QME

, Volume 8, Issue 2, pp 167–205

Switching costs, experience goods and dynamic price competition

Authors

    • Department of Business and EconomicsUniversity of Southern Denmark
    • Faculty of Arts and SciencesSabanci University
Article

DOI: 10.1007/s11129-010-9083-y

Cite this article as:
Doganoglu, T. Quant Mark Econ (2010) 8: 167. doi:10.1007/s11129-010-9083-y

Abstract

I analyze a dynamic duopoly with an infinite horizon where consumers are uncertain about their potential satisfaction from the products and face switching costs. I derive sufficient conditions for the existence of a Markov Perfect Equilibrium(MPE) where switching takes place each period. I show that when switching costs are sufficiently low, the prices in the steady state are lower than what they would have been when they are absent. This result is in contrast to those found in the literature. In the presence of low switching costs competition can be fiercer.

Keywords

Dynamic price competitionExperience goodsMarkov-perfect equilibriumSwitching costs

JEL Classification

C73D21D43L13L14L15

Supplementary material

$$\label{ll} l^*=\frac{1}{2} a^*+{\frac { \left( 2 z(a^*) -\beta \Delta a^* \right) \Delta}{{\Delta}^{2}+2 \beta \Delta s+2 z(a^*) }} $$$$\label{kl} k^*=-a^* $$$$\label{n2l} n_2^*={a^*}^{2} \left( \frac{1}{\Delta}+\frac{1}{2}{\frac {\Delta}{z(a^*) } } \right) , $$$$\label{n1l} n_1^*=-n_{{2}}^*+a^*+{\frac {2 s \left( 2 z(a^*) -\beta a\Delta \right) }{{\Delta}^{2}+2 \beta \Delta s +2 z(a^*) }}, $$$$\label{n0l} n_0^*=\frac{1}{2}\,{\frac { \left( n_{{1}}^*+n_{{2}}^*-a^* \right) \Delta}{ \left( 1- \beta \right)s }}+\frac{1}{4}\,n_{{2}}^*+\frac{1}{2}\,n_{{1}}^* $$$$\label{aeqa} g(a)= a+{\frac { 4 z(a) ^{3} s}{ \left( { \Delta}^{2}+2 z(a) \right) \left( 3 z(a)^{2}-\beta {\Delta}^{2}{a}^{2} \right) }} $$$$p_1^*(x^t) = {\frac {2s\,z(a)^{2} \left( 2\,x^t-1 \right) }{6\,z(a)^{2}+3\,{\Delta}^{2} z(a) -2\,\beta\,{\Delta}^{3}n_{{2}}}}+{\frac {\Delta\, \left( 2\,z(a) -\Delta\,\beta\, \left( n_{{1}}+n_{{2}} \right) \right) }{{\Delta}^{2}+2\,z(a) }}, \label{cp1} $$$$p_2^*(x^t) = -{\frac {2s\,z(a)^{2} \left( 2\,x^t-1 \right) }{6\, z(a)^{2}+3\,{\Delta}^{2} z(a) -2\,\beta\,{\Delta}^{3}n_{{2}}}}+{\frac {\Delta\, \left( 2\,z(a) -\Delta\,\beta\, \left( n_{{1}}+n_{{2}} \right) \right) }{{\Delta}^{2}+2\,z(a) }}. $$$$\label{bellman} V(x^t)=\pi(p_1^*(x^t),p_2^*(x^t),x^t)+\beta V(x^{t+1}). $$$$ p_1^*(x^t)-p_2^*(x^t)=-{\frac {4s\,z(a)^{2} \left( 1-2\,x^t \right) }{6\, z(a)^{2}+3\,{\Delta}^{2} z(a) -2\,\beta\,{\Delta}^{3}n_{{2}}}}. $$$$\label{exps} h=a+{\frac {4s\,z(a)^{2} }{6\, z(a)^{2}+3\,{\Delta}^{2} z(a) -2\,\beta\,{\Delta}^{3}n_{{2}}}}=0. $$$$\label{x2} f_2=n_{{2}}+8\,{\frac {{s}^{2} z \left( a \right)^{3} \left( {\Delta}^{2}+2\,z \left( a \right) \right) }{ \left( 6\, z \left( a \right)^{2}+3\,{\Delta}^{2}z \left( a \right) -2\,\beta\,{\Delta}^{3}n_{{2}} \right) ^{2}}}=0. $$$$\begin{array}{lll} f_1&=&-f_2+h+{\frac { \left( {\Delta}^{2}+2\,\beta\,s\,\Delta+2\,z \left( a^* \right) \right) \left( n_{{1}}+n_{{2}}^*-a^* \right) }{{\Delta}^{2}+2\, z \left( a^* \right) }}\\ &&-\,{\frac {2 s\, \left( 2\,z \left( a^* \right) - \beta\,a^*\Delta \right) }{{\Delta}^{2}+2\,z \left( a^* \right) }}=0. \end{array}$$$$\begin{array}{lll} f_0&=&\frac{f_2}{4}-\frac{h}{2}+\frac{\Delta-s}{2s}(f_1+f_2-h)+ \left( 1-\beta \right) n_{{0}}-{\frac { \left( n_{{1}}^*+n_{{2}}^*-a^* \right) \Delta}{2s}} \\ &&+\frac{1}{4}\left( n_{{2}}^*+2\,n_{{1}}^* \right) \left( 1 -\beta \right)=0. \end{array}$$$$\begin{array}{lll} \rho_1&=&-{\frac {\sqrt {3}\Delta}{\sqrt {\beta} \left( \Delta-2\,\sqrt {3} \sqrt {\beta}s \right) }},\\ \rho_2&=&{\frac {\sqrt {3}\Delta}{\sqrt {\beta} \left( \Delta+2\,\sqrt {3} \sqrt {\beta}s \right) }},\\ \rho_3&=&{\frac {\Delta\, \left( \Delta+2 \right) }{4\beta\,s}}. \end{array}$$$$\begin{array}{lll} {\underline{a}}_1&=&-\frac{\Delta}{\Delta-2\beta s},\\ {\overline{a}}_1&=&\ \ \frac{\Delta}{\Delta+2\beta s},\\ {\underline{a}}_2&=&-\frac{\Delta}{2}+s,\\ {\overline{a}}_2&=&\ \ \frac{\Delta}{2}-s. \end{array}$$$$\begin{array}{lll} \Delta_1&=& -1+s\, \left( 1-\beta \right) +\sqrt { \left( 1+ \left( 1+\beta \right) s \right) ^{2}+4\,s},\\ \Delta_2&=&-1+2\,\beta\,s\, \frac{4-\beta}{3-\beta} + \sqrt { \left( -1+2\,\beta\,s\, \frac{4-\beta}{3-\beta} \right) ^{2}+8\,\beta\,{s}^{2}}, \\ \Delta_3&=&- \left( 1+\beta \right) s+\sqrt {{s}^{2} \left( 1-\beta \right) ^{2}- 4\,s+8\,\beta\,s+4} ,\\ \Delta_4&=&3- \left( 1+\beta \right) s+\sqrt { \left( 3- \left( 1+\beta \right) s \right) ^{2}+4\, \left( 2-s \right) \left(1+ \beta\,s \right) } \end{array}$$$$\tilde n_1 =2 a+{\frac {2 s \left( 2 z(a)-\beta a\Delta \right) }{{\Delta}^{2}+2 \beta \Delta s +2 z(a) }}<n_1^*,$$$$\hat n_1 =2 a+\frac {4 s} {{\Delta} +2 \beta s + 2 }\le \tilde n_1 \le n_1^*,$$$$a_0=-\frac {2 s} {{\Delta} +2 \beta s + 2 }.$$$$a_0-{\underline{a}}_2=-\frac {2 s} {{\Delta} +2 \beta s + 2 }+\frac{\Delta}{2}-s,$$$$g_2(a)={\frac { 4 z(a) ^{3} s}{\left( { \Delta}^{2}+2 \Delta \right) \left( 3 z(a)^{2} -\beta {\Delta}^{2}{a}^{2} \right) }}\ge g_1(a).