Question

Company XYZ has a monopoly on a new product. The product can be manufactured in either of two factories. The costs of production for the two factories are C1=5（Q1）^2 C2=10（Q2）^2 The firm’s demand for the product is P = 500- 4Q. a. (10) Find the quantity produced by factory 1, the quantity produced by factory 2, the total quantity produced, and the price that maximize profit. b. (10) There is an increase in the cost of labor in Factory 1 but the cost of labor in Factory 2 remains the same. How will this increase in labor cost affect the quantity produced in Factory 1? the quantity produced in Factory 2? Total output? Price?

Answer 1

market Demand: P = 500 - 4Q

TR = P x Q = 500Q - 4Q²

MR = dTR/dQ = 500 - 8Q

In factory 1, C1 = 5Q1²

MC1 = dC1/dQ1 = 10Q1, hence Q1 = MC1 / 10 = 0.1MC1

In factory 2, C2 = 10Q2²

MC2 = dC2/dQ2 = 20Q2, hence Q2 = MC2 / 20 = 0.05MC2

Market output: Q = Q1 + Q2, therefore

Q = 0.1MC1 + 0.05MC2

Plugging MC1 = MC2 = MC,

Q = 0.1MC + 0.05MC = 0.15MC

MC = Q / 0.15

The firm maximizes profit when MR = MC, therefore

500 - 8Q = Q / 0.15

75 - 1.2Q = Q

2.2Q = 75

Q = 34  

MC = 34 / 0.15 = 226.67

Q1 = 0.1 x 226.67 = 22.67

Q2 = 0.05 x 226.67 = 11.33

P = 500 - 4 x 34 = 500 - 136 = 364

Higher cost of labor in factory 1 raises MC1, keeping MC2 the same. Therefore, in the equation

Q = 0.1MC1 + 0.05MC2, The first part will increase, leaving the second part the same.

Therefore, market demand (Q) will increase. Output at factory 1 (Q1) will fall and output at factory 2 (Q2) will remain the same. When Q increases, price will fall.