The impact of the road infrastructure endowment on production by Cobb-Douglas production function
This article consists of three parts. In the first one, we will briefly discuss the most important directions of studies on influence of transportation infrastructure, including the road network, on regional development. In the second one, we will present, in more detail, the current of studies which concentrates on the importance of transportation infrastructure in creating the volume of production/income (in various trades and at different levels of data aggregation), using the tools of econometric analysis. The third part was dedicated to presenting results of empirical research made by the author, which analyzes influence of the
“road factor” on the volume of production in Poland across districts (cross-regional analysis)..
1. Research on the influence of transportation infrastructure on economic development of regions – general classification
There are different approaches to scientific research which aims to capture the relationship between transportation infrastructure and economic development (and, in particular, development of regions). The object of research itself has a number of aspects, hence the possibility to adopt various points of view and analysis methods, concentrate on various areas of the interrelationship between expanding transportation infrastructure
(including the road network) and regional development ( e.g.
influence on investment decisions, location of business, efficiency of companies’ functioning, inter- and intraregional trade exchange, municipalities activity, employment, the local community’s economic growth and quality of life etc.
), and use various analysis methods (estimating the econometric model parameters, studying cost functions, thoroughly comparing statistical data, direct surveys).
Thus, the variety of analytic approaches reflects complexity and multitude of aspects of the issue itself as well as ambiguity of the presented views on it
1
.
It is worth briefly looking closer at this differentiation within research. It begins with studying influence of transportation costs on production costs borne by enterprises, by including transportation infrastructure in the production function as a factor determining the production potential at different levels of variables aggregation, and end with considering spatial changes in the directions of trade exchange and in distribution of economic subjects in regional terms, resulting from investments in transportation infrastructure. In econometric analyses, aggregated data are used in the regional section and/or branch section, using which attempts are made to estimate, by estimation of the econometric model based on suitable specification of the production function (mainly the Cobb-Douglas production function), an

increase in the extreme productivity of private capital as a result of improved infrastructure
(or, in wider terms, increased public outgoings). Higher productivity of private capital can stimulate investments in the sector of enterprises. In analyses of different types, location models are used, which consider spatial distribution of private capital and/or employment as the function of infrastructure (its weight often being related to other location factors, such as the labour market, advantages of agglomerations or investment subsidies)
2
. Newer and more detailed methods are based on general equilibrium models, extended by inter-branch and/or inter-regional flows and, together, referring to Leontief’s old input-output model. In these studies, freight flows are usually the function of transportation costs and their changes in time.
In case of disaggregated data (unit data at the corporate level) use is made of, e.g.
studies based on models of revealed preferences, where the structure of individual choices made by enterprises or households is studied as to possible relocation in terms of better accessibility, resulting from expanding transportation infrastructure. This approach is discussed in detail by, e.g.
Kroes and Sheldom, who present its strengths and weaknesses [pp. 11-25].
Another group includes studies based on quasi-experimental methodology, alongside direct surveys and interviews with entrepreneurs. Such research is often descriptive and comparative, i.e.
a comparison is made of the economic situation of the region where improvements in infrastructure had been made ( e.g.
building a new road) with the “control” area with a very similar system of the remaining conditions of economic development, where investments did not take place. It enables concluding about the fact of resulting possible positive changes exclusively or almost exclusively from improvements within transportation connections. This approach was discussed in detail by Isserman [1990]. In turn, direct surveys and interviews consist in interrogating and tracing motives by which companies are guided when choosing location with emphasis on significance of the transportation factor. Questions usually concern current intensity of using transportation infrastructure, the level of satisfaction with services it renders, its relative significance as a determinant of the decision to relocate or expand activity, expected influence of possible infrastructural improvements on the scale of activity, employment etc.
(see Rietveld and Bruinsma [1998, pp. 76-78]).
Due to the variety of research on influence of transport infrastructure on economic development, including the problem of regional development presented here, a clear need arises to systematize the research itself and, in general, order the subject matter. This may be achieved by separating some kind of spheres, levels or specifying ways in which influence of transportation infrastructure on regional development analyzed in the article takes place. The

