Question

Smartphones are differentiated primarily by the operating system they use. An iOS-based iPhone is not identical to an Android-based Samsung. However, consider an Android-based Samsung as virtually identical to an Android-based Google Pixel. They have identical software and assume they are comparable in terms of hardware: battery life, memory storage, camera quality, etc. Assume there are only two Android-based smartphones: Samsung (y_S) and Google Pixel (y_G). Overall demand for Android-based smartphones is D(p) = 1000 – p. Samsung production costs are c_S(y_S)=200y_S, and Google Pixel production costs are c_G(y_G)=200y_G.

1. What is the inverse demand function p(Y), where Y = y_S + y_G? Since profits are respectively are π_S = p(Y)∙y_S – c_S(y_S) and are π_G = p(Y)∙y_G – c_G(y_G), find the reaction functions y_S=R_S(y_G) and y_G=R_G(y_S).

2. Given the two reaction functions you found in part a, what is the Nash equilibrium production bundle y_S and y_G? What is the total market output Y? What is the market price p?

3. Calculate profits π_S and π_G. What is the corresponding consumer surplus CS?

4. Suppose Samsung and Google collude. How much should they each produce? What would be the resulting market price p? How much profits will the two companies make? What would happen to consumer surplus CS?

5. Suppose that instead of two competing companies there were four companies each with the same cost functions. What would happen to market price and profits?

Answer 1

a) Y = 1000 – p

P = 1000-Y = 1000- ys – yg is the inverse demand function

Profit of Samsung = p.ys – cs

Πs = (1000- ys – yg)ys – 200ys

Maximising profit, by FOC, dΠs/dys = 1000 – 2ys – yg -200 = 0

Therefore, 800 – 2ys – yg = 0

Ys = (800-yg)/2 is the BR function of Samsung

Profit of google = p.yg – cg

Πg = (1000- ys – yg)yg – 200yg

Maximising profit, by FOC, dΠg/dyg = 1000 – 2yg – ys -200 = 0

Therefore, 800 – 2yg – ys = 0

Yg = (800-ys)/2 is the BR function of Google

a) For nash equilibrium

Ys = (800-yg)/2 and Yg = (800-ys)/2

Solving for ys and yg

Ys = 400 – yg/2 = 400- (800-ys)/2*2 = 400- (800-ys)/4

Ys = 400 – 800/4 + ys/4

Ys = 400-200 + ys/4

Ys-ys/4 = 200

3ys/4 = 200

Ys = 200*4/3 = 800/3 = 266.67

Ys = 800/3, then yg = (800 – 800/3)/2 = 800/3 = 266.67

So ys = yg = 800/3 in nash equilibrium

Y = ys+yg = 800/3 + 800/3 = 1600/3 = 533.33

P = 1000 – Y = 1000 – 1600/3 = 1400/3 = 466.67

a) Cs = 200ys = 200*800/3 = 160000/3 = 53333.33

Cg = 200yg = 200*800/3 = 160000/3 = 53333.33

Total revenue of Samsung TRs = P.ys = (1400/3)(800/3) = 124444.4

Total revenue of Google TRg = P.yg = (1400/3)(800/3) = 124444.4

Profits:

Πg = TRg – Cg = 124444.4 - 53333.33 = 71111.1

Πs = TRs – Cs = 124444.4 - 53333.33 = 71111.1

consumer surplus = 1/2*Y*P = 0.5*533.33*466.67 = 124444.6

a) If firms collude, they maximise joint profits

Π = TRs + TRg – Cs – Cg

= Pys + Pyg – 200Ys – 200yg

Firms decide the total output together like a monopoly such that Y = ys + yg

So Π = P(ys + yg) – 200(Ys + yg) = P.Y – 200Y = (1000-Y)Y – 200Y

Which is maximised such that d Π/dy = 0

1000 – 2Y – 200 = 0

800 – 2Y = 0

2Y = 800

Y = 400

P = 1000 -400 = 600

with ys + yg = 400, p = 600

firms share Y equally, so that ys = yg = 200

Cs = 200ys = 200*200 = 40000

Cg = 200yg = 200*200 = 40000

Total revenue of Samsung TRs = P.ys = (600)(200) = 120000

Total revenue of Google TRg = P.yg = ((600)(200) = 120000

Πg = TRg – Cg = 120000 – 40000 = 80000

Πs = TRs – Cs = 120000 – 40000 = 80000

consumer surplus = 1/2*y*p = 0.5*400*600 = 120000