A monopoly supplier of music downloads sells to two distinct types of consumer, with the following

Question

monthly demand functions:
type 1: q1=200-2p1
type 2: q2=4005p2
There are 20 Type 1 consumers and 10 Type 2 consumers. The supplier’s marginal cost of production is
40 pence per download, and there are no fixed costs.
1.Suppose the supplier cannot distinguish between the two types of consumer, and must sell to
everyone at a uniform price. Let Q=20q1+10q2 denote the total quantity produced and sold,
and let p denote the uniform price.
a.Write down an expression for the supplier’s total demand function, in the form of an
equation for Q in terms of p.
b.Derive expressions for the supplier’s total revenue and profit functions, in the form of
equations for TR and p in terms of Q.
c.Calculate the supplier’s profitmaximizing value of Q, and the corresponding profit
maximizing value of p.
d.Calculate the total consumer surplus and total producer surplus (monopoly profit) at this
profit-maximizing solution.

Answer 1

To solve this problem, we start with the consumer demands provided for two types of consumers and we will derive the expressions step by step.

1. Total Demand Function

Given the monthly demand functions:

• For type 1: \( q_1 = 200 - 2p_1 \)
• For type 2: \( q_2 = 400 - 5p_2 \)

Since they buy at the same uniform price \( p \), we substitute \( p_1 \) and \( p_2 \) with \( p \):

• \( q_1 = 200 - 2p \)
• \( q_2 = 400 - 5p \)

Now, we calculate the total demand \( Q \):

\[ Q = 20q_1 + 10q_2 \] Substituting for \( q_1 \) and \( q_2 \): \[ Q = 20(200 - 2p) + 10(400 - 5p) \]

Calculating it:

\[ Q = 4000 - 40p + 4000 - 50p \]

Combining terms: \[ Q = 8000 - 90p \]

2. Total Revenue and Profit Functions

Total Revenue \( TR \) is given by price multiplied by quantity \( p \cdot Q \): \[ TR = p \cdot Q = p(8000 - 90p) = 8000p - 90p^2 \]

Profit \( \pi \) is total revenue minus total costs. The total cost \( TC \) is given by marginal cost times quantity \( MC \cdot Q \): \[ TC = 40Q \]

Thus, Profit \( \pi \) is: \[ \pi = TR - TC = (8000p - 90p^2) - 40Q \]

Now we substitute \( Q \) from the total demand function:

\[ TC = 40(8000 - 90p) = 320000 - 3600p \] So, \[ \pi = 8000p - 90p^2 - (320000 - 3600p) \] Thus, \[ \pi = 8000p - 90p^2 - 320000 + 3600p \] Combining terms gives: \[ \pi = (8000 + 3600)p - 90p^2 - 320000 \] \[ \pi = 11600p - 90p^2 - 320000 \]

3. Profit Maximization

To find the profit-maximizing quantity \( Q \) and price \( p \), we differentiate the profit function with respect to \( p \), set the derivative equal to zero, and solve for \( p \):

\[ \frac{d\pi}{dp} = 11600 - 180p = 0 \] \[ 180p = 11600 \] \[ p = \frac{11600}{180} \approx 64.44 \text{ (pence)} \]

Now we calculate \( Q \) when \( p \) is around 64.44: \[ Q = 8000 - 90(64.44) \approx 8000 - 5800 \approx 2200 \]

4. Consumer Surplus and Producer Surplus

To find the consumer surplus and producer surplus:

• Consumer Surplus (CS): The area between the demand curve and the price level. For type 1 consumers, the maximum price they are willing to pay when demand is zero is:

For type 1,

• \( p_1 = 100 \) (setting \( q_1 = 0 \))
• For type 2,
• \( p_2 = 80 \) (setting \( q_2 = 0 \))

Since \( p\) is uniform, the demand curve intersects the price at $64.44, so the CS is:

For type 1: \[ CS_1 = \frac{1}{2} \times (100 - 64.44) \times 20 = \frac{1}{2} \times 35.56 \times 20 = 355.56 \]

For type 2: \[ CS_2 = \frac{1}{2} \times (80 - 64.44) \times 10 = \frac{1}{2} \times 15.56 \times 10 = 77.8 \]

Total Consumer Surplus: \[ CS = CS_1 + CS_2 \approx 355.56 + 77.8 \approx 433.36 \]

• Producer Surplus (PS): Is equal to the profit which is calculated from profit: \[ \pi \approx TR - TC = (p * Q) - MC * Q = 64.44 \cdot 2200 - 40 \cdot 2200 \approx 141768 - 88000 \approx 53768 \]

Thus to summarize:

• Profit-Maximizing Quantity \( Q \): \( 2200 \)
• Profit-Maximizing Price \( p \): \( 64.44 , \text{(pence)}\)
• Total Consumer Surplus \( CS \): \( 433.36 \)
• Total Producer Surplus (Profit) \( PS \): \( 53768 \)