Regional variation in productivity: a study of the Danish economy

Abstract

We develop a model of labor productivity as a combination of capital-labour ratio, vintage of capital stock, regional externalities, and total factor productivity (TFP). The skewness of TFP distribution is related to different growth theories. While negative skewness is consistent with the neo-Schumpeterian idea of catching up with leaders, zero skewness supports the neoclassical view that deviations from the frontier reflect only idiosyncratic productivity shocks. We argue that positive skewness is consistent with an economy where exogenous technology is combined with non-transferable knowledge accumulated in specific sectors and regions. This argument provides the framework for an empirical model based on stochastic frontier analysis. The model is used to analyse regional and sectoral inequalities in Denmark.

Appendices

Appendix 1: measurement of capital stock and vintage

As discussed in the text, a major challenge was in using annual data on investment in buildings, machines and transport at the region and sector level to estimate capital stock and the age of capital. Essentially, our data construction strategy was to obtain annual capital stock estimates by aggregating depreciated values of lagged investment for several preceding years. The depreciation rates used varied by the type of investment as well as with the lag. Vintage of capital for each year at the region ×  sector level was similarly estimated as a weighted average of the age of capital, taking the depreciated value of capital stock as the respective weight for each vintage.

Crucially, the estimation of both capital stock and age of capital relied on obtaining appropriate depreciation rates for different types of capital. For this purpose, we used two alternative methods.

Method 1

In this rather simplistic formulation, we considered fixed depreciation rates of 1/30, 1/10 and 1/6 of initial book value per year for buildings, machines and transport respectively. These fixed depreciation rates approximately matched average annual depreciation rates for aggregate capital stock in Denmark over the period of analysis.

Following this depreciation scheme, we computed total value of capital stock for each sector, region and year (1979–1993). Thus, capital stock, by type of capital, in the year t for region r and sector i was estimated as:

$$ K_{rit}^{(tr)}=\sum\limits_{u=0}^{5}\frac{(6-u)}{6}\cdot I_{ri,t-u} ^{\left(tr\right) };K_{rit}^{\left(ma\right) }=\sum\limits_{u=0}^{9}\frac{(10-u)}{10}\cdot I_{ri,t-u}^ {\left(ma\right) };K_{rit}^{\left(ba\right)}=\sum\limits_{u=0}^{29}\frac{(30-u)}{30}\cdot I_{ri,t-u}^{\left(ba\right)} $$

where K ^(tr) _rit , K ^(ma) _rit and K ^(ba) _rit denote capital stock in transport, machinery and buildings respectively and similarly I ^(tr) _rit , I ^(ma) _rit and I ^(ba) _rit denote investment by type of capital goods.

We used a similar procedure to estimate the average age of capital. For this purpose, a weighted average was used, where the age of investment in each year was simply weighted by its depreciated value in the current year. For example, average age of transport capital in the year t for region r and sector i was estimated as:

$$ a_{rit}^{\left(tr\right)}=\frac{1}{K_{rit}^{\left(tr\right)}}\cdot \sum\limits_{u=0}^{5}\frac{(6-u)}{6}\cdot(u+0.5)\cdot I_{ri,t-u}^{\left(tr\right)} $$

where a ^(tr) _rit is the average age of transport capital (in years). It is assumed that investments during a given year have an average age of 6 months when the accounting year closes. The average age of capital for the other types of capital were computed similarly.

Method 2

The above method is simplistic and may not be satisfactory, in that it assumes a fixed rate of depreciation of capital irrespective of the vintage. It also assumes that capital depreciates at a constant rate rather than a compounded depreciation rule.¹⁹

As an alternative, we collected data (from Danmarks Statistik) on total depreciation and stock of capital at replacement cost, by sector and year for the entire period under study. Using these data, we estimated a nonlinear regression model regressing log of aggregate depreciation on lagged log-investments for several previous years.²⁰

The estimated compounded depreciation rates were the following:

Buildings: 0.018 (past 1–12 years), 0.040 (13–20 years) and 0.084 (21–28 years).

Machinery: 0.000 (past 1–3 years), 0.079 (4–6 years) and 0.372 (7–9 years).

Transport: 0.078 (past 1–3 years) and 0.360 (4–5 years).

For each capital type, we assumed zero depreciation for the current year.

There were several nice features of these estimation results. First, the R ² is 0.988, which is very good given that we have 144 observations and only eight estimated depreciation parameters. The residuals showed some evidence of heteroscedasticity, but autocorrelation was not significant. Second, for each type of capital, the residual undepreciated capital is relatively low. After 28 years, 29% of initial investment in buildings remain; 19% of machinery remain after 9 years; and 22% of transport investment remain undepreciated after 5 years. Third, the a priori expected relationship of increasing depreciation rates with vintage of capital is maintained in the estimates.

Assuringly, the empirical results obtained by adopting the two alternative depreciation rules to compute capital stock and age of capital are very similar. Hence, in the interests of clarity in exposition, we report empirical results only using Method 2. Results using Method 1 are available with the authors.

Appendix 2: additional results and robustness

See Table 3

Table 3 Estimates for alternative models