Markups and bargaining power in tradable and non-tradable sectors

Abstract

This article jointly estimates product and labour market imperfections for narrowly defined sectors in the Portuguese economy for the period 2006–2009, following Roeger (J Polit Econ 103(2):316–330, 1995), Crépon et al. (Ann Econ Stat (79/80):583–610, 2005), Dobbelaere (Int J Ind Organ 22(10):1381–1398, 2004) and Abraham et al. (Rev World Econ 145(1):13–36, 2009). In addition, we propose a criterion for the identification of tradable and non-tradable sectors based on the export-to-sales ratio and compare markups and workers’ bargaining power along this dimension. Our findings suggest that markups are higher in the non-tradable than in the tradable sector but workers’ bargaining power is similar. In addition, there is a significant level of heterogeneity across markets, particularly in the non-tradable sector. Moreover, the article confirms that, if labour market imperfections are disregarded, markups are significantly understated.

Appendices

Appendix 1: Derivation of the underlying model

A standard neoclassical production function for firm i in period t is given by⁸:

$$\begin{aligned} Q=Af(K,L,M) \end{aligned}$$

(4)

where Q stands for real output, A is a technological parameter and K, L and M represent capital, labour and intermediate inputs, respectively. Assuming Hicks-neutral technological progress, the logarithmic differentiation of the production function yields:

$$\begin{aligned} \varDelta {q} = \varepsilon ^{K}\varDelta {k}+\varepsilon ^{L}\varDelta {l}+\varepsilon ^{M}\varDelta {m}+\theta \end{aligned}$$

(5)

where $\theta$ stands for technological progress and $\varepsilon ^{K}$, $\varepsilon ^{L}$ and $\varepsilon ^{M}$ are output elasticities with respect to capital, labour and intermediate inputs, respectively. Under perfect competition, output elasticities with respect to each input match corresponding shares in nominal output, that is:

$$\begin{aligned} \varepsilon ^{\textit{J}} \equiv {\frac{\partial \textit{Q}}{\partial \textit{J}}}\frac{\textit{J}}{\textit{Q}}=\frac{\textit{PJ}^\textit{J}}{\textit{PQ}}\equiv \alpha ^{J} \end{aligned}$$

(6)

where P stands for the deflator of output, $P^J$ are input prices of $J=K,L,M$. Assuming constant returns-to-scale, $(\varepsilon ^{K}+\varepsilon ^{L}+\varepsilon ^{M}=1)$, the Solow (1957) residual (SR) is:

$$\begin{aligned} \mathrm{SR}\equiv \varDelta {q}-(1-\alpha ^{L}-\alpha ^{M})\varDelta {k} -\alpha ^{L}\varDelta {l}-\alpha ^{M}\varDelta {m}=\theta \end{aligned}$$

(7)

Under the assumptions of Hicks-neutral technological progress, constant returns-to-scale and perfect competition in input and outputs market, the Solow residual corresponds exactly to the technological progress. Nevertheless, this is no longer the case if there is imperfect competition in the output market. In this setting, output elasticities become ${\varepsilon }^{J}= {\mu }{\alpha }^{J}$. Replacing the modified output elasticities for each input in the growth accounting equation yields:

$$\begin{aligned} \varDelta {q}=\mu (\alpha ^{L}\varDelta {l}+\alpha ^{K}\varDelta {k}+\alpha ^{M}\varDelta {m})+\theta \end{aligned}$$

(8)

Assuming constant returns-to-scale $(\alpha ^{K}+\alpha ^{L}+\alpha ^{M})\mu =1$, the Solow residual can be rewritten as:

$$\begin{aligned} \mathrm{SR}=\left( 1-\frac{1}{\mu }\right) (\varDelta {q}-\varDelta {k})+\frac{1}{\mu }\theta \end{aligned}$$

(9)

Therefore, the classical price-cost margin can be obtained from the estimate of the parameter $\left( 1-1/\mu \right)$ in Eq. 9. This parameter corresponds to the Lerner index defined as $(P- MgC)/P$ where P and MgC represent the price and marginal cost, respectively. However, technological progress is not observed, and it is likely to be correlated with the input growth rates, which implies that the OLS estimator is inconsistent.