undertaken review of research and the attempt to systematize it is also to help to choose analysis methods for Poland.
We can generally divide the research on the influence of transport infrastructure on regional development into analysing the issue in the following aspects (the similar division is presented by Rietweld and Bruinsma):
1) Accessibility. Changes in the accessbility of the regions, measured by different ratios and sometimes fairly complicated formulas are one of the most important effects of the transport improvements (road upgrading for example)
2) Productivity, production scale and its costs (impact studies). In analysis of this kind different formulations of the production function are used as the analytical tool. To this category of research we can also account those analysing the impact of the changes of transport costs (resulting from transport infrastructure investments) on the productivity, private capital performance and firm's general effectiveness.
3) Location of the economic activity (location studies) and the private sector's propensity to invest. Role of the transport infrastructure in this aspect in relation to other factors stimulating the economic activity on given area is analysed by different methods, ranging from regional economic models to the quality studies like surveys among potential investors, representatives of the local authorities, enterpreneurs and so on.
4) Intra- and interregional trade. Such studies base often on the variations of input-output models (extended Leontief matrices).
It is also worth adding that the listed areas of influence of transportation infrastructure are considerably interrelated and even mutually conditioned. For example, the fact of improving transportation connections positively influencing the marginal efficiency of production factors is not meaningless in the process of taking a decision related to location or relocation of economic entities and is reflected in interregional trade exchange. Further in the article, I will focus on the second mentioned current of research, i.e.
analyses concerning influence of transportation infrastructure on productivity and production volume.
2. Infrastructure and productivity of production factors and costs
The collection of research dealing with problems of transportation infrastructure in terms of the part it plays when defining efficiency of the private sector’s functioning – understood as productivity of production factors, or when specifying the amount of production costs is

essential in the literature on the subject. In studies of this type, infrastructure is included in production factors defining the general level of production, with aggregated data at the regional level. Such factors belong to two groups, i.e.
“rigid” factors of the regional potential, such as occurrence of natural resources, profitable location, sector structure, infrastructure of various kinds (transportation, telecommunications, water and sewage, education, health care services etc.) as well as mobile (subject to the possibility of flow between regions depending on remuneration offered) private production factors, i.e.
human capital and material capital.
This approach is based on simple reasoning that, e.g.
transportation network which meets needs (in terms of quality and quantity) or is improved ( e.g.
increased standard from a onelane road to a fast road or motorway, when occurrence of effects of a specific infrastructural investment is being studied) enables enterprises located in a given area to decrease costs of supply with raw materials as well as distribution of commodities to/from the company
(through cheaper and faster transportation, better organization of logistic systems etc.
). It determines, first of all, the occurrence of economies of scale, and it can entail more effective utilization of private production factors and higher competitiveness of produced goods.
Secondly, it may further contribute to expansion of the company into other markets, and, in consequence, to obtaining additional benefits of scale
3
. Reaction of employment as well as private material investment on changes of production volume triggered by investments in the public sphere is a separate matter. As Rietveld and Bruinsma present it [1998], two kinds of reactions can be distinguished here. First of all, improvement of transportation infrastructure may lead to shifting the point of optimum allocation of labour resources and capital necessary to produce the same quantity of a given good (shift along an isoquant). Secondly, by leading to changing extreme productivities it can increase production, which can lead to changes of demand for work and capital. It may turn out that demand for both production factors falls, and demand for labour grows, whereas demand for capital will fall or vice versa . In all these cases, costs of private production factors will decrease (Rietveld and Bruinsma [1998, pp. 54-
55]). H. Seitz [1985, pp. 123-150] proposes a different illustration of this problem. According to him, with a flexible function of demand for goods, investments in infrastructure diminish marginal costs, enabling a monopolist to increase production and lower the prices. If improvement of infrastructure enables simultaneous “savings” within labour expenditures
( labour saving ) and the production remains at the previous level, a fall in employment will follow. With infrastructure neutrally influencing involvement of the labour factor, employment will grow. Combination of these effects leads to a slight fall in demand for labour.

