Question

The Hamilton Company is a member of a perfectly competitive industry. Like all members of the industry, its total cost function is 2 marks

TC=25,000+150Q+〖3Q〗^2

where TC is the firm’s monthly total cost (in dollars) and Q is the firm’s monthly output.
If the industry is in long-run equilibrium, what is the price of the Hamilton Company’s product?
What is the firm’s monthly output?

Answer 1

In long-run for perfect competition output is on minimum of ATC. By the definition of ATC, we have:

$$ATC=\dfrac{TC}{Q}.$$

To find minimum of ATC we need to get derivative:

$$\dfrac{d}{dQ}(ATC)=\dfrac{d}{dQ}(\dfrac{25000+150Q+3Q^2}{Q})=0,$$ $$3-\dfrac{25000}{Q^2}=0,$$ $$Q=\sqrt{\dfrac{25000}{3}}=91.3$$

In long-run there is no profits for perfect competition, so we can write:

$$TR=TC$$

We can find total cost by substituting Q into the function for TC:

$$TC=25000+150\cdot91.3+3\cdot91.3^2=63702.$$

Finally, we can find the price from the definition of total revenue:

$$TR=PQ,$$ $$P=\dfrac{TR}{Q}=\dfrac{63702}{91.3}=697.7$$

Answer:

(a)-(b) $P=697.7,$ $Q=91.3$.