A monopolist is deciding how to allocate output between two market that are separated geography.demands for the two markets are p1=15-q1 and p2=25-2q2.the monopolist TC is c=5+3(q1+q2).what are price,output,profits,and mr if:a)the monopolist discriminate?
(a) if the monopolist discriminates
Examine the price structures:
more sales mean lower prices
=> there is an upper limit
p1=15-q1 and p2=25-2q2 =>
(q1<15, q2<12.5)
Examine the cost structure:
c=5+3(q1+q2) =>
there is no discrimination of cost for each region.
Assuming the objective is to maximize profit, then we can formulate
total revenue,
tr(q1,q2)=q1*p1(q1)+q2*p2(q2)
=-q1²+15q1+25q2-2q2²
and total profit
tp(q1,q2)=tr(q1,q2)-c(q1,q2)
=tr(q1,q2)-(5+3(q1,q2)
=-q1²+12q1+22q2-2q2²-5
To maximize tp(q1,q2) which contains two variables q1 and q2, we need multi-variable calculus. Luckily, p1 is a simple function of q1, and p2 is a function of q2, and c does not discriminate between q1 and q2, so maximizing independently q1 and q2 works... in this case.
d(tp)/d(q1)=12-2q1=0 => q1=6
d(tp)/d(q2)=22-4q2=0 => q2=5.5
So q1=6, q2=5.5 for maximum total profit where
tp(6,5.5)=91.5
[check by varying q1 and q2 slightly to confirm that the solution gives the maximum profit]
To find marginal revenues,
mr1(q1,q2)=d(tr)/d(q1)=15-2q1
=> mr1(6,5.5)=15-2(6)=3
mr2(q1,q2)=d(tr)/d(q2)=25-4q2
=> mr2(6,5.5)=25-4(5.5)=3
This confirms the solution is optimal since
mr1=mr2 at the given production level.
It would be instructive if you read up on monopolist with price discrimination in your school textbook. If you do not have a text book, try reading articles from the Internet, for example,
http://www.economicsdiscussion.net/monopoly/price-discriminating-monopoly-economics/25670
focus on discrimination of the third degree.
To determine the price, output, profits, and marginal revenue (MR) when a monopolist discriminates between two markets, we need to follow these steps:
Step 1: Calculate the combined demand for the two markets.
To find the combined demand, we need to add the quantity demanded in each market:
Q = Q1 + Q2
Step 2: Determine the marginal revenue (MR) for the combined market.
To find MR, we need to calculate the derivative of the total revenue (TR) function with respect to Q. Since the monopolist discriminates, the MR will be different for each market.
For Market 1:
P1 = 15 - Q1
TR1 = P1 * Q1
MR1 = d(TR1) / dQ1 = 15 - 2Q1
For Market 2:
P2 = 25 - 2Q2
TR2 = P2 * Q2
MR2 = d(TR2) / dQ2 = 25 - 4Q2
Step 3: Set MR equal to MC to find the profit-maximizing quantity.
To maximize profits, a monopolist sets MR equal to marginal cost (MC):
MR1 = MC = 15 - 2Q1
MR2 = MC = 25 - 4Q2
Step 4: Solve the equations to find Q1 and Q2.
Solving the MR equations simultaneously with the MC equation will give us the quantities in each market.
15 - 2Q1 = 5 + 3(Q1 + Q2) [Substituting MC equation]
25 - 4Q2 = 5 + 3(Q1 + Q2) [Substituting MC equation]
Solving these two equations will provide us the values of Q1 and Q2.
Step 5: Calculate the individual prices in each market.
Using the demand equations, we can substitute the obtained quantities into their respective demand functions:
For Market 1:
P1 = 15 - Q1
For Market 2:
P2 = 25 - 2Q2
Step 6: Calculate the total output and total price.
To find the total output, sum up the quantities obtained from each market:
Q = Q1 + Q2
To find the total price, average the prices obtained from each market:
P = (P1 + P2) / 2
Step 7: Calculate total revenue (TR), total cost (TC), and profits.
Total Revenue (TR) = P * Q
Total Cost (TC) = 5 + 3(Q1 + Q2)
Profits = TR - TC
Following these steps will provide you with the price, output, profits, and marginal revenue (MR) when the monopolist is discriminating between the two markets.