Finite purchasing power and computations of Bertrand–Nash equilibrium prices

Abstract

This article considers the computation of Bertrand–Nash equilibrium prices when the consumer population has finite purchasing power. The literal KKT conditions for equilibria contain “spurious” solutions that are not equilibria but can be computed by existing software, even with prominent regularization strategies for ill-posed problems. We prove a reformulated complementarity problem based on a fixed-point representation of equilibrium prices improves computational reliability and provide computational evidence of its efficiency on an empirically-relevant problem. Scientific inferences from empirical Bertrand competition models with explicit limits on individual purchasing power will benefit significantly from our proposed methods for computing equilibrium prices. An analysis of floating-point computations also implies that any model will have finite purchasing power when implemented on existing computing machines, and thus the techniques discussed here have general value. We discuss a heuristic to identify, and potentially mitigate, the impact of computationally-imposed finite purchasing power on computations of equilibrium prices in any model.

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Notes

  1. 1.

    So long as the prices below this limit represent a stationary point for the model containing decisions about only those products; we make this precise in Sect. 4.

  2. 2.

    Conditions (i–ii) imply that \(\psi _{j,p} \equiv 0\) for \(p \ge \iota _*\); thus \(p \in [0,\iota _*]\) can be relaxed w.l.o.g. to any \(p \in \fancyscript{P}\).

  3. 3.

    This initial condition is irrelevant to the results obtained.

  4. 4.

    By taking \(\varepsilon \leftarrow \varepsilon / 10\), until \(\varepsilon < 10^{-6}\), and re-starting at the previous (regularized) solution.

  5. 5.

    A map \(\mathbf {F}\) is contractive on some domain if there exists \(\kappa < 1\) such that \(|| \mathbf {F}(\mathbf {x}) - \mathbf {F}(\mathbf {y}) || \le \kappa ||\mathbf {x}-\mathbf {y}||\) for all \(\mathbf {x},\mathbf {y}\) in that domain; \(\kappa \) is the Lipschitz constant.

  6. 6.

    If \(p_k\) is at or above \(\iota _S\), we set the corresponding row and column in firm f(k)’s Hessian to be zero off the diagonal and \(-1\) on the diagonal.

  7. 7.

    Technically, the model reported by [3] has other factors that enter the price coefficient that we neglect here.

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Acknowledgments

This research was primarily undertaken at Iowa State University, with its support. The author would like to acknowledge helpful comments made by several reviewers, Todd Munson for assistance with the PATH software, and Uday Shanbhag for suggesting Tikhonov regularization. Several anonymous reviewers provided very helpful feedback that shaped the results and presentation.

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Correspondence to W. Ross Morrow.

Appendices

Proofs

Preliminaries

Morrow and Skerlos [26, 27] show that, under Assumption 1, the (continuous) price derivatives of the Mixed Logit choice probabilities can be written \((D_kP_j)(\mathbf {p}) = \delta _{j,k} \lambda _k(\mathbf {p}) - \gamma _{j,k}(\mathbf {p})\) where, as usual, \(\delta _{j,k} = 1\) if \(j = k\) and \(\delta _{j,k} = 0\) otherwise,

$$\begin{aligned} \lambda _k(\mathbf {p})&= \int w_k^\prime (\varvec{\theta },p_k) P_k^L(\varvec{\theta },\mathbf {p}) d\mu (\varvec{\theta }) \quad \text {and}\\ \gamma _{j,k}(\mathbf {p})&= \int w_k^\prime (\varvec{\theta }, p_k)P_k^L(\varvec{\theta },\mathbf {p}) P_j^L(\varvec{\theta }, \mathbf {p}) d\mu (\varvec{\theta }) \end{aligned}$$

Equations (12, 13) specialize these formulae to simulators. We also define

$$\begin{aligned} \hat{\pi }_f^L(\varvec{\theta },\mathbf {p}) = \sum _{j \in \fancyscript{J}_f} P_k^L(\varvec{\theta },\mathbf {p}) (p_j - c_j) \end{aligned}$$

for any \(\mathbf {p}, \varvec{\theta }\).