$$$$\frac{\partial }{\partial a}g_2(a)= -{\frac {8 \beta\, s\, z \left( a \right) ^{2} \left( (3\, z \left( a \right) ^{2}-\beta { \Delta}^{2} {a}^{2})s-{\Delta}^{3}a \right) }{\Delta\, \left(2+ \Delta \right) \left( 3\, z \left( a \right)^{2}-\beta\,{\Delta}^{2 }{a}^{2} \right) ^{2}}}\le 0, $$$$\frac{\partial^2 }{\partial a^2}g_2(a)= {\frac {24 \beta\, {\Delta}^{3} s\,z \left( a \right) \left( z\left( a \right)^{2} +\beta\,{\Delta}^{2}{a}^{2} \right) }{ \left( 2+ \Delta \right) \left( 3\, z \left( a \right)^{2} -\beta\,{\Delta}^{2}{a}^{2} \right) ^{3}}}\ge 0. $$$$\begin{array}{lll} g_3(a)&=& -\frac{g_2(0)-g_2({\underline{a}}_1)}{{\underline{a}}_1}a+g_2(0)\\ &=&-\frac{4}{3}\,{\frac { \left( \Delta+6\,s-2\,\beta\,s \right) \beta\,s\,}{ \Delta\, \left( 3-\beta \right) \left( \Delta+2 \right) }}\ a+\frac{4}{3}\,{ \frac {s}{\Delta+2}} \ge g_2(a), \end{array}$$$$a+g_3(a)=a -\frac{g_2(0)-g_2({\underline{a}}_1)}{{\underline{a}}_1}a+g_2(0)\ge a+g_2(a)\ge g(a).$$$$g_3(0)=g(0)=\frac{4s}{3(\Delta+2)}\ge 0.$$$$a_0+g_3(a_0)=-{\frac {2 s\, \left[ \left( 3-\beta \right) \left( {\Delta}^{2}+ 2\,\Delta-4\,\beta\,s\,\Delta-8\,\beta\,{s}^{2} \right) -4\,\beta\,s\, \Delta \right] } {3\Delta\,\left( 3-\beta \right) \left( \Delta+2 \right) \left( \Delta+2\,\beta\,s+2 \right) }}, $$$$d_1^o=\frac{1}{2}+{\frac {s\, \left( 2\,{x}^{t}-1 \right) }{\Delta}}+{\frac {p_{{2}} -p_{{1}}}{\Delta}},$$$$p_1-p_2=\varrho\ge \frac{\Delta}{2}+s.$$$$d_1^n=\frac{1}{2}+\frac{1}{2}\,{\frac {\Delta\, \left( p_{{2}}-p_{{1}} \right) }{z(a)}}$$$$\phi(a)=\frac{\Delta}{2}+s-\frac{z(a)}{\Delta},$$$$\begin{array}{lll} \phi(a_0)&=&s+\frac{1}{2}\,\Delta-1-\,{\frac {4 \beta\,{s}^{2}}{ \left(\Delta+ 2\,\beta\,s+2 \right) \Delta}}\\ &>&s+\frac{1}{2}\,\Delta-1-\,{\frac {2 \beta\,{s}}{ \left(\Delta+ 2\,\beta\,s+2 \right)}}\\ &=&\frac{1}{2}\,{\frac {{\Delta}^{2} +2\,s\,\Delta\, \left( 1 +\beta \right)+4\,s\, \left(1+ \beta\,s \right) -8\,\beta\,s-4}{\Delta+2\,\beta\,s+2}}=\hat \phi(a_0). \end{array}$$$$\pi_1^d=p_1d_1^o+\beta V_1(0).$$$$p_1^d=\frac{1}{2}\,s\, \left( 2\,{x}^{t}-1 \right) +\frac{1}{4}\,\Delta+\frac{1}{2}\,p_{{2}}^*,$$$$ \psi(a,x^t)=p_1^d-p_2^*-z(a)/\Delta <0. $$$$p_1^d-p_2^*=\frac{1}{2}\,s\, \left( 2\,{x}^{t}-1 \right) +\frac{1}{4}\,\Delta-\frac{l^*}{2}+\frac{a}{2}(1-x^t)$$$$\hat \psi(a)= \frac{s}{2}+\frac{\Delta}{4}-1-\frac{1}{4}a+\frac{2\beta s}{\Delta}a- {\frac { \left( 2 z(a) -\beta \Delta a \right) \Delta}{{\Delta}^{2}+2 \beta \Delta s+2 z(a) }}.$$$$\bar \psi=\frac{s}{2}+\frac{\Delta}{4}-1+\frac { s} {2({\Delta} +2 \beta s + 2 )} -\frac {2 \Delta} {{\Delta} +2 \beta s + 2 }\ge \hat \psi(a).