An international cohort comparison of size effects on job growth

Abstract

The contribution of different-sized businesses to job creation continues to attract policymakers’ attention; however, it has recently been recognised that conclusions about size were confounded with the effect of age. We probe the role of size, controlling for age, by comparing the cohorts of firms born in 1998 over their first decade of life, using variation across half a dozen northern European countries Austria, Finland, Germany, Norway, Sweden and the UK to pin down size effects. We find that a very small proportion of the smallest firms play a crucial role in accounting for cross-country differences in job growth. A closer analysis reveals that the initial size distribution and survival rates do not seem to explain job growth differences between countries, rather it is a small number of rapidly growing firms that are driving this result.

Appendix

1.1 A framework for the decomposition of survivor job growth

Firms at birth (in the present case 1998) are denoted by firm$^{b}$, and jobs at birth by job$^{b}$, so average firm size (measured by jobs per firm) at birth, avjob$^{b}$, can be defined as,

$$\begin{aligned} \hbox {avjob}^{b} = \frac{\hbox {job}^{b}}{\hbox {firm}^{b}} \end{aligned}$$

(3)

and we can denote average firm size for each of the four size-bands by avjob$^{b}_{i}$ where $i$ runs from 1 to 4.

Let us also define a set of shares, firmsh$^{b}_{i}$ , where,

$$\begin{aligned} \hbox {firmsh}^{b}_{i} = \frac{\hbox {firm}^{b}_{i}}{\hbox {firm}^{b}} \end{aligned}$$

(4)

(and, of course, $\sum _{i=1}^{4} \hbox {firmsh}^{b}_{i}=1$)

We can now use the expression for shares to expand the definition of avjob$^{b}$,

$$\begin{aligned} \hbox {avjob}^{b} = \sum _{i=1}^{4} (\hbox {firmsh}^{b}_{i} \times \hbox {avjob}^{b}_{i}) \end{aligned}$$

(5)

Consider next the firms which survive to the “terminal” period (in the present case 2008) firm$^{bs}$. The ratio of survivors to all firms at birth is the survival rate, denoted here by $\delta$,

$$\begin{aligned} \hbox {firm}^{bs}=\delta \times \hbox {firm}^{b} \end{aligned}$$

(6)

We can also define, in a parallel fashion, a survival rate $\delta _{i}$ for each size-band category and use it to re-write the definition of $firmsh$ for the survivors,

$$\begin{aligned} \hbox {firmsh}^{bs}_{i} = \frac{\delta _{i}\times \hbox {firm}^{b}_{i}}{\delta \times \hbox {firm}^{b}} \end{aligned}$$

(7)

So we can write the average firm size for survivors at birth, avjob$^{bs}$, as,

$$\begin{aligned} \hbox {avjob}^{bs} = \sum _{i=1}^{4} (\hbox {firmsh}^{b}_{i}\times \hbox {rsrb}_{i}\times \hbox {avjob}^{bs}_{i}) \end{aligned}$$

(8)

where $\frac{\delta _{i}}{\delta }$ is the between “relative survival ratio” (rsrb$_{i}$).

The survival rate varies within size-bands as well as between size-bands, so we account for this by defining a between “relative survival ratio” effect ($rsrw_{i}$)— the ratio of the average size at birth of survivors in a size-band to the average size at birth of all firms in that size-band,

$$\begin{aligned} \hbox {rsrw}_{i} = \frac{\hbox {avjob}^{bs}_{i}}{\hbox {avjob}^{b}_{i}} \end{aligned}$$

(9)

Combining these two expressions we can write,

$$\begin{aligned} \hbox {avjob}^{bs} = \sum _{i=1}^{4} (\hbox {firmsh}^{b}_{i}\times \hbox {rsrb}_{i} \times \hbox {rsrw}_{i} \times \hbox {avjob}^{b}_{i}) \end{aligned}$$

(10)

Finally, if we define a growth ratio (growth$_{i}$), expressing average firm size in the terminal period (avjob$^{t}_{i}$) as a ratio to the average size of survivors at birth,

$$\begin{aligned} \hbox {avjob}^{t}_{i}= \hbox {avjob}^{bs}_{i} \times \hbox {growth}_{i} \end{aligned}$$

(11)

So we can now write,

$$\begin{aligned} \hbox {avjob}^{t} = \sum _{i=1}^{4} (\hbox {avjob}^{b}_{i} \times \hbox {firmsh}^{b}_{i}\times \hbox {rsrb}_{i} \times \hbox {rsrw}_{i} \times \hbox {growth}_{i}) \end{aligned}$$

(12)

by definition,

$$\begin{aligned} \hbox {growth} = \frac{\hbox {avjob}^{t}}{\hbox {avjob}^{bs}} \end{aligned}$$

(13)

so finally,

$$\begin{aligned} \hbox {growth} = \frac{ \sum _{i=1}^{4} (\hbox {avjob}^{b}_{i} \times \hbox {firmsh}^{b}_{i}\times \hbox {rsrb}_{i} \times \hbox {rsrw}_{i} \times \hbox {growth}_{i})}{ \sum _{i=1}^{4} (\hbox {avjob}^{b}_{i} \times \hbox {firmsh}^{b}_{i}\times \hbox {rsrb}_{i} \times \hbox {rsrw}_{i})} \end{aligned}$$

(14)

and this is the expression which appears in the main text (Table 9).