Proof

(of Lemma 1) (i) Because \(w_j\) is \(C^2\) on \((0,\iota (\varvec{\theta }))\), \(P_j^L(\varvec{\theta },\mathbf {p})\) is \(C^2\) on \((0,\iota (\varvec{\theta }))^J\). \(P_j^L(\varvec{\theta },\mathbf {p})\) is defined and continuous for all \(\mathbf {p} \in \fancyscript{P}^J\) with \(P_j^L(\varvec{\theta },\mathbf {p}) = 0\) if \(p_j \ge \iota (\varvec{\theta })\) by Assumption 1(ii, b, c). Moreover,

$$\begin{aligned} \lim _{p_j \uparrow \iota (\varvec{\theta })} (D_j^pP_k^L)(\varvec{\theta },\mathbf {p}) = \lim _{p_j \uparrow \iota (\varvec{\theta })} \Big [ w_j^\prime (\varvec{\theta },p_j)P_j^L(\varvec{\theta },\mathbf {p}) \Big ] \left( \delta _{j,k} - P_k^L(\varvec{\theta },\mathbf {p}) \right) , \end{aligned}$$

and

$$\begin{aligned} w_j^\prime (\varvec{\theta },p_j)P_j^L(\varvec{\theta },\mathbf {p})&= w_j^\prime (\varvec{\theta },p_j) \frac{ e^{w_j(\varvec{\theta },p_j)+v_j(\varvec{\theta })} }{ e^{\vartheta (\varvec{\theta })} + \sum _{k=1}^J e^{w_k(\varvec{\theta },p_k)+v_k(\varvec{\theta })} }\\&\le \frac{ w_j^\prime (\varvec{\theta },p_j)e^{w_j(\varvec{\theta },p_j)} }{ e^{\vartheta (\varvec{\theta })-v_j(\varvec{\theta })} + e^{w_j(\varvec{\theta },p_j)} } \rightarrow 0 \end{aligned}$$

by Assumption 1(ii, d). Note also that

$$\begin{aligned} (D_l D_k P_j^L)(\varvec{\theta },\mathbf {p})&= D_l \Big [ w_k^\prime P_k^L ( \delta _{j,k} - P_j^L ) \Big ] \\&= \delta _{k,l} w_k^{\prime \prime } P_k^L ( \delta _{j,k} - P_j^L ) + w_k^\prime (D_l P_k^L) ( \delta _{j,k} - P_j^L )\\&\,\quad - w_k^\prime P_k^L (D_l P_j^L) \\&= \delta _{k,l} w_k^{\prime \prime } P_k^L ( \delta _{j,k} - P_j^L ) + w_k^\prime w_l^\prime P_l^L ( \delta _{k,l} - P_k^L ) ( \delta _{j,k} - P_j^L )\\&\,\quad - w_k^\prime P_k^L w_l^\prime P_l^L ( \delta _{j,l} - P_j^L) \\&= \delta _{k,l} \Big ( w_k^{\prime \prime } + ( w_k^\prime )^2 \Big ) P_k^L ( \delta _{j,k} - P_j^L ) - ( \delta _{j,k} + \delta _{j,l} ) w_k^\prime P_k^L P_l^L w_l^\prime \\&\,\quad + 2 w_k^\prime P_k^L P_j^L P_l^L w_l^\prime \end{aligned}$$

(neglecting arguments for notational simplicity). Given this formula and Assumption 1(ii,d), \((D_l D_k P_j^L)(\varvec{\theta },\mathbf {p}) \rightarrow 0\) as either \(p_k,p_l \uparrow \iota (\varvec{\theta })\) regardless of j. Now if \(j \notin \{k,l\}\) and \(p_k,p_l < \iota (\varvec{\theta })\), we have

$$\begin{aligned} (D_l D_k P_j^L)(\varvec{\theta },\mathbf {p})&= P_j^L \Bigg [ 2 w_k^\prime P_k^L P_l^L w_l^\prime - \delta _{k,l} \Big ( w_k^{\prime \prime } + ( w_k^\prime )^2 \Big ) P_k^L \Bigg ]; \end{aligned}$$

thus if \(p_j \uparrow \iota (\varvec{\theta })\), \((D_l D_k P_j^L)(\varvec{\theta },\mathbf {p}) \rightarrow 0\).

(ii) follows from (i) using the Dominated Convergence Theorem. Because \(P_j^L(\cdot ,\mathbf {p})\) is uniformly bounded by one, an integrable function for any \(\mu \), and \(P_j^L(\varvec{\theta },\cdot )\) is continuous, \(P_j\) is continuous with

$$\begin{aligned} \lim _{p_j \uparrow \iota _*} \int P_j^L(\varvec{\theta },\mathbf {p}) d\mu (\varvec{\theta }) = \int \left( \lim _{p_j \uparrow \iota _*} P_j^L(\varvec{\theta },\mathbf {p}) \right) d\mu (\varvec{\theta }) = 0. \end{aligned}$$

