Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach

Abstract

Dynamic pricing of new products has been extensively studied in monopolistic and oligopolistic markets. But, the optimal control and differential game tools used to investigate pricing behavior on markets with a number of firms are not well-suited to model competitive markets with a large number of firms. Using a mean-field game approach, this article develops a setting where numerous firms optimize prices for a new product. We analyze a framework à la Bass with product diffusion and experience effects. The analytical contribution of the paper is to prove the existence and uniqueness of a mean-field game equilibrium, further characterized in terms of mean tendencies and market heterogeneity. We also demonstrate the possible emergence of one or more groups of firms with regards to their pricing strategy. Numerical simulations illustrate how differences in firm experience translate into market heterogeneity in sales and profits. We show that, on a market where the absolute price effect is stronger than the relative price effect, we observe the emergence of two groups of firms, characterized by different prices, sales, and profits. Heterogeneity in firms’ prices and profits is thus compatible with competitive markets.

Appendices

Appendix A: Proof of Lemma 5.4

Proof

General considerations When the overall dynamics ${\overline{X}}_t$ and ${\overline{P}}_t$ is given, the situation enters, formally, the setting of Kalish [54, bottom of page 141] and Clarke et al. [24, Proposition 2 page 523]. Consider the function

$$\begin{aligned} p \in {\mathbb R}_+ \mapsto {\mathscr {L}}(p) = p + \frac{g(p)}{g'(p)}. \end{aligned}$$

(A.1)

Since ${\mathscr {L}}'(p)= 2- \frac{g(p) g''(p)}{(g'(p))^2}$ and hypothesis (H1) holds, we obtain ${\mathscr {L}}'(p)>0$. ${\mathscr {L}}$ is thus a strictly increasing function. Its image is $[\frac{g(0)}{g'(0)},\infty [$. Since $g\ge 0$ and $g'(p)\le 0$, we get $\frac{g(0)}{g'(0)} \le 0$. Thus, the image of ${\mathscr {L}}$ contains ${\mathbb R}_+$.

P

Proof of points 2 and 3

For $r>0$, the optimum price $p^*$ satisfies as in Kalish [54] and Clarke et al. [24]:

$$\begin{aligned} \frac{d p^*_t}{\mathrm{d}t} \cdot {\mathscr {L}}'(p^*_t) < 0, \end{aligned}$$

(A.2)

which together with hypothesis (H1) provide the point 2 of the conclusion.

For $r=0$, one obtains $\frac{d p^*_t}{\mathrm{d}t} {\mathscr {L}}'(p^*_t) =0$ thus $p^*_t$ is constant (point 3 of the conclusion).

Proof of point 1

We introduce the Hamiltonian

$$\begin{aligned} H(t,p,\lambda ) = (p-c(x) + \lambda ) f({\overline{X}}_t,{\overline{P}}_t) g(p). \end{aligned}$$

(A.3)

The optimal price $p^*$ will maximize H with respect to p when $\lambda$ is the adjoint state $\lambda ^*$; recalling that at the optimal solution, $\lambda ^*(t) = \partial _x \Pi ^*(t,x_t)$.

With the market conditions given, the maximization of (A.3) is related to the maximization of $p \mapsto (p-\beta )g(p)$ (with the particular case of interest $\beta =c(x) -\lambda$). Its derivative is $g(p) + (p-\beta )g'(p)$ which is positive when $\beta \ge {\mathscr {L}}(p)$ or, equivalently, ${\mathscr {L}}^{-1}(\beta ) \ge p$ (for $\beta$ in the domain of ${\mathscr {L}}$). Thus $(p-\beta )g(p)$ increases and then decreases; it has a unique maximum attained at ${\mathscr {L}}^{-1}(\beta )$. Going back to the maximization of Eq. (A.3) it results that the optimal price satisfies $p^*_t = {\mathscr {L}}^{-1}(c(x_t) - \lambda _t ) = {\mathscr {L}}^{-1} \left( c(x_t) - \partial _x \Pi ^*(t,x_t) \right)$, which shows that the optimal price is unique when $\Pi ^*(t,x)$ is differentiable with respect to x everywhere (and not necessarily so otherwise).

