Volume 35, Issue 4
Market size effects on long-run demand of a network good
Dai Zusai
Department of Economics, Temple University

Abstract
We consider optimal dynamic pricing under a network externality. We construct the demand dynamic of the network
good from the aggregate best response dynamic of agents who have different adoption costs. When the distribution of
adoption costs is convex, expansion of potential market inevitably enlarges long-run demand.

I would like to thank William H. Sandholm for his encouragement and advice. I am also grateful to Raymond J. Deneckere, Toshihiro
Matsumura, Hideo Owan, Dan Sasaki, Tadashi Sekiguchi, and seminar participants at the University of Wisconsin-Madison, the University ofTokyo (ISS), and the Econometric Society Far Eastern and South Asia Meeting, as well as Prof. Quan Wen, the associate editor, and theanonymous referee of the bulletin, for helpful comments and discussions. I also thank Nathan Yoder for careful reading and editing suggestions.
Financial support from the Richard E. Stockwell Fellowship is gratefully acknowledged.
Citation: Dai Zusai, (2015) ''Market size effects on long-run demand of a network good'', Economics Bulletin, Volume 35, Issue 4, pages
2768-2775
Contact: Dai Zusai - zusai@temple.edu.
Submitted: May 07, 2014. Published: December 13, 2015.

1

Introduction

It is natural to expect expansion of the untapped portion of a market to encourage a supplier
to reach new customers. On this note, we consider the effects of such market expansion on
long-run demand under a network externality. Dhebar and Oren (1985, henceforth cited asDO85) formulate the optimal dynamic pricing policy for a network good, assuming concavityof the optimal control problem. As we will see in detail, expansion of the untapped portionof the market does not affect long-run demand in the concave case. We adopt their model,
while assuming non-concavity in the dynamic governing consumers’ demand. In this setting,
we show that expansion of the potential market inevitably enlarges long-run demand.
To give an economic interpretation of this nonconcavity, we explicitly derive the demand
dynamic as the aggregate best response dynamic of potential customers who have different
costs of adopting the good. It follows that nonconcavity of the dynamic is a consequence of
non-concavity of the distribution of adoption costs. In particular, convexity of the cumulative
distribution function means that the density function is increasing: potential customers arefound at an increasing rate as the supplier targets customers with larger adoption costs.
For example, consider the diffusion of a new innovation or a social network on the Internet.
People who are less familiar with technology are more reluctant to adopt the new productand thus incur greater costs of doing so. The convex distribution of adoption costs meansthat there are fewer geeks than laymen, i.e., that most people are unfamiliar with technology.
It is commonly observed that diffusion of new technology starts with a few early adoptersand then grows exponentially.1
Alternatively, we can think of a social club, pub, or nightclub as a platform where people
gather and enjoy a positive externality from socializing.2 A shopping center can also be
considered as such a platform because the number of customers may affect prices and thevariety of available goods and thus an increase in customers may cause a negative or positive
externality.3 For such a non-virtual platform, transportation costs may be the main sourceof heterogeneity of adoption costs. If a customer’s transportation cost is proportional to
the geometric distance between his home and the location of the platform and potential
customers are located uniformly around the platform, the distribution of the transportation
costs is convex since the area of a circle is proportional to the square of the radius.
Nonconcave dynamic optimization typically results in a bang-bang control. Radner et al.
(2014, henceforth cited as RRS) study the dynamic pricing of a network good both in concave
and nonconcave (convex) cases, while assuming heterogeneity in customers’ sensitivity to the
network externality and linearity of individual gross utility in the platform size.4 (Note thatthey fix the market size and thus do not study size effects.) They suggest nonexistence ofthe deterministic optimal control in the nonconcave case and a need to allow randomizationin the model. Hence, while we a priori restrict the admissible fee levels to the lowest and
1

