolist, as you describe, acts in the same way as nature provides factor land (cannot produce more; markets if MR>0) does the monopolist still obtain a "monopoly rent"? If nature does the same thing as the monopolist, does the outcome become qualitatively different, i.e. would it no longer be a "monopoly rent"?
sba222
CC BY-SA 4.0
2020-11-17T14:35:17.637Z
Since the monopolist would optimally want to (if it were possible) produse more than $q$ units, so would all producers in total under any other market form. At price $P(q)$ and quantity $q$ the monopolist's surplus is equal to the total producer surplus of another market form. Therefore, I would not be so insistent on the term monopoly rent.
Bayesian
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2020-11-17T17:23:38.060Z
I’m not sure I understand how it relates to the question. Why would a monopolist want to supply more than _q_ units? A competitive producer considered in isolation as a price-taker, yes, would want to supply more because her demand curve is flat. But in this question supply is permanently limited so the slope of the demand curve might es well be horizontal, it only matters where it meets the supply curve. So both nature and the monopolist acting in the same way as nature are also price-takers.
sba222
CC BY-SA 4.0
2020-11-17T17:32:36.047Z
If MR(q)>0, the monopolist would like to offer more than $q$ units if there was no capacity constraint. This depends on demand. If demand is indeed flat, monopoly and perfect competition would yield the same result. A monopolist who is a price taker has no market power and cannot distort the allocation, $MR(q')=p$ for all $q'$.
Bayesian
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2020-11-17T17:33:58.970Z
Because such a "monopolist" is a price taker, it would be a perfectly competitive market.
Bayesian
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2020-11-17T17:55:56.020Z
Right, I think the lesson here is that if supply is fixed (vertical supply curve) the demand curve can be assumed to be horizontal. Any supplier will be a price-taker, including a monopolist who cannot distort the outcome. _P_ depends exclusive on the demand curve. Because market entry is not possible any supplier will benefit from rents in excess of marginal cost (zero).
sba222
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2020-11-17T17:58:04.087Z
This is simply not true. You cannot assume demand to be horizontal. Why should that be the case?
Bayesian
CC BY-SA 4.0
2020-11-17T18:04:26.493Z
If the supply is permanently fixed (vertical supply curve) it does not matter what the demand curve looks like. The demand curve other than where it meets the supply curve would be hypothetical.
sba222
CC BY-SA 4.0
2020-11-17T18:08:41.100Z
A monopolist does not have a supply curve. See my answer. If demand is such that $MR(q')=0$ at $q'<q$, the monopolist supplies less than $q$ units. For a general demand curve, a monopolist is not a price taker.
Bayesian
CC BY-SA 4.0
2020-11-17T13:46:29.543Z
2020-11-17T13:46:29.543Z
Bayesian
<p><strong>Answer to Original Question Assuming Limited Supply</strong></p>
<p>In my opinion equating the two is fallacious from outset. The reason for this is that you  can have cases where you have both monopoly and limited supply, only limited supply or only monopoly or neither. In addition, these have different implication for firm firm behavior in general depending on market structure and form of competition.</p>
<p>For example, consider a trivial example of competition under Bertrand duopoly with firm A and B with demand of individual firm assumed to be <span class="math-container">$D_i = Q/2=100-p$</span> (where <span class="math-container">$Q=q_A+q_B$</span>) if prices are <span class="math-container">$p_A = p_B$</span> (e.g. if firms charge exactly the same price they each get half of the market demand). If <span class="math-container">$p_A&gt;p_B$</span> firm A only gets residual demand after  demand of firm B is satisfied (that is in this case <span class="math-container">$D_i = Q-q_j =100-p_i$</span> , and if <span class="math-container">$p_A&lt;p_B$</span>. Moreover, we will assume that marginal costs are equal to <span class="math-container">$c_i=10$</span> for both firms</p>
