2 Methodology
2.1 Theoretical specification
Growth in Average Labour Productivity (ALP) is defined as the growth in average output per unit of labour input (for example, per worker-hour) over a specified period of time. By contrast, growth in another widely cited productivity measure – Multi-Factor Productivity (MFP) – is defined as the increase in output (net of intermediate inputs) that cannot be attributed to increases in the quantity or quality of physical capital and labour, for example, growth in output deriving from more efficient deployment of existing resources.
Thus MFP is evaluated as a residual after taking account of measured growth in other production inputs. As well as capturing improved efficiency in resource utilisation, it also includes the effects of ‘disembodied’ technical change, that is, technical improvements and innovations which are not embodied in measured capital inputs. Other variables which may be picked up by a MFP measure include economies of scale, capacity utilisation and measurement errors of different kinds.
Letting Y denote nominal value added and L labour input, average labour productivity (ALP)for industry i and country k at time t is defined as:
(1) 
Relative labour productivity levels comparing countries k and j can be derived as the ratio of labour productivity for both countries, but with value added Y denominated in a common currency. To achieve the latter it is necessary to multiply value added in j by the ratio of its prices to those in the numeraire country k. Thus relative labour productivity levels are given by:
(2) 
ALP growth between periods t and t-1 can be calculated from equation (1) except that each country’s domestic price indexes are employed to deflate nominal values. Combining levels with growth rates allows calculation of relative labour productivity at each point in time.
In order to estimate relative levels of multi-factor productivity (MFP) in different countries, we use growth accounting methods which have been employed extensively in international comparisons of productivity growth rates and levels, e.g. in Jorgenson, Gollop and Fraumeni (1987), O’Mahony (1999) and O’Mahony and van Ark (2003). The theoretical underpinning for this approach is the neoclassical growth model, with underlying assumptions that all markets are competitive and that all factors in the production process are paid their marginal products, the sum of which exhausts all returns from pursuing those activities. In addition the use of value added to measure output involves the assumption that material input is separable from other inputs in the production function.
Under these assumptions MFP levels in country J relative to country K in industry i can be calculated using the Törnqvist discrete approximation to the Divisia index, given by:
(3) In(MFPij,k)= In(RYij,k) - αij,k In(RLij,k)In(RKij,k)
where RYJ,K denotes value added in country J relative to country K (with nominal output converted to a common currency), RL is relative labour input, RK is relative capital stocks, and αJ,K is the share of labour in value added averaged over the two countries. Assuming constant returns to scale, the weight on capital is one minus labour’s share of value added.
Analogously, comparing periods t and t-1, again letting Y denote real output, L labour and K capital, and dropping the country subscript, the Törnqvist MFP growth index is given by: ;
(4) In(MFPi,t) - In MFPi,t-1) = (In Yi,t - In Yi,t-1) - ϖil,k(In Li,t-1) - (1 - ϖil)(In Ki,t - In Ki,t-1)
whereϖil, is the share of labour in the value of output, averaged across periods t and t-1.
In addition, changes in the quality of labour input in each industry may be estimated by extending the growth accounting method to distinguish labour by skill type with each type weighted by its wage bill share. Hence, assuming there are l types of labour hours (L), a change in aggregate labour input can be estimated as:
(5) 
with weights equal to the share of each labour type in the total wage bill. A measure of labour quality change in each industry can be derived from the difference between dlabour as defined above and the growth in total worker-hours.
2.2 Data sources and measurement issues
For this study we make use of National Accounts data on gross output, value added, and labour inputs in each country while using production censuses such as the Annual Business Inquiry in the UK to obtain more disaggregated information as and when required. Throughout we use National Accounts aggregates as control totals since international conventions are employed in National Accounts measurement and so these data are usually the most internationally comparable.
In order to construct capital stocks series, we make use of capital investment data provided by the UK Office for National Statistics (ONS) and Statistics New Zealand (SNZ). For cross-country comparisons these estimates require assumptions on common sector-specific depreciation rates across the countries in question. Therefore our capital stocks estimates for both New Zealand and the UK are based on US depreciation rates, with assets divided into structures, vehicles, computers, other plant and machinery and intangibles (principally software).[2] Letting c denote types of capital, with I denoting investment and d the (geometric) depreciation rate, capital stocks are measured as:
(6) 
The growth in aggregate capital is then calculated in an analogous manner to aggregate labour as in equation (5) above, with weights equal to the share of each asset type in the total value of capital. This provides a basis for benchmark estimates of relative capital stocks in each country with PPPs for investment goods employed to convert capital to a common currency. Further details are set out in Section 5.1.
The primary sources for data on employment and wages by skill type (proxied by qualifications category) are the Labour Force Survey (LFS) for the UK and the NZ Income Survey and NZ Census of Population and Dwellings. Previous research comparing labour force skills across countries has tended to divide the labour force into three or four categories of formal qualifications and then attempted to match those categories across countries (see, for example, O’Mahony, 1999). This method is sensitive to the allocation of qualifications to the various categories which is fraught with difficulty due to the differences in education and training institutions and formal qualifications systems in each country. Hence in this study our approach is to benchmark on the highest qualifications category (First/Bachelor degree and above) where comparability across countries is at its strongest and then use the ratios of mean wages in other qualification groups relative to mean graduate wages within each country to derive a country-specific measure of labour quality. This approach is explained in detail in Section 5.2.
For conversion of nominal value added to a common currency for sector-level productivity comparisons, one potential source of purchasing power parity (PPP) exchange rate estimates is the ‘expenditure PPPs’ produced by Eurostat and OECD. However, these are designed to capture cross-country differences in standards of living rather than productivity differences and so the goods and services priced frequently include imported goods and do not include prices for intermediate products and services. An alternative source is unit value ratios (UVRs) calculated as sales of products divided by quantities produced. UVRs - which may be described as ‘output PPPs’ -- are clearly closer to the required producer price concept for sector-level comparisons. However, in practice, due to limited availability of quantity data in some sectors, as well as difficulties in matching products, it is necessary to employ a combination of output PPPs and expenditure PPPs, with the latter adjusted for relative trade and transportation margins and for taxes. For this project we make use of a new set of sector-level purchasing power parity (PPP) exchange rate estimates for New Zealand prepared by the Groningen Growth and Development Centre (GGDC) which comprise a mix of UVRs and adjusted expenditure PPPs.[3] Further details of the GGDC methodology are set out in Appendix Section A2.
