Internet Service Pricing: Flat or Volume?

Abstract

In this paper, we study the pricing of Internet services under monopoly and duopoly environments using an analytic model in which a service provider and users try to maximize their respective payoffs. We compare a few popular pricing schemes, including flat, volume-based, two-part, and nonlinear tariffs, with respect to revenue, social welfare, and user surplus. We perform a study of the sensitivity of these schemes to the estimation errors. In the duopoly situation, we formulate a simple normal form game between two service providers and study their equilibrium behaviors. Our main findings include: (1) the flat pricing generates higher revenue than the pure volume pricing when the elasticity of demand is low; (2) the volume-based pricing is better for society and users than the flat pricing regardless of the elasticity; (3) the market is segmented into two when one provider provides flat pricing and another provides volume based pricing.

Appendices

Appendix 1: Analysis of the Two-Part Tariff

In a two-part tariff, a user is charged a price p + β x for usage x. That is, a user pays a fixed price p to join the network plus a price β x proportional to usage.

The user problem (P _user ) in which a user of type v chooses x to maximize her net-utility can be formulated as follows:

$$ ({\bf P}_{{\bf user}}) \quad \quad \hbox{maximize}_{x}\,u(v; x) - [p + \beta x] = v x^\alpha - p - \beta x. $$

The maximizing value of x is

$$ x(v) = \left(\frac{\alpha v}{\beta}\right)^{\frac{1}{(1-\alpha)}}, \quad 0 \le \alpha \le 1. $$

The resulting utility is then u(v; x(v)) given by

$$ u(v; x(v)) = k_1(\alpha) \left(\frac{v}{\beta^\alpha}\right)^{\frac{1}{(1-\alpha)}} - p, $$

where $k_1(\alpha) = \alpha^{\frac{\alpha}{1-\alpha}} (1-\alpha)$. This utility is nonnegative if v ≥ v ₀ where

$$ p = k_1(\alpha) \left(\frac{v_0}{\beta^\alpha}\right)^{\frac{1}{(1-\alpha)}}. $$

(14)

Since v is uniformly distributed, the total traffic in the network is then T where

$$ T = \int\limits_{v_0}^1 x(v) Ndv = \int\limits_{v_0}^1 \left(\frac{\alpha v}{\beta}\right)^{\frac{1}{(1-\alpha)}}\,Ndv $$

(15)

$$ = N k_2(\alpha) \left(\frac{1}{\beta}\right)^{\frac{1}{(1-\alpha)}}\left(1-v_0^\frac{2-\alpha}{1-\alpha}\right), $$

(16)

where $k_2(\alpha) = \alpha^{\frac{1}{1-\alpha}}\left({\frac{1-\alpha}{2-\alpha}}\right)$. The total revenue of the provider is then R where

$$ \begin{aligned} R&= pN(1 - v_0) + \beta T \\ &= pN(1 - v_0) + N k_2(\alpha) \left(\frac{1}{\beta}\right)^{\frac{\alpha}{1-\alpha}} {\left(1 - v_0^{\frac{2-\alpha}{1-\alpha}}\right)}.\\ \end{aligned} $$

In this expression, pN(1 − v ₀) is the fixed part of the price where N(1 − v ₀) users subscribe and pay a flat price p to join; the term β T is the contribution of the usage based price.

Assuming that the network has a total capacity C, the provider aims to maximize R subject to T ≤ C. That is, the provider solves the following problem:

$$ \begin{aligned} ({\bf P}_{{\bf NSP}}) \quad \quad & \\ &\hbox{Maximize}\,r = p(1 - v_0) + \beta t \\ &\hbox{over}\,(p, \beta )\\ &\hbox{subject to}\, t \leq c\,\hbox{and Eq.}\,(14)\\ \end{aligned} $$

where c = C/N, t = T/N, and r = R/N. We normalized the problem (P _NSP ) by a factor of N. To solve this problem, we replace p and t as in (14) and (16) in the expression for R, and we find

$$ r = \frac{ k_1(\alpha) v_0^{\frac{1}{1-\alpha}} (1 - v_0) + k_2(\alpha) (1 - v_0^{\frac{2-\alpha}{1-\alpha}})}{\beta^{\frac{\alpha}{1-\alpha}}}. $$

We can expect the full capacity to be used at the optimal price, so that

$$ k_2(\alpha) \frac{\left(1 - v_0^{\frac{2-\alpha}{1-\alpha}}\right)}{\beta^{\frac{1}{1-\alpha}}} = c, $$

i.e.,

$$ \beta = \left(k_2(\alpha) \frac{\left(1 - v_0^{\frac{2-\alpha}{1-\alpha}}\right)}{c} \right)^{(1-\alpha)}. $$