<p>Let us start by assuming that both firm A and B production cannot exceed 100 i.e.: <span class="math-container">$q_A,q_B \leq 100$</span>. In this case clearly the Nash Equilibrium (NE) will be given at a point where <span class="math-container">$p_A=p_B = c$</span>. The reason for this is that in this case the fixed supply is large enough to satisfy the demand so if any firm would dare to raise prices above <span class="math-container">$p=c$</span> it would lost all the demand to the another firm. Also in this case firms will have zero profit, and quantity sold in the market will be <span class="math-container">$90$</span>.</p>
<p>However, consider monopoly in such situation. Again let us assume demand is the same <span class="math-container">$Q=100-p$</span> (here naturally whole <span class="math-container">$Q$</span> goes to the monopolist) and let us again assume <span class="math-container">$c=10$</span>. Furthermore, let us again assumed supply cannot be higher than <span class="math-container">$100$</span> so <span class="math-container">$q\leq 100$</span> In that case the profit would be given by:</p>
<p><span class="math-container">$$\pi = (100-Q)Q - 10Q $$</span></p>
<p>and it is trivial to see that optimal profit maximizing quantity is <span class="math-container">$ Q^* = 45$</span> and consequently <span class="math-container">$p^*= 55$</span> and <span class="math-container">$\pi^*= 2025 $</span>.</p>
<p>In both cases we have some restriction on supply and in one case we have no profit and <span class="math-container">$Q=90$</span> and in second case we have quite a large monopoly profit and <span class="math-container">$Q=45$</span>.</p>
<p>Now, of course in the above we get this large contrast also because I assumed that the restricted supply is still larger than the maximum demand. However, even if we would make the supply restriction more strict we would get whole range of quantities at which the outcomes between monopoly and Bertrand competition would not be the same. Eventually as we would start restricting quantity further there would be a special case where Bertrand Duopoly and monopoly would have exactly the same outcome. Hence, I won't deny there are special cases where in terms of outcome monopoly and restriction of supply will get you the same result.</p>
<p>But those are special cases not general ones. Generally you cannot equate restriction/limit on supply with monopoly. They even can exist jointly and independently of each other. I mean there are special cases where monopoly charges the same price as perfectly competitive firm (e.g. perfectly elastic demand) but it would be absolutely inappropriate to conclude that there is no difference between monopoly and perfect competition.</p>
<p><strong>Answer to Edit</strong></p>
<p>If you assume that quantity is fixed at some <span class="math-container">$\bar{q}$</span> and it cannot change then it is trivial to prove that generally <span class="math-container">$q$</span> supplied by monopoly wont be equal to the case of fixed supply.</p>
<p>In the monopoly example above we found that monopolist would supply exactly <span class="math-container">$q^*=45$</span> - no more no less. The quantity that monopolist chooses is not random - it is literally engineered to maximize the monopolist profit and to get as much profit as possible.</p>
<p>However, <span class="math-container">$\bar{q}$</span> fixed supply that <em>must</em> be brought to market will almost always give you different outcome as in this case there is no reason to assume nature chose <span class="math-container">$\bar{q}$</span> to maximize anyone's profit. For example, if <span class="math-container">$\bar{q}=10$</span> then price on the market in the example above would be <span class="math-container">$p=90$</span>. Moreover, if there would be multiple firms lets say 10 firms all offering 1 of those <span class="math-container">$q=10$</span> products profit would be just <span class="math-container">$90$</span> per firm, if <span class="math-container">$\bar{q} =60$</span> price would be <span class="math-container">$P=40$</span> and again we assume there is 10 firms individual profits would be <span class="math-container">$240$</span>.</p>
<p>Literally only in most special case with all fixed supply provided by singe firm and fixed supply happening to be exactly <span class="math-container">$q=45$</span> - which is astronomically unlikely to happen at random would you have a special case where market outcomes are identical between restricted supply and monopoly case.</p>

