I know math isn't welcome here, but my professor is a lunatic and I don't know who else to ask.
If the Demand curve is Q=20-p, and MC=4:
1. What is the profit maximizing monopoly price and output?
2. What would be the equilibrium price and output in a perfectly competitive market?
3. What is the dead weight loss from monopolization in this market?
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4. If the monopoly reduces it's MC to $2 what would be the change in surplus, if there is one?
Okay, so for question 2 I know that in perfectly competitive markets MC=MR, and I know that Q=20-p. So for revenue (PQ) I think I take (20-p)p => 20p-p^2 (take derivative for marginal revenue) 20-2p (set equal to 4?) 20-2p=4 => 2p=16 => p=8? Put in the demand equation: 20-8=12? Or should I use elasticity?
I'm pretty sure I'm wrong, and I can't answer questions 3-4 until I find the monopoly price.
So my question is, did I do part 2 correctly, and how do I do part one?
Esuric:2. What would be the equilibrium price and output in a perfectly competitive market?
Condition for a perfectly competitive market: MR = MC = p
So, you already have the answer for the equilibrium price in a perfectly competitive market, which is, 4.
Now it's simple.
It has been given: Q = 20 - p
The equation can be re-written as: p = 20 - Q
Now, for price to be equal to 4(that is for p to be equal to MR and MC), Q has to be equal to 16. So the equilibrium output is 16.
Hence, p = 4; Q = 16.
Esuric:how do I do part one
The conditions are quite the same, except that MR = MC; but not equal to p.
When p = 1; Q = 19; TR = 19 (MR = 19 > MC)
When p = 2; Q = 18; TR = 36 (MR = 17 > MC)
When p = 3; Q = 17; TR = 51 (MR = 15 > MC)
When p = 4; Q = 16; TR = 64 (MR = 13 > MC)
When p = 5; Q = 15; TR = 75 (MR = 11 > MC)
When p = 6; Q = 14; TR = 84 (MR = 9 > MC)
When p = 7; Q = 13; TR = 91 (MR = 7 > MC)
When p = 8; Q = 12; TR = 96 (MR = 5 > MC)
When p = 9; Q = 11; TR = 99 (MR = 3 < MC)
When p = 10; Q = 10; TR = 100 (MR = 1 < MC)
So, the profit maximizing monopoly price would be 8; and the profit maximizing monopoly output would be 12.
First two questions have been previously answered, but I thought I should elaborate for the last two.
Esuric:If the Demand curve is Q=20-p, and MC=4:
My assumption is that MC function is a constant:
MC = 4 + 0q
Esuric: 1. What is the profit maximizing monopoly price and output?
Let R be Revenue, such that R = pq:
q = 20 - p
p = 20 - q
R = (20 - q) q
R = 20q - q^2
MR is the first derivative of the Revenue function:
MR = dR / dq = 20 - 2q
MC = 4
Profit maximization when MR = MC:
20 - 2q = 4
q = 8
Plug optimum q back into Demand function:
p = 20 - 8 = 12
Profit maximization is:
q = 8, p = 12
Esuric: 2. What would be the equilibrium price and output in a perfectly competitive market?
Demand function:
Perfect competition such that MR = MC = p:
q = 20 - 4 = 16
q = 16, p = 4
Esuric: 3. What is the dead weight loss from monopolization in this market?
Loss of consumer surplus because of higher price and less quantity at monopoly than perfect competition.
Monopoly:
Perfect Competition:
Dead weight loss triangle:
Area = 1/2 Bh
Area = 1/2 (16 - 8) (12 - 4) = 32
Dead weight loss:
Dead weight loss is 32.
Esuric: 4. If the monopoly reduces it's MC to $2 what would be the change in surplus, if there is one?
Profit maximization at MC = 2:
MR = 20 - 2q = 2
q = 9
p = 20 - 9 = 11
q = 9, p = 11
Producer surplus at MC = 4:
pq - (MC)q = (12)(8) - (4)(8) = 64
Producer surplus at MC = 2:
pq - (MC)q = (11)(9) - (2)(9) = 81
Net gain in producer surplus:
81 - 64 = 17
Consumer surplus at MC = 4:
Area = 1/2 (8 - 0) (20 - 12) = 32
Consumer surplus at MC = 2:
Area = 1/2 (9 - 0) (20 - 11) = 40.5
Net gain in consumer surplus:
40.5 - 32 = 8.5
Prashanth Perumal: So, the profit maximizing monopoly price would be 8; and the profit maximizing monopoly output would be 12.