Proof of point 4

Intuitively the profit $\Pi ^*$ is increasing with respect to x because a firm with higher initial sales can use the same strategy as a firm with lower initial sales, but its cost will be lower, which allows for higher profit. The rigorous transcription of this idea is as follows: consider two firms with initial sales $x_0^1$ and $x_0^2$, with $x_0^1 \le x_0^2$, p some pricing strategy as in (3.3) and $x^{\tau ,x_0^1}_t$ and $x^{\tau ,x_0^2}_t$ the sales of the firms using the strategy p. Then $\frac{d}{\mathrm{d}t}x^{\tau ,x_0^2}_t = \frac{d}{\mathrm{d}t} x^{\tau ,x_0^1}_t$ and thus $x^{\tau ,x_0^2}_t \ge x^{\tau ,x_0^1}_t$ for any $t\ge \tau$ which shows that $c(x^{\tau ,x_0^2}_t) \le c(x^{\tau ,x_0^1}_t)$ for any $t\ge \tau$. But then, recalling definition (4.3), $\Pi (\tau ,x_0^2,p) \ge \Pi (\tau ,x_0^1,p)$. Taking now the supremum with respect to p and using (3.3) it follows that

$$\begin{aligned} \Pi ^*(\tau ,x_0^2) = \sup _{p} \Pi (\tau ,x_0^2,p) \ge \sup _{p} \Pi (\tau ,x_0^1,p) = \Pi ^*(\tau ,x_0^1), \end{aligned}$$

(A.4)

proving the first part of point 4.

Denote ${\mathscr {H}}(\beta )= \max _{p\ge 0} (p-\beta ) g(p)$. The optimal profit $\Pi ^*$ is the unique viscosity solution (see Crandall and Lions [26], Bressan and Piccoli [8] and Bardi and Capuzzo-Dolcetta [5] for more details) of the following Hamilton–Jacobi–Bellman equation:

$$\begin{aligned}&\partial _{t} \Pi ^*(t,x) + \mathscr {H} \left( c(x) - \partial _x \Pi ^*(t,x) \right) f({\overline{X}}_t,{\overline{P}}_t) - r \Pi ^*(t,x) = 0 \ \end{aligned}$$

(A.5)

$$\begin{aligned}&\Pi ^*(\infty ,x)=0. \end{aligned}$$

(A.6)

In addition, at any point $x_t$ where $\partial _x \Pi ^*(t,x_t)$ exists the optimal price $p^*_t$ of a firm with cumulative sales $x_t$ at time t only depends on t and $x_t$ and satisfies:

$$\begin{aligned} p^*_t= {\mathscr {L}}^{-1} \left( c(x_t) - \partial _x \Pi ^*(t,x_t) \right) \le {\mathscr {L}}^{-1} \left( c(x_t) \right) \le {\mathscr {L}}^{-1} \left( c(x_0) \right) . \end{aligned}$$

(A.7)

Thus, hypothesis (H2) and (H3) allow bounding the profit for $x_0\rightarrow \infty$. For $x_0 \rightarrow 0$ the optimal price can be unbound (since $c(x_0)$ may tend to $\infty$ for $x_0\rightarrow 0$) but the profit will certainly be finite (being inferior to any profit for fixed $x_0 > 0$), which proves the second part of point 4 of the conclusion. $\square$

Appendix B: Proof of Theorem 5.6

Proof

General considerations

Point 3 of Lemma 5.4 gives that for $r=0$, the optimal price $p^*$ of a representative firm is constant. Given ${\overline{X}}_t$, ${\overline{P}}_t$ (not necessarily at equilibrium) the (constant) optimal price $p^*$ of a firm maximizes the profit [obtained from Eq. (4.3) with p constant using (4.1)] :

$$\begin{aligned} \int _0^\infty (p -c(x_t))\frac{d x_t}{\mathrm{d}t} \mathrm{d}t = p \cdot (x_\infty - x_0) - [ I_c(x_\infty )-I_c(x_0)], \end{aligned}$$

(B.1)

where $I_c(\cdot )$ refers to a primitive of the cost function c; that is, $I_c'(y)= c(y)$, for any $y> 0$.

