Evolution of price-cost margins during the troika intervention

Abstract

Market competition is a key driver for an efficient allocation of resources and thus to sustained economic growth and higher aggregate welfare. Portugal implemented significant policy reforms during the difficult period 2010-2016 to improve the level of competition and flexibility in the product and labour markets. This paper measures price-cost margins in 190 markets during these years and the results present a stability pattern while providing evidence of imperfectly competitive markets. This seems to indicate that both the market power of firms and competition did not improve significantly during the period, although there was a reduction in the mark-ups of some non-manufacturing sectors, such as construction and other services. In addition, there was a sizeable decrease of the estimated parameter for labour market frictions in some services sectors, which may be interpreted as the policy reforms leading to a reduction in their workers’ bargaining power.

Appendix

1.1 Derivation of the underlying model

1.1.1 Price-cost margin estimation

We start by assuming a standard neoclassical production function:

$$\begin{aligned} Q=A\textit{f}\left( {K,L,M}\right) \end{aligned}$$

(3)

where Q stands for real output, A is a technological parameter and K,L and M represent capital, labour and intermediate inputs. Additionally, we assume Hicks-neutral technological progress and proceed to the logarithmic differentiation of the production function²⁰ obtaining:

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

(4)

where $\theta$ stands for technological progress, q is the log of output, k,l and m are the logs of inputs and $\epsilon ^k$,$\epsilon ^l$ and $\epsilon ^m$ are output elasticities with respect to capital, labour and intermediate inputs, respectively. Profit maximizing firms operating in competitive output and input markets implies that there is no market power and, thus, the marginal productivity of each input can be replaced by its corresponding price. As a result, output elasticities with respect to each input and the corresponding input shares in nominal output are equivalent:

$$\begin{aligned} \epsilon ^J \equiv \frac{\partial {Q}}{\partial {J}}\frac{J}{Q}= \frac{P_{J}J}{PQ} \equiv \alpha ^J \end{aligned}$$

(5)

where P stands for the output deflator, ${P_{J}}$ is the deflator of input J=K,L and M. Under constant returns to scale, $\epsilon ^K+\epsilon ^L+\epsilon ^M=1$ and assuming a perfect competitive output market where firms aim to maximize profits the (primal) Solow Residual is obtained:

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

(6)

These assumptions imply that the (primal) Solow Residual (SR) is exactly equal to the technological progress parameter. By relaxing the hypothesis of perfect competition in the output market, the SR no longer accurately captures technological progress since output elasticities in the presence of market power are not equivalent to the corresponding market shares. Instead, output elasticities become $\epsilon ^J=\mu \alpha ^J$, where $\mu$ is the mark-up ratio. Using the expression to substitute the output elasticities in the growth accounting equation we obtain:

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

(7)

Additionally, if we assume constant returns to scale $\left( \epsilon ^K+\epsilon ^L+\epsilon ^M\right) =1$, the Solow residual can be rewritten as follows:

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

(8)

The above equation would enable us to obtain the classical price-cost margin from the estimate of the parameter $\left( 1-\frac{1}{\mu }\right)$ which is equivalent to the Lerner index $1-\frac{1}{\mu }=\frac{(P-MC)}{P}$, where P and MC represent the price and marginal cost, respectively. However, the last term in Eq. 8 is unobservable implying that the OLS estimator would be inconsistent given the absence of adequate instruments to solve this problem together with the fact that results tend to be sensitive to the choice of instruments. The use of the firm’s dual optimization problem proposed by Roeger (1995) eliminates the unobserved technological parameter.

Considering the firm’s cost minimization problem (the dual problem) for a given level of output and also using the Shepard’s lemma we obtain the following:

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

(9)

where p represents the log of output price, w, r and $p^m$ are the wages, cost of capital and cost of intermediate inputs, in logarithms. Furthermore, if we assume again imperfect competition in the output market and constant returns to scale, the dual Solow Residual ($SR^d$) is:

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

(10)

Lastly, computing the difference between the primal and the dual Solow residual:

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

(11)

where

$$\begin{aligned} \begin{aligned} SR - SR^d&\equiv \left( \Delta {p}+\Delta {q}\right) -\alpha ^L\left( \Delta {w}+\Delta {l}\right) -\alpha ^M\left( \Delta {p^m}+\Delta {m}\right) - \\&\left( 1-\alpha ^L-\alpha ^M\right) \left( \Delta {r}+\Delta {k}\right) \end{aligned} \end{aligned}$$

(12)

Given the elimination of the unobserved technological progress parameter, the aforementioned inconsistency problem is solved, and the price-cost margin can be consistently estimated by OLS. However, it requires a measure of firms’ cost of capital which is usually subject to some measurement error, implying that the endogeneity problem may still exist in Roeger’s formulation.

1.1.2 Price-cost margin under an imperfect competitive labour market

Thus far, mark-ups were estimated under the hypothesis of perfect competition in the labour market where workers received perfectly competitive wages and, consequently, workers’ bargaining power was absent. However, economic literature shows that the labour market has several significant inefficiencies and there is empirical evidence that mark-ups are significantly underestimated in the context of such assumption. Therefore, the standard approach was modified to account for imperfect competition in the labour market (see e.g. Crépon et al. (2005) and Abraham et al. (2009)). Within an imperfect labour market, we can assume that wages (W) and the number of workers (L) are simultaneously chosen according to a standard efficient bargaining problem which involves sharing the surplus between profit-maximizing firms and workers whose utility comes from employment and wages:

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

(13)

where $\overline{W}$ is the reservation wage (which depends on the alternative wage in the labour market and the unemployment benefits), and $0\le \phi \le 1$ denotes workers’ bargaining power, where a competitive labour market corresponds to $\phi =0$ and $\phi =1$ implies that firm’s surplus is fully transferred to the workers. The first order condition of maximization for L is:

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

(14)

where

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

(15)

In the case of imperfect competition and assuming an isoelastic demand for output $P=Q^{\left( \frac{-1}{\eta }\right) }$, where $\eta$ is the price elasticity of demand, the Lerner Index becomes $\left( 1-\frac{1}{\eta }\right) =\frac{1}{\mu }$. Then, as shown in Amador and Soares (2017), it is possible to derive output elasticity with respect to labour by using the ratio of labour costs on output, and the first order condition stated above:

$$\begin{aligned} \epsilon ^L = \mu \alpha ^L + \mu \frac{\phi }{1-\phi }\left( \alpha ^L-1\right) \end{aligned}$$

(16)

Similarly, the output elasticities with respect to intermediate inputs and capital were adjusted:

$$\begin{aligned} \epsilon ^M = \mu \alpha ^M \end{aligned}$$

(17)

$$\begin{aligned} \epsilon ^K = 1 - \mu \alpha ^M - \mu \alpha ^L - \mu \frac{\phi }{1-\phi }\left( \alpha ^L-1\right) \end{aligned}$$

(18)

Replacing output elasticities in the growth accounting equation and computing the difference between the primal and the dual Solow Residual we obtain the following expression, which includes a new term as a result of the workers’ bargaining power:

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

(19)

where $SR - SR^d$ follows Eq. 12.

The above equation allows us to jointly estimate the mark-up and the workers’ bargaining power. By including the last term accounting for an imperfect labour market, we are able to improve the consistency of our estimates.

1.2 Drescriptive statistics

Table 7 Drescriptive statistics (mean and standard deviation) of the main variables