See Geroski (2000) for a survey on diffusion of innovation over time.
Becker (1991) and Karni and Levin (1994) explain a restaurant’s pricing from a social interactions
perspective.
3
See Beggs (1994) and Smith and Hay (2005) for examples of the theory of shopping centers as platforms
with network externalities.
4
As a result of this linearity, a bang-bang control is the solution in their model in both concave andconvex cases, while the optimal control in the concave version of ours, i.e., DO85, is continuous.
2

highest possible levels, we allow randomization (a mixed strategy) between these. The
optimal pricing policy in our model indeed takes the form of a bang-bang control, setting
the fee at either bound without randomization until demand reaches a limit state. At thelimit state, the fee level is randomized to keep demand unchanged on average.
We find that long-run demand depends on the size of the potential market. Assumingthat the aggregate gross utility of the participants grows at a constant elasticity with thenumber of the participants, there are two distinct cases of the correlation between potential
market size and long-run demand. If aggregate gross utility is concave in platform size andconsequently participation imposes a negative externality on other individuals, the platform
absorbs all agents in the long run when the potential market is small. In the opposite
case, where there is a positive externality, the entire potential market is absorbed when thepotential market size is large.
There is a notable difference in our model from RRS and their variants.5 In our model,
we add an additively separable idiosyncratic term to the utility function, which better represents heterogeneity in transportation costs or outside options. In the forementioned literature, idiosyncrasy enters the utility function in a multiplicative way, which is interpreted
as heterogeneity in sensitivity to network effects. Additionaly separable idiosyncrasy makes
our model consistent with econometric models on discrete choices6 and allow us to justify
the dynamic as the aggregate best response dynamic.
In the next section, we set up the model. In Section 3, we formulate the dynamic
optimization problem and characterize its solution. We study the comparative statics oflong-run demand and clarify the size effects mentioned above in Section 4.
2

The model

There is a mass of agents who decide on whether or not to participate in a platform. Let n̄ be
the total mass of agents, i.e., the potential market size. At each moment in time t ∈ R+ ,an agent gains gross utility v(nt ) from participation in the platform, but incurs adoption
cost φ and is charged a participation fee Pt by the platform owner. An agent receives zeroutility if he does not participate.
Gross utility of participation v : [0, n̄] → R is a function of the mass of current participants
nt . We assume that v is C 2 and strictly concave. Adoption cost φ ∈ [0, φ̄] varies among
agents, and thus we call it an agent’s type. Let F (φ) be the mass of agents whose types
are not higher than φ. We assume that F : [0, φ̄] → R+ is an increasing and C 2 function
with F (0) = 0 and F (φ̄) = n̄. We call the type equal to v(nt ) − Pt the target type andF (v(nt )−Pt ) the target demand at time t. Setting the target type at φ̂t means that the fee
is set at Pt = v(nt ) − φ̂t . We restrict the set of feasible target types to [0, φ̄] and consequently
the set of feasible fee levels to [v(nt ) − φ̄, v(nt )].7 The platform owner’s instantaneous profit
5

See Radner (2003) and Radner and Richardson (2003). DO85 proposes a general model without specifying the functional form of heterogeneity in the utility function and argues multiplicative heterogeneity inthe canonical example.
6
See Anderson et al. (1992).
7
Our assertion that the fee level is bounded above may sound inconsistent with the owner’s profit maximization, because the owner may want to exploit the participants as much as possible. We justify our choiceof the upper bound v(nt ) by considering that, if Pt > v(nt ), it is obvious for every agent that none of them

is Pt nt = {v(nt ) − φ̂t }nt .
In the example of a shopping center, the fee Pt includes not only membership fees orparking fees but also (per-customer) rents paid indirectly from retailers. Incentives to attract
shoppers such as coupons and events are considered to be negative components of Pt . The
adoption cost φ is the transportation cost for an agent to visit the shopping center. Its upperbound φ̄ defines the geographic boundary of the area from which potential customers are
drawn, while n̄ is its population.
We construct the platform demand dynamic from aggregation of the best response dynamic (BRD), as defined in Gilboa and Matsui (1991); Hofbauer (1995). In the BRD, eachagent only occasionally revises his choice of whether or not to participate at a constantfrequency, and, upon receiving a revision opportunity at time t, chooses the myopically optimal action based on the current state (Pt , nt ).8 According to Ely and Sandholm (2005),
the aggregate BRD reduces to9ṅt = F (v(nt ) − Pt ) − nt .