The same is true for \(\lambda _j\) and \(\gamma _{j,k}\) by Assumption 1, applying the almost-sure positivity of \(\vartheta (\cdot )\) and the integrability condition on \(\psi _{j,p}^1\), which implies that \(\left| w_k^\prime (\cdot ,p_k) \right|P_k^L(\cdot ,\mathbf {p})\) is \(\mu \)-integrable (as is easily checked). Thus

$$\begin{aligned} \lim _{p_j \uparrow \iota _*} \int (D_jP_k^L)(\varvec{\theta },\mathbf {p}) d\mu (\varvec{\theta }) = \int \left( \lim _{p_j \uparrow \iota _*} (D_jP_k^L)(\varvec{\theta },\mathbf {p}) \right) d\mu (\varvec{\theta }) = 0. \end{aligned}$$

Validity of the Leibniz rule for second derivatives rests on both integrability conditions, on \(\psi _{j,p}^1\) and \(\psi _{j,p}^2\), given in Assumption 1(iii). \(\square \)

The following Lemma identifies the key to the arguments below: for large enough prices, simulated Mixed Logit choice probabilities are equivalent to simple Logit choice probabilities.

Lemma 8

For any \(\mathbf {p} \in (0,\iota _S)^J\) with \(p_k \in (\iota _{S-1},\iota _S)\) the following hold: (i) \(P_k(\mathbf {p}) = P_k^L(\varvec{\theta }_S,\mathbf {p})\); (ii) \(\lambda _k(\mathbf {p}) = w_k^\prime (\varvec{\theta }_S,p_k)P_k^L(\varvec{\theta }_S,\mathbf {p})\); (iii) \(\gamma _{j,k}(\mathbf {p}) = w_k^\prime (\varvec{\theta }_S,p_k)P_k^L(\varvec{\theta }_S,\mathbf {p})P_j^L(\varvec{\theta }_S,\mathbf {p})\); (iv) \(\gamma _{j,k}(\mathbf {p})/\lambda _k(\mathbf {p}) = P_j^L(\varvec{\theta }_S,\mathbf {p})\); (v) \(P_k(\mathbf {p})/\lambda _k(\mathbf {p}) = 1/w_k^\prime (\varvec{\theta }_S,p_k)\); (vi) \(\zeta _k(\mathbf {p}) = \hat{\pi }_{f(k)}^L(\varvec{\theta }_S,\mathbf {p} ) - 1/w_k^\prime (\varvec{\theta }_S,p_k)\).

The proof follows from the fact that \(P_k^L(\varvec{\theta }_s,\mathbf {p}) \downarrow 0\) and \(w_k^\prime (\varvec{\theta }_s,p_k)P_k^L(\varvec{\theta }_s,\mathbf {p}) \uparrow 0\) as \(p_k \uparrow \iota _{S-1}\) for all \(s \le S-1\), both following from Assumption 1; see also Lemma 1.

Convergence of sample-average approximations

The proof of Lemma 2 starts with an application of Proposition 19 in [35] to prove solutions to Eq. (7) converge to solutions of Eq. (6). This relies the compactness of \([0,\iota _*]^J\) and the applicability of a uniform Law of Large Numbers.

Lemma 9

Suppose \(\iota _* < \infty \) and the samples are i.i.d.. Denote the set of solutions to Eq. (6) by \(\mathfrak {S} \subset [0,\iota _*]^J\), and the set of solutions to Eq. (7) by \(\hat{\mathfrak {S}}_S \subset [0,\iota _S]^J \subset [0,\iota _*]^J\). Then

$$\begin{aligned} \sup \Big \{ \; \inf \big \{ || \mathbf {p}^S - \bar{\mathbf {p}} || : \bar{\mathbf {p}} \in \mathfrak {S} \big \} \; : \; \mathbf {p}^S \in \hat{\mathfrak {S}}_S \; \Big \} \rightarrow 0 \end{aligned}$$

w.p.1 as \(S \rightarrow \infty \).

Proof

This is an application of [35, Proposition 19, p. 411]. We observe that the set of solutions to Eqs. (6, 7) lie in a common compact subset of \(\mathbb {R}^J\), namely \([0,\iota _*]^J\). Let

$$\begin{aligned} \phi (\mathbf {p}) = \mathbb {E}_{\varvec{\theta }}\left[ (\tilde{\nabla }\hat{\pi }^L)(\varvec{\theta },\mathbf {p}) \right] \quad \text {and}\quad \hat{\phi }_S(\mathbf {p}) = \frac{1}{S} \sum _{s=1}^S \left( \tilde{\nabla }\hat{\pi }\right) (\varvec{\theta }_s,\mathbf {p}). \end{aligned}$$

To apply [35, Proposition 19, p. 411], we show the following:

  1. (i)

    \(\phi \) is continuous on \([0,\iota _*]^J\),

  2. (ii)

    w.p.1 for S large enough the set \(\hat{\mathfrak {S}}_S\) is nonempty and \(\hat{\mathfrak {S}}_S \subset [0,\iota _*]^J\),

  3. (iii)

    \(\hat{\phi }_S(\mathbf {p})\) converges to \(\phi (\mathbf {p})\) w.p.1 uniformly on \([0,\iota _*]^J\) as \(S \rightarrow \infty \).

