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Quantitative Marketing and Economics

Erschienen in:

01.12.2014

Who pays for switching costs?

verfasst von: Guy Arie, Paul L. E. Grieco

Erschienen in: Quantitative Marketing and Economics | Ausgabe 4/2014

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Abstract

Earlier work characterized pricing with switching costs as a dilemma between a short-term “harvesting” incentive to increase prices versus a long-term “investing” incentive to decrease prices. This paper shows that small switching costs may reduce firm profits and provide short-term incentives to lower rather than raise prices. We provide a simple expression which characterizes the impact of the introduction of switching costs on prices and profits for a general model. We then explore the impact of switching costs in a variety of specific examples which are special cases of our model. We emphasize the importance of a short term “compensating” effect on switching costs. When consumers switch in equilibrium, firms offset the costs of consumers that are switching into the firm. If switching costs are low, this compensating effect of switching costs causes even myopic firms to decrease prices. The incentive to decrease prices is even stronger for forward looking firms.

Vorheriger Artikel The Joint identification of utility and discount functions from stated choice data: An application to durable goods adoption

Nächster Artikel Economic valuation of product features

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Many empirical studies consider small SC (e.g., Keane 1997; Shy 2002; Shcherbakov 2010; Dubé et al. 2010; Ellickson and Pavlidis2014). We discuss these in more detail below.

Klemperer (1987) considers several alternative two-period models. Of these, the closest to ours is when consumer preferences are uncorrelated between the two periods. In this setting he focuses on symmetric firms and finds that the short term (second period) effect is zero and the first period (investing) effect decreases prices and profits. Our analysis generalizes this result and confirms Klemperer’s conjecture (in the conclusion there) that in asymmetric markets a firms’ reaction depends on its market share.

Pearcy (2014) shows that allowing for consumers to be forward looking can also reverse the qualitative effect of SC on symmetric markets.

Aside from small SC, the main assumption in our analysis is that the consumers’ purchasing decision is myopic. With the exception of Pearcy (2014), this assumption is common to the previous studies mentioned above, and relaxing it is an important avenue for future research.

Roughly this result requires that the firm has less than average market share.

In this paper, we consider so-called “transactional” switching costs, which are most commonly addressed in the literature. In this framework, a cost must be paid each time consumers switch products. Nilssen (1992) contrasts this type of switching costs with “learning” switching costs, where customers may costlessly switch between products they have already learned how to use.

Klemperer (1995) argues that SC are likely to play an important role in many areas of economics, including industrial organization and international trade.

Equivalently, one could normalize utility without loss of generality such a consumer in loyalty group j derives γ less utility from all goods except j.

In particular, this rules out network effects.

It is also possible to analyze alternative laws of motion that might allow for the introduction of new consumers, or experimentation by consumers who might try a new good, but remain loyal to their earlier purchase with some positive probability. Both of these extensions have the effect of reducing firms’ investment incentive, but do not change the qualitative results of our model.

This normalization is for convenience and the model can be easily generalized to accommodate firm-specific marginal costs.

Moreover, there is no reason to believe that restricting ourselves to stationary strategies will ensure a unique equilibrium as firms may detect defections from market shares.

The propositions below formalize dominance using bounds on market shares and HHI.

The exact price effect is derived for a monopolist in Section 3.4.

Note that consumers in group x are less price sensitive if $D_{p\gamma }^{x}\geq 0$.

Similar results for the Salop demand system are provided in Appendix C.

For other demand structures, the interaction between SC and price sensitivity may not cancel when firms are symmetric. For example, switchers may become more price sensitive while the price sensitivity of loyals may not change—e.g., add a term $-\gamma _{p} p_{jt} \mathbf {1}\left [ j\neq k\right ]$ to $\bar {u}$ in Eq. 3.2.

While the proof relies on the quasi-linearity of utility, Appendix C provides a similar result for linear models.

We thank a referee for suggesting this.

It can be shown that with n symmetric market leaders, the same upper bound applies as $J \to \infty $

We focus on the case in which the SC also applies when switching to/from the outside good. Our base case is reasonable for many industries. For example, switching across cable TV, satellite, IP-TV and outside “broadcast” television requires learning the channel layout, acquiring and installing the necessary equipment, and ordering or canceling the relevant services even when switching to or from broadcast. We briefly discuss the alternative where switching to the outside good does not incur SC at the end of this section.

This is illustrated and discussed further using the numerical example in Fig. 1.

