What causes the megabyte price drop in the mobile industry?

Abstract

Mobile industry is characterized by a sharp fall in megabyte price which highly benefits to consumers. This article aims to identify the main parameters that lead to such a fall and shows that the growth of traffic is by far the main cause. It proposes a parametric model that explains the growth of traffic from investment. Using a 20-countries wireless market dataset to calibrate the model, it shows that investment actually drives the exponential growth of traffic. As the growth of revenues are much lower, the price of megabyte decreases sharply. The role of competition is ambiguous. On the one hand it reduces margin and thus prices, on the other hand, as the relationship between investment and competition turns to be inverted-U shaped, it may reduce investment and therefore slow down the fall in unit price.

Appendix

1.1 Proof of Equation (2)

Let us denote $j=i-\delta$, from Eq. (1), using the approximation of investment $I_{t}=I_{t_{0}}(1+\lambda )^{t-t_{0}},$ the traffic writes:

$$\begin{aligned} T_{t}=A_{0}\alpha I_{t_{0}}(1+\lambda )^{-t_{0}}\left[ \sum _{i=0}^{t}\left( 1+\theta \right) ^{i}\left( 1+\lambda \right) ^{i}-\sum _{j=0}^{t-\delta }\left( 1+\theta \right) ^{j}(1+\lambda )^{j}\right] \end{aligned}$$

We know that $\left( 1+\theta ^{\prime }\right) =\left( 1+\theta \right) \left( 1+\lambda \right) ,$ thus $\theta ^{\prime }=\theta +\lambda +\theta \lambda$. Notice that $I_{0}=I_{t_{0}}(1+\lambda )^{-t_{0}},$ the expression becomes: $T_{t}=A_{0}I_{0}\alpha \left[ \sum _{i=0}^{t}\left( 1+\theta ^{\prime }\right) ^{i}-\sum _{j=0}^{t-\delta }\left( 1+\theta ^{\prime }\right) ^{j}\right] .$ According to the sum of the terms of a geometric sequence: $T_{t}=A_{0}I_{0}\alpha \left( 1+\theta ^{\prime }\right) ^{t}\left( \frac{\left( 1+\theta ^{\prime }\right) -\left( 1+\theta ^{\prime }\right) ^{1-\delta }}{\theta ^{\prime }}\right)$. At time $t_{0},$ the initial traffic is $T_{t_{0}}=A_{0}I_{0}\alpha \left( 1+\theta ^{\prime }\right) ^{t_{0}}\left( \frac{\left( 1+\theta ^{\prime }\right) -\left( 1+\theta ^{\prime }\right) ^{1-\delta }}{\theta ^{\prime }}\right) .$ As a result, traffic at time $t$ writes: $T_{t}=T_{t_{0}}\left( 1+\theta ^{\prime }\right) ^{t-t_{0}}.$ This is Eq. (2).

1.2 Distributions $\frac{\mu _{t}}{I_{t}}$ and $\varepsilon _{t}$

Shapiro–Wilk test. Null hypothesis: the distribution is normal. Probability of null hypothesis $>$ 0.1, it can not be rejected. Kolmogorov–Smirnov test. Probability of null hypothesis $>$ 0.1, it can not be rejected.

1.3 Proof of Equation (6)

From Eq. (4): $up_{t}=\frac{C_{t}\left( 1+\theta ^{\prime }\right) ^{-(t-t_{0})}}{T_{t_{0}}(1-L)_{t}}$

$$\begin{aligned} \ln (up_{t})=\ln (C_{t})-\ln (1-L)_{t}-\ln (T_{t_{0}})-(t-t_{0})\ln (1+\theta ^{\prime }) \end{aligned}$$

For $t_{f}=2012$ and $t_{0}=2006$

$$\begin{aligned} \ln (up_{t_{f}})=\ln (C_{t_{f}})-\ln (1-L)_{t_{f}}-\ln (T_{t_{f}})-6\ln (1+\theta ^{\prime }) \end{aligned}$$

Same manner: $\ln (up_{t_{0}})=\ln (C_{t_{0}})-\ln (1-L)_{t_{0}}-\ln (T_{t_{0}})$

Thus $\ln (up_{t_{f}})-\ln (up_{t_{0}})=\ln (C_{t_{f}})-\ln (1-L)_{t_{f}}-6\ln (1+\theta ^{\prime })-\ln (C_{t_{0}})+\ln (1-L)_{t_{0}}$.

We know that $\left( 1+\theta ^{\prime }\right) =\left( 1+\theta \right) \left( 1+\lambda \right) ,$ therefore $\ln \left( \frac{up_{t_{f}}}{ up_{t_{0}}}\right) =\ln \left( \frac{C_{t_{f}}}{C_{t_{0}}}\right) -\ln \left( \frac{\left( 1-L\right) _{t_{f}}}{\left( 1-L\right) t_{0}}\right) -6\ln (1+\theta )-6\ln (1+\lambda )$.

This is Eq. (6).

1.4 Figure 4, y-axis

Dynamic effect has two parts, on the one hand, the impact of regular investment according to the rate of technical progress, $\theta$, in the other hand, the impact of the growth in investment, $\lambda$. Dynamic effect is written $\ln \left( \frac{De_{t_{f}}}{De_{t_{0}}}\right) =-\left( t_{f}-t_{0}\right) \left[ \ln \left( 1+\theta \right) +\ln (1+\lambda ) \right] .$ Denoting the impact of regular investment $De\theta ,$ and the impact of the growth in investment $De\lambda ,$ such that $\ln \left( \frac{ De\theta _{t_{f}}}{De\theta _{t_{0}}}\right) =-\left( t_{f}-t_{0}\right) \ln \left( 1+\theta \right)$ and $\ln \left( \frac{De\lambda _{t_{f}}}{ De\lambda _{t_{0}}}\right) =-\left( t_{f}-t_{0}\right) \ln \left( 1+\lambda \right) ,$ it can be written: $\ln \left( \frac{De_{t_{f}}}{De_{t_{0}}} \right) =\ln \left( \frac{De\theta _{t_{f}}}{De\theta _{t_{0}}}\right) +\ln \left( \frac{De\lambda _{t_{f}}}{De\lambda _{t_{0}}}\right) .$ The contribution of the growth of investment to the change in unit price in CAGR is $CAGR_{De\lambda (t_{f}-t_{0})}=e^{\frac{1}{\left( t_{f}-t_{0}\right) } \ln \left( \frac{De\lambda _{t_{f}}}{De\lambda _{t_{0}}}\right) }-1=\frac{1}{ 1+\lambda }-1=\frac{-\lambda }{1+\lambda }.$

1.5 Erlang’s internet formula

Erlang’s internet formula: $P_{c}(A,N)=\frac{\frac{A^{N}}{N!}\frac{N}{N-A}}{ 1+A+\cdots +\frac{A^{N-1}}{(N-1)!}+\frac{A^{N}}{N!}\frac{N}{N-A}}$ with $A=\frac{ D}{c}$ and $N=\frac{C}{c}$ where $D$ is the overall traffic demand, $C$ is the capacity and $c$ is the peak rate allowed by the network. $P_{c}$ is the probability of congestion.