Question

Suppose the Weyland-Yutani Corporation is the world’s leading supplier of androids. We’ll begin by assuming they have a monopoly on the production of these androids (which are largely used as personal attendants or soldiers). Because of this monopoly, the quantity of androids that WYC chooses to sell has a direct effect on the price of these androids. This relationship is given by the inverse demand curve:

P=120-Q_WYC

Where P is the price (in thousands of dollars) and Q is the quantity of androids produced. Assume DTC has simple total cost function:

TC=12Q_WYC

In other words (they have no fixed costs and a variable cost of 12 (hundred) dollars per android with the following marginal cost and marginal revenue:

MR=120-2Q_WYC

And

MC=12

a) Write down Weyland-Yutani’s profit function, then find the profit-maximizing quantity, price, and profit earned at that level of output.

Frustrated with WYC’s monopoly on androids, several cybernetics researchers and venture capitalists collectively set up a rival company called Skynet to act as a competitor to WYC. The new inverse demand curve is:

P=120-Q_WYC-Q_SKY

Assume each firm has the same total cost function and marginal cost as listed in part a.

Assume each firm’s marginal revenue is given by:

MR_WYC=120-2Q_WYC-Q_SKY

and

MR_SKY=120-2Q_SKY-Q_WYC

b) Write down each firm’s reaction curve equation.

c) What is the Nash Equilibrium quantity produced by each firm (in previous examples we did this in terms of price)

d) What is the market price associated with these NE quantites?

e) How much profit does each firm earn at the NE quantities and market price?

f) Comparing your answer from part a to parts b-e should convince you that the entrance of Skynet into the market has made buyers of androids better off. For five bonus points, quantify how much better off consumers are: what is the exact difference in price and the number of androids on the market before and after Skynet?

Answer 1

a. Given P=120-Qwyc, we can find the total function revenue of Weyland-Yutani Corporation as below :

TRwyc = P*Qwyc = (120-Qwyc)Qwyc = 120Qwyc- (Qwyc)²

We are given the Total Cost (TCwyc) = 12Qwyc

PROFITwyc = TRwyc – Tcwyc

PROFITwyc = 120Qwyc- (Qwyc)² - 12Qwyc

PROFITwyc = 108Qwyc - (Qwyc)²                                  {Profit Function}

Since Weyland-Yutani Corporation has a monopoly power, it will maximize its profits by producing that level of quantity where marginal revenue is equal to marginal cost :

MR = MC

120-2Qwyc = 12

120 – 12 = 2Qwyc

108 = 2Qwyc

Qwyc = 108/2

Qwyc = 54

Using the above optima level of quantity in the inverse demand function we get :

P=120-Qwyc

P = 120 – 54

Price (P) = 66

Profits earned at the optimal level of output be :

PROFITwyc = 108Qwyc - (Qwyc)² = 108(54) – (54)² = 5832 – 2916 = 2916

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b. Given Inverse demand function of the Duopoly as : P=120-Qwyc-Qsky

MRwyc=120-2Qwyc-Qsky

MRsky=120-2Qsky-Qwyc             

MRwyc =MRsky = 12

Reaction curve for WYC

At the profit maximization condition: MRwyc = MC

120-2Qwyc-Qsky = 12

120 – 12 – Qsky = 2Qwyc

108 – Qsky = 2Qwyc

54 – 0.5Qsky = Qwyc

Reaction curve for SKY

At the profit maximization condition: MRsky = MC

120-2Qsky-Qwyc = 12

120 – 12 – Qwyc = 2Qsky

108 – Qwyc = 2Qsky

54 – 0.5Qwyc = Qsky

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c. To find the Nash Equilibrium quantities for each firm we will first use the reaction function of SKY in the reaction function of WYC:

54 – 0.5Qsky = Qwyc

54 – 0.5(54 – 0.5Qwyc) = Qwyc

54 – 27 + 0.25Qwyc = Qwyc

27 = Qwyc – 0.25Qwyc

27 = 0.75Qwyc

27/ 0.75 = Qwyc

Qwyc = 36

Using the value of Qwyc in putting it in the reaction function of SKY

54 – 0.5Qwyc = Qsky

54 – 0.5(36) = Qsky

54 – 18 = Qsky

Qsky = 36

Thus Nash equilibrium quantities is given by : Qwyc = Qsky = 36

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d. The market price associated with these quantities can be calculated with new inverse demand curve :

P=120-Qwyc-Qsky

P = 120 – 36 – 36

P = 48

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e. To find the profits for each firm we need to know the TR and TC for each firm.

TCwyc = 12Qwyc = 12*36 = 432

TCsky = 12Qsky = 12*36 = 432

TRwyc = P*Qwyc = (120-Qwyc-Qsky)Qwyc = 120Qwyc – (Qwyc)² – (Qsky)(Qwyc)

                                = 120(36) – (36)² – 36x36 = 4320 – 1296 – 1296 = 1728

       

TRsky = P*Qsky = (120-Qwyc-Qsky)Qsky = 120Qsky – (Qsky)(Qwyc) - (Qsky)²

                                       = 120(36) – 36x36 – (36)² = 4320 – 1296 – 1296 = 1728

PROFITwyc = TRwyc – TCwyc = 1728 – 432 = 1296

PROFITsky = TRsky – TCsky = 1728 – 432 = 1296

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f. After the entry of Skynet the consumers have to pay lower price as compared to the situation when only WYC had the monopoly. The price they had to pay earlier was 66, but with entry of SKYnet, the market price got lowered to 48. Thus an amount of 66 – 48 = 18 is saved by the consumers.

The number of androids sold in the market before Skynet entry was 54, however the amount of market quantity available increased with the entry of Skynet to Qwc + Qsky = 36 + 36 = 72. Hence, an amount of 72 –54 = 18 more androids are available in the market now.