Question

Consider a duopoly, i.e., a market with only two brands A and B that produce a certain good. Assume that the following pattern holds true among consumers in this market: 60% of those that bought brand A on their previous purchase and 20% of those that bought brand B on their previous purchase will buy brand A on their next purchase.

(a) Find the transition matrix. What will the market shares of each brand be in the long-term?

(b) For each brand, find the expected time until a given customer will purchase the same brand.

(c) For each brand, find the expected time until a given customer will purchase the opposite brand.

Answer 1

A. Let's form the transition matrix.

                       To
                  A           B
From           A  0.60       0.40
               B  0.20       0.80

Assuming that in the long-run the system reaches an equilibrium [x₁, x₂] where

[x₁, x₂] = [x₁, x₂]P and x₁ + x₂ = 1

we have that

x1 = 0.60x1 + 0.20x2  (1) 

x2 = 0.40x1 + 0.80x2  (2) 

and  x1 + x2 = 1      (3) 

From (3) we have that x₂ = 1-x₁

so substituting into (1) we get

x₁ = 0.60x₁ + 0.40(1-x₁)

x₁ = 0.60x₁ + 0.40 - 0.40x₁

(1-0.6+0.4)x₁ = 0.4

x₁ = 0.4/0.8 = 0.5

x₂ = 1 - 0.5 = 0.5

Hence the long-run market shares are 50% each for A & B.

b. For Brand A, recurrence time = 1/ Long term mkt share = 1/0.5 = 2

For Brand B too, recurrence time = 1/ 0.5 = 2

So for both the brands, on an average we will expect the expected time a given customer will purchase the same brand is 2 periods.

c. As the recurrence period and long run mkt share for the both the brands is same, the expected time until a given customer will purchase the opposite brand is 1/ Long run mkt share = 1/0.5 = 2