CC BY-SA 4.0
2020-11-17T18:42:07.647Z
Apologies I don't think the question was clear. I'm only referring to the special case where supply is absolutely and permanently fixed.
sba222
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2020-11-17T18:48:38.553Z
@Steve222 if that is the case then answer is even more trivial. It all depends on where the supply is fixed in that case you can have any outcome depending on q and most of them will not be equal to monopoly ones
1muflon1
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2020-11-17T19:30:28.610Z
@Steve222 I added edit to my q addressing the fixed q case
1muflon1
CC BY-SA 4.0
2020-11-17T17:42:51.100Z
2020-11-17T19:50:06.437Z
2020-11-17T19:50:06.437Z
1muflon1
<p>The assumption are that</p>
<ul>
<li>supply is absolutely and permanently fixed at <em>q</em></li>
<li>average and marginal costs are zero</li>
<li>demand always exceeds supply (P ≥ 0)</li>
</ul>
<p>The graph should look the same in both cases: a vertical supply curve and P depending exclusively on demand.</p>
<p>A monopolist single supplier would be able to choose a price-quantity combination to obtain a rent. In <em>this</em> case, however, the single supplier is a price-taker. So following the definition she would not be considered a monopolist. Instead as MR never decends to MC the supplier would want to maximise supply. In fact, it does not seem to matter whether the fixed supply is <em>distributed</em> by a single or multiple suppliers.</p>
<p>Yet at the same time because supply is fixed and MC=0 the supplier/s generate a return in excess of the costs needed to bring the factor into production: an economic rent. Could this economic rent be understood to be due to market power due to lack of competition as firms cannot enter the market (Mankiw)?</p>
<p>If the latter is the case, does that imply that the rent derived from a supply limited by nature may be subject to market power in the same sense?</p>