(i) and (iii) follow from [35, Proposition 7, p. 363], given the pointwise norm-integrability of \((\tilde{\nabla }\hat{\pi }^L)(\cdot ,\mathbf {p}) : \fancyscript{T} \rightarrow \mathbb {R}^J\) and the pointwise continuity of \((\tilde{\nabla }\hat{\pi }^L)(\varvec{\theta },\cdot ) : [0,\iota _*]^J \rightarrow \mathbb {R}^J\) which both follow from Assumption 1. See also the remarks on pg. 412 of [35]. (ii) holds because, as proved in Lemma 3, \(\iota _S\mathbf {1} \in \hat{\mathfrak {S}}_S\) for all S and any samples \(\{\varvec{\theta }_s\}_{s=1}^S\) and thus \(\hat{\mathfrak {S}}_S\) is nonempty. \(\square \)

Corollary 4

Suppose \(\iota _* < \infty \) and the samples are i.i.d.. Let \(\{ \mathbf {p}_s \}_{s=1}^\infty \) be a sequence of solutions to Eq. (7) as \(S \rightarrow \infty \). Then, w.p.1, any limit point \(\bar{\mathbf {p}}\) of \(\{ \mathbf {p}_s \}_{s=1}^\infty \) is a solution to Eq. (6).

Proof

W.p.1. any limit point \(\bar{\mathbf {p}}\) of \(\{ \mathbf {p}_s \}_{s=1}^\infty \) lies in the closure of \(\mathfrak {S}\), by Lemma 9. But \(\mathfrak {S}\) is the set of zeros of the continuous normal map \(\mathbf {p} - \varPi (\mathbf {p}) + \phi ( \varPi (\mathbf {p}) )\) where \(\varPi \) is the Euclidean projection onto \([0,\iota _*]^J\) [8, 34]. Thus \(\mathfrak {S}\), the pre-image of a closed set under a continuous map, is closed and \(\bar{\mathbf {p}} \in \mathfrak {S}\). \(\square \)

Profit derivatives vanish

Proof

(of Lemma 3) This result is really a corollary to Lemma 1: Eq. (4) shows that as \((D_k\hat{\pi }_f)(\mathbf {p}) \rightarrow 0\) as \(p_k \uparrow \iota _* < \infty \) because \(P_k(\mathbf {p}) \downarrow 0\) and \((D_kP_j)(\mathbf {p}) \rightarrow 0\) for all \(j \in \fancyscript{J}_f\). This also implies that \((D_k\hat{\pi }_f)(\mathbf {p}) = 0\) if \(p_k \ge \iota _S\), as can be checked. Similar logic holds for the second derivatives. \(\square \)

We also note that Assumption 1(iii) ensures the norm-integrability of \((\tilde{\nabla }\hat{\pi }^L)(\cdot ,\mathbf {p}) : \fancyscript{T} \rightarrow \mathbb {R}^J\) when \(\iota _* < \infty \). Note that (neglecting arguments for notational simplicity)

$$\begin{aligned} \left| (D_k\hat{\pi }_f^L) \right|&\le \left| w_k^\prime \right| P_k^L \left| p_k - c_k \right| + \left| w_k^\prime \right| P_k^L \sum _{j \in \fancyscript{J}_{f(k)}} P_j^L \left| p_j - c_j \right| + P_k^L \\&\le 2\left( \max _{j \in \fancyscript{J}_{f(k)}} \left| p_j - c_j \right| \right) \left| w_k^\prime \right|P_k^L + P_k^L; \end{aligned}$$

\(\max _{j \in \fancyscript{J}_{f(k)}} \left| p_j - c_j \right|\) is fixed, \(P_k^L\) is integrable, and so the question becomes one of integrability of \(\left| w_k^\prime \right|P_k^L\). Again it is sufficient assume integrability of \(\psi _{k,p_k}\) (Assumption 1(iii)) as this is an upper bound for \(\left| w_k^\prime (\cdot ,p_k) \right|P_k^L(\cdot ,\mathbf {p})\). This, in part, implies that spurious solutions exist for the true Mixed Logit model as well as for simulators of them.

