Further results on monopolistic competition, markup pricing and the business cycle in Switzerland

Abstract

This paper investigates how firms’ market power affects the price level. Based on a small macro-model it is shown empirically that firms have structural markup pricing power and take advantage of favourable business cycle fluctuations. To this aim, a multivariate time series model with double integrated variables is estimated. Thereby a model-based business cycle indicator can be derived. Its information content is confronted with survey data giving rise to what is going to be called semantic cross validation approach.

Appendix

An extension of the basic economic model

We need to show that inflation can be linked to the markup. Define P _t =ωP _t ^m+(1−ω)P _t ^c with 0≤ω≤1 and P _t ^m, P _t ^c are the prices in the monopolistic and the perfect competition sector of the economy, respectively. The procedure is as follows. It will be shown that the demand elasticity in the monopolistic sector is affected by changes in P _t ^c. By assumption the firms in the competitive sector are price takers and hence are not affected by P _t ^m.¹¹ Then, given that the same absolute change in the elasticity has a different impact on profits depending on the sign of the change, it is argued that the expected profit is maximised by choosing a price level that deviates from the one given in Eq. 2. Thus, the first and second derivatives of profits with respect to the demand elasticity need to be calculated and the two situations, one positive and the other with negative expectation errors are to be compared to one another. Finally, since P _t is a weighted average of P _t ^m and P _t ^c and labour is assumed to have the same price everywhere in the economy it can be argued that the price setting behaviour of the monopolistic sector is reflected in the economy wide aggregates of prices and wages as it is done in the main text.

The demand elasticity and P_t ^c

First, the behavioral equations are re-stated:

$$\begin{array}{*{20}c} {{\text{production}}}{Q_{t} = Q{\left( {L_{t} } \right)},}{\frac{{\partial Q}}{{\partial L_{t} }} > 0,} \\ {{\text{demand}}}{P^{m}_{t} = P^{m} {\left( {Q{\left( {L_{t} } \right)},P^{c}_{t} } \right)},}{\frac{{\partial P^{m} }}{{\partial L_{t} }} = \frac{{\partial P^{m} }}{{\partial Q}}\frac{{\partial Q}}{{\partial L_{t} }},} \\ {{\text{inverse demand}}}{Q_{t} = Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)},}{\frac{{\partial Q}}{{\partial P^{m}_{t} }} < 0,\;\frac{{\partial Q}}{{\partial P^{c}_{t} }} \gtreqless 0,} \\ {{\text{demand elasticity}}}{\eta _{t} = \eta {\left( {P^{m}_{t} ,Q_{t} } \right)},}{\eta ^{{ - 1}}_{t} , = \frac{{\partial P^{m} }}{{\partial Q}}\frac{{Q_{t} }}{{P^{m}_{t} }}\;{\text{with}}\;\eta _{t} < - 1.} \\ \end{array} $$

The optimal solution conditional on P _t ^c is equivalent to Eq. 2:

$$ P^{m}_{t} = {\left( {1 + \eta ^{{ - 1}}_{t} } \right)}^{{ - 1}} a_{1} W^{{a_{2} }}_{t} \tau ^{{ - a_{3} }}_{t} , $$

(15)

We need to know the impact of P _t ^c on η _t :

$$ \begin{aligned} \eta _{t} = \frac{{\partial Q}} {{\partial P^{m} }}\frac{{P^{m}_{t} }} {{Q_{t} }} \\ = P^{m} {\left( {Q_{t} ,P^{c}_{t} } \right)}\frac{{\partial Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}} {{\partial P^{m} {\left( {Q_{t} ,P^{c}_{t} } \right)}Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}}, \\ \end{aligned} $$

