Entanglement Between Demand and Supply in Markets with Bandwagon Goods


Whenever customers’ choices (e.g. to buy or not a given good) depend on others choices (cases coined ‘positive externalities’ or ‘bandwagon effect’ in the economic literature), the demand may be multiply valued: for a same posted price, there is either a small number of buyers, or a large one—in which case one says that the customers coordinate. This leads to a dilemma for the seller: should he sell at a high price, targeting a small number of buyers, or at low price targeting a large number of buyers? In this paper we show that the interaction between demand and supply is even more complex than expected, leading to what we call the curse of coordination: the pricing strategy for the seller which aimed at maximizing his profit corresponds to posting a price which, not only assumes that the customers will coordinate, but also lies very near the critical price value at which such high demand no more exists. This is obtained by the detailed mathematical analysis of a particular model formally related to the Random Field Ising Model and to a model introduced in social sciences by T.C. Schelling in the 70’s.

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  1. 1.

    In cases not studied here, σ=0 (homogeneous IWP distribution), or σ=∞ (fat tails), one would measure quantities in units of either J or H.

  2. 2.

    In the argument of a function, we make use of “;” to separate quantities which have to be considered as parameters, here j and \(\hat{p}\), from those which appear as variables, here η.

  3. 3.

    In economics, the demand curve denotes the graph price vs. quantity, but it is meant to represent the demand function, the quantity as function of the price—hence the inverse demand function denotes the price as function of the quantity.

  4. 4.

    Here from −∞ to +∞. In the case where the IWP is defined on an interval so that the normalized variable x lies in [x m ,x M ], Γ takes the finite values Γ(η=0)=−x M and Γ(η=1)=−x m .

  5. 5.

    For multi-modal pdfs, the maximum number of such extrema is 1 plus the number of modes of the pdf [40]. Note that in the context of the RFIM, most studies consider the unimodal, Gaussian, case, or the bimodal case with H i H.

  6. 6.

    One can note that a standard decreasing marginal cost would lead, in the large N limit, to a total cost that can be neglected (compared to ). Taking into account a total cost that depends on would require an analysis of finite size effects, which is out of the scope of the present paper.

  7. 7.

    In the case of a distribution with a fat tail, one can show that for a population of N customers, with N large but finite, a maximum of the profit is obtained by selling a single unit of the good (η=1/N) at the customer with the largest willingness to pay, the price being then (at least) of order N.


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We are grateful to David Martimort for helpful suggestions. We thank Annick Vignes for useful comments and for a critical reading of the manuscript. We gratefully acknowledge the anonymous referees for their comments.

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Corresponding author

Correspondence to Jean-Pierre Nadal.

Additional information

This work was part of the project ELICCIR supported by the joint programme “Complex Systems in Human and Social Sciences” of the French Ministry of Research and of the CNRS, and of the project DyXi supported by the Programme SYSCOMM of the French National Research Agency, the ANR (grant ANR-08-SYSC-008). M.B.G., J.-P.N. and D.P. are CNRS members.



A.1 Demand Equilibria from the Function \(\mathcal{D}\)

The demand equilibria \(\eta^{d}(\hat{p},j)\) are the solutions to (6). Graphically, at a given value of j, they correspond to the intersection(s) of the function \(y=\mathcal{D}(j;\eta)\) with the horizontal line \(y=\hat{p}\). Plots of \(\mathcal{D}(j;\eta)\) against η for different values of j are presented in Fig. 9. According to Eq. (8), these curves, shifted vertically by the value h (the average IWP of the population) give the demand curves like those represented in Fig. 1.

Fig. 9

For the case of a logistic distribution of the IWP, graph of the function \(\mathcal{D}\) for j=1<j B (classical behavior at low social influence or high mean willingness to pay), j=j B ≈2.21 (curve passing through the point B) and j=5>j B (multiply valued function at large social influence; dashed: unstable equilibria) (Color figure online)

A.2 Profit Maximization

A.2.1 An Effective Supply Function

There are different equivalent ways of performing the profit maximization when social interactions are present. One is to consider that η, and consequently the profit, are functions of p (as in [32]): η=η d(p), and π(p)= d(p); another one is to consider that p and the profit are functions of η, p=p d(η), and π(η)=p d(η)η. Here we follow a reasoning that leads (obviously) to the same results, but deals symmetrically with the variables p and η. The interest of this approach is that it puts forward an analogy between the demand and the supply equations.

