Part III: Calculate the Following Questions by Using the Necessary Steps
(4 pts each)
1) A monopolist is deciding how to allocate output between two markets that are 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, profits, and MR if:
a) The monopolist can price discriminate?
b) The law forbids (prohibits) charging different prices in the two regions?
2. Suppose you are the manager of a watch-making firm operating in a competitive market. Your cost of production is given by C = 100 + Q2, where Q is the level of output and C is total cost.
a) If the price of watches is birr 60, how many watches should you produce to maximize profit?
b) What will your profit level be?
c) At what minimum price will you produce a positive output?
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Answers
To answer these questions, we will use economic principles and formulas to calculate the desired values.
Question 1a) - Price Discrimination:
To maximize profit through price discrimination, the monopolist charges different prices in each market. However, since the question doesn't specify the quantities of output in each market, we need to find the optimal allocation.
1. Determine the monopoly's MR (Marginal Revenue) for each market:
The MR for each market can be found by taking the derivative of the demand functions with respect to Q1 and Q2, respectively.
For market 1:
P1 = 15 - Q1
MR1 = d(P1 * Q1) / dQ1
= Q1 * dP1 / dQ1 + P1
= Q1 * (-1) + 15
= 15 - Q1
Similarly, for market 2:
P2 = 25 - 2Q2
MR2 = d(P2 * Q2) / dQ2
= Q2 * dP2 / dQ2 + P2
= Q2 * (-2) + 25
= 25 - 2Q2
2. Equate the MR for each market with the marginal cost (MC) to find the optimal quantities:
Since the total cost function is given as C = 5 + 3(Q1 + Q2), the MC can be calculated as the derivative of the total cost function with respect to Q1 and Q2.
For Q1:
MC1 = dC / dQ1
= 3
Similarly, for Q2:
MC2 = dC / dQ2
= 3
Setting MR1 = MC1 and MR2 = MC2, we can solve the two equations to find the quantities Q1 and Q2.
3. Calculate the prices and profits:
After finding the optimal quantities, substitute the values into the demand functions to determine the prices in each market.
For market 1: P1 = 15 - Q1
For market 2: P2 = 25 - 2Q2
Profit can be calculated as:
Profit = (P1 - MC1) * Q1 + (P2 - MC2) * Q2
Question 1b) - Price Discrimination Prohibited:
In this scenario, charging different prices in the two regions is forbidden by law. As a result, the monopolist must set a single price for both markets.
To find the equilibrium price and quantity, we need to equate the total quantity demanded to the quantity supplied by the monopolist.
1. Determine the total demand for the two markets:
The total demand is the sum of the individual demands for each market, which can be calculated as:
Total Demand = Q1 + Q2
2. Find the equilibrium condition:
At equilibrium, the quantity supplied by the monopolist equals the total quantity demanded.
Q1 + Q2 = Quantity Supplied
3. Calculate the equilibrium price and quantity:
Substitute the equilibrium condition into the demand functions to determine the equilibrium price.
For the quantity supplied, use the total quantity supplied by the monopolist, which is Q1 + Q2.
Profit can then be calculated as:
Profit = (P - MC) * (Q1 + Q2)
Question 2:
a) To maximize profit, the watch-making firm in the competitive market should produce at the quantity where marginal cost (MC) equals marginal revenue (MR).
Since the cost of production is given by C = 100 + Q^2, the MC is the derivative of the cost function:
MC = dC / dQ
= 2Q
Since we are operating in a competitive market, the price is equal to the marginal revenue (MR).
To find the optimal quantity, equate MR to MC:
MR = MC
P = 2Q
Substitute the price of 60 into the MR equation:
60 = 2Q
Q = 30
The firm should produce 30 watches to maximize profit.
b) The profit level can be calculated as the difference between total revenue (TR) and total cost (TC):
Total Revenue (TR) = Price * Quantity
Total Revenue (TR) = 60 * 30
Total Cost (TC) = 100 + Q^2
Total Cost (TC) = 100 + 30^2
Profit = Total Revenue (TR) - Total Cost (TC)
c) To determine the minimum price at which the firm produces a positive output, set the total cost equal to zero and solve for Q:
Total Cost (TC) = 100 + Q^2
0 = 100 + Q^2
Q^2 = -100 (Since Q cannot be negative, there's no real solution here)
Hence, the minimum price required to produce a positive output is not achievable based on the given cost function.