Consumer surplus is calculated using compensating variation (CV). In the single firm model this is:

$$CV=-\frac{1}{\alpha}\left( \log\left( e^{\delta_{0}}+e^{\delta_{1} +\alpha\cdot p-\gamma}\right) -\delta_{0}\right). $$

For high switching-cost values, this method is actually a generous calculation as it assumes that all consumers can stay with the outside good “for free.” The main alternative, expenditure variation, (EV) would actually be negative for higher SC as loyal consumers are paying a very high price to keep themselves from switching.

Other symmetric specifications do not add qualitative insight. Specifically, in a symmetric duopoly, no firm can dominate the market regardless of δ. Thus, increasing δ ₁ and δ ₂ provides no additional insight and is not presented

We use value function iteration to approximate symmetric MPEs. The Mathematica software code is available from the authors. Our simulation used a 361 point grid for the state (share) space and process stops when the change in the value over all states is smaller than a threshold, which was set at 10⁻⁶. We verify that the optimal policy is unique by verifying that the best response function is quasi-concave and use the optimal pricing strategy to “play out” the equilibrium strategies starting at various states until a steady state is obtained. A unique steady state was obtained for all parameter values that we consider. For higher SC values, we do not expect a steady state exists as firms profit from ’invest then harvest’ cycles that take advantage of consumer myopia. Indeed, our equilibrium for high SC (roughly γ>2.7) did not converge to a steady state.

For expositional simplicity, we subtract 𝜖 in this section rather than add it as in Eq. 3.1. This is without loss of generality.

To see this note that consumers purchase the firm’s product if it offers the higher utility, that is the loyal consumer’s purchase if δ−α⋅p−𝜖 _i>−γ and non-loyal consumers purchase if δ−α⋅p−γ−𝜖 _i>0. Integrating these expressions over 𝜖 _i leads to the demand equation.

Under the quasi-linear model, the assumption is equivalent to log concavity in Φ, i.e., $({\Phi }^{\prime })^{2}\ge {\Phi } \cdot {\Phi }^{\prime \prime }$.

An older version of the paper derived the exact value for $\frac {dp_{1}}{d\gamma }$:

$$ \frac{dp_{1}}{d\gamma=0} =-\frac{F^{1}_{\gamma}}{2D^{1}_{p_{1}} +p_{1} \cdot D^{1}_{p_{1} p_{1}} } $$

However, this does not provide any additional insights; the proof is available from the authors. Note that the denominator must be negative by the standard second order condition.

We use steady-state non-SC market shares as a convenient proxy for the underlying firm quality

In a logit setting this requires a quality difference of five, which is over three standard deviations above the mean in the idiosyncratic term. In other words, if the firm would sell at the outside good price, it would have a share of over 0.95.

We have experimented with alternative specifications and found the results to be robust and indicative of the key insights. The upper bound of $\bar {\gamma }=2$ reflects the upper bound for the steady state to exist. Analysis available from the authors shows that the true upper bound is slightly above $\bar {\gamma }=2$ and decreases with δ. Reducing δ (i.e. making the market more competitive) allows increasing $\bar {\gamma }$

Here, because all price-setting firms are symmetric and the game is in strategic complements, can drop the strategic interaction term because we know it will only re-enforce the incentives of the leading terms.

The numerator for the derivative of the RHS wrt s _J is

$$\begin{array}{@{}rcl@{}} \left(1-6s_{J}\right)\left(1+3{s_{J}^{2}}\right)-6s_{J}\left(s_{J}-3{s_{J}^{2}}\right) &=&1-6s_{J}+3{s_{J}^{2}}-18{s_{J}^{3}}-6{s_{J}^{2}}+18{s_{J}^{3}}\\ &=&1-6s_{J}-3{s_{J}^{2}}. \end{array} $$

This quadratic expression is positive at s _J=0 and negative at $s_{J}=\frac {1}{3}$ (the upper bound). So need to take the maximizing point,

$$s_{J}=\frac{-6\pm\sqrt{36+12}}{6}=-1+\frac{4\sqrt{3}}{6}=-1+\frac{2}{\sqrt{3}}\;. $$

By first-order effect we mean that the analysis does not consider the firm’s reaction to it’s rival price change.

A similar analysis can be applied to other models of linear demand, but the qualitative results are the same. In particular, both Rhodes (2014) and Shin et al. (2009) consider a Hotelling model. The Salop model was chosen to highlight the additional endogenous asymmetry caused by SC when there are more than two firms in the market.

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Titel: Who pays for switching costs?
verfasst von: Guy Arie
Paul L. E. Grieco
Publikationsdatum: 01.12.2014
Verlag: Springer US
Erschienen in: Quantitative Marketing and Economics / Ausgabe 4/2014
Print ISSN: 1570-7156
Elektronische ISSN: 1573-711X
DOI: https://doi.org/10.1007/s11129-014-9151-9