In order to maximize the monopolist’s profit, let us define

$$ \varPsi(\eta,p) \equiv p^d(\eta) - p. $$

The equation Ψ(η,p)=0 defines a curve Ψ in the plane {η,p} along which π has to be maximized. Be

$$ {\bf v}(\eta,p) \equiv(v_\eta,v_p)=(\partial\varPsi/ \partial p,- \partial\varPsi/ \partial\eta) $$

a vector tangent to the curve Ψ=0 at the point (η,p).

The maximization of (11) along Ψ imposes that the directional derivative of π vanishes,

$$ ({\bf v} \cdot{\nabla}) \pi\equiv v_{\eta} \frac{\partial\pi }{\partial\eta} + v_p \frac{\partial\pi}{\partial p} = 0, $$

to guarantee that the profit is an extremum. If the maximum is reached inside the support of Γ(η), it must also satisfy the second order condition

$$ ({\bf v} \cdot{\nabla}) \biggl(v_{\eta} \frac{\partial\pi}{\partial\eta} + v_p \frac{\partial\pi}{\partial p}\biggr)\leq0. $$

For finite range pdfs (compact support), one has to check whether the maximum maximorum lies on one of the boundaries of the support. Remark: had we chosen to consider η as a function of p, that is π(p)= d(p) as done in [32], this stability condition would read

$$\frac{d^2 \pi}{d p^2} \leq0. $$

Introducing the components of \(\bf v\) given by Eq. (A.2)


into the first order condition (A.3) gives

$$ p^d(\eta)=p^s(\eta) $$

where p d(η) is the inverse demand given by Eq. (8), and

$$ p^s(\eta) \equiv-\eta\frac{dp^d(\eta)}{d\eta}, $$

which, making use of the expression (8) of p d, is also given by

$$ p^s(\eta) = -\eta\mathcal{D}'(j;\eta) = \eta\bigl[ \varGamma'(\eta)-j\bigr] . $$

Using (A.5), (A.6) and (A.8), the second order condition (A.4) reads

$$ \biggl[- \frac{\partial}{\partial\eta} - \frac{d p^d(\eta)}{d\eta} \frac {\partial}{\partial p}\biggr] \bigl[-p + p^s(\eta)\bigr] \leq 0, $$

and this can be rewritten as

$$ \frac{d }{d\eta} \bigl[p^d(\eta) - p^s(\eta)\bigr] \leq0. $$

We may consider p s(η) given by (A.8) as an effective inverse supply function, although it is clearly not a true one, since it is defined by the monopolist’s optimization program, itself based on the knowledge of the demand function. Nevertheless, it has all the properties of an inverse supply function, and the market equilibrium can be understood from the equality (A.7) between demand and (effective) supply. Note that from (A.8) the positivity of the supply price p s is equivalent to having the inverse demand decreasing with η (or equivalently, from (A.9)), to have the demand at a stable equilibrium, that is \(\mathcal{D}'(j; \eta) \le0\).

A.2.2 Behavior of \(\widetilde{\varGamma}\)

In order to have a full analogy between the Supply and Demand problems, on needs \(\widetilde{\varGamma}(\eta)\), defined by Eq. (15), to be monotonically increasing from −∞ to +∞ when η goes from 0 to 1, and to have a single inflexion point. One can easily check that, for ‘usual’ pdfs such as the logistic or the Gaussian, \(\widetilde{\varGamma}\) is a monotonic, strictly increasing function of η for all 0<η<1. For an arbitrary (smooth enough) pdf, one can show that \(\widetilde{\varGamma}\) is a monotonically increasing function for η>η B , as well as for η small enough whenever the pdf has a finite variance. It remains the possibility to have a non monotonic behavior of \(\widetilde{\varGamma}\) in some small intermediate range, η not too small and not too close to η B .