The analytic method in the current research described above is usually based on different forms of the production function, which enables an econometric analysis of influence of infrastructure and other related production factors on economic growth. The “traditional” factors of production function is therefore expanded by “potential factors”, i.e
. creating production base, first of all - transportation infrastructure. Ultimately, causes of disproportion between regions with a similar supply of mobile factors (variables), coming from the private sector and with different infrastructure are identified. The analytic formulation is a production quasi-function, which (for sector “ i”
in region “ r
”) accepts the following general form in this research:
Q ir
 f ir
( L ir
, K ir
, IA r
,...
IN r
) , where:
Q ir
- value added in sector i , region r (supply),
L ir
- employment in sector i , region r (supply),
K ir
- private capital in sector i , region r,
IA r
,...
IN r
- supply or infrastructure expenditures of different types in region r .
The Cobb-Douglas (C-D) production function is usually used to specify this function when carrying out its possible estimation. The C-D function was used to specify the model of influence of infrastructure on the production volume in the research of, among others, Biehl
[1986] (research for the whole of European Community), Blum [1982, pp. 151-168],
Andersson, Anderstig, Harsman [1989] (research for Sweden), Nijkamp [1986, pp. 1-21]
(Netherlands) or Prud'homme [1996, pp. 37-47] (France), and, for non-European countries,
Aschauer [1989a, pp. 177-200] (USA), Mayor [1973, pp. 157-186] as well as Sasaki,
Kunihisa, Sugiyama, [1995, pp. 143-154] (Japan).
Optionally, the translogarithmic (trans-log) production function (as a matter of fact, a transformation of the C-D function) was used to specify the model related to influence of transportation infrastructure on productivity. Research with its use was made by, among others, Seitz [1995, pp. 121-141] (Germany), Costa, Ellson, Martin [1987, pp. 419-437]
(USA), Deno [1988, pp. 400-411] (also USA). Moreover, there were cases when the usual linear function was applied: Kawashima, Straszak, Wagle [1978, pp. 183-201] (Japan).
Data used in estimation of such models differ as to the spatial level (national, regional, local etc.
) sector (different branches of industry), time series or their type itself (different combinations of data are used).

A vital part in such research is played by, e.g.
Aschauer’s works [1989 a,b,c], esp. by repeatedly quoted in literature, “ Is public infrastructure productive?” The work, published in
1989, triggered a further research popularized in literature as the Public Infrastructure
Hypothesis and opened a field to further research at the national and regional level and to a wider discussion about the role of public investments in creating national income. Aschauer proved that the fall in the private sector productivity in the USA in 1970s and 1980s can be partially explained by a general fall in government investments. In 1950—1970, the annual productivity growth ( total factor productivity ) in the USA was 2% on average, to fall to
0.08% annually in 1971-1985. Aschauer argues that besides other possible causes ( i.e.
growth in oil prices, mass migrations of workers from agriculture to other branches of industry), a very large part was played here by the fall in outlays on public capital, and, first of all, on infrastructure, whose growth rate was 4.1% on average in 1950—1970, 1.6% between 1971—
80, and 0.7% in 1981—85.
Achauer’s analysis includes a number of dependencies, analyzed both in relation to the whole national economy and sector-wise. In this article, however, I will focus on this part which strictly concerns the importance of outlays on public infrastructure in creating income in the USA in the period in question. Estimated flexibility of productivity in relation to the coefficient public/private capital is 0.4, i.e.
a 1% growth of the coefficient value: public expenditures (exc. military spendings) to material private investments results in increasing capital productivity by 0.4% 4 . As to estimating the total function of productivity of factors, it was shown that a 1% increase of public outlays contributes to an increase in productivity of
0.37% to 0.49% depending on the sample studied (analyzed set of branches).
To summarize, it can be stated that the factor of transportation infrastructure in the regional production function can be interpreted in various ways. First of all, infrastructure is perceived as a constant factor limiting development in the sense that the shape of this function must be extinguishing whereas variable factors have falling marginal productivity, down to a breakthrough, when expanding the network opens new prospects for expansion of the private sector, entailing productivity growth of factors found in there. In a different approach, corresponding to a different input situation, we have infrastructure allowing for expansion
(from the research start point) – it does not represent limitation, as it is far from exhausting flow capabilities, logistic capabilities etc . In such a situation, the production function could assume the shape of a logistic curve, which after crossing its inflexion point would become an

