# Is there a qualitative difference between the rent derived from a supply fixed by nature and single supplier required to supply a fixed quantity?

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?

• 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 Nov 17 '20 at 13:33
• I changed "fixed" to "limited" – sba222 Nov 17 '20 at 13:46
• Why would the outcomes between the two be same in most cases? – csilvia Nov 17 '20 at 18:00
• @csilvia What do you mean? – sba222 Nov 17 '20 at 18:06
• @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 Nov 17 '20 at 18:39

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'$$.

• 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 Nov 17 '20 at 14:16
• 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 Nov 17 '20 at 14:35
• 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 Nov 17 '20 at 17:23
• 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 Nov 17 '20 at 17:32
• Because such a "monopolist" is a price taker, it would be a perfectly competitive market. – Bayesian Nov 17 '20 at 17:33

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.

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.