Proofs for Sect. 4

Proof

(of Lemma 4) By Lemma 8, for any \(\mathbf {p}\) with \(p_k \in (\iota _{S-1},\iota _S)\),

$$\begin{aligned} \zeta _k(\mathbf {p}) = \sum _{j \in \fancyscript{J}_f} \left( \frac{\gamma _{j,k}(\mathbf {p})}{\lambda _{k}(\mathbf {p})} \right) (p_j - c_j) - \left( \frac{P_k(\mathbf {p})}{\lambda _{k}(\mathbf {p})} \right) = \hat{\pi }_f^L(\varvec{\theta }_S,\mathbf {p}) + \frac{1}{|w_k^\prime (\varvec{\theta }_S,p_k)|}. \end{aligned}$$

Assumption 1(ii,c) and the Fundamental Theorem of Calculus require that \(|w_k^\prime (\varvec{\theta }_S,p_k)| \uparrow \infty \) as \(p_k \uparrow \iota _S\) and hence \(1/|w_k^\prime (\varvec{\theta }_S,p_k)| \rightarrow 0\) as \(p_k \uparrow \iota _S\). Thus \(\zeta _k(\mathbf {p}) \rightarrow \hat{\pi }_f^L(\varvec{\theta }_S,\mathbf {p})\) as \(p_k \uparrow \iota _S\). Because the right-hand side is well-defined for all \(\mathbf {p} \in [0,\iota _S]\), so is \(\zeta _k\). \(\square \)

Proof

(of Lemma 5) We ignore competition in the proof, assuming a single firm offers all products. This doesn’t affect the proof, but it does ease the notation.

(i) Suppose that \(\mathbf {p} \in [0,\iota _*]^J\) is an isolated local maximizer of \(\hat{\pi }\). Then there exists a neighborhood \(\fancyscript{U}\) of \(\mathbf {p}\) such that \(\hat{\pi }(\mathbf {p}) > \hat{\pi }(\mathbf {q})\) for all \(\mathbf {q} \in \fancyscript{U} \cap [0,\iota _*]^J\). Particularly, if \(j \in \fancyscript{J}^*(\mathbf {p})\), then \(\hat{\pi }(\mathbf {p}) > \hat{\pi }(\mathbf {p}-\delta \mathbf {e}_j)\) for sufficiently small \(\delta > 0\). The Mean Value Theorem states that for any such \(\delta \), there exists \(\varepsilon \in [0,\delta ]\) such that

$$\begin{aligned} 0 < \hat{\pi }(\mathbf {p}) - \hat{\pi }(\mathbf {p}-\delta \mathbf {e}_j) = \delta (D_j\hat{\pi })( \mathbf {p} - \varepsilon \mathbf {e}_j) = \delta \lambda _j( \mathbf {p} - \varepsilon \mathbf {e}_j) \Big ( \iota _* - \varepsilon - c_j - \zeta _j( \mathbf {p} - \varepsilon \mathbf {e}_j) \Big ) \end{aligned}$$

Because \(\lambda _j( \mathbf {p} - \varepsilon \mathbf {e}_j) < 0\), this inequality implies \(\iota _* - \varepsilon - c_j - \zeta _j( \mathbf {p} - \varepsilon \mathbf {e}_j) < 0\). Thus,

$$\begin{aligned} 0 \ge \liminf _{p_j \uparrow \iota _*} \Big [ p_j - c_j - \zeta _j( \mathbf {p} ) \Big ] = \lim _{p_j \uparrow \iota _*} \Big [ p_j - c_j - \zeta _j( \mathbf {p} ) \Big ]. \end{aligned}$$

If \(j \in \fancyscript{J}^\circ (\mathbf {p})\), then \(p_j - c_j - \zeta _j(\mathbf {p}) = 0\) follows from the simultaneous stationarity condition [27].

(ii) Let \(\mathbf {p}\) satisfy the hypotheses of (ii), which imply that for \(j \in \fancyscript{J}^*(\mathbf {p})\), \(q_j - c_j - \zeta _j(\mathbf {q}) < 0\) for all \(\mathbf {q}\) in some neighborhood of \(\mathbf {p}\).

For any \(\mathbf {s}\), \(|| \mathbf {s} || = 1\), \(\mathbf {p} + \mathbf {s} \in [0,\iota _*]^J\), and any \(\delta \in (0,1)\), there exists \(\varepsilon \in [0,\delta ]\) such that

$$\begin{aligned} \hat{\pi }(\mathbf {p}) - \hat{\pi }(\mathbf {p} + \delta \mathbf {s})&= \delta (\nabla \hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s})^\top \mathbf {s} \\&= \delta \sum _{j \in \fancyscript{J}^\circ (\mathbf {p})} (D_j\hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s})s_j + \delta \sum _{j \in \fancyscript{J}^*(\mathbf {p})} (D_j\hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s})s_j \\&= \delta \sum _{j \in \fancyscript{J}^\circ (\mathbf {p})} \Big ( (D_j\hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s}) - (D_j\hat{\pi })(\mathbf {p}) \Big ) s_j \\&\,\quad + \delta \sum _{j \in \fancyscript{J}^*(\mathbf {p})} (D_j\hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s})s_j \end{aligned}$$