(1)

We make the following assumptions on the functions F and V .
Assumption 1. F ′′ > 0.
Assumption 2. Define the aggregate gross utility of participants V : R+ → R by V (n) =v(n)n. V ′ (n)/n is decreasing and satisfies(2 + ρ)φ̄V ′ (n)V ′ (n)
>
> lim
.
n→∞
n→0
n
n̄
n
lim

Assumption 1 means more high types than low types; because of this, the platformdemand dynamic exhibits non-concavity and, rather, strict convexity. Assumption 2 is toguarantee the existence and uniqueness of long-run demand under optimal dynamic pricing.
3

Dynamic optimization

The platform owner chooses the fee schedule {Pt }t∈R+ , or equivalently, the target type{φ̂t }t∈R+ , to maximize the discounted sum of profits under the platform demand dynamic
receives positive net payoff from participation; thus, every agent immediately exits from the platform andnt jumps to 0. What is crucial for our results is this upper bound assumption; without it, there may not be
any optimal price.
8
Both DO85 and RRS extend the models to allow an agent’s choice to be based on a subjective assessmentof the platform size, which takes a form of a weighted average between the current platform size and sometime-invariant assessment of the would-be platform size: in our term, the latter is F (φ) for a type-φ agent
in DO85, and a constant ω commonly held for all agents in RRS. Such extension may be possible for ourmodel but is not discussed in this paper because our focus is on implications of non-concavity.
9
We should notice aggregability of the dynamic: the evolution of nt relies only on the aggregate statent (and Pt ) but not on the composition of participants over different types. For aggregability, DO85 (§2.5)
assumes that sequence of participation is always perfectly sorted according to the degree of the net benefit
from participation; then, anyone with a lower type than F −1 (nt ) participates in the platform and anyone
with a greater type does not. But such sorting cannot be expected if individual agents’ revision processes areinertial and independent to each other. Zusai (2015) argues that aggregability is not generally guaranteed.
However, Ely and Sandholm (2005) verify aggregability of the BRD with additive heterogeneity.

(1). If we had assumed concavity of both V and F , we could find the optimal dynamic
pricing scheme as a solution of an optimal control problem that is essentially the same as
the problem in DO85:
Z ∞e−ρt {v(nt ) − φ̂t }nt s.t. (1).
max{(φ̂t ,nt )}t∈R+

0Here ρ > 0 is the discount rate. Under the optimal pricing scheme, the target type changes
continuously with current demand; see DO85. Demand converges globally and asymptotically to the long-run demand n$ = min{n̄, n̂}, where n̂ is the unique solution toV ′ (n̂) − F −1 (n̂) − (1 + ρ)n̂F ′ ◦ F −1 (n̂)

= 0.

(2)

Change in the agents’ population affects the long-run demand n$ only if it changes the
distribution of the types [0, F −1 (n$ )]. That is, a population increase in an area from which
the platform has not yet attracted customers does not stimulate the platform to attract themand grow further.10
Convexity, as assumed in Assumption 1, implies that the optimal policy is a bang-bang
control in which the target type takes either the lowest type 0 or the highest type φ̄. Furthermore, RRS suggests that the optimal control may switch between the two extremes
randomly.11
Thus, we a priori restrict the feasible set of the target types to a binary set {0, φ̄}, whileallowing the owner to randomize the target type, i.e., to play a mixed strategy. The owner’s
maximization problem is(Z ∞φ̂t = φ̄, ṅt = n̄ − nt with prob. qt ,(3)e−ρt {v(nt ) − φ̂t }nt dt s.t.
max E{(φ̂t ,nt )}t∈R+
φ̂t = 0, ṅt = −nt
with prob. 1 − qt .
0Here, qt is the probability with which the target type is set to φ̄ at time t. With co-statevariable µ ∈ R, the corresponding Hamiltonian isH(n, q; µ) = {v(n) − q φ̄}n + µ(qn̄ − n).
The solution of (3) is thus characterized by the following first-order conditions:
if n̄µ > φ̄n,
= 1q ∈ [0, 1] if n̄µ = φ̄n,(4a)µ̇ = (1 + ρ)µ + q φ̄ − V ′ (n).
0if n̄µ < φ̄n,
10

