A monopolist is deciding how to allocate output between two markets that separated geographically. Demands for the two markets are P1=15-Q1 and P2=25-2Q2.The monopolist's TC is C=5+3(Q1+Q2).What are PRICE,OUTPUT,PROPISTS, MR if
a)the monopolist can price discrinate?
b)the law forbids(prohabits) charging in different prices in the two regions?
answer
MR =2Q1
tel me the answer
To find the price, output, profit, and marginal revenue (MR) in both scenarios, we need to follow these steps:
Step 1: Determine the monopolist's profit-maximizing condition:
To maximize profit, the monopolist should produce where Marginal Cost (MC) equals Marginal Revenue (MR) and set the output level accordingly.
Step 2: Find the output level:
For both scenarios, we'll calculate the output level using the profit-maximizing condition. We'll solve the MC = MR equation and use the corresponding outputs to calculate the prices, profit, and MR.
Let's calculate the values for each scenario:
a) When the monopolist can price discriminate:
Step 1: Profit-maximizing condition (MC = MR):
MC = 3(Q1 + Q2)
Since the monopolist can price discriminate, MR in each market is equal to the demand curve for that market.
In Market 1:
MR1 = P1 = 15 - Q1
In Market 2:
MR2 = P2 = 25 - 2Q2
Step 2: Find the output level:
Set MC equal to MR1 and MR2:
3(Q1 + Q2) = 15 - Q1 (Equation 1)
3(Q1 + Q2) = 25 - 2Q2 (Equation 2)
Solve Equations 1 and 2 simultaneously to find Q1 and Q2 (the output levels).
Q1 = 3
Q2 = 4
Step 3: Calculate prices, profit, and MR:
Using the output levels obtained above, substitute them back into the demand equations to find the prices in each market:
In Market 1:
P1 = 15 - Q1 = 15 - 3 = 12
In Market 2:
P2 = 25 - 2Q2 = 25 - 2(4) = 17
To calculate the profit (PROFIT), subtract total costs (TC) from total revenue (TR):
PROFIT = TR - TC
TR = P1 * Q1 + P2 * Q2
TC = 5 + 3(Q1 + Q2)
Plug in the values:
TR = (12 * 3) + (17 * 4) = 36 + 68 = 104
TC = 5 + 3(3 + 4) = 5 + 3(7) = 5 + 21 = 26
PROFIT = 104 - 26 = 78
Using the output levels, prices, and cost, we can find MR by taking the derivative of TR with respect to Q:
TR = P1 * Q1 + P2 * Q2
TR = (15 - Q1) * Q1 + (25 - 2Q2) * Q2
TR = 15Q1 - Q1^2 + 25Q2 - 2Q2^2
MR1 = 15 - 2Q1
MR2 = 25 - 4Q2
b) When the law forbids charging different prices in the two regions:
Step 1: Profit-maximizing condition (MC = MR):
The monopolist cannot discriminate prices, so both markets have the same demand function:
MR = P = 15 - Q1 = 25 - 2Q2
Step 2: Find the output level:
Set MC equal to MR:
3(Q1 + Q2) = 15 - Q1 = 25 - 2Q2
Solve the equation to find Q1 and Q2:
Q1 = 4
Q2 = 1/2
Step 3: Calculate the prices, profit, and MR:
Substitute the output levels back into the demand equations to find the prices:
P1 = 15 - Q1 = 15 - 4 = 11
P2 = 25 - 2Q2 = 25 - 2(1/2) = 25 - 1 = 24
Profit (PROFIT) = TR - TC
TR = P1 * Q1 + P2 * Q2
TC = 5 + 3(Q1 + Q2)
Plug in the values:
TR = (11 * 4) + (24 * 1/2) = 44 + 12 = 56
TC = 5 + 3(4 + 1/2) = 5 + 15/2 = 5 + 7.5 = 12.5
PROFIT = 56 - 12.5 = 43.5
MR can be calculated by taking the derivative of TR with respect to Q:
TR = P * Q
TR = (15 - Q1) * Q1 + (25 - 2Q2) * Q2
TR = 15Q1 - Q1^2 + 25Q2 - 2Q2^2
MR = 15 - 2Q1 = 25 - 4Q2
So, MR1 = 15 - 2Q1 = 15 - 2(4) = 7
MR2 = 25 - 4Q2 = 25 - 4(1/2) = 25 - 2 = 23
In summary:
a) When the monopolist can price discriminate:
- Q1 = 3, Q2 = 4
- P1 = 12, P2 = 17
- PROFIT = 78
- MR1 = 7, MR2 = 23
b) When the law forbids charging different prices:
- Q1 = 4, Q2 = 1/2
- P1 = 11, P2 = 24
- PROFIT = 43.5
- MR1 = 7, MR2 = 23