(17)

Substituting this expression for β into r, we find

$$ r = \left(\frac{c}{k_2(\alpha)}\right)^\alpha \frac{ k_1(\alpha) v_0^{\frac{1}{1-\alpha}} (1 - v_0) + k_2(\alpha) (1 - v_0^{\frac{2-\alpha}{1-\alpha}})}{\left(1 - v_0^{\frac{2-\alpha}{1-\alpha}}\right)^\alpha }. $$

This expression is increasing in c, which justifies our assumption that the full network capacity is used at the optimal price. Note that r is a function of a single variable v ₀ and is unimodular with a unique maximizer $v_0^*$. Figure 7a shows the optimal $v_0^*$ that maximizes r for $\alpha \in [0,1].$ When α is close to zero, $v_0^*$(α) is close to 0.5, and it increases to 1 when $\alpha \rightarrow 1$. When α = 0.5, $v_0^*$ ≈ 0.623. Substituting this value into the expression for β of (17) and then for p of (14), we can compute β and p as shown in Fig. 7b. Note that the optimal price parameters (p, β) that maximize the revenue r depend heavily on the shape of the utility function that is parameterized by α. The fixed price p decreases in α from 0.5 to around 0.25 while the unit charge β increases from 0 to 0.94. Figure 7c shows the revenue r and the social welfare w as defined by

$$ w = \int\limits_{v_0}^1 v \sqrt{x(v)}\,dv. $$

When $\alpha=\frac{1}{2},$ we find that

$$ \beta \approx \frac{0.25}{\sqrt{c}}\quad \hbox{and}\quad p \approx 0.39 \sqrt{c}. $$

The optimal revenue is then found to be

$$ r \approx 0.40 \sqrt{c}. $$

Thus, the optimal pricing for a user is

$$ p + \beta x = 0.39 \sqrt{c} + \frac{0.25}{\sqrt{c}} x $$

and a fraction 1 − v ₀ ≈ 37.7 % of the users subscribe to the service.

Fig. 7

Two-part tariff: a the revenue-maximizing $v_0^*$; b the optimal price parameters β and p, c revenue and social welfare

Appendix 2: Analysis of Flat Pricing

Flat pricing is one of the most common pricing schemes. Under flat pricing, users would like to generate an infinite amount of traffic. There are two reasons why this does not happen in practice. First, the network capacity is limited, and the users experience a drop in service quality as they saturate the network. Second, users have other things to do than use the network, and the time they spend on the network has an opportunity cost that may be viewed as a parasitic volume-based cost. In this paper, we only model the first effect. Because the capacity of the network is limited, so is the traffic of each user. Users with a larger utility for the traffic still use the network more than others. To model this effect, we consider that users face a drop in quality, such as an increase in latency, when they increase their traffic. This quality drop is a disutility that depends on the total traffic. Thus, at the equilibrium, this drop in quality has some common derivative δ with respect to the degree of utilization of every user. Thus, the derivative of the utility vx ^α must be equal to the derivative of the latency at the equilibrium. Hence,

$$ \frac{v\alpha}{x^{(1-\alpha)}} = \delta,\quad \hbox{so that}\quad x = x(v) := \left(\frac{\alpha v}{\delta}\right)^{\frac{1}{(1-\alpha)}}. $$

The value of δ is such that the total traffic is less than C. The net-utility of a user of type v is then

$$ v x(v)^\alpha - p = k_3(\alpha) \left(\frac{v}{\delta^\alpha}\right)^\frac{1}{1-\alpha} - p, $$

where $k_3(\alpha)= \alpha^{\frac{\alpha}{1-\alpha}}$. In this analysis, the shadow price δ is introduced to constrain the traffic, but the disutility of the loss in quality is negligible as soon as the traffic is slightly less than the capacity C. Thus, a user of type v joins the network if his net-utility is positive, i.e., if v ≥ v ₀, where

$$ p = k_3(\alpha) \left(\frac{v_0}{\delta^\alpha}\right)^{\frac{1}{(1-\alpha)}}. $$

(18)

The difference from the previous analysis is that the revenue now consists only of the fixed price. That is,

$$ R = Np (1 - v_0). $$

Thus, the provider chooses p to solve the following problem:

$$ \begin{aligned} &\hbox{Maximize}\,r = p(1 - v_0)\\ &\hbox{over} \; p\\ &\hbox{subject to}\,t = c\,\hbox{and Eq. (18)} \end{aligned} $$

