Question

Assume that SpaceX has a variable cost of $50 million for each Falcon 9 launch, so that VC(y)=50∙y, and y is the number of launches per year. Assume that overhead costs, the yearly fixed costs involved in operating SpaceX regardless of launches, is $1 billion, so that FC=1000 (since 1 billion is 1000 million). Yearly demand for space launches (of satellites, predominantly) is D(p)=50/n – 0.2p, where n is the number of aerospace companies serving the market.

a) Assume SpaceX is the only aerospace company serving this market, so that n=1. What is the inverse demand function, p(y)? What is the cost function c(y)=VC(y)+FC?

b) Having profits be π = p(y)∙y – c(y), what is the profit maximizing output level? What is the corresponding market price?

c) Calculate the monopolist’s profit and producer surplus. What is the consumer surplus? What is the deadweight loss?

d) Given these conditions (VC, FC, D(p)), what would happen if a competitor enters the market, so that n=2? Respond in terms of profitability of both the incumbent and the entering firm.

e) Quantify what would need to change in terms of VC, FC or D(p) in order for this not to be a natural monopoly.

Answer 1

a).

Here the demand for space launches is given by, => Y = 50/n – 0.2*P, where n=1.

=> Y = 50 – 0.2*P, => 0.2*P = 50 – Y, => P = 50/0.2 – Y/0.2, => P(Y) = 250 – 5*Y.

The total cost schedule is the sum of “fixe cost” and “variable cost”.

=> C(Y) + FC + VC(Y) = 1,000 + 50*Y, => C(Y) = 1,000 + 50*Y.

b).

The profit function is given by, => A = P(Y)*Y – C(Y).

=> A = (250 - 5*Y)*Y – (1,000 + 50*Y) = 250*Y - 5*Y^2 – 1,000 - 50*Y. So, the profit maximization require “dA/dY = 0”.

=> 250 - 10*Y – 50 = 0, => 10*Y = 200, => Y = 200/10 = 20, => Y=20. So, the market price is “P=250-5*Y = 150”. So, the profit maximizing price and quantity are “P=$150” and “Y=20” respectively.

c).

The profit of the monopolist is, => A = P*Y – C = 150*20 – (1,000 + 50*20) = 3,000 – 2,000 = 1,000, => A = $1,000. Consider the following fig.

Here the “consumer surplus” is given by the area “APmB = 0.5*(250-150)*20 = $1,500”, “producer surplus” is given by the area “PmCEmB = (150-50)*20 = $2,000”,

The “dead weight loss” is given by the area “BEcEm = 0.5*(40-20)*(150-50) = $1,000”.

d).

Let’s assume a new firm enter into the industry, => the new demand schedule face by incumbent is given by.

Y = 50/n – 0.2*P, where n=2.

=> Y = 25 – 0.2*P, => 0.2*P = 25 – Y, => P = 25/0.2 – Y/0.2, => P(Y) = 125 – 5*Y.

The profit function of incumbent is given by, => A = P(Y)*Y – C(Y).

=> A = (125 - 5*Y)*Y – (1,000 + 50*Y) = 125*Y - 5*Y^2 – 1,000 - 50*Y. So, the profit maximization require “dA/dY = 0”.

=> 125 - 10*Y – 50 = 0, => 10*Y = 75, => Y = 75/10 = 7.5, => Y=7.5. So, the market price is “P=125-5*Y = 87.5”. So, the profit maximizing price and quantity are “P=$87.5” and “Y=7.5” respectively. The profit of the incumbent is, => A = P*Y – C = 87.5*7.5 – (1,000 + 50*7.5) = 656.25 – 1,375 = (-718.75), => A= (- $718.75) < 0.

So, the incumbent is making loss. If the entrant also having same types of cost function, => as the entrant enter into the industry the market demand will be shared between them equally and both will starts making loss.