Using the definition of $Z=Z_{{\overline{X}}_t,{\overline{P}}_t}$, which is fixed and integrating (4.1) with respect to time, yields $x_\infty = x_0 + Z g(p)$. To determine the optimal price $p^*=p^{Z,*}(x_0)$ that maximizes Eq. (B.1), we define the profit function J:

$$\begin{aligned} J(p;x_0,Z):= p Z g(p) - [ I_c(x_0 + Z g(p))-I_c(x_0)]. \end{aligned}$$

(B.2)

Note that J is increasing with respect to $x_0$. By differentiating J with respect to p, we obtain $Z(p^*g'(p^*)+ g(p^*)) = c(x_0 + Z g(p^*)) Z g'(p^*)$. Divide now both terms by $Z g'(p^*)$, to obtain $p^*+ g(p^*)/g'(p^*) = c(x_0 + Z g(p^*))$, which can also be written as ${\mathscr {L}}(p^*)= c(x_0 + Z g(p^*))$. Hence, after inverting the function ${\mathscr {L}}$ and indicating the explicit dependence on $x_0$, we obtain:

$$\begin{aligned} p^{Z,*}(x_0) = {\mathscr {L}}^{-1}\left[ c(x_0 + Z g(p^{Z,*}(x_0))) \right] \le {\mathscr {L}}^{-1}\left( c(x_0 )\right) . \end{aligned}$$

(B.3)

Note that at this time $p^{Z,*}(x_0)$ is not necessarily unique. However, we can show that $p^{Z,*}(x_0)$ is decreasing with respect to $x_0$; indeed consider $x_0^1 < x_0^2$. The optimum price being constant means that we must have for any $t \ge 0$

$$\begin{aligned} x_t^{x_0^1,*} < x_t^{x_0^2,*}. \end{aligned}$$

(B.4)

Otherwise, some point $\bar{t}$ exists with $x_{\bar{t}}^{x_0^1,*} = x_{\bar{t}}^{x_0^2,*}$, and thus, a non-constant price can be constructed that is optimal for the firm starting in $x_0^2$. However, (B.4) implies $x_\infty ^{x_0^1,*} \le x_\infty ^{x_0^2,*}$, which also implies

$$\begin{aligned} p^{Z,*}(x_0^1) = {\mathscr {L}}^{-1}\left[ c(x_\infty ^{x_0^1,*}) \right] \ge {\mathscr {L}}^{-1}\left[ c(x_\infty ^{x_0^2,*}) \right] = p^{Z,*}(x_0^2). \end{aligned}$$

(B.5)

A closer analysis reveals that $p^{Z,*}(x_0)$ is in fact strictly monotonic. Thus $p^{Z,*}(x_0)$ is strictly decreasing with respect to $x_0$, has left and right limits (which are optimums) and any other (optimal) value is between these limits.

On the other hand, the function

$$\begin{aligned} (x_0,Z) \mapsto J^*(x_0,Z)=\sup _{p \ge 0} J(p,x_0,Z), \end{aligned}$$

(B.6)

is Lipschitz thus differentiable almost everywhere with respect to $x_0$; in any point of differentiability of $J^*(x_0,Z)$ we have that $p^{Z,*}(x_0)$ is uniquely given by [compare with formula (A.7)]

$$\begin{aligned} p^{Z,*}(x_0)= {\mathscr {L}}^{-1}\left[ c(x_0) - \partial _{x_0} J^*(x_0,Z) \right] . \end{aligned}$$

(B.7)