The second equality holds simply because \((D_j\hat{\pi })(\mathbf {p}) = 0\) for all \(j \in \fancyscript{J}^\circ (\mathbf {p})\). Let \(\mathbf {r}\) be defined as follows: \(r_j = s_j\) if \(j \in \fancyscript{J}^\circ (\mathbf {p})\) and \(r_j = 0\) if \(j \in \fancyscript{J}^*(\mathbf {p})\). Then

$$\begin{aligned} \hat{\pi }(\mathbf {p}) - \hat{\pi }(\mathbf {p} + \delta \mathbf {s}) \!&=\! \delta \Big ( (\nabla \hat{\pi })(\mathbf {p} \! +\! \varepsilon \mathbf {s}) - (\nabla \hat{\pi })(\mathbf {p}) \Big )^\top \mathbf {r} \! +\! \delta \sum _{j \in \fancyscript{J}^*(\mathbf {p})} (D_j\hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s})s_j. \end{aligned}$$

We examine both terms in this last equation, starting with the first. Note that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left[ \frac{ ( (\nabla \hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s}) - (\nabla \hat{\pi })(\mathbf {p}) )^\top \mathbf {r} }{ \varepsilon } \right] = \mathbf {s}^\top (D\nabla \hat{\pi })(\mathbf {p}) )\mathbf {r} = \mathbf {r}^\top (D\nabla \hat{\pi })(\mathbf {p}) )\mathbf {r} < 0 \end{aligned}$$

The last inequality holds because all of rows and columns in \((D\nabla \hat{\pi })(\mathbf {p}) )\) with indices \(j \in \fancyscript{J}^*(\mathbf {p})\) vanish (Lemma 3), and the inequality follows from sub-problem local optimality. Thus, for sufficiently small \(\delta \) (and thus \(\varepsilon \)),

$$\begin{aligned} \Big ( (\nabla \hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s}) - (\nabla \hat{\pi })(\mathbf {p}) \Big )^\top \mathbf {r} < 0. \end{aligned}$$

We examine the second term:

$$\begin{aligned} \sum _{j \in \fancyscript{J}^*(\mathbf {p})} (D_j\hat{\pi })(\mathbf {p} + \varepsilon \mathbf {s})s_j&= \sum _{j \in \fancyscript{J}^*(\mathbf {p})} s_j \lambda _j(\mathbf {p} + \varepsilon \mathbf {s}) \Big ( \iota _* - p_j - c_j - \zeta _j(\mathbf {p} + \varepsilon \mathbf {s}) \Big ) \\&= \sum _{j \in \fancyscript{J}^*(\mathbf {p})} \left| s_j \right| \left| \lambda _j(\mathbf {p} \! +\! \varepsilon \mathbf {s}) \right| \Big ( \iota _* \! - \! p_j \! -\! c_j \! -\! \zeta _j(\mathbf {p} \! +\! \varepsilon \mathbf {s}) \Big ) \! \le \! 0, \end{aligned}$$

where equality holds only if \(s_j = 0\) for all \(j \in \fancyscript{J}^*(\mathbf {p})\). The absolute values follow from the fact that \(s_j \le 0\) for \(j \in \fancyscript{J}^*(\mathbf {p})\), \(\lambda _j(\mathbf {p} + \varepsilon \mathbf {s}) \le 0\) for \(\varepsilon > 0\). Thus \(\hat{\pi }(\mathbf {p}) - \hat{\pi }(\mathbf {p} + \delta \mathbf {s}) < 0\) for small enough \(\delta > 0\). \(\square \)

Proof

(of Corollaries 1 and 2) The only additional observation that must be made is that strict complementarity implies that \(p_j - c_j - \zeta _j(\mathbf {p}) < 0\) for \(j \in \fancyscript{J}^*(\mathbf {p})\). \(\square \)

Proof

(of Lemma 6) (i) \(\mathbf {z}\) is a continuous extension of \(\varvec{\zeta }\), by the proof given for Lemma 4. Continuous differentiability follows from a comparison of the formulae

$$\begin{aligned} (D_l\zeta _k)(\mathbf {p})&= \delta _{k,l} \left( \frac{w_k^{\prime \prime }(\varvec{\theta }_S,p_k)}{w_k^\prime (\varvec{\theta }_S,p_k)^2} \right) - (D_l\hat{\pi }_{f(k)}^L)(\varvec{\theta }_S,\mathbf {p}) \\ (D_lz_k)(\mathbf {p})&= \delta _{k,l} \varOmega _k - (D_l\hat{\pi }_{f(k)}^L)(\varvec{\theta }_S,\mathbf {p}) \end{aligned}$$

both valid for \(p_k\) in a neighborhood of \(\iota _S\), as \(p_k \uparrow \iota _S\) and \(p_k \downarrow \iota _S\) respectively.

