Misallocation, productivity and fragmentation of production: the case of Latvia

Abstract

This paper evaluates misallocation of resources in Latvia during 2007–2014 using firm-level data. I find that misallocation of resources increased before 2010 and declined afterwards. Initially, output distortion was the major source of misallocation, while the importance of capital distortions increased after the financial crisis. Determinants of changes in allocation efficiency may include growing competition in domestic markets, tighter credit supply and legal issues. However, I show that fragmentation of production induces bias to the estimates of firm-specific distortions, leading to the overestimation of gains from reallocation. Thus, in the absence of inter-firm trade data, the conclusions on misallocation should be treated with caution.

Appendices

Appendix A: Two-stage production framework

A.1 Both stages of production performed in one firm

Production process includes two stages and both stages occurs in one firm. The production function looks the following way:

$$\begin{array}{*{20}{l}}Y_{si} = & A_{2,si}K_{2,si}^{\alpha _{2,s}}L_{2,si}^{\beta _{2,s}}M_{2,si}^{1 - \alpha _{2,s} - \beta _{2,s}}{\mathrm{ = }}A_{2,si}K_{2,si}^{\alpha _{2,s}}L_{2,si}^{\beta _{2,s}}\\ &\times\left( {A_{1,si}K_{1,si}^{\alpha _{1,s}}L_{1,si}^{\beta _{1,s}}M_{1,si}^{1 - \alpha _{1,s} - \beta _{1,s}}} \right)^{1 - \alpha _{2,s} - \beta _{2,s}},\end{array}$$

(A1)

where $M_{2,si}{\mathrm{ = }}A_{1,si}K_{1,si}^{\alpha _{1,s}}L_{1,si}^{\beta _{1,s}}M_{1,si}^{1 - \alpha _{1,s} - \beta _{1,s}}$. The profit maximisation problem evolves into

$$\begin{array}{*{20}{l}}\pi _{si} = & \left( {1 - \tau _{Ysi}} \right)P_{si}Y_{si} - \left( {1 + \tau _{Ksi}} \right)R_s\left( {K_{1,si} + K_{2,si}} \right)\\ & - \left( {1 + \tau _{Lsi}} \right)w_s\left( {L_{1,si} + L_{2,si}} \right) - \\ & - P_{1,s}^MM_{1,si}\mathop{\longrightarrow}\limits^{{L_{1,si},L_{2,si},K_{1,si},K_{2,si},M_{1,si}}}\max .\end{array}$$

(A2)

Firm i maximises profits subject to production function and demand constraints given by Eqs. (A1) and (3). This leads to the following first order conditions:

$$\begin{array}{l}\left( {1 - \tau _{Ysi}} \right)\frac{{\sigma _s - 1}}{{\sigma _s}}P_{si}\frac{{\alpha _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)Y_{si}}}{{K_{1,si}}} = \left( {1 + \tau _{Ksi}} \right)R_s\\ \left( {1 - \tau _{Ysi}} \right)\frac{{\sigma _s - 1}}{{\sigma _s}}P_{si}\frac{{\alpha _{2,s}Y_{si}}}{{K_{2,si}}} = \left( {1 + \tau _{Ksi}} \right)R_s\\ \left( {1 - \tau _{Ysi}} \right)\frac{{\sigma _s - 1}}{{\sigma _s}}P_{si}\frac{{\beta _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)Y_{si}}}{{L_{1,si}}} = \left( {1 + \tau _{Lsi}} \right)w_s\\ \left( {1 - \tau _{Ysi}} \right)\frac{{\sigma _s - 1}}{{\sigma _s}}P_{si}\frac{{\beta _{2,s}Y_{si}}}{{L_{2,si}}} = \left( {1 + \tau _{Lsi}} \right)w_s\\ \left( {1 - \tau _{Ysi}} \right)\frac{{\sigma _s - 1}}{{\sigma _s}}P_{si}\frac{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)Y_{si}}}{{M_{1,si}}} = P_{1,s}^M\end{array}$$

(A3)

After re-arranging one can express capital, labour and output distortions as a function of observable variables:

$$1 + \tau _{Ksi} = \frac{{P_{1,s}^MM_{1,si}\left( {\alpha _{2,s} + \alpha _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)} \right)}}{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)R_s\left( {K_{1,si} + K_{2,si}} \right)}};$$

(A4)

$$1 + \tau _{Lsi} = \frac{{P_{1,s}^MM_{1,si}\left( {\beta _{2,s} + \beta _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)} \right)}}{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)w_s\left( {L_{1,si} + L_{2,si}} \right)}}.$$

(A5)

$$1 - \tau _{Ysi} = \frac{{\sigma _s}}{{\sigma _s - 1}}\frac{{P_{1,s}^MM_{1,si}}}{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)P_{si}Y_{si}}}.$$

(A6)

A.2 Outsourcing first stage of production

Assume that output is still produced in two stages, but the first stage of production is outsourced to a different firm. Instead of producing M_2,si itself, the firm i buys it at the price P^M_2,s. The production function transforms into:

$$Y{\prime}_{si} = A_{2,si}K{\prime}_{2,si}^{\alpha _{2,s}}L{\prime}_{2,si}^{\beta _{2,s}}M{\prime}_{2,si}^{1 - \alpha _{2,s} - \beta _{2,s}},$$