$$$$-\frac{1}{4}\,{\frac { \left( 2\,z \left( a^* \right) +{\Delta}^{2} \right) \left( 4\, z \left( a^* \right)^{2}-\beta\,{\Delta}^{2 }{a^*}^{2} \right) }{\Delta\, z \left( a^* \right)^{3}}}<0, $$$$l(a)-\tilde n_1=-\frac{3}{2} a+{\frac { \left( 2 z(a) -\beta \Delta a \right) \left(\Delta-2s\right)}{{\Delta}^{2}+2 \beta \Delta s+2 z(a) }}\ge 0,$$$$R(p_2,s)=\frac{\left( 2 z \left( a \right)^{2}-\beta\,{\Delta}^{2}{a}^{2} \right) p_{ {2}}}{{4\, z \left( a \right) ^{2}-\beta\,{\Delta}^{2}{a}^{2}}} +\frac { 2\left( 2\,z \left( a \right) -\beta \Delta a\right) z \left( a \right) ^{2}\Delta} {({\Delta}^{2}+2\,s\, \beta\,\Delta+2\,z \left( a \right) )(4\, z \left( a \right) ^{2}-\beta\,{\Delta}^{2}{a}^{2})}, $$$$\frac{\left( 2 z \left( a \right)^{2}-\beta\,{\Delta}^{2}{a}^{2} \right) }{{4\, z \left( a \right) ^{2}-\beta\,{\Delta}^{2}{a}^{2}}}\ge 0. $$$$\rho_4=-\frac{\sqrt{2} \Delta}{\sqrt{\beta}\Delta-2\sqrt{2}\beta s}.$$$$\label{MCS1a} \frac{\partial}{\partial s}R(p_2,s)=\frac{\frac{\partial^2}{\partial s \partial p_1}\Pi_{1}(p_1,p_2,s)} {-\frac{\partial^2}{\partial {p_1}^2} \Pi_{1}(p_1,p_2,s)}, $$$$\mbox{sign}\bigg[\frac{\partial}{\partial s}R(p_2,s)\bigg]=\mbox{sign}\bigg[{\frac{\partial^2}{\partial s \partial p_1}\Pi_{1}(p_1,p_2,s)} \bigg]. $$$$\begin{array}{lll} \frac{\partial}{\partial p_1}\Pi_{1}(p_1,p_2,s)&=&d_1^o+d_1^y+p_1\left(\frac{\partial d_1^o}{\partial p_1}+ \frac{\partial d_1^y}{\partial p_1}\right) \\ &&+\beta \left(n_1^*+2 n_2^*d_1^y\right)\frac{\partial d_1^y}{\partial p_1} \label{foc11a}. \end{array} $$$$\begin{array}{lll} \frac{\partial^2}{\partial s \partial p_1}\kern-.5pt\Pi_{1}\kern-.5pt(\kern-.5pt p_1,p_2,s\kern-.5pt)&=&\frac{\partial d_1^o}{\partial s}\kern-.5pt+\kern-.5pt \frac{\partial d_1^y}{\partial s}\kern-.5pt+\kern-.5pt \frac{\partial d_1^y}{\partial a} \frac{\partial a}{\partial s} \kern-.5pt+\kern-.5ptp_1\left(\frac{\partial^2 d_1^o}{\partial s \partial p_1}\kern-.5pt+\kern-.5pt \frac{\partial^2 d_1^y}{\partial s\partial p_1}\kern-.5pt+\kern-.5pt \frac{\partial^2 d_1^y}{\partial a\partial p_1}\frac{\partial a}{\partial s}\right) \\ &&+\beta \left(\frac{\partial n_1^*}{\partial s}+\frac{\partial n_1^*}{\partial a}\frac{\partial a}{\partial s}\right) \frac{\partial d_1^y}{\partial p_1}+\beta n_1^*\left(\frac{\partial^2 d_1^y}{\partial s \partial p_1} +\frac{\partial^2 d_1^y}{\partial a \partial p_1}\frac{\partial a}{\partial s}\right) \\ &&+2\beta \kern-1pt\left(\kern-.5pt\frac{\partial n_2^*}{\partial s}\kern-.5pt+\kern-.5pt\frac{\partial n_2^*}{\partial s}\frac{\partial a}{\partial s}\kern-.5pt\right)\kern-.5pt d_1^y\frac{\partial d_1^y}{\partial p_1}\kern-.5pt+\kern-.5pt2\beta n_2^* \kern-1pt\left(\kern-.5pt\frac{\partial d_1^y}{\partial s}\kern-.8pt+\kern-.8pt \frac{\partial d_1^y}{\partial