(4b)

Let n̂0 be the solution of (2) before change in F . Suppose that, with arbitrary small ε > 0, the densityof type-φ agents changes only if φ ∈ [F −1 (n̂0 ) + ε, φ̄]. (It may or may not keep φ̄ or n̄.) This does not change
F −1 (n̂0 ) or F ′ ◦ F −1 (n̂0 ) at all and thus n̂0 is still the solution of (2) after this change in F . As long as n̄
does not change too much, it does not change which of n̄ and n̂0 is smaller; thus, the long-run demand n$remains the same.
11
Radner and Richardson (2003) verify that such a random extreme pricing scheme like our solution canbe justified as an approximation of a pricing scheme where price oscillates arbitrarily quickly. Accordingto Roth and Sandholm (2013), the BRD in a continuum of agents is also an approximation of the finite
population dynamic. If the number of agents was finite, the platform size would change discretely and thus
it would oscillate around the limit point in the continuous dynamic. Then, price would also oscillate. We
should also note that price in our solution is not randomized until the platform size reaches the limit n∞ .

s=0

μ

μ

s>0

s>0

s=0


̄
ϕ

s<0

0

̄ϕ / n̄nñn(a) ñ < n̄ ⇒ n∞ = ñ

0

̄
ϕ /̄
n

s<0
n


ñ

n

(b) ñ > n̄ ⇒ n∞ = n̄Figure 1: Phase diagram

The optimal policy of q switches when (n, µ) crosses the line µ/n = φ̄/n̄ in the phase
diagram. The proof of the next lemma shows that it switches only from 1 to 0 if n is below
the threshold ñ and only from 0 to 1 if it is above ñ. This implies the existence of a saddle
path to (ñ, µ̃) with µ̃ = φ̄ñ/n̄.
Lemma 1. Suppose that assumptions 1 and 2 hold. There exists a unique ñ ∈ (0, +∞) s.t.
(2 + ρ)φ̄ñ = n̄V ′ (ñ).

(5)

There is a saddle path to (ñ, µ̃) if ñ < n̄; otherwise, there is a path converging to (n̄, φ̄).
Proof. Define the function s : R2 → R by s(n, µ) := n̄µ − φ̄n. According to (4a), the optimal
q is q = 1 if s > 0 and q = 0 if s < 0. Note that the derivative of s is Ds(n, µ) = (−φ̄, n̄).
Now we investigate how the value of s changes when s(n, µ) = 0.
Let vq (n, µ) be the transition vector (ṅ, µ̇) at (n, µ) with q ∈ [0, 1]:

 

n̄
qn̄ − n.
= v0 (n, µ) + q
vq (n, µ) :=(1 + ρ)µ + q φ̄ − V ′ (n)φ̄Notice that Ds(n, µ)vq (n, µ) = Ds(n, µ)v0 (n, µ) for any q. Thus we have>ṡ = Ds(n, µ)v0 (n, µ) = (2 + ρ)φ̄n − n̄V ′ (n)<if s > 0,
if s = 0,
if s < 0.

(5) implies ṡ = 0 if s = 0 and n = ñ. Assumption 2 guarantees the unique existence of suchñ in (0, +∞); in addition, it implies ṡ < 0 if s ≤ 0 and n < ñ and ṡ > 0 if s ≥ 0 and n > ñ.

