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Monopoly: Marginal Cost and Long Run Equilibrium

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Ajax cleaning Products is a medium sized operating in an industry dominated by one large firm Tile King; Ajax produces a multi-headed tunnel wall scrubber that is similar to a model produced by Title King to avoid the possibility of price war. The price charged by Title King is $ 20,000.

Ajax has the following short-run cost curve:

TC = 800,000 – 5,000Q + 100Q^2

a) Compute the marginal cost curve for Ajax


Marginal Cost (MC) = dTC/dQ

Since the derivative of a constant = 0,
MC = -5,000 + 200Q

b) Given Ajax pricing strategy, what is the marginal revenue function for Ajax?

Since Ajax is pricing as if it were a perfectly competitive firm, then, it’s price would equal its marginal revenue, thus:

P = MR = $20,000

c) Compute the profit maximizing level of output for Ajax:

A profit maximizing level of output would be achieved at the point where MR = MC

We thus have to set both functions equal each other and then solve for Q.

20,000 = -5000 + 200Q

Solving for Q:

20000 + 5000 = 200Q

25,000 = 200Q

25,000/200 = Q

125 = Q

125 is the profit maximizing quantity that Ajax would have to produce in order to realize the maximum profit possible.

d) Compute Ajax’s total dollar profits:

profit = TR – TC

Profit = (20,000*Q) – (800,000 – 5000Q + 100Q^2)

Profit = (20,000*125) – (800,000 – 5000*125 + 100*125^2)

Profit = 2,500,000 – (800,000 – 625,000 + 1,562,500)

Profit = 2,500,000 – 1,737,500

Profit = $762,500
4. Unique creations hold a monopoly position in the production and sale of magnometers the cost function facing Unique is estimated to be Marginal cost of unique is $ 20.00 as a marginal cost is the cost that is incurred due to the increase of the cost divided by increase in the quantity. Answer:

Suppose that unique produces 10 units
Total cost: 100,200
20 Units produced: 100, 400
Increase in cost =8 100,400-100,200 =200
Increase in quantity = 20-10 = 10
Marginal cost is: 200/ 10 =20
6. Wyandotte chemical company sells various chemicals to the automobile industry Wyandotte currently sells 30,000 gallons of polyol per year at an average price of 15 per gallon. Fixed costs = 90,000

Variable costs = 180,000

The operations research department has estimated that a 15 percent increase in output would not affect fixed costs but would reduce average variable costs by 60 cents per gallon. The market department has estimated the arc elasticity of demand for polyol to be -2.0.

*How much would Wyandotte have to reduce the price of polyol to achieve a 15% increase in the quantity sold?
*Evaluate the impact of such a price cut on total revenue, total cost, and total profits.

Chapter 12: Problems 1, 2(b), and 5(b)
Assume that two companies (A and B) are duopolists who produce identical products. Demand for the products is given by the following linear demand function:
p = 600 – Qc − Qd
where Qc and Qd are the quantities sold by the respective firms and P is the selling price. Total cost functions for the two companies are
TCc = 25,000 + 100Qc
TCD = 20,000 + 125QD

Assume that the firms act independently as in the Cournot model (i.e., each firm assumes that the other firm’s output will not change).
a. Determine the long-run equilibrium output and selling price for each firm.
b. Determine the total profits for each firm at the equilibrium output found in Part (a)
Determine the long run equilibrium output and selling price for each firm. Solution:

Q = Qc + Qd

For Company C

TRc = P*Qc

= (600 – Qc – Qd)Qc

= 600 – Qc2 – Qd*Qc

MRc = 600 – 2Qc – Qd

MCc = 100

At long run equilibrium, MR = MC

600 – 2Qc – Qd = 100

2Qc + Qd = 500————– (1)

For Company D

TRd = P*Qd

= (600 – Qc – Qd)Qd

= 600 – Qc*Qd – Qd2

MRd = 600 – 2Qd – Qc

MCd = 125

At long run equilibrium, MR = MC

600 – 2Qd – Qc = 125

Qc + 2Qd= 475——— (2)

Solving the two linear equation (1) and (2), we get

Qc = 175 and Qd = 150

Putting these values in price equation,

P = 600 – 175-150

P = $275

b. Determine the total profits for each firm at the equilibrium output found in Part A 5. Alchem (L) is the price leader in the polyglue market, all 10 other manufactures (follower (f) firms) sell polyglue at the same price as Alchem. Established price and supplies the remainder of the demand itself. Total demand for poly-glue I given by the following function (Qr = QL + QF)

P = 20,000? – 4QR
Alchem marginal cost function for manufacturing and selling poly-glue is MCL = 5000 + 5QL

The aggregate marginal cost function for the other manufacturing of poly-glue is ∑〖MCf=2000+4Qf〗
To maximize profits how much poly-glue should Alchem produce and what price should it charge? P* $9,666.70, Q* = 666.7
*What is the total market demand for polyglue at the price established by Alchem in Part (a)
*How much of total demand do the follower firms supply?

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