Pricing algorithms in oligopoly with decreasing returns

Abstract

Pricing algorithms are computerized procedures a seller may use to adapt instantaneously its price to market conditions, including to prices quoted by its rivals. These algorithms are related to the extensive use of web-collectors which contribute in many industries to identifying the best price. In such settings, price competition operates between algorithms, no longer between executives of brick and mortar companies. In this context, the question is to know how implicit forms of collusion may arise between the sellers. This paper is aimed at discussing this conceptual issue in a price-setting homogeneous product oligopoly with decreasing returns to scale where algorithms implement matching policies. Using fixed point argument, we find a family of equilibrium prices encompassing Cournot and Pareto efficient solutions, if matching is allowed upward and downward. Dynamical stability is studied in the linear demand constant return case. When matching operates only for price undercutting, this family is extended up to a bottom value of the market price, close to the Walrasian price. Pricing algorithms may solve the Bertrand–Edgeworth paradox.

Appendix

Proof of theorem 3

Let us consider an oligopoly state $(p^{0},q^{0})$ with $\sum \nolimits _{k=1}^{n}q_{k}^{0}=D(p^{0})$ and $q_{k}^{0}>0.\$Starting from this particular point, the price $p_{i}$ that maximizes firm i ’s profit solves the following program:

$$\begin{aligned} \left\{ \begin{array}{l} \max _{p_{i}}\left[ p_{i}(q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0}))-C_{i}(q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0}))) \right] \\ q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0}))\ge 0, \end{array} \right. \end{aligned}$$

(28)

The Lagrangian of this program is

$L_{i}=p_{i}(q_{i}^{0}+$ $\gamma _{i}(D(p_{i})-D(p^{0})))-C_{i}(q_{i}^{0}+$ $\gamma _{i}(D(p_{i})-D(p^{0})))+\lambda _{i}[q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0}))],$ where $\lambda _{i}$ is the Kuhn and Tucker multiplier associated with the positivity constraint; the first-order conditions are

$$\begin{aligned} \begin{array}{c} (q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0})) +\gamma _{i}D^{\prime }(p_{i})\left[ p_{i}-C_{i}^{\prime }(q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0}))\right] \\ +\lambda _{i}\gamma _{i}D^{\prime }(p_{i})=0, \\ \lambda _{i}[q_{i}^{0}+\gamma _{i}(D(p_{i})-D(p^{0}))]=0,\pi _{i}\ge 0. \end{array} \end{aligned}$$

(29)

The oligopoly equilibrium is defined by a vector $\left( p^{*},q_{i}^{*},\lambda _{i}^{*},\right)$ satisfying conditions (29) with $p^{0}=p^{*}$ and for any $i=1,\ldots ,n,$ namely

$$\begin{aligned} q_{i}+\gamma _{i}D^{\prime }(p)\left[ p-C_{i}^{\prime }(q_{i})\right] +\lambda _{i}\gamma _{i}D^{\prime }(p) & = 0. \end{aligned}$$

(30)

$$\begin{aligned} \lambda _{i}q_{i} & = 0,\lambda _{i}\ge 0. \end{aligned}$$

(31)

Then the equilibrium price and quantities are characterized by the conditions:

$$\begin{aligned} q_{i}+\gamma _{i}D^{\prime }(p)\left[ p-C_{i}^{\prime }(q_{i})\right] & = 0, \text { if }q_{i}>0, \end{aligned}$$

(32)

$$\begin{aligned} p-C_{i}^{\prime }(0) & \le 0,\text {if }q_{i}=0. \end{aligned}$$

(33)

$$\begin{aligned} D(p) & = \sum \limits _{i=1}^{n}q_{i}. \end{aligned}$$

(34)

Let ${A\!\!\!/}_{n}=\left\{ r\text { st.}.q_{i}^{r}\ge 0,i=1,\ldots ,r\right\} ,$ with $r\le n.\$Clearly, $A_{n}\ne \emptyset ,$ as $1\in A_{n}.$ Then $q^{r^{*}}$ always exists. Since $q_{r^{*}}^{r^{*}}\ge 0,$ we have $p^{r*}\ge C_{r^{*}}^{\prime }(0).$ Let us prove by contradiction that $p^{r^{*}}\le C_{r^{*}+1}^{\prime }(0)$. Assume that it is not true, namely

$$\begin{aligned} p^{r^{*}}>C_{r^{*}+1}^{\prime }(0). \end{aligned}$$

(35)

