Monopolist Pricing with Two Consumer Groups: Students and General Consumers

To find the monopolist's optimal price, we need to find the point where the monopolist's marginal cost equals the marginal revenue. The monopolist's marginal cost is given by the variable cost function c(q) = 2q.

To find the monopolist's marginal revenue, we need to find the derivative of the total revenue function. The total revenue function is the product of the price and the quantity sold, which is R(q) = pq. Taking the derivative with respect to q gives us the marginal revenue function, which is MR(q) = p.

Since the monopolist sets the same price for both groups of consumers, we can set the marginal revenue equal to the marginal cost to find the optimal price:

p = 2q

Now we can solve this equation to find the monopolist's optimal price. Substituting the demand functions, we have:

10 - p + 40 - 2p = q

Combining like terms, we get:

50 - 3p = q

Now we can substitute this equation back into the equation p = 2q to solve for p:

p = 2(50 - 3p)

Simplifying, we have:

p = 100 - 6p

Bringing the p terms to one side, we get:

7p = 100

Dividing both sides by 7, we find:

p = 100/7 ≈ 14.29

So the monopolist's optimal price is approximately $14.29.

To find the consumer surplus, we need to integrate the demand functions from 0 to the quantity sold at the optimal price. The quantity sold is given by the equation q = 50 - 3p, so we need to integrate the demand functions from 0 to 50 - 3p.

The consumer surplus for group 1 is given by the integral of D1(p) from 0 to 50 - 3p:

∫[0,50-3p] (10 - p) dp

Integrating, we get:

[10p - (p^2/2)] from 0 to 50 - 3p

Plugging in the limits of integration, we have:

[10(50 - 3p) - ((50 - 3p)^2/2)] - [0 - (0^2/2)]

Simplifying, we get:

[500 - 30p - (2500 - 300p + 9p^2/2)] - 0

Combining like terms, we have:

[500 - 30p - 2500 + 300p - 9p^2/2]

Simplifying further, we get:

-9p^2/2 + 270p - 2000

The consumer surplus for group 2 is given by the integral of D2(p) from 0 to 50 - 3p:

∫[0,50-3p] (40 - 2p) dp

Integrating, we get:

[40p - p^2] from 0 to 50 - 3p

Plugging in the limits of integration, we have:

[40(50 - 3p) - ((50 - 3p)^2)] - [0 - (0^2)]

Simplifying, we get:

[2000 - 120p - (2500 - 300p + 9p^2)]

Combining like terms, we have:

-9p^2 + 420p - 500

Now we can add the consumer surplus for group 1 and group 2 to get the total consumer surplus:

-9p^2/2 + 270p - 2000 + -9p^2 + 420p - 500

Combining like terms, we have:

-9p^2/2 + 690p - 2500

To find the producer surplus, we need to calculate the area between the price and the marginal cost curve. Since the marginal cost curve is given by the variable cost function c(q) = 2q, the producer surplus is given by the integral of 2q from 0 to the quantity sold at the optimal price:

∫[0,50-3p] 2q dq

Integrating, we get:

q^2 from 0 to 50 - 3p

Plugging in the limits of integration, we have:

(50 - 3p)^2 - 0

Simplifying, we get:

2500 - 300p + 9p^2

So the producer surplus is 2500 - 300p + 9p^2.

To summarize:

• The monopolist's optimal price is approximately $14.29.
• The consumer surplus is -9p^2/2 + 690p - 2500.
• The producer surplus is 2500 - 300p + 9p^2.