Let us now inquire about the variation of $p^{Z,*}(x_0)$ with respect to Z: consider $Z_2 > Z_1$ and the optimal dynamics $x_t^{x_0,*}$ of a firm starting in $x_0$; at time $t=0$ the price $p^{Z_2,*}(x_0)$ represents a maximum of the profit $J(x_0,Z_2=\int _0^\infty f({\overline{X}}_t,{\overline{P}}_t) \mathrm{d}t,p)$ with respect to p, while at a later time $t\ge 0$ it will also be an optimum of $J(x_0,\int _t^\infty f({\overline{X}}_t,{\overline{P}}_t) \mathrm{d}t,p)$. Choosing the time t for which $Z_1= \int _t^\infty f({\overline{X}}_t,{\overline{P}}_t) \mathrm{d}t$ we obtain $p^{Z_2,*}(x_0) = p^{Z_1,*}(x_t^{x_0}) < p^{Z_1,*}(x_0)$, where we used the monotonicity of $p^{Z,*}(x_0)$ with respect to $x_0$; thus, we obtain that $p^{Z,*}(x_0)$ is strictly decreasing with respect to Z.

In summary, $x_0 \mapsto p^{Z,*}(x_0)$ is strictly decreasing; thus, although it is not continuous, the set of discountinuity points is at most countable. The same holds for $Z \mapsto p^{Z,*}(x_0)$.

To avoid lengthy technical developments, we will only present the reminder of the proof for the situation when $\mu _0$ admits a density $\rho _0(x_0)$, i.e., $\mu _0 = \rho _0(x_0) d x_0$.

When $\mu _0$ does not admit a density, then it may occur that for a set A (with $\mu _0(A)> 0$) of firms $\omega \in A$, the price p is not necessarily unique, which means that two firms starting from the same $x_0$ can choose different prices. In mathematical terms, p depends not only on the cumulative sales $X_0(\omega )$ of the firm $\omega$ but also on the firm $\omega$ itself: $p=p(\omega ,X_0(\omega ))$; in this case ${\mathscr {E}}(\cdot )$ is a set-valued map and the proof has to use more general fixed point theorems (see Remark 5.10).

Moreover, to avoid degenerate settings, we also assume that the cost function $c(\cdot )$ is always strictly positive.

Proof of point 2a

Since each firm has constant optimal price, the equilibrium mean price ${\overline{P}}^\dagger _t$ will also be constant:

$$\begin{aligned} {\overline{P}}^\dagger _t = {\mathbb E}\left[ P^\dagger _t \right] = {\mathbb E}\left[ P^\dagger _0 \right] . \end{aligned}$$

(B.8)

In addition, hypothesis (H4) ensures that ${\overline{X}}^\dagger _t \in [0,N_0]$ and ${\overline{X}}^\dagger _t\rightarrow N_0$ for $t\rightarrow \infty$.P

Proof of point 1

The proof of existence relies on the remark that to each possible ${\overline{X}}_t$ and ${\overline{P}}_t$ a number Z corresponds. We will find the Z that gives an equilibrium. Note first that any equilibrium Z can be bounded by some universal constant $Z_{max}^{\mu _0} := \frac{N_0- {\overline{X}}_0}{ {\mathbb E}\left[ g({\mathscr {L}}^{-1}\left( c(X_0 )\right) ) \right] }$; thus, it is enough to look for an equilibrium in the interval $[0, Z_{max}^{\mu _0}]$. Let $Z \in [0, Z_{max}^{\mu _0}]$; then one can associate to it the optimal price $p^{Z,*}(x_0)$ maximizing $J(p,x_0,Z)$ (with respect to p). This price is unique for almost all $x_0$; for $Z_n \rightarrow Z$ any accumulation point of $\{ p^{Z_n,*}(x_0), n\ge 1 \}$ is necessarily an optimum for $J(p,x_0,Z)$, thus by uniqueness $p^{Z,*}(x_0) = \lim \limits _{n\rightarrow \infty } p^{Z_n,*}(x_0)$ a.e. in $x_0$. The mapping $Z \in [0, Z_{max}^{\mu _0}] \mapsto {\mathscr {E}}(Z) = \frac{N_0-{\overline{X}}_0}{ {\mathbb E}\left[ g(p^{Z,*}(X_0)) \right] } \in [0, Z_{max}^{\mu _0}]$ is thus well defined and continuous; it must have a fixed point (according to Brower’s theorem) that corresponds to an equilibrium.

