Question

When Apple introduced the iPod in 2001, the iPhone in 2007 and the iPad in 2010, it substantially innovated respectively the multimedia player, the smartphone and the tablet industry. Consider the demand for an innovative Apple product y as D(p) = 1000 – p, and a cost function of c(y)=200∙y.

1. At first consider Apple having a monopoly in this market. What is the inverse demand curve p(y)? With profits being π = p(y)∙y – c(y), what is the profit maximizing quantity y? What is the corresponding price p? What is profits π and what is the consumer surplus CS?

2. Soon after another firms enters with a virtually identical product, and a leader-follower dynamic ensues. Overall industry output is Y = y₁ + y₂, where y₁ is the leader Apple product and y₂ is the follower competitor product. The inverse demand curve you derived in part a is now p(Y) and Apple’s profits are π₁ = p(Y)∙y₁ – c₁(y₁) and the competitor’s profits are π₂ = p(Y)∙y₂ – c₂(y₂), where c₂(y₂) = 200y₂. Find the competitor’s reaction function y₂=R₂(y₁).

3. Being the industry leader, Apple is well aware of how the follower will react once Apple decides its production capacity. What is Apple’s optimal production level y₁? Substitute R₂(y₁) into Y and maximize π₁ with respect to y₁. How does this y₁ compare to the output level you found in part a? How does the leader quantity decision making differ from a monopolist?

4. Given y₁ you determined in part c, what is y₂? Plug y₁ into R₂(y₁). Given y₁ and y₂, what is Y? What is the market price p and how does it compare to what you found in part a? What are profits π₁ and π₂? What is the consumer surplus now, and how does it compare to the one you found in part a?

Answer 1

D(p) = 1000 – p

c(y)=200∙y

a. For inverse demand curve:

y=D(p)

y=1000-p

p= 1000-y Inverse demand curve

At first consider Apple having a monopoly in this market.

For MR:

Calculate TR= py= (1000-y)y= 1000y-y2

Now differentiate TR with respect to y for MR

MR=1000-2y

For MC:

Differentiate C(y) with respect to y:

MC= 200

Now Condition for profit maximization:

MR=MC

1000-2y=200

2y=800

y*=400

p*=1000-y= 1000-400= 600

Profit= p.y-C(y)= 600*400-200(400)= 160000

Consumer surplus= 1/2*(Price(when y=0)-p*)y*= 1/2(1000-600)400= 80000

a. After entry of a new firm:

p= 1000-y1-y2

Where  y1 is the leader Apple product and y2 is the follower competitor product.

c2(y2) = 200y2

c1(y1) = 200y1

For competitor reaction curve:

π2 = p(Y)∙y2 – c2(y2)=(1000-y1-y2)y2-200y2= 1000y2-y1y2-y22-200y2

Differentiate it with respect to y2:

dπ2/dy2= 1000-y1-2y2-200=0

2y2= 800-y1

y2= (800-y1)/2 Reaction curve of competitor

a. For apple's optimal production:

π1 = p(Y)∙y1 – c1(y1)=(1000-y1-y2)y1-200y1= 1000y1-y1y2-y12-200y1

Put value of y2 for reaction curve of competitor:

π1 =1000y1-y1[(800-y1)/2 ]-y12-200y1= 1000y1-800y1/2 +y12/2 -y12-200y1

Differentiate with respect to y1

dπ1/dy1= 1000-400+2y1/2-2y1-200=0

400= 2y1-y1

y1=400

y1 remain same because it is a leader and it knows that how the follower will react once Apple decides its production capacity. So it chooses the maximum it can produce.

a. Given y1=400, Put this value into reaction curve of competitor:

y2=800-400 / 2= 400/2= 200

Here, Y**= y1+y1= 400+200= 600

p**=1000-600= 400

π1 = p(Y)∙y1 – c1(y1)= 400*400 - 200*400= 80000

π2 = p(Y)∙y2 – c2(y2)= 400*200-200*200= 40000

Consumer surplus= 1/2*(Price(when y=0)-p**)Y**= 1/2(1000-400)600= 180000

The consumer surplus increases from the monopoly situation because due to competition price decreases which cause the difference between willing to pay and price to rise.