Let $q_{i}=h_{i}(p)$ solution of relation (9). Clearly, $h_{i}^{\prime }(p)=-(1-\gamma D^{\prime }C_{i}^{\prime \prime })/(\gamma D^{^{\prime \prime }}(p-C_{i})+\gamma D^{\prime })\ge 0.$ Let us define $f(p)=D(p)-\sum \limits _{i=1}^{r}h_{i}(p)$ and $\varphi (p)=f(p)+\gamma _{r^{*}+1}D^{\prime }(p)\left[ p-C_{r^{*}+1}^{\prime }(f(p)\right] .$ We have $f(p^{r^{*}})=0$ and $h_{i}(p^{r^{*}})\ge 0,i=1,\ldots ,r.$ We have $\varphi (p^{r^{*}})=\gamma _{r^{*}+1}D^{\prime }(p^{r^{*}}) \left[ p^{r^{*}}-C_{r^{*}+1}^{\prime }(0\right] \le 0,$ thanks to assumption (35). Clearly, we have $f^{\prime }<0$ and then $f(C_{r^{*}+1}^{\prime }(0))\ge f(p^{r^{*}})=0.$ Consequently, $\varphi (C_{r^{*}+1}^{\prime }(0))=f(C_{r^{*}+1}^{\prime }(0))\ge 0.$ In addition $\varphi ^{\prime }(p)=f^{\prime }(p)+\gamma _{r^{*}+1}D^{\prime \prime }(p)\left[ p-C_{r^{*}+1}^{\prime }(f(p)\right] +\gamma _{r^{*}+1}D^{\prime }(p)\left[ 1-C_{r^{*}+1}^{\prime \prime }(f(p)f^{\prime }(p)\right] \le 0.$ Thanks to the intermediate value theorem, there exists a value $p^{a}\in \left[ C_{r^{*}+1}^{\prime }(0),p^{r^{*}}\right]$ such that $\varphi (p^{a})=0.$ By definition, $p^{a}=$ $p^{r^{*}+1}\in \left[ C_{r^{*}+1}^{\prime }(0),p^{r^{*}} \right] .$ Let us prove by contradiction that $h_{i}(p^{r^{*}+1})\ge 0,i=1,\ldots r^{*}$. If $h_{i}(p^{r^{*}+1})<0,$ as $h_{i}(p^{r^{*}})\ge 0,$ applying again the intermediate value theorem exhibits a value ${\tilde{p}}\in \left[ p^{r^{*}+1},p^{r^{*}}\right] ,$ such that $h_{i}( {\tilde{p}})=0,$ i.e. ${\tilde{p}}=C_{i}^{\prime }(0).$ Hence, $C_{r^{*}+1}^{\prime }(0)\le p^{r^{*}+1}\le C_{i}^{\prime }(0),$ which is impossible according to (8). To summarize, we have $q_{i}^{r^{*}+1}=h_{i}(p^{r^{*}+1})\ge 0,i=1,\ldots ,r^{*}$ and $q_{r^{*}+1}^{r^{*}+1}=f(p^{r^{*}+1})\ge 0.$ This contradicts that $r^{*}$ is defined as the maximum of $\left\{ r\text { st. }q_{i}^{r}\ge 0,i=1,\ldots ,r. \right\} .$

Finally, we have $p^{r^{*}}\le C_{r^{*}+1}^{\prime }(0)\le C_{i}^{\prime }(0),i=r^{*}+1,\ldots ,n.$ Then according to (33), we state $q_{i}^{*}=0,$ for $i=r^{*}+1,\ldots ,n.$ Putting $q_{i}^{*}=q_{i}^{r^{*}}\ge 0,$ for $i=1,\ldots ,r^{*}$ completes the full characterization of the $\gamma$-equilibrium.