Let us now express the derivative of \(\widetilde{\varGamma}\) in term of the pdf f(x) and its cumulative G(x)≡1−F(x). One has η=G(x). From the definition of \(\widetilde{\varGamma}\), one has

$$\frac{ d \widetilde{\varGamma}}{d\eta} = -2 \frac{ dx }{d\eta} - G(x) \frac{ d^2x }{d\eta^2 }. $$

Now \(\frac{ dx }{d\eta}=1/G'(x)\), and thus \(\frac{ d^{2}x }{d\eta^{2} } = -\frac{G''}{G'^{3}}\). It follows that one can write

$$\frac{ d \widetilde{\varGamma}}{d\eta} = -\frac{G^3}{G'^3} \biggl[ -\frac{G''}{G^2} + 2 \frac{G'^2}{G^3} \biggr] = -\biggl[\frac{G}{G'}\biggr]^3 \biggl[ \frac{ d^2}{dx^2 } \frac{ 1}{G(x) } \biggr] $$

which means

$$ \frac{ d \widetilde{\varGamma}}{d\eta} = \biggl[\frac{1-F(x)}{f(x)}\biggr]^3 \biggl[ \frac{ d^2}{dx^2 } \frac{ 1}{1-F(x) } \biggr] . $$

The right hand side is positive if and only if

$$ \frac{ d^2 }{dx^2 } \frac{ 1 }{1-F(x) } > 0. $$

Hence \(\widetilde{\varGamma}\) is a monotonic, strictly increasing function of η for all 0<η<1 iff the above property is true for every x belonging to the support of f.

This condition (A.13) is not a very stringent one. After some algebra it may be shown that it is equivalent to impose that, in the absence of externalities (j=0), the demand satisfies

$$ \frac{ d^2 }{ dp^2 } \frac{ 1 }{\eta^d(p)} > 0. $$

This is a weaker condition than a convexity condition, d 2logη d(p)/dp 2<0, frequently assumed in economics.

A.3 Numerical Illustrations

Figures 10 present examples of possible situations met by the monopoly. They correspond to systems with parameters j=3.5>j B (for the logistic, j B ≈2.21) and the three different values of h, indicated on Fig. 4, that fall within the region of uncertain outcome for the \(\eta^{s}_{+}\) strategy. The curves \(\mathcal{D}\) and \(\widetilde{\mathcal{D}}\) in figures (a), (c) and (e) are typical of any smooth pdf in the region with multiple solutions of the profit optimization. They show the constructions allowing to determine the optimal prices. The lines y=−h ch are represented: for the values of h considered the optimal strategy corresponds to the high-η s branch. Optimal prices are given by the difference \(\mathcal{D - \widetilde{\mathcal{D}}}\) at the value of η where the line y=−h intersects the curve \(\widetilde{\mathcal{D}}\). The introductory price p is indicated whenever it is viable (i.e. positive). In particular, when h=−1.5 introductory prices allowing to get rid of the coordination problem do not exist. Figures (b), (d) and (f) present the corresponding demand and supply prices, \(p^{d}=h+\mathcal{D}\) and p s (the optimal price corresponds to p s=p d), and the profits π=ηp s. In (a), (b) the only optimal strategy is \(\eta^{s}_{+}\). In (c), (d) both strategies \(\eta^{s}_{-}\) and \(\eta^{s}_{+}\) fall inside the multi-valued demand region. The profit of targeting \(\eta^{s}_{-}\), which does not need coordination, is smaller than the profit corresponding to the introductory price p , and the latter may allow to drive the system to the \(\eta^{s}_{+}\) equilibrium of optimal profit. If h=−1.5, figures (e), (f), both strategies fall in the coordination region, and there is no possibility to get rid of the problem through introductory prices.

