Question

You've decided to sell sweatshirts for the upcoming game. You project that at a price of

​$36.48

per​ sweatshirt, you will be able to sell

120

shirts. In order to sell

290

​sweatshirts, you would need to lower the price to

​$24.41

per sweatshirt. In order to produce the​ sweatshirts, it costs you

​$2.15

per sweatshirt on top of fixed costs of

​$380.

What price should you charge in order to earn maximize your​ profit?

Assume linear demand.

Answer 1

Let,

P₁ = $36.48 and Q₁ = 120 units

P₂ = $24.41 and Q₂ = 290 units

The linear demand function is given by:

P - P₁ = (P₂ - P₁/Q₂ - Q₁) * (Q - Q₁)

P - 36.48 = (24.41-36.48/290-120) * (Q - 120)

P - 36.48 = -0.071(Q - 120)

P - 36.48 = -0.071Q + 8.52

P = 45 - 0.071Q

TR = P*Q = 45Q - 0.071Q²

MR = dTR/dQ = 45 - 0.142Q

The total costs are given by:

TC = 2.15Q + 380

MC = dTC/dQ = 2.15

Profit maximization will occur at:

MR = MC

45 - 0.142Q = 2.15

0.142Q = 42.85

Q = 301.76 units ~ 302 units

P = 45 - (0.071*301.76)

P = $23.57