Complete question:
A monopolist is deciding how to allocate output between two geographically separated markets (East Coast and Midwest). Demand and marginal revenue for the two markets are: P1 = 15 - Q1MR1 = 15 - 2Q1P2 = 25 - 2Q2MR2 = 25 - 4Q2. The monopolist’s total cost is C = 5 + 3(Q1 + Q2 ).
What are price, output, profits, marginal revenues, and dead-weight loss
(i) if the monopolist can price discriminate?
(ii) if the law prohibits charging different prices in the two regions?
Solution:
Through price control, the monopolist selects quantity in each sector in such a manner that total income of each business is equivalent to total expense. The marginal cost is equivalent to three (the slope of the overall cost curve).
In the first market
15 - 2Q1 = 3, or Q1 = 6.
In the second market
25 - 4Q2 = 3, or Q2 = 5.5
Substituting into the respective demand equations, we find the following prices for the two markets : P1 = 15 - 6 = $9 and P2 = 25 - 2(5.5) = $14.
Noting that the total quantity produced is 11.5, then
π = ((6)(9) + (5.5)(14)) - (5 + (3)(11.5)) = $91.5.
The monopoly dead-weight loss in general is equal to
DWL = (0.5)(QC - QM)(PM - PC ).
Here, DWL1 = (0.5)(12 - 6)(9 - 3) = $18 and
DWL2 = (0.5)(11 - 5.5)(14 - 3) = $30.25.
Therefore, the total dead-weight loss is $48.25.
Without pricing disparity, the monopoly holder would demand a single price for the whole sector. To optimize income, we find that the total revenue is equivalent to the total expense. Using demand calculations, we note that the complete market curve is kinked to Q = 5:
P=25-2Q, if Q≤518.33-0.67Q, if Q5 .
This implies marginal revenue equations of MR=25-4Q, if Q≤518.33-1.33Q, if Q5
With marginal cost equal to 3, MR = 18.33 - 1.33Q is relevant here because the marginal revenue curve “kinks” when P = $15.
To determine the profit-maximising quantity, equate marginal revenue and marginal cost: 18.33 - 1.33Q = 3, or Q = 11.5.
Substituting the profit-maximizing quantity into the demand equation to determine price :P = 18.33 - (0.67)(11.5) = $10.6.
With this price, Q1 = 4.3 and Q2 = 7.2.
(Note that at these quantities MR1 = 6.3 and MR2 = -3.7).
Profit is(11.5)(10.6) - (5 + (3)(11.5)) = $83.2.
Dead-weight loss in the first market is DWL1 = (0.5)(10.6-3)(12-4.3) = $29.26.