̄n V ' (n)n̄ V ' (n)(2+ρ) ̄
ϕ
0

n 0

ñ

(a) V ′′ < 0; b < 1 in Example 1(2+ρ) ̄ϕñn(b) V ′′ > 0; b ∈ (1, 2) in Example 1

Figure 2: Determination of threshold size ñ from (5)If s(n, µ) = 0 and n < ñ, then s changes only from positive to negative when (n, µ) crosses
s = 0. That is, the optimal q switches from 1 to 0. After this switch, s keeps decreasing
and thus (n, µ) never crosses s = 0. By the same token, if (n, µ) crosses s = 0 at n > ñ, qswitches only from 0 to 1 and s changes from negative to positive and keeps increasing.
If ñ < n̄, there is a saddle path converging to (ñ, µ̃) in finite time, as shown in Figure 1.
Otherwise, there is a path converging to (n̄, φ̄). Notice that, with q = 1, nt converges to n̄
only asymptotically and thus the path does not reach (n̄, φ̄) in finite time.
To select a solution path, we impose the transversality condition
lim e−ρt nt µt = 0.

t→∞

(6)

If ñ < n̄, the saddle path is the only solution of the FOCs (4) that also satisfies the TVC(6). So is the path to (n̄, φ̄) when ñ ≥ n̄. On any other solution paths of (4), µ diverges to∞ or −∞ faster than ρ after sufficiently long time has passed; these paths violate (6).
ñ is the threshold between q = 1 (pure strategy φ̂ = φ̄) and q = 0 (pure strategy φ̂ = 0);
call it the threshold demand, distinguished from the long-run demand n∞ := limt→∞ nt .
Theorem 1. Suppose assumptions 1 and 2 hold. In the long run, demand converges to ñ
defined in (5) if ñ < n̄ and to n̄ if ñ ≥ n̄ . That is, long-run demand is n∞ = min{ñ, n̄}.
4Market size effects

From (5), it is obvious that the threshold demand ñ depends on the potential market sizen̄. As is suggested by Figure 2, ñ increases with n̄ (and decreases when φ̄ or ρ increases).
In contrast to the concave case, expansion of the potential market triggers larger demand in
the long run no matter where it occurs.
Corollary 1. Suppose assumptions 1 and 2 hold. As potential market size n̄ increases, sodoes long-run demand n∞ .
Proof. Notice that, if n̄ ≤ ñ, n∞ = n̄ and thus the claim is immediate. We show ñ increases
with n̄. Assumption 2 and (5) imply
 V ′ (n) ′n̄V ′′ (ñ) − (2 + ρ)φ̄V ′′ (ñ)ñ − V ′ (ñ)
0>
=
,
=
n
ñ2
n̄ñ
n=ñ

n

n

n= n
̄

̄ ( ̄n ) n= ̄n
n= N̄ ( ̄n )
n= N

n
̄

0

̄n

0

(a) b < 1(b) b ∈ (1, 2)

Figure 3: The relationship between long-run demand n∞ (the bold line) and potential market sizen̄.

and thus n̄V ′′ (ñ) − (2 + ρ)φ̄ < 0. From this, implicit differentiation of (5) yieldsdñ
V ′ (ñ)=> 0.
dn̄
(2 + ρ)φ̄ − n̄V ′′ (ñ)
In the following example, we specify the aggregate gross utility function V and calculate
long-run demand numerically from (5).
Example 1 (Constant elasticity). Let V be V (n) = Anb with A, b > 0. This functional form
implies that the aggregate gross utility of the participants grows at a constant elasticity bwith the platform size n. Assumption 2 reduces to b ≤ 2. (5) is solved by1/(2−b)Abn̄.
ñ = Ñ (n̄) :=(2 + ρ)φ̄Ñ is convex in n̄ if b ∈ (1, 2) and concave if b < 1. The relationship between potential
market size n̄ and long-run demand n∞ is significantly different in these two cases, as seenin Figure 3.
Corollary 2. Assume constant elasticity of V , i.e., V (n) = Anb with A > 0 and b ∈ (0, 2).
Consider the case b < 1. Regardless of initial demand n0 , demand nt converges to n̄, andthus the platform eventually absorbs all agents in the economy, when potential market size n̄
is sufficiently small.
If b ∈ (1, 2), this happens when potential market size n̄ is sufficiently large.
Notice that individual participant’s gross utility from participation is v(n) = V (n)/n =Anb−1 . Thus, additional participation causes a negative externality if b < 1; otherwise, itcauses a positive externality.

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