Two Questions

[Archimedes]

1. For the production function y= f(x1, x2) = Min{2x1, 5x2} with input prices w1, w2, derive the cost minimizing input demands and the long-run total cost function.

2. A monopolist firm operates in a market where the demand for its product is given by Q = 36 - P/4 where Q and P are the quantity demanded and the price respectively. The firms cost curve is given by C(Q) = 2Q² + 4.
(a) Find the profit maximizing price, quantity and profit level for the monopolist.
(b) Calculate the Consumer Surplus, the Producer Surplus and the Deadweight Loss associated with the monopoly.

No idea what to do

[gerry87]

Originally Posted by Archimedes

No idea what to do

2. A monopolist firm operates in a market where the demand for its product is given by Q = 36 - P/4 where Q and P are the quantity demanded and the price respectively. The firms cost curve is given by C(Q) = 2Q² + 4.
(a) Find the profit maximizing price, quantity and profit level for the monopolist.
(b) Calculate the Consumer Surplus, the Producer Surplus and the Deadweight Loss associated with the monopoly.

a)
Monopolist sets it's price, so we want to find the price that maxamises profit

Q = 36 - P/4
TC = C(Q) = 2Q² + 4.
TC = C(36 - P/4)= 2(36 - P/4)² + 4

Profit = PQ - TC
Profit = 36P - P²/4 - (2(36 - P/4)² + 4)
Profit = 36P - P²/4 - (2592 - 36P + P²/8 + 4)
Profit = 36P - P²/4 - 2592 + 36P - P²/8 - 4
Profit = 72P - 3P²/8 - 2596

Max Profit = d(Profit)/dP = 72 - 6P/8 = 0 (set equal to zero to find max/min and solve for P)

P = 96

I think they like you to check to make sure it's a max and not a min by getting the 2nd derivative and seeing if it's positive or negative.

d2(Profit)/dP2 = -6/8, it's negative which means it's a maximium.

So you've got P, sub it in to find Q = 36 - 96/4 = 12

Then putting P into your profit function
Profit = 72(96) - 3(96)²/8 - 2596 = 860

b)



In the graph pm is monopoly price, pc is perfect competition price, A is consumer surplus, C is producer surplus, D + E are deadweight loss just find the area of the triangles.

eg
When P=96, Q = 12
When P =144, Q=0 use these to find the triangle (.5*bast*height)

The height is 144-96 = 48
Base is 12 - 0 = 12

Area = .5*(12)*(48) = 288 = Consumer Surplus

do the same for the rest.

Not entirely sure what q1 is asking.

[Archimedes]

Thanks man, huge help. How do I find the competitive Quantity and Price? Dont I need these to calculate the deadweight loss and producer surplus?