extinguishing curve and then the situation becomes similar to the first one – characteristic for a passive attitude of authorities (who do not begin appropriate investments on time).
Many authors criticize the method based on the regional production function, among other things because it disregards advantages obtained by consumers. Besides, aggregated data on which estimations are made enable only ex-post and not ex-ante analyses, that is, according to some researchers, they have a limited use in terms of forecasting capabilities.
Yet another kind of reservation is expressed by Gramlich [1994, pp. 1045-1066], who emphasizes that richer regions (belonging to the category of economic "core" - developed centres) are usually equipped with better infrastructure and have bigger funds for investments, which naturally implies higher positive correlation between infrastructure and productivity or production level. Therefore, it is not known to what extent a high level of economic development can be related to infrastructure (implying effective transportation, better accessibility etc.
), and to what extent it results from joint influence of other factors, such as agglomeration effects, absorptive sales market, abundance of capital, availability on urbanized areas, usually better educated staff etc.
3. Econometric analysis of influence of road infrastructure on production growth and management efficiency in regions with the use of the Cobb-Douglas production function
This part of article is devoted to estimation of parameters of the econometric model based on the production theory with the use of the popular and simple case of the Cobb-Douglas production function 5 . The analysis made in this clause is modelled on similar western research, which belongs to one of essential currents of estimating the role of infrastructure (as material public capital – bases, foundations for economic activity), including transportation infrastructure, in manufacturing production
6
.
Most studies based on application of the production function, as it was said above, belong to Public Infrastructure Hypothesis . Its representatives include D. A. Aschauer, A.
Munell, K. Mera and D. Biehl. Following inclusion of transportation infrastructure in the set of factors defining the production volume in a given region (given trade etc.
), the author carried out an estimation (using the ordinary least squares method) of three econometric models, showing differently economic significance of the “road factor” in Poland. In order to obtain more exact results, the analysis was made in the district spatial-level (poviat) for the years 2000-2004 (statistical data necessary to carry out the estimation are accessible for such a period)
7
. The first model (3.a) describes elasticity of sold production of the industry (in millions PLN) in all administrative districts (poviats) in Poland on production factors, and so:

traditionally accepted labour and capital, and, specifically for the questions under analysis, density of the road network. The second and third models (3.b and 3.c) have a similar character, i.e.
they both are an attempt to estimate the influence of the road factor on productivity of the remaining production factors (capital and labour). Model 3.b estimates elasticity of capital productivity, and 3.c – of labour productivity on the road factor. Source data come from statistics of the Central Statistical Office – the variable being explained is aggregated data on the “value of sold production” (in millions PLN), and the explanatory variables are: “number of workers” (labour factor), “gross value of fixed assets” (capital factor) as well as “density of the district (poviat) road network and communal road network with hardened surface” within individual administrative districts (poviats).
The research required gathering and appropriately, in terms of the goal of the research, selecting a very large number of statistical data, and then introducing them as a basis for the intended estimation. On the other hand, certain imperfections of statistical information provided by the Central Statistical Office is a cause for some concern. For example, in many cases, registered lengths of roads fell in subsequent years
8
, moreover several administrative districts (poviats) in relation to which administrative changes (joining them, dividing them, creating new units etc.) were made in the period under examination, had to be excluded from observation.
First model - regression function, based on the Cobba-Douglas production function has the following formulation: n n



 
L G n

 in the linarised form where: Y ti
- production sold in the year t (t=2000,2001,2002,2003) in poviat " i " ( i

1 ,..., 290 ) ,
ST ti
- capital (fixed assets),
L ti
– labour (number of workers),
GD ti
- road density
Below the results of estimation of the 5.3 a. regression function by ordinary least squares
(OLS) was presented ln Y
ˆ it
 
1 , 65

0 , 76 ln ST ti

0 , 28 ln L ti

0 , 056 ln GD ti
(5.3.a)
(0,35) (0,19) (0,38) (0,38)
R
2 = 0,706 F(3,1048) = 835,6
In this formulation, coefficient with variables “capital” (ln ST ti
) and “labour” (ln L ti
) are positive, i.e.
0.76% and 0.28%, and so they reach values very similar to the ones presented in

the classic model formulation describing dependence of the production volume on production factors. It testifies to general correctness of the model, and also to good selection of coefficients themselves to the estimation carried out in the light of the theory of economy.
Regularity of the model is testified to by a high value of the determination coefficient
R
2
. For all parameters, there is a positive evaluation of average estimation errors – they are low, and so it can be stated that the coefficients are estimated precisely. Verification based on t-Student test also shows statistical significance of parameters (at the significance level of
10%), which contributes to the fact that all variables, including GD ti
, explains the volume of the production sold in industry in district (poviat) section Y . it
The coefficient at the variable “density of the road network” (ln GD ti
) is positive, and so flexibility of production in relation to road infrastructure is positive. 1% increase in the length of roads increases the production value by approx. 0.056%. This result therefore shows positive implications of having a good road network endowment for formation of the value of sold production of enterprises at the level of administrative districts (poviats) in Poland. It may prove that transport infrastructure endowment is, at least in the static meaning, the factor of creating potential for development of economic activity in a given area. On the other hand, a question arises as to how this element of economic environment will dynamically affect the production growth, and so whether or not and to what extent the length of roads itself and its changes in Poland influence productivity of the remaining factors – labour and capital.
3.2. Influence of the road factor on productivity of labour and capital
At the second stage of research based on the Cobb-Douglas production function an attempt was made to check whether the factor of road infrastructure affects productivity of the remaining factors of production, i.e.
capital and labour. Both models presented below (3.b and
3.c) have a logarithmic and linear form, but here, in order to make calculations more precise from the point of view of changes in time, zero-one variables were added to show influence of individual years.
Model 3.b. assumes such specification which allows checking influence of the road infrastructure endowment (length of roads in km) on productivity of the capital unit. ln Y ti
 ln ST ti
 