Also note that ${\mathscr {E}}(Z) \ge \frac{N_0-{\overline{X}}_0}{g(0)} > 0$. The strict monotony of $p^{Z,*}(x_0)$ with respect to Z implies that ${\mathscr {E}}(Z)$ is decreasing with respect to Z and thus the fixed point is unique.

Proof of point 2b

Now take $Z^{\dagger }$ as the equilibrium value and $p^\dagger (\cdot )= p^{Z^\dagger ,*}(\cdot )$; note that

$$\begin{aligned} x^{\dagger ,x_0}_t = x_0 + g(p^\dagger (x_0)) \frac{{\overline{X}}^\dagger _t-{\overline{X}}_0}{ {\mathbb E}\left[ g(p^\dagger (X_0)) \right] }, \ X_t^\dagger = X_0 + g(P^\dagger _0) \frac{{\overline{X}}^\dagger _t-{\overline{X}}_0}{ {\mathbb E}\left[ g(P^\dagger _0) \right] }. \end{aligned}$$

(B.9)

To prove that ${\mathbb V}(X_t)$ is increasing with respect to t, we will use an alternative formula for the variance of a random variable Y:

$$\begin{aligned} {\mathbb V}(Y) = \frac{1}{2} \left\{ \int _{\Omega } \int _{\Omega } \Big [(Y(\omega _1)-Y(\omega _2) \Big ]^2 {\mathbb P}(d \omega _1) {\mathbb P}(d \omega _2) \right\} . \end{aligned}$$

(B.10)

Or, formula (B.9) and the monotonicity of $p^{Z,*}(x_0)$ with respect to $x_0$ show that for any $y_1 = X_0(\omega _1),y_2=X_0(\omega _1) \ge 0$:

$$\begin{aligned} | X_t(\omega _1)-X_t(\omega _2) | = |x^{\dagger ,y_1}_t-x^{\dagger ,y_2}_t| \ge |y_1-y_2| = |X_0(\omega _1)-X_0(\omega _2) |, \end{aligned}$$

(B.11)

which implies, using (B.10), that ${\mathbb V}(X_t) \ge {\mathbb V}(X_0)$.

Proof of point 2c

We now analyze the past cumulative revenue ${\mathbb V}\left[ \int _{0}^{t} P^\dagger _s \dot{X}^\dagger _s ds \right]$. Recalling that with firm’s price being constant in time, the past cumulative revenue is simply $P^\dagger _0(X_t^\dagger -X_0)$; equation (B.9) shows that ${\mathbb V}(P^\dagger _0(X_t^\dagger -X_0)) = {\mathbb V}\left[ P^\dagger _0 g(P^\dagger _0) \right] \frac{ ({\overline{X}}^\dagger _t-{\overline{X}}_0)^2}{ {\mathbb E}\left[ g(p^\dagger (X_0)) \right] ^2 }$ which is increasing because ${\overline{X}}^\dagger _t-{\overline{X}}_0$ is positive increasing. The conclusion on the Lorenz curve follows from the remark that cumulative revenue $P^\dagger _0 (X^\dagger _t-X_0)$ is the product of a time-dependent real constant ${\overline{X}}^\dagger _t-{\overline{X}}_0$ and a time-independent random variable $\frac{ P^\dagger _0 g(P^\dagger _0) }{ {\mathbb E}\left[ g(P^\dagger _0) \right] }$.