Think Blue: Profit maximization is: q = 8, p = 12
There seems to be two valid solutions to Question #1, since the Revenue function is quadratic, namely (8, 12) and (12, 8) for (q, p).
However (8, 12) has a MR of +4 while (12, 8) has a MR of -4 from an increasing q on function p[q] from left to right.
Because MC is +4, then of necessity MR must be +4 as well, thus (8, 12) is the more likely solution.
Think Blue: Dead weight loss triangle: Area = 1/2 Bh Area = 1/2 (16 - 8) (12 - 4) = 32
I still don't understand that, if you could just elaborate a little.
Esuric: Think Blue: Dead weight loss triangle: Area = 1/2 Bh Area = 1/2 (16 - 8) (12 - 4) = 32 I still don't understand that, if you could just elaborate a little.
It would be much more easier if I can graph it. Basically, deadweight loss is when the monopolist deprives the consumer of extra goods at lower prices, with a loss of consumer surplus and no gain of producer surplus.
In other words, deadweight loss is when the consumer surplus is extinguished, while the monopolist gains no additional benefit.
Here is a graph that can explain how things works:
Here at the rectangle of quantity interval [0, Qm] and price interval [Pc, Pm], the monopolist gains the benefit of a higher price, while the consumer looses the benefit of a lower price. But because the gain in producer surplus exactly offsets the loss in consumer surplus at quantity Qm, the surplus is merely transferred.
In other words, the surplus is conserved, with no net gain or loss in surplus, such that the surplus is transferred from one party to the other.
However for the triangle at quantity interval [Qm, Qc] and price interval [Pc, Pm], the consumer looses consumer surplus from less goods at higher prices. But since the consumer surplus does not transfer to the monopolist, this surplus is extinguished with a net loss of surplus.
Furthermore, for the triangle below Pc (shaded in yellow) at [Qm, Qc], the monopolist looses a producer surplus since the monopolist does not benefit from selling the extra goods at the competitive price.
Thus both the shaded yellow triangles above and below Pc are considered deadweight loss for the consumer and the monopolist, respectively.
Because the MC is a constant for the problem presented, the graph actually looks like this:
Note: Ignore the numbers in the graph above, since they're not relevant to the problem.
Because the MC curve is horizontal, there is no deadweight loss to the monopolist at [Qm, Qc], but there is still a deadweight loss to the consumer.
For this particular problem, the monopolist makes zero profit at the competitive price at any quantity.
To calculate the deadweight loss triangle to the consumer, do the following:
Area = 1/2 (Pm - Pc) (Qc - Qm)
Plug in the profit maximizing quantities and prices for monopoly and perfect competition:
Area = 1/2 (16 - 8) (12 -4) = 32
Hope this helps, and once you're finished with your homework, as an antidote, read the chapter on monopoly in Man, Economy, and State by Murray Rothbard.
Thanks, this was a big help.
Esuric: I know math isn't welcome here, but my professor is a lunatic and I don't know who else to ask. If the Demand curve is Q=20-p, and MC=4: 1. What is the profit maximizing monopoly price and output? 2. What would be the equilibrium price and output in a perfectly competitive market? 3. What is the dead weight loss from monopolization in this market? Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 4. If the monopoly reduces it's MC to $2 what would be the change in surplus, if there is one? Okay, so for question 2 I know that in perfectly competitive markets MC=MR, and I know that Q=20-p. So for revenue (PQ) I think I take (20-p)p => 20p-p^2 (take derivative for marginal revenue) 20-2p (set equal to 4?) 20-2p=4 => 2p=16 => p=8? Put in the demand equation: 20-8=12? Or should I use elasticity? I'm pretty sure I'm wrong, and I can't answer questions 3-4 until I find the monopoly price. So my question is, did I do part 2 correctly, and how do I do part one?
fnord
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