and hence,

$$ \begin{aligned} \frac{{\partial \eta }} {{\partial P^{c} }} = \frac{{\partial P^{m} {\left( {P^{c}_{t} } \right)}}} {{\partial P^{c} }}\frac{{\partial Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}} {{\partial P^{m} {\left( {Q_{t} ,P^{c}_{t} } \right)}Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}} + P^{m} {\left( {P^{c}_{t} } \right)}\frac{{\partial {\left( {\frac{{\partial Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}} {{\partial P^{m} {\left( {P^{c}_{t} } \right)}Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}}} \right)}}} {{\partial P^{c} }} \\ = \frac{1} {{Q_{t} }}\frac{{\partial P^{m} }} {{\partial P^{c} }}\frac{{\partial Q}} {{\partial P^{m} }} + P^{m} {\left( {P^{c}_{t} } \right)}\frac{{\partial {\left( {\frac{{\partial Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}} {{\partial P^{m} {\left( {P^{c}_{t} } \right)}}}\frac{1} {{Q{\left( {P^{m} {\left( {P^{c}_{t} } \right)}} \right)}}}} \right)}}} {{\partial P^{c} }} \\ = \frac{1} {{Q_{t} }}{\left[ {\frac{{\partial Q}} {{\partial P^{c} }} + P^{m} {\left( {P^{c}_{t} } \right)}\frac{{\partial ^{2} Q}} {{\partial P^{m} \partial P^{c} }} - \frac{{P^{m} {\left( {P^{c}_{t} } \right)}}} {{Q_{t} }}\frac{{\partial Q}} {{\partial P^{m} }}\frac{{\partial Q}} {{\partial P^{c} }}} \right]} \\ = \frac{1} {{Q_{t} }}{\left[ {\frac{{\partial Q}} {{\partial P^{c} }} + P^{m} {\left( {P^{c}_{t} } \right)}\frac{{\partial ^{2} Q}} {{\partial P^{m} \partial P^{c} }} - \eta _{t} \frac{{\partial Q}} {{\partial P^{c} }}} \right]} \\ = \frac{1} {{Q_{t} }}{\left[ {\frac{{\partial Q}} {{\partial P^{c} }}{\left( {1 - \eta _{t} } \right)} + P^{m} {\left( {P^{c}_{t} } \right)}\frac{{\partial ^{2} Q}} {{\partial P^{m} \partial P^{c} }}} \right]}. \\ \end{aligned} $$

(16)

Profit maximisation with perfect foresight

Next, we need to show that profits negatively depend on η _t . This is fairly simple and can be confirmed in any textbook. It is nevertheless given here again for completeness of the exposition. In Eq. 2 the profit maximising price is given in terms of the elasticity. Taking first and second derivatives with respect to η _t produces:

$$ \frac{{\partial P^{m} }} {{\partial \eta }} = \frac{1} {{{\left( {1 + \eta _{t} } \right)}^{2} }} > 0 $$

(17)

$$ \frac{{\partial ^{2} P^{m} }} {{{\left( {\partial \eta } \right)}^{2} }} = \frac{{ - 2}} {{{\left( {1 + \eta _{t} } \right)}^{3} }} > 0 $$

(18)

due to η _t <−1. This result does not change when we are looking at the variables in logs and therefore, the arguments hold also for the log–linear econometric model.

Maximising expected profit

After aggregating the results 16–18 we can evaluate what firms do when they do not know the realisation of P _t ^c at the time they are planning production for the period t. Naturally, firms will formulate expectations at t−1 conditional on the information set I _t−1 about P _t ^c which we denote E _t−1(P _t ^c). Based on this value, the optimal price setting is given by Eq. 2. However, even if E _t−1 (P _t ^c−E _t−1(P _t ^c))=0, firms have an incentive to deviate from Eq. 2 if the variance of the expectation error, is sufficiently large. To see this, notice that in the neighborhood of the optimal price according to Eq. 2 changes in the elasticity will have asymmetric effects on the profits. When demand becomes less elastic (Δη _t >0), then profits will rise more than they would fall if demand would turn more elastic by the same absolute value. From Eq. 16 we know that deviations of P _t ^c from its expected value can be regarded as expectation errors of η _t which we denote by ϱ _t . Therefore, applying Eq. 18 and using a second order Taylor approximation of P ^m around ϱ _t =0 gives rise to

$$ P^{m} {\left( {\varrho _{t} } \right)} = P^{m} {\left( {E_{{t - 1}} {\left( {\eta _{t} } \right)}} \right)} + \frac{{\partial P^{m} }} {{\partial \eta }}\varrho _{t} + \frac{1} {2}\frac{{\partial ^{2} P^{m} }} {{{\left( {\partial \eta } \right)}^{2} }}\varrho ^{2}_{t} . $$