a}\frac{\partial a}{\partial s}\kern-.5pt\right)\kern-1pt\frac{\partial d_1^y}{\partial p_1} \\ &&+2\beta n_2^* d_1^y \left(\frac{\partial^2 d_1^y}{\partial s \partial p_1}+\frac{\partial^2 d_1^y}{\partial a \partial p_1}\frac{\partial a}{\partial s}\right) \label{derp1s}. \end{array} $$$$\frac{\partial a}{\partial s}=\frac{-\frac{\partial g(a)}{\partial s}}{\frac{\partial g(a)}{\partial a}}\doteq -\frac{4}{3(\Delta +2)}.$$$$ {\ensuremath{\frac{\partial {d_1^o}}{\partial {s}}}}=\frac{2x^t-1}{\Delta}\doteq 0 \qquad {\ensuremath{\frac{\partial^2 {d_1^o}}{\partial {s} \partial {p_1}}}}=0\doteq 0. $$$$\begin{array}{lll} {\ensuremath{\frac{\partial {d_1^o}}{\partial {s}}}}=-\frac{\beta \Delta a(p_1-p_2)}{z(a)^2} \doteq 0 &\qquad& {\ensuremath{\frac{\partial{d_1^y}}{\partial {a}}}}=-\frac{\beta \Delta s (p_1-p_2)}{z(a)^2} \doteq 0\\ {\ensuremath{\frac{\partial^2 {d_1^y}}{\partial {s} \partial {p_1}}}}=\frac{\beta \Delta a}{z(a)^2} \doteq 0 &\qquad& {\ensuremath{\frac{\partial^2 {d_1^y}}{\partial {a} \partial {p_1}}}}=\frac{\beta \Delta s}{z(a)^2} \doteq 0 \end{array}$$$${\ensuremath{\frac{\partial {d_1^y}}{\partial {p_1}}}}=-\frac{\Delta}{z(a)}\doteq -\frac{1}{2}.$$$$ {\ensuremath{\frac{\partial {n_2^*}}{\partial {s}}}}=\frac{\beta \Delta a^3}{z(a)^2}\doteq 0 \qquad {\ensuremath{\frac{\partial {n_2^*}}{\partial {a}}}}=\frac{2a}{\Delta}+\frac{\Delta a^2}{2z(a)} \doteq 0. $$$$\begin{array}{lll} {\ensuremath{\frac{\partial {n_1^*}}{\partial {s}}}}&=&-{\ensuremath{\frac{\partial {n_2^*}}{\partial {s}}}}+2\,{\frac { \left( 2\,z \left( a \right) -\beta\,a\Delta \right) \Delta\, \left( 2+\Delta \right) }{ \left( {\Delta}^{2}+2\,\beta\,s\, \Delta+2\,z \left( a \right) \right) ^{2}}}-8\,{\frac {\beta\,s\,a}{{ \Delta}^{2}+2\,\beta\,s\,\Delta+2\,z \left( a \right) }},\\ &=&\frac{4}{\Delta+2}, \end{array}$$$$\begin{array}{lll} {\ensuremath{\frac{\partial {n_1^*}}{\partial {a}}}}&=&-{\ensuremath{\frac{\partial {n_2^*}}{\partial {a}}}}+1-2\,{\frac {\beta\,s\,\Delta\, \left( \left( 4\,s+\Delta \right) \left( \Delta+2\,\beta\,s \right) +2\,\Delta \right) }{ \left( { \Delta}^{2}+2\,\beta\,s\,\Delta+2\,z \left( a \right) \right) ^{2}}} ,\\ &=&1. \end{array}$$$$\begin{array}{lll} \frac{\partial^2}{\partial s \partial p_1}\Pi_{1}(p_1,p_2,s)&=&\beta (\frac{\partial n_1^*}{\partial s}+ \frac{\partial n_1^*}{\partial a}\frac{\partial a}{\partial s}) \frac{\partial d_1^y}{\partial p_1}\\ &=&\beta\left(\frac{4}{\Delta+2}+(1)\left(-\frac{4}{3(\Delta +2)}\right)\right)\left(-\frac{1}{2}\right)\\ &=&-\frac{4 \beta}{3(\Delta+2)}<0. \end{array}$$

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