CC BY-SA 4.0
2020-11-17T13:33:57.333Z
In what way do you think that supply is fixed in a monopoly? A monopolist supplies based on the demand curve. If demand curve changes so will the supply. How do you consider it *fixed*?
Dayne
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2020-11-17T13:46:13.350Z
I changed "fixed" to "limited"
sba222
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2020-11-17T18:00:14.417Z
Why would the outcomes between the two be same in most cases?
csilvia
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2020-11-17T18:06:25.537Z
@csilvia What do you mean?
sba222
CC BY-SA 4.0
2020-11-17T18:39:30.990Z
@Steve222 like I don't get it why would you ever think they should be the same? Monopoly, can choose any quantity that maximizes profit. When quantity is restricted even monopoly my be forced to choose quantity that does not maximize profit - monopoly makes profit by exploiting its ability to manipulate prices not fixing quantities. If quantity is fixed below monopoly quantity then monopolist will not even be able to maximize profits. If it is above the profit maximizing quantity monopolist will maximize profit by choosing different q then what is the restriction
csilvia
CC BY-SA 4.0
2020-11-17T18:42:15.107Z
I guess that when you just say that the you have market where supply magically happens to be fixed at monopoly profit max quantity you would get the same result d'oh, but what are chances of that ever happenin
csilvia
CC BY-SA 4.0
2020-11-17T18:46:39.467Z
I will propose to close the question and restate it
sba222
CC BY-SA 4.0
2020-11-17T18:52:31.510Z
@Steve222 I am sorry but if question has As with positive votes even if 5 users would vote to close the Q would not be automatically deleted. That would require moderator intervention- and this is not valid reason for such action. In addition I think that would be quite unfair to both Bayesian and me since we took some serious effort in answering your Q you are of course free to ask another Q if you make clear that it is not duplicate because of fixed quantity it won’t be closed as duplicate
1muflon1
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2020-11-17T19:07:46.047Z
OK I might restate it then. I think the majority opinion here is that the question is not useful in this form.
sba222
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2020-11-17T21:27:06.123Z
This new edit has contradiction "A monopolist single supplier would be able to choose a price-quantity combination to obtain a rent."  -but you just assumed Q is fixed. Note monopolist cannot arbitrary choose price - monopolists *manipulates* price by changing $q$ if you assume $q$ is fixed price is fixed. Also what about case where supply is in aggregate fixed at some Q =100 but you have 20 firms instead of monopoly? Outcome simply wont be same all these firms will have smaller share of $\pi$ and as n goes to infinity  any rent goes to 0
1muflon1
CC BY-SA 4.0
2020-11-17T21:35:03.500Z
Yes _q_ is fixed. Any number of suppliers would have the same cost and be price-takers. Their presumed market power derives from the supply being absolutely and permanently fixed.
sba222
CC BY-SA 4.0
2020-11-17T21:40:47.627Z
@Steve222 I dont understand. 1. terminology-wise a price taker has no market power, so I am very confused about why you state that they now have market power. 2. Also market power is not derived from supply being fixed market power means firm can change price by changing its quantity.
1muflon1
CC BY-SA 4.0
2020-11-17T22:02:53.797Z
What I'm trying to understand is if the supplier of a fixed-quantity good has market power? Or what else might be the origen of the rent she obtains from supplying the fixed-quantity good?
sba222
CC BY-SA 4.0
2020-11-17T22:07:42.357Z
@Steve222 no fixed supply does not guarantee market power by itself. In fact it does not even guarantee profit. Consider this example demand is equal to $q=100-p$ supply is fixed at 50. Firms have zero marginal cost $MC=0$ but that does not necessarily mean they have no cost at all - if supply is fixed marginal cost are zero just because $q$ is constant not because there would be no cost so if we assume costs are  equal to 2500 there is zero profit in the above case as $\pi = (100-50)50-2500 = 0$.
1muflon1
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2020-11-17T22:10:51.243Z
Average and marginal costs are zero
sba222
CC BY-SA 4.0
2020-11-17T22:15:19.597Z
@Steve222 ok even if average costs are zero let me just choose different numbers suppose supply is fixed as $Q=100$ then $p=0$ and profit is zero as well. In addition in Bertrand competition even if $P$ is such that demand exceeds supply as long as it does not exceeds supply by sufficiently large amount you wont get monopoly result. Furthermore, of course now you can edit your question and add some more assumptions until you will arrive at your desired conclusion. $x^2$ and $x^3$ are the same functions at $x \in {0;1}$ but elsewhere not - generally fixed supply does not equate to monopoly
1muflon1
CC BY-SA 4.0
2020-11-17T22:18:32.847Z
Lastly, even if we would have demand such $Q=100-p$ fixed quantity at $50$ and have zero marginal and average costs if there is sufficiently large number of firms  the profits will be zero. In this case if there would be single firm profit would be $\pi= (100-50)50= 2500$ but fixed supply does not necessary mean only 1 single firm exist. If there exists $n$ firms all owning one n'th of the fixed supply $Q/n$, then profit of individual firm will be $\pi/n$ which goes to 0 as n goes to infinity - also this does not violate no-entry there might already be any arbitrary n firms at the start
1muflon1
<monopoly><competition><artificial-scarcity>
Is there a qualitative difference between the rent derived from a supply fixed by nature and single supplier required to supply a fixed quantity?
sba222
# Is there a qualitative difference between the rent derived from a supply fixed by nature and single supplier required to supply a fixed quantity?