Substituting the expressions for p and δ in r, we find that

$$ r = \left(\frac{c}{k_2(\alpha)}\right)^\alpha \frac{ k_3(\alpha) v_0^{\frac{1}{1-\alpha}} (1 - v_0)}{\left(1 - v_0^{\frac{2-\alpha}{1-\alpha}}\right)^\alpha }, $$

is a unimodular function of v ₀ with a single maximizer $v_0^*$. We can find $v_0^*$ numerically in a similar way as before and the results are shown in Fig. 8a. As 1 − v ₀ is the subscription ratio, we observe that the ratio decreases with higher α. The optimal flat price $p^*$ can be computed using (18) and is shown in Fig. 8b. The flat price $p^*$ increases quite rapidly when α ≥ 0.8. Figure 8c shows the revenue and the social welfare of the flat pricing scheme, which can be computed in a similar way. When $\alpha =\frac{1}{2},$ v ₀ ≈ 0.760. It follows that $p \approx 0.67 \sqrt{c}$ and $r \approx 0.32 \sqrt{c}$. Thus, under flat pricing, 24 % of the users join the service and the revenue is about 80 % of that of the two-part tariff.

Fig. 8

Flat pricing: a the revenue-maximizing $v_0^*$; b the optimal flat price p, c revenue and social welfare

Appendix 3: Analysis of Volume-Based Pricing

In volume-based pricing, the net utility of a type v user is

$$ v x^\alpha - \beta x $$

so that she chooses a usage $x = x(v) := \left(\frac{\alpha v}{\beta}\right)^{\frac{1}{(1-\alpha)}}$ and her utility

$$ k_1(\alpha) \left(\frac{v}{\beta^\alpha}\right)^{\frac{1}{(1-\alpha)}} $$

is always positive, where $k_1(\alpha) = \alpha^{\frac{\alpha}{1-\alpha}} (1-\alpha)$. The total traffic in the network is then

$$ T = \int\limits_{v_0}^1 x(v) Ndv = N k_2(\alpha) \left(\frac{1}{\beta}\right)^{\frac{1}{(1-\alpha)}}, $$

where $k_2(\alpha) = \alpha^{\frac{1}{1-\alpha}}\left({\frac{1-\alpha}{2-\alpha}}\right).$ To meet the capacity constraint, one chooses β so that T = C, i.e.,

$$ k_2(\alpha) \left(\frac{1}{\beta}\right)^{\frac{1}{(1-\alpha)}} = c $$

that gives

$$ \beta = \left(\frac{k_2(\alpha)}{c}\right)^{1-\alpha}. $$

(19)

It follows that the revenue is

$$ R = Nr = \beta T = \beta Nc, $$

so that

$$ r = \beta c = \left({k_2(\alpha)}\right)^{1-\alpha} c^\alpha $$

(20)

$$ = \alpha \left[\left({\frac{1-\alpha}{2-\alpha}}\right) \right]^{1-\alpha} c^{\alpha}. $$

(21)

The subscription ratio of volume-based pricing is 100 % regardless of α because the net-utility of any user is nonnegative. Figure 9b, c show the price parameter β, the revenue r and the social welfare w of volume-based pricing.

Fig. 9

Volume-based pricing: a the revenue-maximizing $v_0^*$; b price parameter β, c revenue and social welfare

Appendix 4: Computing the Nonlinear Tariff

Let ϕ(x) be the nonlinear price for x units of traffic and let $\beta(x) = \phi^{\prime}(x)$. The traffic x(v) of type v is given by

$$ x(v) = \left(\frac{\alpha v}{\beta}\right)^\frac{1}{1-\alpha}, $$

(22)

when β(x) = β.

Let D(β, x) be the demand profile defined as follows:

$$ D(\beta,x) = Prob[ x(v) \ge x | \beta(x) = \beta], $$

and let it represent a fraction of users who are willing to pay $\beta \cdot dx$ for using marginal traffic [x, x + dx]. Assuming the uniform distribution of v, we have

$$ D(\beta,x) = Prob( v^\frac{1}{1-\alpha} \ge \left(\frac{\beta}{\alpha}\right)^\frac{1}{1-\alpha} x )\\ $$

(23)

$$ = Prob(v \ge \left(\frac{\beta}{\alpha}\right) x^{1-\alpha})\\ $$

(24)

$$ = \left( 1 - \left(\frac{\beta}{\alpha}\right) x^{1-\alpha} \right)^+. $$

(25)

The marginal price schedule β(x) that maximizes the marginal revenue (β − λ) D(β, x) using marginal traffic [x, x + dx] is given by

$$ \begin{aligned} \beta(x) &= arg\max_{\beta} \left( (\beta-\lambda) \cdot D(\beta,x) \right)\\ &= arg\max_{\beta} \left((\beta-\lambda) \left( 1 - \left(\frac{\beta}{\alpha}\right) x^{1-\alpha} \right)^+ \right).\\ \end{aligned} $$

Here, λ is chosen such that T = C is satisfied.