Proof of point 2d

Recall that the Gini coefficient of a real variable B is

$$\begin{aligned} G_B = \frac{ \int _{\Omega } \int _{\Omega } |B(\omega _1)-B(\omega _2)| {\mathbb P}(d\omega _1) {\mathbb P}(d\omega _2) }{2{\mathbb E}\left[ B \right] }. \end{aligned}$$

(B.12)

Thus

$$\begin{aligned} G_{X_t^\dagger }= & {} \frac{ \int _{\Omega } \int _{\Omega } |X_t^\dagger (\omega _1)-X_t^\dagger (\omega _2)| {\mathbb P}(d\omega _1) {\mathbb P}(d\omega _2)}{2{\mathbb E}\left[ X_t \right] } \nonumber \\= & {} \frac{ \int _{\Omega } \int _{\Omega } \left| X_0^\dagger (\omega _1)-X_0^\dagger (\omega _2) + \left[ g(P^\dagger _0(\omega _1)) - g(P^\dagger (\omega _2)) \right] \frac{{\overline{X}}^\dagger _t-{\overline{X}}_0}{ {\mathbb E}\left[ g(P^\dagger _0) \right] } \right| {\mathbb P}(d\omega _1) {\mathbb P}(d\omega _2)}{2{\mathbb E}\left[ X_t \right] }.\nonumber \\ \end{aligned}$$

(B.13)

But since $y_1-y_2$ and $g(p^\dagger (y_1)) -g(p^\dagger (y_2))$ have the same sign and $\frac{{\overline{X}}^\dagger _t-{\overline{X}}_0}{ {\mathbb E}\left[ g(P^\dagger _0) \right] } >0$ we obtain:

$$\begin{aligned}&\left| X_0^\dagger (\omega _1)-X_0^\dagger (\omega _2) + \left[ g(P^\dagger _0(\omega _1)) - g(P^\dagger (\omega _2)) \right] \frac{{\overline{X}}^\dagger _t-{\overline{X}}_0}{ {\mathbb E}\left[ g(P^\dagger _0) \right] } \right| = \left| X_0^\dagger (\omega _1)-X_0^\dagger (\omega _2) \right| \nonumber \\&+ \left| g(P^\dagger _0(\omega _1)) - g(P^\dagger (\omega _2)) \right| \frac{{\overline{X}}^\dagger _t-{\overline{X}}_0}{ {\mathbb E}\left[ g(P^\dagger _0) \right] } \end{aligned}$$

(B.14)

and after some computations,

$$\begin{aligned} G_{X^\dagger _t} = \frac{{\mathbb E}\left[ X_0 \right] }{{\mathbb E}\left[ X^\dagger _t \right] } G_{X_0} + \left( 1- \frac{{\mathbb E}\left[ X_0 \right] }{{\mathbb E}\left[ X^\dagger _t \right] }\right) G_{g(P^\dagger _0)}, \end{aligned}$$

(B.15)

hence the conclusion follows. $\square$

Appendix C: Proof of Corollaries

Proof of Corollary 5.7

Proof

The proof of Corollary 5.7 uses the same arguments as those developed in “Appendix B” to prove that the variance of cumulative demand ${\mathbb V}(X_t^\dagger )$ increases over time (point 2b of Theorem 5.6). Formula (B.9) and the monotony of ${\overline{X}}^\dagger _t-{\overline{X}}_0$ and ${\overline{X}}^\dagger _\infty -{\overline{X}}_t^\dagger$, respectively guarantee that the variance of past cumulative demand ${\mathbb V}(X_t^\dagger -X_0)$ increases over time, while the variance of future cumulative demand ${\mathbb V}(X_\infty ^\dagger -X_t^\dagger )$ decreases over time. $\square$

Proof of Corollary 5.8

Proof

The proof of Corollary 5.8 is similar to that of Point 3 of Theorem 5.6 in “Appendix B.” $\square$

Proof of Corollary 5.9

Proof

In the General considerations of the proof of Theorem 5.6, we shown that the optimal equilibrium price $p^\dagger (x_0)$ is strictly decreasing with respect to $x_0$, has left and right limits (which are optimums). The monotonicity of $p^\dagger (\cdot )$ implies (by the Darboux–Froda theorem) that it can only have at most a countable number of discontinuities. Each such discontinuity corresponds to a non-differentiability point of $\Pi ^\dagger (\cdot )$, which proves the claims of the Corollary. $\square$