(Asked by: username_0 on 2020-11-17)

The assumption are that

* supply is absolutely and permanently fixed at *q*
* average and marginal costs are zero
* demand always exceeds supply (P ≥ 0)

The graph should look the same in both cases: a vertical supply curve and P depending exclusively on demand.

A monopolist single supplier would be able to choose a price-quantity combination to obtain a rent. In *this* case, however, the single supplier is a price-taker. So following the definition she would not be considered a monopolist. Instead as MR never decends to MC the supplier would want to maximise supply. In fact, it does not seem to matter whether the fixed supply is *distributed* by a single or multiple suppliers.

Yet at the same time because supply is fixed and MC=0 the supplier/s generate a return in excess of the costs needed to bring the factor into production: an economic rent. Could this economic rent be understood to be due to market power due to lack of competition as firms cannot enter the market (Mankiw)?

If the latter is the case, does that imply that the rent derived from a supply limited by nature may be subject to market power in the same sense?

**Tags:** monopoly, competition, artificial-scarcity
> **Comments:**
> **username_1**: In what way do you think that supply is fixed in a monopoly? A monopolist supplies based on the demand curve. If demand curve changes so will the supply. How do you consider it *fixed*?  
> **username_0**: I changed "fixed" to "limited"  
> **username_2**: Why would the outcomes between the two be same in most cases?  
> **username_0**: @csilvia What do you mean?  
> **username_2**: @Steve222 like I don't get it why would you ever think they should be the same? Monopoly, can choose any quantity that maximizes profit. When quantity is restricted even monopoly my be forced to choose quantity that does not maximize profit - monopoly makes profit by exploiting its ability to manipulate prices not fixing quantities. If quantity is fixed below monopoly quantity then monopolist will not even be able to maximize profits. If it is above the profit maximizing quantity monopolist will maximize profit by choosing different q then what is the restriction  
> **username_2**: I guess that when you just say that the you have market where supply magically happens to be fixed at monopoly profit max quantity you would get the same result d'oh, but what are chances of that ever happenin  
> **username_0**: I will propose to close the question and restate it  
> **username_3**: @Steve222 I am sorry but if question has As with positive votes even if 5 users would vote to close the Q would not be automatically deleted. That would require moderator intervention- and this is not valid reason for such action. In addition I think that would be quite unfair to both Bayesian and me since we took some serious effort in answering your Q you are of course free to ask another Q if you make clear that it is not duplicate because of fixed quantity it won’t be closed as duplicate  
> **username_0**: OK I might restate it then. I think the majority opinion here is that the question is not useful in this form.  
> **username_3**: This new edit has contradiction "A monopolist single supplier would be able to choose a price-quantity combination to obtain a rent."  -but you just assumed Q is fixed. Note monopolist cannot arbitrary choose price - monopolists *manipulates* price by changing $q$ if you assume $q$ is fixed price is fixed. Also what about case where supply is in aggregate fixed at some Q =100 but you have 20 firms instead of monopoly? Outcome simply wont be same all these firms will have smaller share of $\pi$ and as n goes to infinity  any rent goes to 0  
> **username_0**: Yes _q_ is fixed. Any number of suppliers would have the same cost and be price-takers. Their presumed market power derives from the supply being absolutely and permanently fixed.  
> **username_3**: @Steve222 I dont understand. 1. terminology-wise a price taker has no market power, so I am very confused about why you state that they now have market power. 2. Also market power is not derived from supply being fixed market power means firm can change price by changing its quantity.  
> **username_0**: What I'm trying to understand is if the supplier of a fixed-quantity good has market power? Or what else might be the origen of the rent she obtains from supplying the fixed-quantity good?  
> **username_3**: @Steve222 no fixed supply does not guarantee market power by itself. In fact it does not even guarantee profit. Consider this example demand is equal to $q=100-p$ supply is fixed at 50. Firms have zero marginal cost $MC=0$ but that does not necessarily mean they have no cost at all - if supply is fixed marginal cost are zero just because $q$ is constant not because there would be no cost so if we assume costs are  equal to 2500 there is zero profit in the above case as $\pi = (100-50)50-2500 = 0$.  
> **username_0**: Average and marginal costs are zero  
> **username_3**: @Steve222 ok even if average costs are zero let me just choose different numbers suppose supply is fixed as $Q=100$ then $p=0$ and profit is zero as well. In addition in Bertrand competition even if $P$ is such that demand exceeds supply as long as it does not exceeds supply by sufficiently large amount you wont get monopoly result. Furthermore, of course now you can edit your question and add some more assumptions until you will arrive at your desired conclusion. $x^2$ and $x^3$ are the same functions at $x \in {0;1}$ but elsewhere not - generally fixed supply does not equate to monopoly  
> **username_3**: Lastly, even if we would have demand such $Q=100-p$ fixed quantity at $50$ and have zero marginal and average costs if there is sufficiently large number of firms  the profits will be zero. In this case if there would be single firm profit would be $\pi= (100-50)50= 2500$ but fixed supply does not necessary mean only 1 single firm exist. If there exists $n$ firms all owning one n'th of the fixed supply $Q/n$, then profit of individual firm will be $\pi/n$ which goes to 0 as n goes to infinity - also this does not violate no-entry there might already be any arbitrary n firms at the start  