1.2 The decomposition of the Austrian growth ratios

The average job/firm at birth, for survivors at birth, and survivors at age 10 can be written as the sum of weighted average jobs/firm ($wavjob$) overs size-bands. So the difference between birth, survivors at birth and survivors at age 10 can be written as differences in the weighted average terms. As we can see from Table 3 of the paper, the first pair of differences depend on the effect of the two relative survival rates, whilst the second pair depend only on relative growth rates.

In general,

$$\begin{aligned} \Delta (a\times b) \equiv \Delta a \times b + \Delta b \times a + \Delta a \times \Delta b \end{aligned}$$

(15)

Using Eq. (1), we can calculate the difference—wavjob$^{bs}$ less wavjob$^{b}$—as the sum of terms (by size-band) involving: $\Delta \hbox {avjob}^{b}$ (avjob$^{bs}$ less avjob$^{b}$) and $\Delta \hbox {firmsh}^{b}$ (firmsh$^{bs}$ less firmsh$^{b}$). The results of this calculation are shown in panel (a) of the table. Similarly, we can calculate the difference—wavjob$^{t}$ less wavjob$^{bs}$—as the sum of terms (by size-band) involving: $\Delta \hbox {avjob}^{bs}$ (avjob$^{bs}$ less avjob$^{b}$).³⁰ The results of this calculation are shown in panel (b) of the table.

Although the interpretation of the results in panel (a) of the table is complicated by the fact that some entries are positive and others negative, nevertheless the overall pattern seems quite clear. The effects of the “between” survival ratio—which drives the difference in column (2)—is considerably more important than the effects of the “within” survival ratio in column (1). Indeed, the only figure of any size in column (1) is that for the smallest size-band and, remember from Table 4 in the paper, this is the only “within” ratio of any size). The interpretation of the results in panel (b) is more straightforward since we only have the growth terms to consider, and the finding is very clear-cut: it is the growth rate of the 1–4 size-band which has very much the largest effect (Table 10).

1.3 The decomposition of the size-band 1–4 growth ratio

The strategy here follows along similar lines, as the “principal decomposition”, using where possible the same notation. Since all the firms and jobs being referred to here originate from the 1–4 size-band this subscript has been suppressed, and since we are now concerned only with 2008 survivors, by definition, the stock of firms at birth and in 2008 is the same, so the “survivor” superscript ($bs$) is no longer necessary. However, we do need to distinguish size-bands at birth from size-bands in 2008, these will be denoted by $b$ for birth and $t$ for 2008.

Let us define a set of shares which record the proportions of surviving firms from size-band 1–4 in each “destination” size-band ($i$), mob$_{i}$ , where,

$$\begin{aligned} \hbox {mob}_{i} = \frac{\hbox {firm}^{t}_{i}}{\hbox {firm}^{t}} \end{aligned}$$

(16)

(and, of course, $\sum _{i=1}^{4} \hbox {mob}_{i}=1$)

We can now use the expression for shares to expand the definition of avjob$^{t}$,

$$\begin{aligned} \hbox {avjob}^{t} = \sum _{i=1}^{4} (\hbox {mob}_{i} \times \hbox {avjob}^{t}_{i}) \end{aligned}$$

(17)

We are interested in the growth of firms, so we can divide by size at birth ($\hbox {avjob}^{b}$),

$$\begin{aligned} \frac{\hbox {avjob}^{t}}{\hbox {avjob}^{b}} = \sum _{i=1}^{4}\frac{ (\hbox {mob}_{i} \times \hbox{avjob}^{t}_{i}}{\hbox{avjob}^{b}}) \end{aligned}$$

(18)

Now expanding the denominator on the right hand side we can re-write the expression as,

$$\begin{aligned} \frac{\hbox {avjob}^{t}}{\hbox {avjob}^{b}} = \sum _{i=1}^{4} \left( \hbox {mob}_{i} \times \frac{\hbox {avjob}^{t}_{i}}{\hbox {avjob}^{b}_{i}} \times \frac{\hbox {avjob}^{b}_{i}}{\hbox {avjob}^{b}}\right) \end{aligned}$$

(19)

The second term on the right hand side is the ratio of avjob in 2008 to avjob at birth for a destination size-band, so it can be interpreted as the size-band-specific growth rate gr$_{i}$. The third term is the ratio of avjob for firms in a destination size-band to the average size of 1–4 size-band firms at birth, so it is a variety of “selection” effect, denoted sel$_{i}$. So we have,

$$\begin{aligned} \hbox {gr}_{i}=\frac{\hbox {avjob}^{t}_{i}}{\hbox {avjob}^{b}_{i}} \end{aligned}$$

(20)

and,

$$\begin{aligned} \hbox {sel}_{i}=\frac{\hbox {avjob}^{b}_{i}}{\hbox {avjob}^{b}} \end{aligned}$$

(21)

Now re-writing the expression,

$$\begin{aligned} \frac{\hbox {avjob}^{t}}{\hbox {avjob}^{b}} = \sum _{i=1}^{4} (\hbox {mob}_{i} \times \hbox {gr}_{i} \times \hbox {sel}_{i}) \end{aligned}$$

(22)

and this is the expression which appears in the main text (Table 11).

Table 9 Job growth decomposition: birth to 2008, Austria, Finland, Germany, Norway, Sweden and UK

Table 10 The decomposition of the Austrian growth ratios

Table 11 Austria, Finland, Germany, Norway, Sweden and UK: contributions of 1–4 size-band at birth to job growth ratio by destination (2008) size-band