3  Labour productivity accounting

This section outlines two widely used methods for decomposing aggregate and industry labour productivity growth into contributions from firm entry, exit and continuation.

Consider the level of aggregate or industry labour productivity which is defined as the sum of the share weighted levels of firm labour productivity:[6]

(5)      ,

where denotes the labour input share[7] of firm in year and the labour productivity of firm in year . Note that .

To compute the change in aggregate or industry labour productivity consider two years, and . The level of aggregate labour productivity in is the sum of share weighted labour productivity of continuing and exiting firms (ie, ) and the level of aggregate labour productivity in is the sum of share weighted labour productivity of continuing and entering firms (ie, ).

The change in aggregate or industry labour productivity is found by taking the difference in the level of labour productivity between and :

(6)     .

To compute aggregate or industry labour productivity growth it is necessary to divide equation by the level of aggregate or industry labour productivity in .

After some manipulation, it is possible to rewrite equation in the following two ways:

(7)    

and

(8)     .

A bar indicates a time average over and . There is no unique decomposition of labour productivity as defined by equation (6). In particular, the choice of weights and the benchmark parameter in equations (7) and (8) is arbitrary. Equation is the decomposition proposed by Foster, Haltiwanger and Krizan (1998) (hereafter FHK), while equation is the decomposition outlined by Griliches and Regev (1995) (hereafter GR).

The first component of the FHK decomposition measures the contribution to labour productivity growth arising from the labour productivity growth of continuing firms. This component is often termed the ‘within’ component as it measures the change in labour productivity that occurs within a firm. The second component measures the contribution to labour productivity growth from changes in continuing firms’ size, as measured by their labour input shares, after accounting for continuing firms’ labour productivity relative to aggregate labour productivity. This component is called the ‘between’ component as it captures changes in labour input shares between firms. The between component makes a positive contribution to aggregate labour productivity growth when continuing firms that have above average labour productivity experience an increase in their labour input share. The third component, which is like a covariance or cross-product term, measures the interaction between changes in continuing firms’ labour productivity and changes in their shares. The fourth and fifth components measure the contribution to labour productivity growth from entering and exiting firms, once accounting for firms’ labour productivity relative to aggregate labour productivity.

A disadvantage of the FHK decomposition is it is prone to measurement error and can generate spurious results in the cross term. It can also suffer from spurious effects associated with transitory changes in labour use and output. The GR decomposition is less sensitive to measurement error and is more appealing because of its symmetry in using time averages (Balk and Hoogenboom-Spijker 2003). However, components of the GR decomposition can be interpreted in a similar manner to the components of the FHK decomposition.

The first component of the GR decomposition measures the contribution to aggregate labour productivity growth from changes in the labour productivity of continuing firms. This is the ‘within’ component of the GR decomposition. The second component measures the contribution from changes in continuing firms’ shares once accounting for a firm’s labour productivity relative to average labour productivity. The third and fourth terms measure the contribution to aggregate labour productivity growth from entering and exiting firms once controlling for firm labour productivity relative to aggregate labour productivity.

Baldwin and Gu (2003) have suggested the FHK and GR decompositions should be modified so that continuing and entering firms’ labour productivity is compared relative to the average labour productivity of exiting firms. The rationale for comparing entering firms’ labour productivity relative to the average labour productivity of exiting firms is that entering firms replace exiting firms in the turnover process. When the labour productivity of continuing and entering firms is compared relative to the average labour productivity of exiting firms, the FHK decomposition becomes:

(9)     .

The first and third components of the of the FHK decomposition relative to the average labour productivity of exiting firms (hereafter FHK(X)) are identical to the first and third components of the FHK decomposition. The second component of the FHK(X) decomposition measures changes in continuing firms’ market shares once accounting for a firm’s labour productivity relative to the average labour productivity of exiting firms. The fourth component of the FHK(X) decomposition measures the contribution to aggregate labour productivity growth of entering firms relative to the productivity of exiting firms.

The GR decomposition relative to the average labour productivity of exiting firms becomes:

(10)     .

The first component of the GR decomposition relative to the average labour productivity of exiting firms (hereafter GR(X)) is identical to the first component of the GR decomposition. The second and third components measure the contribution from changes in continuing firms’ labour input shares and the labour productivity of entering firms after adjusting for the average labour productivity of exiting firms.