Fig. 10

(a), (c) and (e): functions \(\mathcal{D}(j; \eta)\) and \(\widetilde{\mathcal{D}}(j; \eta)\). The construction that determines the profit extrema is illustrated. (b), (d) and (f): price and profit vs. η showing the extrema of π. When h>h ch ≈−2: the large-η strategy is the optimum. The dotted lines correspond to the non-economic demand between η L and η U , where \(\mathcal{D}\) has positive slope (Color figure online)

A.4 Monopolist’s Phase Diagram: Details

A.4.1 Domain of Multiple Pricing Strategies

Like for the demand, there is for the supply a bifurcation at critical value of j defined by

$$ j_A=\widetilde{\varGamma}'(\eta_A)/2,\quad \mbox{with } \eta_A \equiv\arg\min_{\eta} \widetilde{ \varGamma}'(\eta), $$

beyond which there are multiple solutions for the supply. One can show that η A η B and j A j B .

If j<j A both curves \(\mathcal{D}\) and \(\widetilde{\mathcal{D}}\) are monotonically decreasing functions of η. Eq. (12) has a single solution η s for each couple j, h. Given any j(<j A ), if h increases continuously from a very small value (highly negative) to a large positive one, the optimal price—which by (16) is the difference between \(\mathcal{D}\) and \(\widetilde{\mathcal{D}}\)—decreases and the fraction of buyers increases, both monotonically.

For jj A , there is a finite range of values of h where \(\widetilde{\mathcal{D}}(j; \eta)\) presents two extrema: a minimum at η (j)<η A and a maximum at η +(j)>η A . They satisfy \(\widetilde{\mathcal{D}}'(j; \eta_{\pm}) = 0\). Correspondingly, the profit presents two relative maxima at the intersections of y=−h with the branches of \(\widetilde{\mathcal{D}}(j; \eta)\) that have negative slope. Notice that the additional intersection at an intermediate value of η with the branch having \(\widetilde{\mathcal{D}}'(j; \eta) > 0\) corresponds to a minimum of the profit. The profit’s absolute maximum has to be determined numerically, through comparison of the profit relative maxima.

As an example, plots of \(\widetilde{\mathcal{D}}(j; \eta)\) and \(\mathcal{D}(j; \eta)\) for a logistic pdf and a value j>j B are presented in Fig. 10. We have explicitly indicated the values of \(\mathcal{D}(j; \eta)\) corresponding to the unstable equilibria η∈[η L (j),η U (j)]. The price construction for a particular value of h is exhibited.

The values of η +(j) and η (j) are determined following the same steps as for the customer’s model, and have the same form as Eqs. (10) which define η L (j) and η U (j), but with \(\widetilde{\varGamma}(\eta)\) and \(\hat{\jmath}\) instead of Γ(η) and j respectively. Introducing the values η +(j) and η (j) into (12) we obtain:

$$ h_{\pm}(j) = -\widetilde{ \mathcal{D}}\bigl(j; \eta_{\pm}(j)\bigr). $$

In the plane {j,h} the lines h=h +(j) and h=h (j) represented in Fig. 4 are the boundaries of a region where the profit has multiple (sub)optima. These boundaries merge at the point A that satisfies simultaneously \(\widetilde{\mathcal{D}}'(j; \eta)=0\), and \(\widetilde{\mathcal{D}}''(j; \eta)= 0\), that is

$$ A \equiv\bigl\{ j_A , h_A \equiv\widetilde{\varGamma}( \eta_A)-\eta_A \widetilde{\varGamma}'( \eta_A) \bigr\}. $$

It may be easily checked that d[h (j)−h +(j)]/dj=2[η +(j)−η (j)], meaning that, like in the customers problem, the width of the region with multiple extrema increases with j because η <η +.

A.4.2 Null Price Boundary

When j>j B both \(\mathcal{D}\) and \(\widetilde{\mathcal{D}}\) present positive slopes, but for different ranges of η. As already stated, both curves cross each other at η L and η U , with \(\widetilde{\mathcal{D}}'(\eta_{L})>0\) and \(\widetilde{\mathcal{D}}'(\eta_{U})<0\). Thus, at η L the profit is a minimum. Since η <η L , for any η s<η , \(\widetilde{\mathcal{D}} < \mathcal{D}\): the optimal prices of the low-η s strategies are positive in all the range of {h,j} values for which it exists.