In this context, Roeger (1995) proposed an alternative approach that considers the firm’s dual optimization problem, i.e., the cost minimization for a given level of output. The first order conditions, along with Shepard’s lemma and assuming that markups are constant, yield:

$$\begin{aligned} \varDelta {p} = \varepsilon ^{L}\varDelta {w}+\varepsilon ^{K}\varDelta {r}+\varepsilon ^{M}\varDelta {p^{m}}-\theta \end{aligned}$$

(10)

Assuming perfectly competitive output markets, production and cost shares coincide and the equation above can be rewritten as:

$$\begin{aligned} \varDelta {p} = \alpha ^{L}\varDelta {w}+\alpha ^{K}\varDelta {r}+\alpha ^{M}\varDelta {p^{m}}-\theta \end{aligned}$$

(11)

Under competitive markets, the dual Solow residual ($\mathrm{SR}^d$) matches the technological progress as in the primal problem of the firm:

$$\begin{aligned} -\mathrm{SR}^d\equiv \varDelta {p}-\alpha ^{L}\varDelta {w}-\alpha ^{K}\varDelta {r}-\alpha ^{M} \varDelta {p^{m}}=\theta \end{aligned}$$

(12)

However, under imperfect competition, the dual Solow residual becomes:

$$\begin{aligned} -\mathrm{SR}^d = \left( 1-\frac{1}{\mu }\right) (\varDelta {p}-\varDelta {r})-\frac{1}{\mu }\theta \end{aligned}$$

(13)

Assuming imperfect competition in the output market and constant returns-to-scale, the $\mathrm{SR}^d$ is written as:

$$\begin{aligned} -\mathrm{SR}^d\equiv \varDelta {p}-\alpha ^{L}\varDelta {w}-\alpha ^{K}\varDelta {r}-\alpha ^{M} \varDelta {p^{m}}=\left( 1-\frac{1}{\mu }\right) (\varDelta {p}-\varDelta {r})-\frac{1}{\mu }\theta \end{aligned}$$

(14)

Finally, adding the Solow residuals under primal and dual approaches (Eqs. 9 and 14), it is possible to write:

$$\begin{aligned} \mathrm{SR}-\mathrm{SR}^d=\left( 1-\frac{1}{\mu }\right) \left[ (\varDelta p+ \varDelta q)-(\varDelta r+ \varDelta k)\right] + u \end{aligned}$$

(15)

where $\mathrm{SR}-\mathrm{SR}^d\equiv (\varDelta p+\varDelta q)-\alpha ^{L}(\varDelta w+\varDelta l)-\alpha ^{M}(\varDelta p^m+\varDelta m)-(1-\alpha ^{M}-\alpha ^{L})(\varDelta r+\varDelta k)$.

In Eq. (15), the term generating the endogeneity problem referred above is eliminated and OLS allows for a consistent identification of the markup ratio.

There is a large body of empirical evidence arguing that labour markets are not perfectly competitive. Under an efficient Nash bargaining negotiation, wages (W) and the number of workers (L) are simultaneously chosen. Following McDonald and Solow (1981):

$$\begin{aligned} \max _{L,W} \Omega =\left[ (W-\overline{W})L\right] ^{\phi }({\textit{PQ}}-{\textit{WL}})^{(1-\phi )} \end{aligned}$$

(16)

where $\overline{W}$ is the reservation wage and $1\ge \phi \ge 0$ represents the workers’ bargaining power, with $\phi =0$ corresponding to competitive labour markets (right-to-manage model) and $\phi =1$ to a total appropriation of the firm’s surplus by the workers.⁹ The first order conditions with respect to W and L are, respectively:

$$\begin{aligned} W= & {} (1-\phi )\overline{W} + \phi \frac{{\textit{PQ}}}{L} \end{aligned}$$

(17)

$$\begin{aligned} W= & {} (1-\phi )\frac{\partial ({\textit{PQ}})}{\partial L}+\phi \frac{{\textit{PQ}}}{L} \end{aligned}$$

(18)

where:

$$\begin{aligned} \frac{\partial ({\textit{PQ}})}{\partial {\textit{L}}} = \frac{\partial {\textit{Q}}}{\partial L}\left[ \frac{\partial P}{\partial Q} Q+P\right] = \frac{{\textit{P}}}{\mu }\frac{\partial {\textit{Q}}}{\partial {\textit{L}}} \end{aligned}$$