---

## 2 answers

### Answer by username_4 on 2020-11-17. Score: 3

A monopolist prices by setting marginal revenue equal to marginal cost. The marginal revenue depends on demand.

If you have a fixed supply in the sense that the quantity offered is always some $q\in \mathbb R$ independent of the price, then you define a market equilibrium such that the demanded quantity at the equilibrium price must be equal to $q$. As a dictator you may also impose that the entire quantity is traded at price zero and then somehow ration the excess demand, but then --some might argue that-- a secondary market would form such that the consumers with the highest willingness-to-pay end up with the good anyway.

If you have a monopolist who can produce up to $q$ units at cost zero (put differently, this monopolist already has $q$ goods and production is impossible), this monopolist would either trade the entire capacity if at $q$ marginal revenue > 0 OR trade up to $q' units where marginal revenue is zero at quantity $q'$.

> **Comments:**
> **username_0**: The human monopolist obtains a "monopoly rent". If the monopolist, as you describe, acts in the same way as nature provides factor land (cannot produce more; markets if MR>0) does the monopolist still obtain a "monopoly rent"? If nature does the same thing as the monopolist, does the outcome become qualitatively different, i.e. would it no longer be a "monopoly rent"?  
> **username_4**: Since the monopolist would optimally want to (if it were possible) produse more than $q$ units, so would all producers in total under any other market form. At price $P(q)$ and quantity $q$ the monopolist's surplus is equal to the total producer surplus of another market form. Therefore, I would not be so insistent on the term monopoly rent.  
> **username_0**: I’m not sure I understand how it relates to the question. Why would a monopolist want to supply more than _q_ units? A competitive producer considered in isolation as a price-taker, yes, would want to supply more because her demand curve is flat. But in this question supply is permanently limited so the slope of the demand curve might es well be horizontal, it only matters where it meets the supply curve. So both nature and the monopolist acting in the same way as nature are also price-takers.  
> **username_4**: If MR(q)>0, the monopolist would like to offer more than $q$ units if there was no capacity constraint. This depends on demand. If demand is indeed flat, monopoly and perfect competition would yield the same result. A monopolist who is a price taker has no market power and cannot distort the allocation, $MR(q')=p$ for all $q'$.  
> **username_4**: Because such a "monopolist" is a price taker, it would be a perfectly competitive market.  
> **username_0**: Right, I think the lesson here is that if supply is fixed (vertical supply curve) the demand curve can be assumed to be horizontal. Any supplier will be a price-taker, including a monopolist who cannot distort the outcome. _P_ depends exclusive on the demand curve. Because market entry is not possible any supplier will benefit from rents in excess of marginal cost (zero).  
> **username_4**: This is simply not true. You cannot assume demand to be horizontal. Why should that be the case?  
> **username_0**: If the supply is permanently fixed (vertical supply curve) it does not matter what the demand curve looks like. The demand curve other than where it meets the supply curve would be hypothetical.  
> **username_4**: A monopolist does not have a supply curve. See my answer. If demand is such that $MR(q')=0$ at $q'<q$, the monopolist supplies less than $q$ units. For a general demand curve, a monopolist is not a price taker.  

---

### Answer by username_3 on 2020-11-17. Score: 2

**Answer to Original Question Assuming Limited Supply**

In my opinion equating the two is fallacious from outset. The reason for this is that you can have cases where you have both monopoly and limited supply, only limited supply or only monopoly or neither. In addition, these have different implication for firm firm behavior in general depending on market structure and form of competition.