In contrast, at the other side of the customers’ unstability gap [η L ,η U ], \(\widetilde{\mathcal{D}}(\eta_{U})=\mathcal{D}(\eta_{U})\) and since η +<η U , the boundary of the high-η s strategy (at h=h +(j)) corresponds to a negative price. Notice that in the range η +<η s<η U where the monopolist’s relative maximum has a negative price, the customers demand is unstable and is not expected to exist at equilibrium. Thus an equilibrium with high demand only exists for η>η U (j). The line h=h U (j) is the null-price line (that is actually the line where P=C) on the high-η manifold: it sets a lower bound to the values of h for which the monopolist’s high-η strategy is viable.

A.4.3 Bifurcation Point B

In the plane {j,h} the point B defined by:

$$ B \equiv\{j_B, h_B \equiv-\hat{p}_B \}, $$

is analogous to the point B defined for the customers phase diagram (see Sect. 2). It belongs both to the line h +(j) and to the null-price line. The corresponding fraction of buyers is η B . In the Appendix, Sect. A.4.4 we study with some details the vicinity of the singular points A and B in the monopolist’s phase diagram.

A.4.4 Vicinity of the Singular Points A and B

The bifurcation point A plays the same role, in the monopolist’s phase diagram, as the bifurcation B in the Demand phase diagram [32, 40]. The singular behavior at this apex A is obtained in the very same way. Developing in the vicinity of the A, at which \(\widetilde{\varGamma}''(\eta_{A})=0\), we obtain expressions for the boundaries of the multiple extrema region that are similar to those of the demand phase diagram, but with \(\widetilde{\varGamma}\) in the place of Γ, \(\hat{\jmath}\) instead of j and \(\hat{\epsilon}\equiv2 \epsilon\) instead of ϵ.

It is interesting to consider more in details the vicinity of the (monopolist’s) point B, where one has both p=0 and marginal stability for the ‘+’ solution. Near B, for j>j B and/or h>h B , the ‘+’ solution gives a small price value, and a value of η close to η B . Like for the demand [40], we can expect a similar behavior for the monopolist’s solution: a linear increase of the price and a singular, square root, behavior for η. Indeed at first non trivial order in ϵ one gets

$$ \mbox{for } j=j_B, 0 < \epsilon\equiv h-h_B \ll 1 {:} \quad \left \{ \begin{array}{@{}l} p_{+} = \epsilon, \\[2pt] \eta_{+} = \eta_B + \sqrt{\frac{2}{\eta_B \varGamma'''(\eta_B)}} \epsilon^{1/2}, \\[2pt] \varPi_{+} = \eta_B \epsilon+ \sqrt{\frac{2}{\eta_B \varGamma'''(\eta_B)}} \epsilon^{3/2}. \end{array} \right . $$

And similarly,

$$ \mbox{for } h=h_B,\ 0 < \epsilon\equiv j-j_B \ll 1 {:}\quad \left \{ \begin{array}{@{}l} p_{+} = \eta_B \epsilon\\[2pt] \eta_{+} = \eta_B + \sqrt{\frac{2}{\varGamma'''(\eta_B)}} \epsilon^{1/2} , \\[2pt] \varPi_{+} = \eta_B^2 \epsilon+ \eta_B \sqrt{\frac{2}{\varGamma'''(\eta_B)}} \epsilon^{3/2}. \end{array} \right . $$

The ‘+’ solution appears at B through a continuous transition for the profit Π, with a discontinuous jump for η (from 0 to η B ), and then a square-root behavior. The latter is specific to the point B. Indeed, one can perform a similar expansion in the vicinity of the null price line.