(19)

The two conditions above determine de contract curve $\partial ({\textit{PQ}})/\partial L=\overline{W}$, which does not depend on the negotiated wage, i.e., the contract curve under a Nash bargaining negotiation is vertical. In addition, any combination of W and L in the contract curve are above the labour demand. By rewriting Eq. (18), it possible to see that the true market power of the firm is evaluated at the reservation wage and not at the negotiated wage:

$$\begin{aligned} \mu = \frac{{\textit{P}}}{\overline{W}}/\left( \partial {{\textit{Q}}}/{\partial {\textit{L}}}\right) \end{aligned}$$

(20)

Assuming imperfect competition and an isoelastic output demand ${\textit{P}}={\textit{Q}}^{-\frac{1}{\eta }}$, where $\eta$ is the price elasticity of demand, then $1/\eta$ is the Lerner index and $(1-1/\eta )=1/\mu$. Next, using the ratio of labour costs on output and Eq. (18), it is possible to write:

$$\begin{aligned} \frac{{\textit{WL}}}{{\textit{PQ}}}= \frac{{\textit{L}}}{{\textit{PQ}}}\left[ (1-\phi )\frac{{\textit{P}}}{\mu }\frac{\partial {\textit{Q}}}{\partial {\textit{L}}}+\phi \frac{{\textit{PQ}}}{{\textit{L}}}\right] \end{aligned}$$

(21)

Therefore, the adjusted output elasticities with respect to labour, intermediate inputs and capital become, respectively:

$$\begin{aligned}&\displaystyle \varepsilon ^{L}=\mu \alpha ^{L}+\mu \frac{\phi }{1-\phi }(\alpha ^{L}-1) \end{aligned}$$

(22)

$$\begin{aligned}&\displaystyle \varepsilon ^{M}={\mu }\alpha ^{M} \end{aligned}$$

(23)

$$\begin{aligned}&\displaystyle \varepsilon ^{K}=1-{\mu }\alpha ^{M}-{\mu }\alpha ^{L}-\mu \frac{\phi }{1-\phi }(\alpha ^{L}-1) \end{aligned}$$

(24)

Considering the adjusted output elasticities, the primal and dual Solow residual become, respectively:

$$\begin{aligned} \mathrm{SR}= & {} \left( 1-\frac{1}{\mu }\right) (\varDelta q-\varDelta k)+\left( \frac{\phi }{1-\phi }\right) (\alpha ^{L}-1)\left[ \varDelta l-\varDelta k)\right] +\frac{1}{\mu }\theta \qquad \end{aligned}$$

(25)

$$\begin{aligned} -\mathrm{SR}^d= & {} \left( 1-\frac{1}{\mu }\right) (\varDelta p-\varDelta r)+\left( \frac{\phi }{1-\phi }\right) (\alpha ^{L}-1)\left[ \varDelta w-\varDelta r)\right] -\frac{1}{\mu }\theta \qquad \end{aligned}$$

(26)

Therefore, allowing for imperfect competition in the labour market and under constant returns-to-scale, the modified Roeger (1995) approach is:

$$\begin{aligned} \mathrm{SR}-\mathrm{SR}^d= & {} \left( 1-\frac{1}{\mu }\right) [(\varDelta p+\varDelta q)-(\varDelta r+ \varDelta k)]\nonumber \\&+\,\frac{\phi }{(1-\phi )}(\alpha ^{L}-1)[(\varDelta l+\varDelta w)-(\varDelta r+\varDelta k)] + u\quad \quad \end{aligned}$$

(27)

Appendix 2: Detail on tradable and non-tradable sectors

See Table 4.

Table 4 Classification of markets within each 1 and 2 digit NACE industry

Appendix 3: Robustness analysis

See Tables 5–8.

Table 5 Average price-cost margin under imperfect labour markets (%)

Table 6 Average workers’ bargaining power (%)

Table 7 Average price-cost margin under perfect labour markets (%)

Table 8 Average price-cost margin bias by assuming competitive labour markets (p.p.)