Question

1. Competition:

Assume:

    Q_d = 625 -5p

    Q_s = 175 +5P

    TC = 1 + 5Q + 4Q²

Find: Profit max P, Q, TR, TC, profit.

2. Monopoly:

Assume:   P = 45 - .5Q

                  TC = 3Q2 + 15Q -12

Find: Profit max P, Q, TR, TC, profit and elasticity.

Please solve both questions. Please show all work

Answer 1

1. At equilibrium, Qd = Qs

Or, 625 - 5P = 175 + 5P

Or, 10P = 450

Or, P = 45

A perfectly competitive firm produces at the point where price=MC in order to maximize profit.

Here, TC = 1 + 5Q + 4Q²

Or, MC = d(TC)/dQ = 5 + 8Q

And, we calculated that Market price = $45

Therefore, setting price = MC we get,

5 + 8Q = 45

Or, 8Q = 40

Or, Q = 5

Therefore, TR = P*Q = $(45*5) = $225

TC = 1 + (5*5) + 4(5*5) = $(1+25+100) = $126

Profit = TR - TC = $(225 - 126) = $99

2. A profit maximizing Monopoly produces and the point where marginal revenue equals marginal cost and sets its profit-maximizing price at the point where profit maximizing output lies on the demand curve.

Here, Demand equation is given as : P = 45 - 0.5Q

Or, TR = P*Q = 45Q - 0.5Q²

Or, MR = d(TR)/dQ = 45 - Q

And TC = 3Q² + 15Q -12

Or, MC = d(TC)/dQ = 6Q + 15

Therefore, setting MR = MC we get,

45 - Q = 6Q + 15

Or, 7Q = 30

Or, Q = (30/7)

Now from demand equation we get, P = 45 - 0.5(30/7) = 45 - (15/7) = 300/7

TR = P*Q = $(9000/49) = $183.67 (approx)

TC = 3(30/7)² + 15(30/7) -12 = (2700/49) + (450/7) - 12 = $107.39 (approx)

Profit = TR - TC = $(183.67 - 109.39) = $74.28

Now we can use mark-up formula to calculate elasticity:

(P - MC)/P = -(1/Ed)

At the profit maximizing output, MC = 6(30/7)+15 = (180/7)+15 = (285/7)

So, -(1/Ed) =( (300/7) - (285/7)) / (300/7) = (15/7)/(300/7) = (15/300) = (1/20) = 0.05

Or, - 1/Ed = 0.05

Or, Ed = -(1/0.05) = -20

Therefore, elasticity (Ed) = -20 for this monoply.