Consider a point on this line with j>j B . The corresponding value of η is the solution η 0(j) of j=j 0(η), and the value of h is h 0(j)≡h 0(η 0(j)). Then for h=h 0(j)+ϵ, 0<ϵ≪1, expansion of p=p s(η)=p d(η) at first non trivial order in ϵ gives

$$ \mbox{for } j > j_B,\ 0 < \epsilon\equiv h-h_0(j) \ll 1 {:}\quad \left \{ \begin{array}{@{}l} p_{+} = \epsilon, \\[2pt] \eta_{+} = \eta_0(j) + \frac{\epsilon}{\eta_0(j) \varGamma''(\eta_0(j))}, \\[2pt] \varPi_{+} = \eta_0(j) \epsilon+ \frac{\epsilon^2}{\eta_0(j) \varGamma''(\eta_0(j))}. \end{array} \right . $$

On sees on the above expansion for η how the singular behavior at j=j B appears: as j approaches \(j_{B}^{+}\), η 0(j)→η B , hence Γ″(η 0(j)) tends to zero, and thus the coefficient of ϵ in the expansion of η diverges.

A.4.5 Correspondence Between the Demand and the Supply Branches

The fact \(\eta^{s}_{-}\) and \(\eta^{s}_{+}\), the low-η and large η solutions for the seller, do correspond to η values falling on, respectively, the low-η and large η branches of the demand, is shown more formally here.

Any solution \(\eta^{s}_{-}\) corresponding to the low-η branch for the supply lies also on the low-η branch of the demand, because \(\eta^{s}_{-} < \eta_{-} < \eta_{L}\) (see the left-hand side Fig. 10).

Similarly, since the zero-price line corresponds to \(h_{0}(j)=-\hat{p}_{U}(j)\), any viable (that is with p s≥0) solution \(\eta^{s}_{+}\) on the high-η branch for the seller also lies on the high-η branch of the demand, i.e. \(\eta^{s}_{+} \geq\eta_{U}\).

A.5 Monopolist’s Low-η Branch

One can make the analysis of the low-η branch more precise. Let us define η m (j)<η L (j) such that

$$ \mathcal{D}(j;\eta_m)= \hat{p}_U(j), $$

which is the low-η demand at the border \(\hat{p}_{U}(j)\) of the customers’ multiple equilibria region. For η m <η<η L the customers low-η equilibrium coexists with the one at large-η. Be

$$ h_m(j) \equiv-\widetilde{ \mathcal{D}}(j,\eta_m)=-\hat{p}_U(j)-\eta_m \mathcal{D}'(j;\eta_m). $$

Depending on j, η m (j) may be smaller or larger than η (j). Be j C (>j B ) the value of j for which these two values of η are equal: η m (j C )=η (j C ) and consequently h C h m (j C )=h (j C ). It verifies

$$ \mathcal{D}\bigl(j;\eta_m(j_C)\bigr) - \widetilde{\mathcal{D}}\bigl(j;\eta_m(j_C)\bigr) = \hat{p}_U(j_C)+h_-(j_C). $$

In words, j C is the value of j at which the (low-η s) optimal strategy corresponds to a fraction of buyers η and the corresponding price is exactly equal to \(\hat{p}_{U}(j_{C})+h_{-}(j_{C}) \).

If j>j C , η m <η . Then, for h m <h<h the (low-η s) relative maximum lies at \(\eta_{-}^{s}\) with \(\eta_{m} < \eta_{-}^{s} < \eta_{-}\), i.e. inside the region with \(\hat{p}_{L} < \mathcal{D} < \hat{p}_{U}\), where the customers’ system has multiple equilibria. As shown in Appendix A.6, this maximum never gives the absolute maximum of the profit, which will be on the high-η branch. In particular one can thus conclude that the high-η strategy becomes optimal before h reaches h m , that is

$$ h_{ch}(j) < h_m(j). $$

In addition to the \(\eta_{-}^{s}\) solutions, if \(h_{m} < - \hat{p}_{L}\) there exists the relative maximum at the margin η L of the multiple equilibria region. Although it does not correspond to an optimal pricing strategy, for completeness let us point out that this end point becomes a minimum for j>j D defined by

$$ h_D \equiv h_m(j_D) = - \hat{p}_L(j_D), $$

because for larger values of j, \(h_{m} < - \hat{p}_{L}\).

