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ICE Princess25 [194]
2 years ago
11

The quantity of a product demanded by consumers is a function of its price. The quantity of one product demanded may also depend

on the price of other products. For example, if the only chocolate shop in town (a monopoly) sells milk and dark chocolates, the price it sets for each affects the demand of the other. The quantities demanded, q1​ and q2​, of two products depend on their prices, p1 and p2, as follows:
q1​=150−2p1​−p2
q2​=200−p1​−3p2​.​

If one manufacturer sells both products, how should the prices be set to generate the maximum possible revenue? What is that maximum possible revenue?
Mathematics
1 answer:
Hitman42 [59]2 years ago
5 0

Answer:

Both prices should be set to #25.

The maximum revenue is #4375

Step-by-step explanation:

Given

q_1 = 150-2p_1-p_2

q_2 = 200-p_1-3p_2

Start by calculating the total revenue (R):

R = p_1q_1 + p_2q_2

R = p_1(150-2p_1-p_2) + p_2(200-p_1-3p_2)

R = 150p_1-2p_1^2-p_1p_2 + 200p_2-p_1p_2-3p_2^2

Collect and solve like terms

R = 150p_1+ 200p_2-2p_1^2-2p_1p_2 -3p_2^2

Differentiate with respect to pi and to p2, respectively

\frac{dR}{dp_1} = 150 -4p_1 - 2p_2

\frac{dR}{dp_2} = 200 -2p_1 - 6p_2

Equate both to 0, to get the critical point

150 -4p_1 - 2p_2 = 0

200 -2p_1 - 6p_2 = 0

Solve for p1 in 150 -4p_1 - 2p_2 = 0

4p_1 = 150 - 2p_2

p_1 = 37.5 - 0.5p_2

Substitute p_1 = 37.5 - 0.5p_2 in 200 -2p_1 - 6p_2 = 0

200 - 2(37.5 - 0.5p_2) - 6p_2 = 0

200 - 75 - p_2 - 6p_2 = 0

125 - 7p_2 = 0

-7p_2 =-125

p_2 = 25

Substitute p_2 = 25 in p_1 = 37.5 - 0.5p_2

p_1 = 37.5 - 0.5 * 25

p_1 = 37.5 - 12.5

p_1 = 25

So, we have:

p_1 = p_2 = 25

This implies that the prices should be set to #25.

The maximum possible revenue is:

R = 150p_1+ 200p_2-2p_1^2-2p_1p_2 -3p_2^2

R = 150 * 25 + 200 * 25 -2 * 25^2 - 2 * 25 * 25 - 3 * 25^2

R = 4375

The maximum revenue is #4375

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