Question

4. You know that million-dollar idea you had for that unique product everyone needs but no one has ever made before? It's your lucky day: you've figured out how to manufacture it. You know with all your heart that if you set your price at p, annual demand for your revolutionary product will be Qa(p) = 50,000 – 40p. However, you're admittedly more of an ideas person at this point; the fixed costs of your proposed method of manufacturing are kind of expensive. Your total cost to produce a quantity Q per year will be TC(Q) = 1,000,000 + 200. Believe it or not, you just don't have that kind of startup cash. In order to pitch your idea to potential investors, you need to convince them of how profitable this market will be, once you invent it. Fortunately, you can tell them that your product is so revolutionary, you will be a monopolist in its market; there are no close substitutes for this incredible thing. Now you need to figure out what your expectations are for your fledgling firm. (a) What is your total revenue function? What is marginal revenue, MR? (b) What is marginal cost, MC? (c) What are the profit maximizing level of production, Q*, and the associated price,p*? (d) What are your profits if you are producing Q*?

Answer 1

a) TR = pQ = [(50,000 - Q)/40] x Q = (50000Q - Q2)/40

MR = dTR/dQ = (50,000 - 2Q)/40

b) MC = dTC/dQ = 20

c) Equilibrium condition is where MR = MC

(50,000 - 2Q)/40 = 20

50,000 - 2Q = 20 x 40

50,000 - 2Q = 800

2Q = 50,000 - 800

2Q = ,49,200

Q* = 24,600 units

P = [(50,000 - Q)/40] = (50,000 - 24,600)/40

P = 25,400/40

P* = $ 6350

d) Profit = TR - TC = PQ - (1000,000 + 20Q)

= 6350 x 24600 - 1000,000 - 20 x 24600

= 156210,000 - 1000,000 - 492,000 = $ 154718000