(ii) Because \(\varvec{\varPhi }\) is coercive, it cannot have a zero with infinite components.

(iii) Suppose \(\varvec{\varPhi }(\mathbf {p}) = \mathbf {0}\). If \(j \in \fancyscript{J}^\circ (\mathbf {p})\), then \(\mathrm {proj}(p_j) = p_j\) and

$$\begin{aligned} 0 = \varPhi _j(\mathbf {p}) = p_j - c_j - \zeta _j(\mathbf {p}). \end{aligned}$$

If \(j \in \fancyscript{J}^*(\mathbf {p})\), then \(\mathrm {proj}(p_j) = \iota _S\) and

$$\begin{aligned} 0 = \varPhi _j(\mathbf {p})&= p_j - c_j - \varOmega _j(p_j - \iota _S) - \hat{\pi }_f^L(\varvec{\theta }_S,\mathbf {p}) \\&= \big ( 1 - \varOmega _j ) p_j + \varOmega _j \iota _S - c_j - \hat{\pi }_f^L(\varvec{\theta }_S,\mathbf {p}) \\&= \big ( 1 - \varOmega _j ) p_j - \iota _S + \varOmega _j \iota _S + \iota _S - c_j - \hat{\pi }_f^L(\varvec{\theta }_S,\mathbf {p}) \\&= \big ( 1 - \varOmega _j ) ( p_j - \iota _S ) + \varphi _j(\mathrm {proj}(\mathbf {p})) \end{aligned}$$

where \(\varvec{\varphi } = \mathbf {p} - \mathbf {c} - \varvec{\zeta }(\mathbf {p})\). Because \(p_j \ge \iota _S\) and \(1-\varOmega _j > 0\), \(\varphi _j(\mathrm {proj}(\mathbf {p})) \ge 0\). Conversely, suppose

$$\begin{aligned} \mathbf {0} \le \mathbf {p} \le \iota _S\mathbf {1} \quad \perp \quad \varvec{\varphi }(\mathbf {p}). \end{aligned}$$

Again, if \(j \in \fancyscript{J}^\circ (\mathbf {p})\), \(\varphi _j(\mathbf {p}) = \varPhi _j(\mathbf {p}) = 0\). If \(j \in \fancyscript{J}^*(\mathbf {p})\), then \(\varphi _j(\mathbf {p}) \le 0\); thus, take

$$\begin{aligned} q_j = \iota _S - \frac{ \varphi _j(\mathbf {p}) }{1 - \varOmega _j} \end{aligned}$$

Because \(1 - \varOmega _j > 0\) and \(\varphi _j(\mathbf {p}) \le 0\), \(q_j \ge \iota _S\). Moreover, if we complete the definition of \(\mathbf {q}\) by letting \(q_j = p_j\) for all \(j \in \fancyscript{J}^\circ (\mathbf {p})\), then \(\varPhi _j(\mathbf {q}) = 0\) for all \(j \in \fancyscript{J}^\circ (\mathbf {p})\) and

$$\begin{aligned} \varPhi _j(\mathbf {q}) = q_j - c_j - \varOmega _j( q_j - \iota _S) - \hat{\pi }_f^L(\varvec{\theta }_S,\mathbf {q}) = (1-\varOmega _j)(q_j - \iota _S) - \varphi _j(\mathbf {p}) = 0. \end{aligned}$$

\(\square \)

Proof

(of Corollary 3) The key piece of the proof is the existence result; we use the Poincare–Hopf theorem [22, 36]. Because \(\hat{\pi }_{f(k)}^L(\varvec{\theta }_S,\cdot )\) is bounded [23, 25], there is a large enough \(\varepsilon _k > 0\) such that \(\varPhi _k(\mathbf {p}) > 0\) for all \(\mathbf {p}\) with \(p_k = \iota _S + \varepsilon _k\). Moreover, when \(\mathbf {p} \ge \mathbf {c}\) and \(p_k = c_k\), \(\varPhi _k(\mathbf {p}) < 0\) (as can be checked). Thus the Poincare-Hopf theorem applies on \([\mathbf {c},\iota _S\mathbf {1}+\varvec{\varepsilon }]\): the sum of the indices of all zeros of \(\varvec{\varPhi }\) over \([\mathbf {c},\iota _S\mathbf {1}+\varvec{\varepsilon }]\) must be one. Thus there is at least one zero of \(\varvec{\varPhi }\) on this set, and, equivalently, at least one solution to Eq. (16).

