Answer to Question #191635 in Accounting for mahlet

Question

Suppose that the firm operates in a perfectly competitive market. The market price of his

product is$10. The firm estimates its cost of production with the following cost function:

TC=10q-4q2+q3

A. What level of out put should the firm produce to maximize its profit?

B. Determine the level of profit at equilibrium.

C. What minimum price is required by the firm to stay in the market?

Answer 1

Solution:

A.). What level of output should the firm produce to maximize its profit?

In a perfectly competitive market, a firm will maximize its profit at the point where P = MR = MC. Under perfect competition MR = P.

P = $10

First, derive the MC:

MC = derivative of the Total Cost (TC) in relative to quantity

TC = 10q - 4q² + q³

MC =$\frac{\partial TC} {\partial q} ​ = 10 – 8q + 3q2$

MC = 10 – 8q + 3q²

Therefore, to derive the level of output that maximizes the firm’s profit: set P = MC

10 = 10 – 8q + 3q²

3q² – 8q – 20 = 0

Solve for quantity:

When you factor to get the roots (zeros).

q = 4.239, q = -1.573

We take the positive value:

q = 4.239

The level of output that the firm should produce to maximize its profit: q = 4.239

 

B.). Determine the level of profit at equilibrium.

Profit = Revenue – Total Cost

Revenue = P $\times$ Q = 10 $\times$ 4.239 = 42.39

Total Cost = 10q - 4q² + q³ = 10(4.239) – 4(4.239)² + 4.239³

= 42.39 – 71.88 + 76.17

= 42.39 + 76.17 – 71.88

Profit/Loss = -4.29

 

C.). What minimum price is required by the firm to stay in the market?

The minimum price is equal to the minimum AVC.

Derive AVC:

AVC = $\frac{VC}{q}$

VC = 10q - 4q² + q³

AVC = $\frac{10q - 4q^{2} + q^{3} }{q}$

AVC = 10 – 4q + q²

The shutdown price occurs at the minimum of the average variable cost (AVC), a point where MC = AVC

Set MC = AVC to derive the shutdown output:

MC = AVC

MC = 10 – 8q + 3q²

10 – 8q + 3q² = 10 – 4q + q²

– 8q + 3q² = – 4q + q²

3q² – q² – 8q + 4q = 0

2q² – 4q = 0

2q (q – 2) = 0

q = 2

Substitute q in either the MC or AVC to derive the minimum price:

MC = 10 – 8q + 3q² = 10 – 8(2) + 3(2²) = 10 – 16 + 12 = 10 + 12 – 16 = 6

The minimum price required by the firm to stay in the market = $6