(A7)

where Y^′ _si , K^′_2,si, L^′_2,si and M^′_2,si denote the real output, capital, labour and intermediate inputs in the case of outsourced first-stage. As before, the firm faces downward sloping demand:

$$P{\prime}_{si}{\mathrm{ = }}P_s\left( {\frac{{Y_s}}{{Y{\prime}_{si}}}} \right)^{\frac{1}{{\sigma _s}}},$$

(A8)

where P^′ _si stands for the output price of the firm i under outsourcing. Profit maximisation problem is similar to Eq. (5):

$$\begin{array}{*{20}{l}}\pi {\prime}_{si} = & \left( {1 - \tau _{Ysi}} \right)P{\prime}_{si}Y{\prime}_{si} - \left( {1 + \tau _{Ksi}} \right)R_sK{\prime}_{2,si}- \left( {1 + \tau _{Lsi}} \right)\\ & \times w_sL{\prime}_{2,si} - P_{2,s}^MM{\prime}_{2,si}\mathop{\longrightarrow}\limits^{{L{\prime}_{2,si},K{\prime}_{2,si},M{\prime}_{2,si}}}\max.\end{array}$$

(A9)

After solving the maximisation problem, the true capital, labour and output distortions should be derived as

$$1 + \tau _{Ksi} = \frac{{\alpha _{2,s}}}{{1 - \alpha _{2,s} - \beta _{2,s}}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{R_sK{\prime}_{2,si}}};$$

(A10)

$$1 + \tau _{Lsi} = \frac{{\beta _{2,s}}}{{1 - \alpha _{2,s} - \beta _{2,s}}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{w_sL{\prime}_{2,si}}};$$

(A11)

$$1 - \tau _{Ysi} = \frac{{\sigma _s}}{{\sigma _s - 1}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)P{\prime}_{si}Y{\prime}_{si}}}.$$

(A12)

When data on firm-to-firm trade is unavailable and most of the firms in the industry still perform both stages of production, the researcher does not observe α_2,s and β_2,s. Instead, researcher observes $\alpha _s = \alpha _{2,s} + \alpha _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)$ and $\beta _s = \beta _{2,s} + \beta _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)$ from the data. As a result, capital, labour and output distortions are going to be evaluated using Eqs. (A4)–(A6), creating a bias:

$$\begin{array}{l}1 + \tau {\prime}_{Ksi} = \frac{{\alpha _s}}{{1 - \alpha _s - \beta _s}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{R_sK{\prime}_{2,si}}} = \frac{{\alpha _{2,s} + \alpha _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)}}{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{R_sK{\prime}_{2,si}}} = \\ = \left( {\frac{1}{{1 - \alpha _{1,s} - \beta _{1,s}}} + \frac{{\alpha _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)}}{{\alpha _{2,s}\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)}}} \right)\left( {1 + \tau _{Ksi}} \right);\end{array}$$

(A13)

$$\begin{array}{l}1 + \tau {\prime}_{Lsi} = \frac{{\beta _s}}{{1 - \alpha _s - \beta _s}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{w_sL{\prime}_{2,si}}} = \frac{{\beta _{2,s} + \beta _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)}}{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{w_sL{\prime}_{2,si}}} = \\ = \left( {\frac{1}{{1 - \alpha _{1,s} - \beta _{1,s}}} + \frac{{\beta _{1,s}\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)}}{{\beta _{2,s}\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)}}} \right)\left( {1 + \tau _{Lsi}} \right);\end{array}$$

(A14)

$$\begin{array}{*{20}{l}}1 - \tau {\prime}_{Ysi} & = \frac{{\sigma _s}}{{\sigma _s - 1}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{\left( {1 - \alpha _s - \beta _s} \right)P{\prime}_{si}Y{\prime}_{si}}}\\ & = \frac{{\sigma _s}}{{\sigma _s - 1}}\frac{{P_{2,s}^MM{\prime}_{2,si}}}{{\left( {1 - \alpha _{1,s} - \beta _{1,s}} \right)\left( {1 - \alpha _{2,s} - \beta _{2,s}} \right)P{\prime}_{si}Y{\prime}_{si}}} = \\ & = \frac{{1 - \tau _{Ysi}}}{{1 - \alpha _{1,s} - \beta _{1,s}}},\end{array}$$

(A15)

where τ′ _Ksi , τ′_Lsi, and τ′ _Ysi are estimated capital, labour and output distortions in the case of outsourced first stage of production.

Appendix B: Figures

Figures 6 and 7.

Fig. 6

Relationship between relative output distortions (1 – τ* _Ysi )/(1 – τ _Ys ) and several firms’ characteristics in 2014. Source: Latvia’s firm-level database, author’s calculations. Notes: Firm-level output distortions are estimated using Eq. (22), while the average industry output distortions—Eq. (15). Higher value corresponds to the larger output distortion relative to the industry average. Obtained by the kernel-weighted local polynomial smoothing

Fig. 7

Alternative estimates of gains from reallocation to aggregate gross output. Source: Latvia’s firm-level database, author’s calculations. Notes: Gains to total gross output are determined as 100 × (Y*/Y – 1). Contributions for output, capital and labour distortions are estimated by eliminating variation in respective distortion. The residual is attributed to interactions between various distortions