0
 
1
(ln L ti
 ln ST ti
)
 
2
(ln D ti
 ln ST ti
)
 t


2000
 t
T t
 i


1
 i
G i
  ti
(5.3.b) where: t = 2000, 2001, 2002, 2003 (individual years of analysis),


,
0

1
...- estimated parameters, remaining marks as above
The ordinary least squares method was used for estimation. The results of estimation are in table 3.
Tab. 3. Wyniki estymacji (MNK) modelu (M2) variable Parameter's estimation
Estimation
Bias
constant -1,901 0,656 ln L ti
- ln ST ti
0,546 0,188
Ln D ti
- ln ST ti 0,312 0,157
2001 0,041 0,010
2002 0,047 0,018
2003 0,131 0,023
Statistics of t-Student
-2,90
2,90
1,99
3,94
2,70
5,63
The results of estimation of model 3.b show positive, comparatively high and statistically significant elasticity of (material) capital productivity amounting to 0.31% in relation to the road infrastructure endowment per unit of this capital. Moreover, flexibility of productivity of the capital unit in relation to the size of expenditures of labour per capital unit (amounting to
0.54%) also turned out to be important in such a formulation.
The last estimated model was 3.c, where influence of the size of road infrastructure per worker on the labour productivity (with added suitable, as in the previous model, zero-one variables) was checked. The final form of the model is (marks as above): ln Y ti
 ln L ti
 
0
 
1
(ln ST ti
 ln L ti
)
 
2
(ln D ti
 ln L ti
)
 t


2000
 t
T t
 i


1
 i
G i
  ti
(5.3.c)
Estimation of parameters with the OLS method gave the following results, given in table 4.
Tab. 4. Wyniki estymacji (MNK) modelu (5.3.b) variable Parameter's estimation
Estimation
Bias constant 1,560 1,978
Ln ST ti
- ln L ti
0,149 0,079
Ln D ti
- ln L ti
-0,277 0,114
2001 0,031 0,012
2002 0,041 0,019
2003 0,118 0,026
Statistics of t-studenta
0,79
1,88
-2,44
2,64
2,11
4,49
The results of this estimation show, in turn, negative, amounting to - 0.27%, and statistically important flexibility of productivity of workers in relation to the size of road

infrastructure per worker. Also, flexibility of productivity of workers turned out to be important in relation to the size of outlays of material capital per worker, but the value of this flexibility is comparatively low and amounts to less than 0.15%.
The results of research made suit the general character of conclusions from studies on influence of transportation infrastructure on economy - in particular, due to their ambiguity.
All models were estimated on the basis of identical panel data and brought rather different, although not conflicting results.
In case of model 3.a, influence of the road factor, for which density of the network was assumed here, was positive and statistically important. It is worth pointing out that positive values (3.b) were obtained for elasticity of productivity of material capital used in the production process in relation to road infrastructure (length of roads). In Polish conditions, it confirms the thesis described in the literature about growing productivity of private capital owing to infrastructural investments. In turn, negative flexibility of productivity of labour in relation to road infrastructure can mean that road investments are not directly reflected in employment growth in the place of carrying out the investment. However, mobility of labour force rises, and transportation costs decrease, which means transferring production and employment to regions where production enterprises are already located.
Another observation from the analysis made in this subclause is related to the desirable direction of further research. Namely, it seems that in the situation of imperfection of data and certain gaps in statistical reports (lack of the Central Statistical Office’s data aggregated at the level of administrative districts (poviats) or provinces (voivodships) about, e.g.
total financial expenditures on extension of local roads in Poland) and due to the small scale of road investment in Poland so far, the best way of economic analysis of the role of infrastructural investments is to carry out case study research in the micro-scale, and so regarding concrete undertakings. In the future, when longer time series of data become available, a similar analysis should be more credible, and therefore it should be extended additionally by the sector analysis, i.e.
taking into consideration structural changes occurring in a given region as a result of investments in road infrastructure.
Agnieszka Domańska
Graduate School of Economics - Higher School of Commerce and International Trade in Warsaw
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