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Question

Suppose a monopolist’s total cost function is given by 𝑐 = 0.004𝑞3 + 30𝑞 + 7000, and the revenue function is 𝑟 = 𝑞(1200 − 6𝑞), where 𝑐 is measured in dollars and 𝑞 is measured in tonnes per week.

1. Determine marginal cost when 𝑞 = 50.
2. Express profit (𝜋) as a function of 𝑞.
3. Determine the profit-maximising level of output and the corresponding maximum profit.

Justify your answers.

Answer 1

𝑐(q) = 0.004𝑞3 + 30𝑞 + 7000

1) c(50)=0.004(50)^3+30x50+7000=500+1500+7000=9000$

2)Express profit (𝜋) as a function of 𝑞. we know this formula 𝑟 = 𝑞(1200 − 6𝑞) so r=𝜋

𝜋= 𝑞(1200 − 6𝑞) solve this equation by x , 𝜋=1200q-6q^2 ⇔-6q^2+1200q-𝜋=0⇔

q=(-1200± √1200^2-4.(-6)𝜋)/-12 ⇔ q=100± √6(60000-𝜋)/6

3)𝑟 = 𝑞(1200 − 6𝑞) ⇔r=-6q^2+1200q the vertex is (m,n) m=-b/2a and n=r(m)

m=-1200/-12=100 the r(100)=60,000 so q=100 tones and max profit is r=$60,000

c(100)=$14,000 ,r(100)=$60,000 so 60,000-14,000=$56,000 profit