Question

5. (10 points) Jack and Annie are the only sellers of otters in a three-state area. The inverse market demand for otters is given by P = 100-0.5Q, where Q = the total quantity offered for sale in the marketplace. Specifically, Q = 4 + 4A, where q, is the amount of otters offered for sale by Jack and qa is the amount offered for sale by Annie. Both Jack and Annie can produce otters at a constant marginal and average total cost of $20. a. Graph the market demand curve. What would be the prevailing price and quantity if this industry were controlled by a monopolist? b. Suppose that Jack solves part (a) and announces that he will bring half of the monopoly quantity to market each day. What would be the profit-maximizing quantity for Annie to produce? c. Given your answers to (b), what will the industry quantity and final price of otters be? How much profit will Annie earn? Jack? d. Suppose that Jack observes Annie's output from part (b). Find Jack's residual demand and marginal revenue curves, and determine if Jack should adjust his output in response to Annie's choice of A. What will the new price of otters be? e. Is the outcome you found in part (d) an equilibrium outcome? How do you know?

Answer 1

Part A

The profit maximizing quantity of monopolist will be,

MR= MC

That is,

100 - Q = 20

Q = 100 – 20 = 80

If Q=80 then price is,

P= 100 – 0.50

= 100 – 0.5 x 80 = 100 – 40 = $60

Then the market demand curve is,

Price 100 - МС \MR 80 100 .. 200 Quantity of otters

Part B

Annie's profit maximizing quantity is,

Here we need to find Annie's residual demand using jack's quantity in the market inverse demand. That is,

P= 100 -0.5QQJ+QA)

  

= 100 – 0.540 +QA)

  

= 100 - 20 - 0.5QA

  

= 80 – 0.5QA

Now the residual marginal revenue for Annie is,

MR= 80 - QA

So, the profit maximization quatity is,

MR= MC

80 - QA = 20

QA = 80 - 20 = 60

That means Annie's profit maximizing quantity is 60

Part C

The quantity produced by both Annie and Jack is 100. So, the final price of otter is,

P= 100 – 0.50

  

= 100 – 0.5 x 100

  

= 100 - 50 = $50

Now, Annie's profit is,

= (P - ATC) XQ

= (50 - 20) x 60

= (30) x 60 = $1,800

And, Jack's profit is,

= (P - ATC) XQ

= (50 – 20) x 40

= (30) x 40 = $1,200

Part D

Here the residual demand curve for Jack is,

P= 100 – 0.5(Q: +QA)

  

= 100 – 0.5(Q1 + 60)

  

= 100 – 0.5Q1 - 30

  

= 70 – 0.5QJ

Now the residual marginal revenue curve for Jack is,

MR = 70 - QJ

The profit maximizing level of output of Jack is,

MR= MC

70 - QJ = 20

QJ = 50

In contrast with 40 units he announced to produce, Jack can choose to produce 50. Then together both Jack and Annie would produce 110. Now the market price at quantity 110 is,

P= 100 – 0.50

= 100 – 0.5 x 110

= 100 - 55 = $45

Now, Annie's profit is,

= (P - ATC) XQA

= (45 – 20 x 60

= (25) x 60 = $1,500

And Jack's profit is,

= (P - ATC) XQJ

= (45 – 20 x 50

= (25) x 50 = $1,250

Part E

Here we need to solve counot equilibrium. That is,

P= 100 – 0.5(Q: +QA)

The marginal revenue for Jack is,

MR = 100 - Q1 -0.5QA

In equilibrium,

= 100 – Q1 -0.5QA = 20

= 80 – 0.5QA

Similarly, for Annie,

MR = 100 – 0.5QJ - QA

= 80 – 0.5Qj

Now,

QJ = 80 – 0.5(80 – 0.501)

  

= 5333

Similarly,

,

QA = 53.33

Now the equilibrium price is,

P= 100 – 0.5(Q: +QA)

  

= 100 -0.5153.33 + 53.33)

  

= 100 -0.5(106.66)

  

= 100-53.33 = $46.66

So we can conclude that the equilibrium found in part D is not an equilibrium outcome.