The claim regarding positive markups (\(p_k > c_k\) for all k) is proven for interior solutions in [26, 27]. Because \(\iota _S > \max _k c_k\), this applies immediately to solutions that might have some components equal to \(\iota _S\). The claim regarding strict complementarity is proved by Lemma 5.

The claim that every firm offers some product with a price less than \(\iota _S\) follows from the fact that \(\varPhi _j(\mathbf {p}) = ( 1 - \varOmega _j ) \iota _S - c_j > \iota _S - c_j > 0\) for any \(j \in \fancyscript{J}_f\) when \(p_j = \iota _S\) for all \(j \in \fancyscript{J}_f\). \(\square \)

Proof

(of Lemma 7 (i) We show that \(\varvec{\varPhi }\) is linear for prices above the limit on purchasing power: if \(p_k,p_k+\varepsilon > \iota _S\),

$$\begin{aligned} ( p_k + \varepsilon - c_k - z_k(\mathbf {p}+\varepsilon \mathbf {e}_k) ) - ( p_k - c_k - z_k(\mathbf {p}) ) = (1-\varOmega _k)\varepsilon \end{aligned}$$

This is a close analogue to the normal map, which is obviously linear over changes in \(\mathbf {p}\) that do not change \(\varPi (\mathbf {p})\).

(ii) Note that, if \(p_k > \iota _S\) and \(p_k + \delta > \iota _S\),

$$\begin{aligned} \left| z_k(\mathbf {p} + \delta \mathbf {e}_k) - z_k(\mathbf {p}) \right| = \left| \varOmega _k \right| \left|\delta \right| = \left|\varOmega _k \right| || \mathbf {p} + \delta \mathbf {e}_k - \mathbf {p} ||. \end{aligned}$$

More generally, for any pair \(\mathbf {p},\mathbf {q}\) with \(\varPi (\mathbf {p}) = \varPi (\mathbf {q})\),

$$\begin{aligned} || \mathbf {c} + \mathbf {z}(\mathbf {p}) - \mathbf {c} - \mathbf {z}(\mathbf {q}) ||_\infty \le \max \{ \left|\varOmega _k \right| : p_k > \iota _S \} || \mathbf {p} - \mathbf {q} ||_\infty \end{aligned}$$

Thus if \(\left|\varOmega _k \right| < 1\), \(\mathbf {c} + \mathbf {z}\) is contractive over any pair \(\mathbf {p},\mathbf {q}\) with \(\varPi (\mathbf {p}) = \varPi (\mathbf {q})\). In the BLP model of Example (2)

$$\begin{aligned} \frac{w_k^{\prime \prime }(\varvec{\theta },p)}{w_k^\prime (\varvec{\theta },p)^2} = \frac{- \alpha /(\iota _S - p_k)^2}{\alpha ^2 / (\iota _S - p_k)^2 } = - \frac{1}{ \alpha } \end{aligned}$$

and thus \(\left| \varOmega _k \right| = \left|\alpha \right|^{-1}< 1\). The model in Example (3) has \(\left| \varOmega _k \right| = 0\):

$$\begin{aligned} \frac{w_k^{\prime \prime }(\varvec{\theta },p)}{w_k^\prime (\varvec{\theta },p)^2}&= - \left( \frac{ 2 }{\alpha \iota (\varvec{\theta })} \right) (\iota (\varvec{\theta }) - p) \rightarrow 0 \quad \text {as}\quad p \uparrow \iota (\varvec{\theta }). \end{aligned}$$

The normal map has this stronger property for any utility model: if \(\varPi (\mathbf {p}) = \varPi (\mathbf {q})\),

$$\begin{aligned} || \mathbf {c} + \varvec{\zeta }(\varPi (\mathbf {p})) - \mathbf {c} - \varvec{\zeta }(\varPi (\mathbf {q})) ||_\infty = || \varvec{\zeta }(\varPi (\mathbf {p})) - \varvec{\zeta }(\varPi (\mathbf {p})) ||_\infty = 0; \end{aligned}$$

this is, of course, an entirely general property of the fixed-point equation generated by the normal map for a generic MCP. \(\square \)

Complete numerical results tables

Tables 2, 3, and 4 detail the results obtained in the simulations discussed in Sect. 5.

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Morrow, W.R. Finite purchasing power and computations of Bertrand–Nash equilibrium prices. Comput Optim Appl 62, 477–515 (2015). https://doi.org/10.1007/s10589-015-9743-7

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Keywords

  • Bertrand–Nash equilibrium prices
  • Mixed complementarity problems
  • Ill-posed problems
  • Finite purchasing power
  • Mixed logit models