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The Manchester City Region The Case for Agglomeration Economies Appendices Annex 1: Definition and measurement of productivity and agglomeration economies In this annex we discuss measurement of labour and total factor productivity in more detail. We then provide details on the econometric approach that we use to estimate agglomeration externalities. Finally, we describe the underlying data, including descriptive statistics and the construction of the agglomeration measures. Labour productivity Labour productivity for firm i at time t is constructed as: VAit Labour Productivityit = (A1) Lit where VA is value added and L is employment. As discussed in the text, labour productivity might be a misleading measure of overall productivity if some other inputs can be substituted for labour. It does, however, have lower data requirements than TFP and so may be easier to calculate for the purposes of estimating agglomeration economies. That said, complications still arise with respect to the measurement of output (e.g. what if a firm produces several different goods?) and inputs (e.g. should the measurement of labour be adjusted if workers are of different quality?). Total Factor Productivity TFP captures the combined efficiency of inputs in producing output. Conceptually, just as for labour productivity, TFP captures the ratio of output to inputs. If Y is an appropriate index of outputs and X an appropriate index of inputs, then TFP is defined simply as: Yit TFPit = (A2) X it As with labour productivity, complications arise if firms produce different outputs. In addition, for TFP, one must address the tricky question of how to aggregate many different inputs to give some composite measure of inputs. See below for further discussion. The resulting measure is not in any meaningful units, but can be used to assess differences in productivity across firms or regions, and changes in productivity over time. To reiterate, despite these technical difficulties, TFP is economists’ preferred measure of productivity. Conceptually, TFP improvements are changes associated with shifts or changes in the ‘production function’ that transforms inputs (e.g. capital and labour) into output. Imagine that a firm 2 uses capital and labour to produce output. The production function Y=F(K,L) then summarises the amount of output that the firm can produce with given amounts of capital and labour. What if we want to allow for the possibility that the efficiency with which capital and labour get translated into output might vary across firms or across time? The simplest way to do this is to think of multiplying the production function by some parameter A that captures, for example, the effects of technology. That is Y=AF(K,L). Then increases in A imply increases in TFP. Increases to capital (K) and labour (L) do not imply increases in TFP, although increasing capital holding the amount of labour constant will lead to increases in labour productivity, as discussed above. Another important conceptual difference is between TFP on the one hand, and returns to scale on the other. If there are constant returns to scale, then doubling all inputs will double output. If there are increasing returns to scale, doubling all inputs more than doubles output. With decreasing returns to scale, doubling all inputs results in less than double the output. For economists, the returns to scale in production are captured by the functional form of F(K,L) leaving TFP to capture firm’s ability to produce more output with a given quantity of inputs. While it is conceptually easy to separate out these two effects, it is harder to do so in practice. Data issues To estimate TFP researchers need information on quantities of outputs and inputs. Unfortunately, data sets usually report only values of output and inputs. That is, quantities times prices. To get back to quantities, one needs information on prices. These are hardly ever available at the firm level and so researchers resort to the use of industry wide deflators. This ignores individual firm level variation in prices, including spatial variation. More generally, using standard deflators does not allow one to adjust for differences in quality across firms. These effects are important when thinking about spatial variations in productivity. If prices of output are higher in areas with more access to economic mass then we will overstate their productivity advantage. If wages and the costs of producing are higher then we will understate their productivity advantage. It is hard to know which effect might dominate. For goods that are easily tradable across space you would expect the latter effect to dominate. For non-tradable goods higher local prices usually reflect higher local costs so you would expect the effects to cancel out. Overall, this suggests there are reasons to think our results will tend to understate productivity differentials. A second data issue arises from the fact that there are two measures of output available to researchers – gross output and value added. Value added is equal to gross output minus intermediate inputs. The conventional wisdom is that gross output is a more appropriate measure in cases where there may be shifts in input mixes in response to changes in prices. Unfortunately using gross 3 output requires us to control for material inputs and the missing values on this variable would reduce the number of observations used for estimation. In this study we will use value added. Fortunately, in practice, there is usually a strong correlation between TFP from value added and from gross output production functions. There are numerous ways to measure TFP and the appropriateness of these methods is the subject of an ongoing debate in the economics literature. Factor share based TFP Perhaps the best known method of measuring TFP is to calculate the Solow residual. TFP = y − x s (A3) it it ∑k ikt ikt where yit is output at firm i at time t, xikt is the quantity of factor k used by the firm, and sikt is the factor share. Factor shares are calculated based on revenue shares (e.g. the wage bill expressed as a fraction of the total revenue of the firm). The Solow residual provides a measure of TFP if firms use inputs efficiently (i.e. they maximise outputs for a given set of inputs and given technology), factors are paid their marginal products and there are constant returns to scale. Corrections are available if some of these assumptions are not met, but require additional data. Regression based TFP An alternative approach, which allows for increasing returns to scale is to estimate so called, regression based TFP. It is calculated by collecting data on output and inputs for a large number of firms, then deriving TFP from : TFP = y − β x (A4) it it ∑k kt ikt where the coefficients βkt are estimated by statistical regression techniques. In this report, we adapt this regression based approach to calculate the impact of agglomeration economies on TFP. Estimating agglomeration economies A standard approach to estimating agglomeration economies (and the effect of other variables that influence TFP) is to estimate the production function directly with total factor productivity modelled as a function of variables related to city/region size, for example population, employment, density or effective density. The unit of observation might be the firm, the city-industry or the city. Regardless, in general terms, this involves estimating a relationship of the form: 4 ⎛⎞labour,,capital localisation,urbanisation, Valueadded = f ⎜⎟ (A5) ⎝⎠localskills,,cityindicators timeindicators,otherfactors For example, in the case of the firm-level analysis carried out in this report: value-added is calculated as the (natural logarithm) of revenue minus materials costs (deflated by relevant price indices); labour and capital are plant-specific (log) employment and (deflated) capital costs;1 localisation is an agglomeration measure based on (log) local own-sector employment; urbanisation is a measure of agglomeration based on (log) local employment in all (other) sectors; localskills is the local skilled share of employment. The location effects (city region or region) are estimated by including dummy variables, which take the value 1 if a plant is located in a particular location, 0 otherwise. We provide more details below. The major drawback of the ‘production function’ approach is that it is very demanding in terms of data. Obtaining measures of capital stock at the city/region level is particularly problematic. An alternative approach is to work with labour productivity: ⎛⎞localisation,,urbanisation Valueaddedperworker = f ⎜⎟ (A6) ⎝⎠localskills,,cityindicators timeindicators,otherfactors We have discussed the problems of using labour productivity instead of TFP above. Investigation showed that these problems are present in the data that we use thus justifying our focus on TFP. Finally, we could work with the relationship between wages and employment/population. In this case, the general form of the relationship is: ⎛⎞labour,,capital localisation,urbanisation, Wage = f ⎜⎟ (A7) ⎝⎠localskills,,cityindicators timeindicators,otherfactors In the UK case, detailed wages by sector are actually harder to obtain than TFP and labour productivity estimates so we focus in our research on TFP. Data As discussed in the text, our source of firm data is the ARD. The IDBR is used to construct measures of localisation and urbanisation, while measures of skills are taken from the LFS. The source for transport networks is described in the text. Value added and capital stock need to be deflated and to do this we use sectoral value added and inputs deflators from EU Klems (2008). Capital is measured with the perpetual inventory 5 method. Labour is number of employees. We also rerun our results using the total wage bill rather than number of employees. This did not change our broad findings. The most disaggregated level at which the ARD provides information on value added and employment is the Reporting Unit level.
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