To summarize, for j<j C and h<h(j) the low-η s optimal strategy the low-η s strategy (j C <j<j D ) and h>h m the profit has a minimum at the boundary η L whereas for j>j D .

Figure 11 summarizes the results for the low-η manifold in the case of a logistic distribution.

Fig. 11

Supply phase diagram: details of the low-η strategy

A.6 Behavior of the Profit Near the Boundaries

In addition to maxima obtained from the solutions of the 1st and 2nd order equations, as discussed above, there may exist maxima of the profit at extreme (boundary) values: the profit π(p) may have a (possibly local) maximum at some of the boundary values, p=0, p=∞, and, in the domain of h and j values for which the demand η d(p) has two branches, at the maximal value \(p_{U}= h + \hat{p}_{U}(j)\) for which the high-η demand exists, and the minimal value \(p_{L}= \max(0, h + \hat{p}_{L}(j) )\) for which the low-η demand exists. Let us consider the behavior of the profit π(p)= d(p) as a function of the price p.

A.6.1 Behavior of the Profit Near p=0 and Near p=∞

One can check that, for IWP distributions with no fat tailFootnote 7 (that is with finite variance σ), the profit is decreasing to 0 as p goes to ∞: the large p limit is always a minimum of the profit.

From the definition of the profit as π(p)= d(p), one has

$$ \frac{d \pi}{dp} = \eta^d(p)+ p \frac{d \eta^d}{dp} $$

Hence at p=0 one has \(\frac{d \pi}{dp} = \eta > 0\): this boundary is always (unsurprisingly) a minimum of the profit.

A.6.2 End Points of the High and Low η Branches

For j>j B , the demand has two branches, \(\eta^{d}_{L}\) and \(\eta^{d}_{U}\), hence the function π(p) has itself two branches (see Fig. 8) which we denote by π L and π U respectively. Necessarily, at least one maximum of the profit exists on each branch.

If \(h + \hat{p}_{L}(j) < 0\), that is when the low-η branch of the demand exists already at p=0, the above argument for the boundary p=0 applies: the profit increases along this branch when p is increased from p=0 (going through a maximum at some p>0 before decreasing to zero as p→∞).

Now we know that the derivative of the demand at each one of the boundaries, \(\hat{p}_{U}(j) \equiv\mathcal{D}(j; \eta_{U}(j))\) and \(\hat{p}_{L}(j) \equiv\mathcal{D}(j; \eta_{L}(j))\), is singular, \(\frac{ d\eta^{d}(p) }{ dp }=- \infty\) (since \(\frac{ dp^{d}(\eta) }{ d\eta}=0\), and the demand decreases with p). It follows that, for \(h > -\hat{p}_{U}(j)\), at \(p_{U}= h + \hat{p}_{U}(j)\), \(\frac{d \pi_{U}}{dp} = -\infty\), and for \(h > - \hat{p}_{L}(j)\), at \(p_{L}= h + \hat{p}_{L}(j)\), \(\frac{d \pi_{L}}{dp} = -\infty\). Hence the boundary on the high-η branch, p U , is always a minimum of the profit, whereas for \(h > - \hat{p}_{L}(j)\), p L is always a maximum of the profit on the low-η branch of the demand. Remark: for h<h m , on the low η branch there is a maximum for some price greater than p L : increasing the price from p L , the profit decreases, goes through a minimum, increases up to a maximum and decreases again, going to zero as p→∞.

A.6.3 Non-optimality of the Low-η Profit for p L pp U

The maximum of π L at p L for \(h > - \hat{p}_{L}(j)\), however, is always a (strictly) local maximum, as it is the case for any maximum of the profit that would exist on the low-η branch of the demand for a price p L pp U .