For example, consider a trivial example of competition under Bertrand duopoly with firm A and B with demand of individual firm assumed to be $D\_i = Q/2=100-p$ (where $Q=q\_A+q\_B$) if prices are $p\_A = p\_B$ (e.g. if firms charge exactly the same price they each get half of the market demand). If $p\_A>p\_B$ firm A only gets residual demand after demand of firm B is satisfied (that is in this case $D\_i = Q-q\_j =100-p\_i$ , and if $p\_A. Moreover, we will assume that marginal costs are equal to $c\_i=10$ for both firms

Let us start by assuming that both firm A and B production cannot exceed 100 i.e.: $q\_A,q\_B \leq 100$. In this case clearly the Nash Equilibrium (NE) will be given at a point where $p\_A=p\_B = c$. The reason for this is that in this case the fixed supply is large enough to satisfy the demand so if any firm would dare to raise prices above $p=c$ it would lost all the demand to the another firm. Also in this case firms will have zero profit, and quantity sold in the market will be $90$.

However, consider monopoly in such situation. Again let us assume demand is the same $Q=100-p$ (here naturally whole $Q$ goes to the monopolist) and let us again assume $c=10$. Furthermore, let us again assumed supply cannot be higher than $100$ so $q\leq 100$ In that case the profit would be given by:

$$\pi = (100-Q)Q - 10Q $$

and it is trivial to see that optimal profit maximizing quantity is $ Q^\* = 45$ and consequently $p^\*= 55$ and $\pi^\*= 2025 $.

In both cases we have some restriction on supply and in one case we have no profit and $Q=90$ and in second case we have quite a large monopoly profit and $Q=45$.

Now, of course in the above we get this large contrast also because I assumed that the restricted supply is still larger than the maximum demand. However, even if we would make the supply restriction more strict we would get whole range of quantities at which the outcomes between monopoly and Bertrand competition would not be the same. Eventually as we would start restricting quantity further there would be a special case where Bertrand Duopoly and monopoly would have exactly the same outcome. Hence, I won't deny there are special cases where in terms of outcome monopoly and restriction of supply will get you the same result.

But those are special cases not general ones. Generally you cannot equate restriction/limit on supply with monopoly. They even can exist jointly and independently of each other. I mean there are special cases where monopoly charges the same price as perfectly competitive firm (e.g. perfectly elastic demand) but it would be absolutely inappropriate to conclude that there is no difference between monopoly and perfect competition.

**Answer to Edit**

If you assume that quantity is fixed at some $\bar{q}$ and it cannot change then it is trivial to prove that generally $q$ supplied by monopoly wont be equal to the case of fixed supply.

In the monopoly example above we found that monopolist would supply exactly $q^\*=45$ - no more no less. The quantity that monopolist chooses is not random - it is literally engineered to maximize the monopolist profit and to get as much profit as possible.

However, $\bar{q}$ fixed supply that *must* be brought to market will almost always give you different outcome as in this case there is no reason to assume nature chose $\bar{q}$ to maximize anyone's profit. For example, if $\bar{q}=10$ then price on the market in the example above would be $p=90$. Moreover, if there would be multiple firms lets say 10 firms all offering 1 of those $q=10$ products profit would be just $90$ per firm, if $\bar{q} =60$ price would be $P=40$ and again we assume there is 10 firms individual profits would be $240$.

Literally only in most special case with all fixed supply provided by singe firm and fixed supply happening to be exactly $q=45$ - which is astronomically unlikely to happen at random would you have a special case where market outcomes are identical between restricted supply and monopoly case.

> **Comments:**
> **username_0**: Apologies I don't think the question was clear. I'm only referring to the special case where supply is absolutely and permanently fixed.  
> **username_3**: @Steve222 if that is the case then answer is even more trivial. It all depends on where the supply is fixed in that case you can have any outcome depending on q and most of them will not be equal to monopoly ones  
> **username_3**: @Steve222 I added edit to my q addressing the fixed q case  

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