We are now equipped with a number of qualitative predictions about the market share dynamics under the stochastic benchmark. In the following, these predictions are confirmed by means of simulation. The reason why analytical results on the patterns of market share evolution are difficult to obtain even for the simple stochastic benchmark case will become clear below, when the shape of the invariant distribution of market shares is discussed.
The following Figures show simulations of dynamic market share patterns under the stochastic benchmark for different parameter values ρ and λ, and approximations of the invariant distribution of market shares for each set of parameter values. For every parameter set, a different characteristic pattern of market share evolution is obtained. The invariant distributions are also computed by simulation, but with a much higher number of periods.98 Note, that the invariant distribution shows the frequency of each possible state, but it does not reveal the patterns of transition between the states, e.g., whether a high market share of one firm is often maintained for many consecutive periods, or whether market shares tend to fluctuate between the extremes.
20 40 60 80 100t
0 0.2 0.4 0.6 0.8 1 nt
98 usually between 105 and 106 periods; for the market share grid of the approximated invariant distribution I used a uniform step size of 0.01
Figure 3a: Simulated evolution of market shares, stoch. benchmark; λ = 0.95, ρ = 0, α = 1
0.2 0.4 0.6 0.8 1 n
2 4 6 8 density
Figure 3b: distribution of market shares, stoch. benchmark; λ = 0.95, ρ = 0, α = 1
Figure 3a shows a simulated evolution of market shares for λ=0.95, ρ=0, and α= . 1 Using (37) and (14), we obtain the fix point n**=0.997, and l n( **) 0.944= . By (38), k= 4 consecutive market share losses are necessary to destroy firm 1’s dominant position if n1= . 1 Market shares near the extremes are, thus, predicted to occur with high frequency, and a firm with a dominant position is likely to maintain this position for many consecutive periods. The predictions are confirmed by Figure 3a.
Figure 3b shows the invariant distribution for the same parameter values, and illustrates that it has a highly irregular shape. The highest peaks at the extremes in Figure 3b correspond to the fix point computed above. The irregular shape of this distribution gives an idea why quantitative results are hard to obtain analytically.
20 40 60 80 100t
0 0.2 0.4 0.6 0.8 1 nt
Figure 4a: Simulated evolution of market shares, stoch. benchmark; λ = 0, ρ = 0.5, α = 1
0.2 0.4 0.6 0.8 1 n
0.2 0.4 0.6 0.8 1 1.2 1.4 density
Figure 4b: Invariant distribution of market shares, stoch benchmark; λ = 0, ρ 0.5, α = 1
Figure 4a is a simulation for λ = , 0 ρ =0.5, and α= . Using (40), we obtain the fix point 1 , and . According to (41), a single loss of market share is always sufficient to destroy a firm’s dominant position. Therefore, if a firm obtains a dominant position, it is unlikely to maintain this position for several consecutive periods. Figure 4b shows the approximated invariant distribution for
* 2 / 3
n = l n( ) 1/ 3* =
λ= and 0 ρ =0.5. Interestingly, for these parameter values, the invariant distribution is uniform.99 All market shares in the open interval (0,1) occur with the same frequency. However, if ρ differs only slightly from 1/2, highly irregular invariant distributions are obtained again (not shown).
20 40 60 80 100t
0 0.2 0.4 0.6 0.8 1 nt
Figure 5a: Simulated evolution of market shares, stoch. benchmark; λ = 0, ρ 0.95, α = 1
0.2 0.4 0.6 0.8 1n
1 2 3 4 5 density
Figure 5b: Invariant distribution of market shares, stoch. benchmark; λ = 0, ρ 0.95, α = 1
Figure 5a shows a simulation for λ = , 0 ρ =0.9, and α= . According to the results of 1 Section 3.3.1, market shares near and should occur with high frequency, and this is confirmed. Figure 5b shows the approximated invariant distribution.
The tendency towards even splits of the market is confirmed. Unlike for smaller values of
* 0.53
n = l n( ) 0.47* =
ρ ,
99 The small bumps are due to numerical imprecision. Unlike the ones in Figure 3b, they become smaller as the number of periods is raised.
a regular shape is now obtained. The comparison of Figures 3a and 5a illustrates that the presence of word-of-mouth consumers and consumers of the ignorant type can lead to markedly different market share dynamics. When there is a large fraction of word-of-mouth consumers in the population, market shares tend to be skewed, and a firm with a large customer base tends to maintain the dominant position in the market for many consecutive periods. When most consumers are of the ignorant type, market shares tend to fluctuate around 1/2.
25 50 75 100 125 150 175 200t 0
0.2 0.4 0.6 0.8 1 nt
Figure 6a: Simulated evolution of market shares, stoch. benchm.; λ = 0.49, ρ = 0.49, α = 1
0.2 0.4 0.6 0.8 1n
0.5 1 1.5 2 2.5 3 3.5 density
Figure 6b: Invariant distribution of market shares, stoch. benchm.; λ = 0.49, ρ = 0.49, α = 1
Figure 6a shows a simulation with word-of-mouth consumers and ignorant types. Note, that the sum of λ and ρ must be strictly below one, for otherwise, a firm that obtains a market share of 1 will maintain it forever.100 The simulated process shows no clear tendency towards even or skewed distributions of market shares, but the approximated invariant distribution in Figure 6b reveals that skewed distributions occur with higher frequency than even ones.
If the frequency of purchases α is below one, the dynamic evolution of the state variable is not identical to the evolution of the realized market shares. The latter is more volatile than the former if α < . This is illustrated in Figure 7a that shows a simulation for the stochastic 1
100 This is because nobody finds out about the offer of a firm with no customer base if there are no searchers.
benchmark, for λ=0.95, ρ =0, and α=0.2. Figure 7b shows the approximated invariant distribution of realized market shares for these parameter values.
20 40 60 80 100t
0 0.2 0.4 0.6 0.8 1 nt
Figure 7a: Simulated evolution of state and market shares, stoch. benchmark; α = 0.95, ρ = 0, α = 0.2 (thick curve: evolution of the state, thin: realized market shares)
0.2 0.4 0.6 0.8 1 n
0.5 1 1.5 2 2.5 3 3.5 4 density
Figure 7b: Invariant distribution of market shares, stoch. b.m.; α = 0.95, ρ = 0, α = 0.2
The thin curve in Figure 7a, that represents firm 1’s realized market shares, oscillates around the thick curve that shows the evolution of the state . The comparison of Figures 7a and 3a, that were obtained for the same parameters except
n
α, shows that the state and the realized market shares tend to be less extreme for α=0.2 than for α= , but the volatility of realized 1 market shares is higher for α =0.2. This is because for α= , market shares are near 0 or 1 1 most of the time, and step sizes are smaller near the extremes. The comparison of Figures 7b and 3b illustrates that the invariant distribution becomes regular shaped when the failure rate α is reduced (this holds for large values of λ). As a consequence, if the iteration procedure introduced in Section 3.2 fails to converge for large values of δ and λ when α= , it may 1 converge when α is reduced.
3.4. MARKET SHARE DYNAMICS UNDER FIRMS’ STRATEGIC INTERACTION