Indeed, consider a value p in this range, p L pp U . For this price the demand has two possible values, \(\eta^{d}_{L}(d)\) on the low-η branch (with d=ph), and \(\eta^{d}_{U}(d)\) on the high-η branch. Since \(\eta^{d}_{L}(d) < \eta^{d}_{U}(d) \), the profit \(p \eta^{d}_{L}(d)\) is strictly smaller than \(p \eta^{d}_{U}(d)\), that is the profit for the same price obtained with the high-demand, and the later is itself smaller (or equal) to the maximal profit associated to the high-η branch.

In the range h +<h<h , an alternative way to see the fact that the profit at p L is never the optimal one is the following, for smooth enough pdfs. The profit Π(η) is a continuous function of η with two relative maxima, one at η<η , the other at η>η +, and a minimum in between. One can show that η <η L <η +, that is \(\mathcal{D}(j; \eta)\) hits \(\widetilde{\mathcal{D}}(j; \eta)\) at a value η L where \(\widetilde{\mathcal{D}}\) is increasing with η. Indeed, at η=η L , \(\mathcal{D}'=0\), hence \(\widetilde {\mathcal{D}}' = \eta_{L} \mathcal{D}''\), and \(\mathcal{D}''(j;\eta_{L}) = - \varGamma''(\eta_{L}) > 0\) (Γ is concave on [0,η B ], see [40]). Since η <η L <η +, η L is in between the two maxima, so that the profit is always lower at η L than at the global maximum (the largest of the two relative maxima).

In addition, the maximum of the profit at p L is not a stable equilibrium: any small deviation of price below the value p L would make the demand jump on the high-demand branch.

In conclusion, there is no optimal strategy for the seller corresponding to prices at the boundaries. We have also seen that the low-η profits, when obtained with a price in the coexistence domain, never give an absolute maximum of the profit (see Fig. 8 for an example).

A.6.4 Non-Optimality of the Low-η Profit at η=η L

If \(-\hat{p}_{L} < h < h_{-} \), since η <η L , there are two intersections of y=−h with \(y=\widetilde{\mathcal{D}}(j; \eta)\), one at \(\eta_{-}^{s} < \eta_{-}\) where \(\widetilde{\mathcal{D}}\) has negative slope, the other at a larger value of η where \(\widetilde{\mathcal{D}}' > 0\). Only the first one corresponds to a relative maximum of the profit, the other one is a minimum. If η is increased beyond this minimum, the profit increases continuously and reaches a relative maximum at η L . This relative maximum does not satisfy the conditions (12) and (13) because it lies at the border of the domain η∈[0,η L ] where the low-η customers’ equilibrium exists. The corresponding price is \(p^{s}=h+\hat{p}_{L} > 0\), so that the profit is \(\pi (\eta_{L})=\eta_{L} (h+\hat{p}_{L})\). If \(\pi(\eta_{L}) > \pi(\eta^{s}_{-})\) the absolute maximum of the profit lies in the high-η s branch, because the profit is a monotonically increasing function of η from the minimum at η<η L up to its maximum at \(\eta^{s}_{+}>\eta_{+}\) and a +, reached on the high-η branch where \(\widetilde{\mathcal{D}}'(j; \eta)>0\). Conversely, if \(\pi(\eta_{L}) < \pi(\eta^{s}_{-})\) the absolute maximum is at either \(\eta^{s}_{-}\) or \(\eta^{s}_{+}\) but clearly not at η L . In other words, the extremum at the boundary η L is never a winning strategy and the monopolist should never post the corresponding price. For smaller values of h, \(h < - \hat{p}_{L}\) the slope of \(\widetilde {\mathcal{D}}(j; \eta)\) for η∈[0,η L ] is negative: there is no possible maximum at the border η L . For these values of h again there is a single optimum on the low-η s branch, at \(\eta^{s}_{-}\).

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Gordon, M.B., Nadal, JP., Phan, D. et al. Entanglement Between Demand and Supply in Markets with Bandwagon Goods. J Stat Phys 151, 494–522 (2013). https://doi.org/10.1007/s10955-012-0660-1

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