After taking expectations we obtain

$$ \begin{aligned} E_{{t - 1}} {\left( {P^{m} {\left( {\varrho _{t} } \right)}} \right)} = P^{m} {\left( {E_{{t - 1}} {\left( {\eta _{t} } \right)}} \right)} + \frac{{\partial P^{m} }} {{\partial \eta }}E_{{t - 1}} {\left( {\varrho _{t} } \right)} + \frac{1} {2}\frac{{\partial ^{2} P^{m} }} {{{\left( {\partial \eta } \right)}^{2} }}E_{{t - 1}} {\left( {\varrho ^{2}_{t} } \right)} \\ = P^{m} {\left( {E_{{t - 1}} {\left( {\eta _{t} } \right)}} \right)} + \frac{1} {2}\frac{{\partial ^{2} P^{m} }} {{{\left( {\partial \eta } \right)}^{2} }}Var{\left( {\varrho _{t} \left| {I_{{t - 1}} } \right.} \right)}. \\ \end{aligned} $$

(19)

Thus, when considering the uncertainty about P _t ^c the optimal price deviates from the solution in Eq. 2 by $\frac{1}{2}\frac{{\partial ^{2} P^{m} }}{{{\left( {\partial \eta } \right)}^{2} }}Var{\left( {\left. {\varrho _{t} } \right|I_{{t - 1}} } \right)}$ which is according to Eq. 16 linear in the variance of P _t ^c and therefore related to inflation. In other words, independent of the reasons for variations in P _t ^c firms will have an incentive to set prices above the solution in Eq. 2 which further enhances inflation. The next paragraph will establish the relation between inflation and the markup.

Summary of the empirical hypotheses

From Eq. 18 follows that changes in the markup trigger changes of the price level in the same direction. Shocks to the markup may arise in P _t ^c and can have a positive, a negative, or no impact on the markup as Eq. 16 implies. However, irrespective of whether P _t ^c has risen or fallen and whether uncertainty is allowed for or not, the induced variation in η _t means a larger value for P _t ^m than it would have been observed without a shock to P _t ^c (see Eq. 19). Moreover, the larger the variation in the shock to P _t ^c, and hence inflation, the larger the variance of η _t and the larger the increase in P _t ^m.

There are thus two channels by which inflation and the markup are linked. The first is the variance of shocks to η _t which always leads to a rise in P _t ^m. Since changes in the overall price level, namely due to changes in P _t ^c may be responsible for variations in η _t inflation will again result. Taken together, this implies a positive relation between inflation and the markup.

The second channel is provided by Eq. 16. For $\frac{{\partial \eta }} {{\partial P^{c} }} < 0$ initial changes in the overall price level will lead to a decline in η _t and hence in P _t ^m. Thus, inflationary impulses are not reinforced but muted. Naturally, for $\frac{{\partial \eta }} {{\partial P^{c} }} > 0$ the opposite holds.

The actual relationship between inflation and the markup, is therefore given by the relative importance of the two channels and the actual sign of $\frac{{\partial \eta }} {{\partial P^{c} }}$. In fact, there may be a positive relation, a negative, or no relation at all. In the empirical model these possibilities are represented by γ<0, γ>0, or γ=0, respectively. We do not engage in further scrutinising the theoretically correct sign of this value but leave it as an empirical question.

Auxiliary analyses

Table 4 Univariate unit root tests