Solving the equation gives

$$ \beta(x) = \frac{\alpha}{2 x^{1-\alpha}} + \frac{\lambda}{2}. $$

The price schedule $\phi(x) = \int_0^x \beta(x) dx$ is then

$$ \phi(x)= \frac{x^\alpha}{2} + \frac{\lambda}{2} x. $$

(26)

The net-utility of a user of type v with traffic x is then

$$ v x^\alpha - \phi(x) = (v-\frac{1}{2}) x^\alpha - \frac{\lambda}{2} x. $$

Such a user would purchase a quantity x(v) of traffic where

$$ x(v) = \left\{ \begin{array}{ll} 0,& {\hbox{if}\,v \le \frac{1}{2};} \\ \left((v-\frac{1}{2}) \frac{2\alpha}{\lambda}\right)^\frac{1}{1-\alpha}, &\hbox{otherwise.} \end{array} \right. $$

Total traffic is then

$$ T = \int\limits^1_{\frac{1}{2}} \left(\left(v-\frac{1}{2}\right) \frac{2\alpha}{\lambda}\right)^\frac{1}{1-\alpha} N dv =\left( \frac{2\alpha}{\lambda}\right)^\frac{1}{1-\alpha} \left(\frac{1-\alpha}{2-\alpha}\right)\left(\frac{1}{2}\right)^\frac{2-\alpha}{1-\alpha}. $$

Since T = C = Nc, we find that

$$ \lambda = \frac{\alpha}{2^{1-\alpha}} \left(\frac{1-\alpha}{2-\alpha} \cdot \frac{1}{c} \right)^{1-\alpha}. $$

(27)

The revenue r is given by

$$ \begin{aligned} r &= \int\limits_0^\infty D(x)\beta(x) dx \\ &= \int\limits_0^\infty \left( 1 - \left(\frac{\beta}{\alpha}\right) x^{1-\alpha} \right)^+ \left(\frac{\alpha}{2 x^{1-\alpha}} + \frac{\lambda}{2}\right) dx \\ &= \frac{1}{4\lambda^\frac{\alpha}{1-\alpha}} \left( \alpha^\frac{\alpha}{1-\alpha} - \frac{1}{2-\alpha} \alpha^\frac{1}{1-\alpha} \right) \\ &= \frac{1}{2^{1-\alpha}} \left( \frac{1-\alpha}{2-\alpha}\right)^{1-\alpha}c^{\alpha}.\\ \end{aligned} $$

The social welfare w of the nonlinear tariff is given by

$$ w := \frac{W}{N} = \int\limits_0^1 vx(v)^\alpha = \frac{2^\alpha (3-\alpha)}{4} \left( \frac{1-\alpha}{2-\alpha}\right)^{1-\alpha} c^\alpha. $$

Appendix 5: Proof of Proposition 1

Proposition 1

The social welfare of the three linear tariffs is given by

$$ w := \frac{W}{N} = w^{*} \left( 1-v_0^\frac{2-\alpha}{1-\alpha} \right) ^{1-\alpha}. $$

(28)

And that of the nonlinear tariff is given by

$$ w = w^{*} \left(\frac{2^\alpha (3-\alpha)}{4}\right). $$

Proof

We consider the linear tariffs first. As a user of type v generates $x(v) = \left(\frac{\alpha v}{\beta}\right)^{\frac{1}{1-\alpha}}$ to maximize her utility,⁸ substituting x(v) for $\left(\frac{\alpha v}{\beta}\right)^{\frac{1}{1-\alpha}}$ in W, we have

$$ w := \frac{W}{N} = \left( \frac{\alpha}{\beta}\right)^\frac{\alpha}{1-\alpha} \int^1_{v_0} v^{\frac{1}{1-\alpha}} dv $$

(29)

$$ = \left( \frac{\alpha}{\beta}\right)^\frac{\alpha}{1-\alpha} \left(\frac{1-\alpha}{2-\alpha} \right) \left( 1-v_0^\frac{2-\alpha}{1-\alpha} \right) $$

(30)

$$ = \left( \frac{1-\alpha}{2-\alpha}\right)^{1-\alpha} \left( 1-v_0^\frac{2-\alpha}{1-\alpha} \right) ^{1-\alpha} c^\alpha $$

(31)

$$ = w^{*} \left( 1-v_0^\frac{2-\alpha}{1-\alpha} \right) ^{1-\alpha}. $$

(32)

We replace β with (.6) to obtain identity (31). In the case of the nonlinear tariff, refer to the last part of Appendix 4. Algebraic calculations after replacing x(v) in w leads to the desired result. $\square$