Question

The original, first generation Apple Watch was released in 2015. Although the original Apple Watch was designed to compete with various activity trackers at the time, the Apple Watch was sufficiently different for us to assume that there were no close substitutes in 2015.

At the launch Tim Cook announced that the Apple Watch would sell for $500 in the United States and Europe. At that price, q was 1,000, where q was number of watches sold in thousands. An industry group determined that the marginal and average cost of producing the Apple Watch was constant at $100.

1. How much profit did Apple make at the launch of the Apple Watch, given the price and quantity sold (that is, given p=500 and q=1,000)?

In the hope of maximizing its profit, Apple commissioned an economist to estimate demand for the Apple Watch across the U.S. and Europe. The economist determined that the inverse demand function was:

p=600-0.1q.

2. Given the estimated inverse demand, what single price should Apple charge for the Apple Watch in order to maximize its profit?
3. Given that Apple changed the price of the Apple Watch based on the economist’s recommendation, how much profit did Apple make at the new price?

As it turned out, the economist’s estimation actually allowed Apple to distinguish between customers in the United States and customers in Europe, and they were in fact different. These differences were expressed in their different inverse demand functions:

The U.S. market:                              p^A=900-0.25q^A

The European market:                   p^E=400-(1/6)q^E

4. Given the differences across the two groups of customers and assuming that resale across the two continents is impossible, what price should Apple charge in the United States and Europe, respectively?

Given this (group) price discrimination strategy, how much profit did Apple make in each of the two markets? What was total profits earned?

Answer 1

A. How much profit did Apple make at the launch of the Apple Watch, given the price and quantity sold (that is, given p=500 and q=1,000)?

Given price of apple watch p =$500 and quantity sold q = 1000 at the time of launch, the profits can be determined using the below formula.

At time of launch = Total Revenue – Total Cost

Total revenue (TR) = p * q   = 500 * 1000 = 500000

Given constant average cost (AC) = $100, we can find the total cost (TC):

AC = TC/q

TC = AC * q

TC = 100 *1000

TC = 100000

Thus using the profit formula above,

Profit = TR – TC = 50000 – 100000 = 400000

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In the hope of maximizing its profit, Apple commissioned an economist to estimate demand for the Apple Watch across the U.S. and Europe. The economist determined that the inverse demand function was: p=600-0.1q.

A. Given the estimated inverse demand, what single price should Apple charge for the Apple Watch in order to maximize its profit?

Given the inverse demand function, we can find the Total and the marginal revenue function:

Total Revenue (TR) = p*q

= (600-0.1q) *q

= 600q – 0.1q2

Marginal Revenue (MR) = Change in TR / Change in q

MR = dTR /dq {Differentiating TR function w.r.t q]

MR = 600 – 0.2q

A profit maximizing firm produces till that point where MR = MC, we are given constant marginal cost = $100

600 – 0.2q = 100

600 – 100 = 0.2q

500 = 0.2q

q = 500/0.2

q = 2500   {Profit maximizing quantity}

Thus, putting value of q in the inverse demand function we get the profit maximizing price level:

p=600-0.1q

p = 600 – 0.1(2500)

p = 600 – 250

p = $350

B. Given that Apple changed the price of the Apple Watch based on the economist’s recommendation, how much profit did Apple make at the new price?

At the new price, Total Revenue = New Price * New Quantity

TR = 350 *2500

TR = 875000

Total cost is calculated as TC = 100000

Profits at New Prices = TR – TC

= 875000 – 100000

= 775000

Thus with the estimated demand curve, the price falls as compared to the launch price. Quantity sold and the profits increased.      

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As it turned out, the economist’s estimation actually allowed Apple to distinguish between customers in the United States and customers in Europe, and they were in fact different. These differences were expressed in their different inverse demand functions:

The U.S. market:                              pA=900-0.25qA

The European market:                   pE=400-(1/6)qE

A. Given the differences across the two groups of customers and assuming that resale across the two continens is impossible, what price should Apple charge in the United States and Europe, respectively?   Given this (group) price discrimination strategy, how much profit did Apple make in each of the two markets? What was total profits earned?

Step1: Find MR functions for US and Europe

For U.S

TR(A) = pA*qA

TR(A)= (900-0.25qA)qA

TR(A)= 900qA – 0.25qA2

Differentiating TR(A) with respect to qA we get MR(A)

MR(A) = dTR(A)/d(qA)

MR(A) = 900 – 0.5qA

For Europe

TR(E) = pE*qE

TR(E) = (400-(1/6)qE)qE

TR(E) = 400qE – 1/6qE2

Differentiating TR(E) with respect to qE we get MR(E)

MR(E) = dTR(E)/d(qE)

MR(E) = 400 – (2/6)qE

MR(E) = 400 – (1/3)qE

Step2: Equate MR for both countries to the conatant MC = 100 to find the profot maximizing level of quantities in each country

For U.S

MR(A) = MC

900 – 0.5qA = 100

900 – 100 = 0.5qA

800 = 0.5qA

800/0.5 =qA

qA = 1600

For Europe

MR(E) = MC

400 – (1/3)qE = 100

400- 100 = (1/3)qE

300 = (1/3)qE

300 * 3 = qE

qE = 900

Step 3 : Putting the optimal quantities for each country in their respective demand functions we get the prices in USA and Europe

Prices Charged in US: pA =900-0.25qA = 900 – 0.25(1600) = 900 – 400 = $500

Prices Charged in Europe: pE =400-(1/6)qE = 400 – 1/6(900) = 400 – 150 = $250

Step 4: Finding the total revenues for each market using the total revenue function derived in Step 1 and subtracting the total costs to find the profits in each market

For US:

TR(A)= 900qA – 0.25qA2

TR(A) = 900 * 1600 – 0.25 (1600)2

TR(A) = 1440000 – 0.25 *2560000

TR(A) = 1440000 – 640000

TR(A) = 800000

TC = 100000

Profit from US = TR(A) – TC = 800000 – 100000 = 700000

For Europe:

TR(E) = 400qE – 1/6qE2

TR(E) = 400*900 – (1/6)(900)2

TR(E) = 360000 - (1/6)(810000)

TR(E) = 360000 – 135000

TR(E) = 225000

TC = 100000

Profits from Europe = TR(E) – TC = 225000 – 100000 = 125000

Total profits earned = Profits in US + Profits in Europe

Total profits earned = 700000 + 125000 = 825000

Thus, by using the strategy of price discrimination, the firm is able to earn higher profits.