# Decomposition Analysis: By Population Subgroup or by Income Source

The idea underpinning decomposition analysis is that the overall income distribution can be described in terms of a relatively small number of constituent elements, and hence changes in the overall distribution can be analysed in terms of changes in those elements. One can either classify the total population into groups according to individuals' and households' characteristics and then examine the distributions within and between groups, or one can break down total income into its sources and look at the distributions of each type of income and the relationships between them. Although the two approaches are conceptually distinct, they yield similar conclusions about recessions if individuals are classified into groups according to the principal income source of their household (as is done in the Atkinson-Brandolini model).

The idea that different sources of income can be associated with different groups of individuals within the population is used by Muriel and Sibieta (2009) in their analysis of the distributional impact of UK recessions. For example, they state that 'the economic literature gives a reasonably clear answer as to which groups' living standards are likely to be most cyclical, and hence worst affected by recessions—we expect to see strong effects of recessions on the incomes of working-age individuals, but weaker effects on individuals who are retired or who are not strongly attached to the labour force' (2009: 14).

To elaborate, suppose that individuals can be classified into one of three groups according to their household's main income source: (a) 'rentier households' whose main income is from financial assets (including self-employed professionals; rich individuals living off income from the stock market; rich pensioners living off occupational and private pensions); (b) working households whose income is from employment income; and (c) non-working households whose income is largely from the state (unemployed working- age people; pensioners with income only from a state retirement pension). There is an income distribution for each class with mean income highest for rentier households and lowest for non-working households; the density functions for the classes—summarizing the concentration of persons at each point along the income range—are fa(y), fb(y), and fc(y). The density function for the incomes of the population overall, fy), is a weighted average of the subgroup distributions:

where pj is the fraction of the population in group j, and pa + pb + pc = 1.

It follows that recessions have impacts on the overall distribution through several channels. There may be changes in the population shares of each group: a rise in unemployment corresponds to an increase inpc and a fall inpb. Other things being equal, this change shifts the income distribution for the population as a whole to the left: average living standards decline and absolute poverty increases. The impact on overall inequality is unclear because the net effect also depends on the location and shape of the distribution for each subgroup (summarized by its density function).

Recessions induce changes in the location and shape of the income distribution for each group. For example, there may be a shift to the left in the rentier households' distribution associated with declines in stock prices and interest rates on financial assets. There may also be a shift to the left in the income distribution among working households combined with a greater dispersion associated with reductions in work hours or pay cuts among employees in some occupations and little change in other occupations. How the income distribution among non-employed households is affected depends on the nature of changes to benefits and state retirement pensions (and the taxes that finance them). If benefits are not uprated at the same rate as earnings increase (e.g. because of fiscal consolidation measures introduced as a consequence of a recession), the growing gap between the incomes of unemployed and working households increases inequality (between-group inequality is greater).

There are further complications. The discussion has conflated distributions among individuals with distributions among households (our interest). The distributional impact of a general rise in the unemployment rate depends on the extent to which job loss is correlated within multi-adult households—is there a rise in the share of households with no work at all or simply a change in the shares of single- and dual-earner households? The discussion has also ignored behavioural responses to the first-round changes, but they may also play important roles.

Income loss as a consequence of the recession may lead people to alter their living arrangements, for instance. Greater unemployment may lead more young people to return to live with their parents, and unrelated adults may be more likely to share accommodation to benefit from economies of scale. Formerly retired workers may return to the labour market to offset the impact of lower interest rates on their income from assets. The decomposition framework implicitly assumes that the total population is fixed in number, but a recession may also induce more people to leave a country and fewer to enter— though one would expect the distributional impact to be small for most countries since the numbers of people moving is relatively small.

If one employs the subgroup decomposition approach to analyse the impact of recessions, the key elements that are tracked over time are the sizes of the various subgroups, the distributions within each subgroup, and the income gaps between the subgroups. By contrast, the income source decomposition approach characterizes the channels by which a recession has effects on income inequality in terms of changes in three sets of elements: the share of each type of income in total income, the inequality of each income type, and the correlations between the income sources. (Explicit formulae are provided in Chapter 2, drawing on the work of Shorrocks 1982a, b.) If one describes the shapes of the distributions of the income sources using parametric functional forms, one can also derive expressions relating the overall poverty rate to the level of the poverty line and the parameters summarizing distributional shape. (See e.g. Gottschalk and Danziger 1985 who use displaced lognormal distributions.)

The distribution of household income is typically much more equal than the inequality of any one of its constituent sources: see for example Jenkins (1995: table 6) for UK examples. For instance, although income from investments and savings is very unequally distributed in most countries, its share in total income is relatively small, hence moderating the disequalizing contribution of this source. Income taxes are broadly progressive in relation to employment income and this has an equalizing effect. However, when looking at the impact of macroeconomic downturn, it is the relative size of the changes in the decomposition elements that is of particular relevance.

Most decomposition analyses show that employment income typically makes a larger contribution to household income inequality than does every other source (Chapter 2 below confirms this). This suggests that the distributional impact of a recession is largely driven by what happens to the contribution of income from the labour market, but this is not the only relevant channel. On the one hand, the share of labour income typically falls in macroeconomic downturns, because of greater unemployment. This may have an equalizing impact because less weight is given to an income source that comprises a relatively large share of total household income. But, on the other hand, the combined share of all other income sources must rise, which increases inequality if sources with increased shares are those that are more unequally distributed than employment income (e.g. income from investments and savings). The net effect on overall household income inequality depends on the precise nature of the recession, and the policy responses to it (which may change the cash transfers received and taxes paid). For example, does the share of investment income fall or rise, and how much does the share of income from cash transfers increase? The inequalities of each income source may also change: for example, if there are reductions in work hours for middle- and lower-paid workers but no changes for the higher-paid salar- iat, the inequality of employment income will increase, and this has a dis- equalizing impact on the household income distribution.

Formal analysis of the distributional impact of recessions using inequality decomposition formulae (by group or income source) is rare, largely because few countries have income data for a sufficiently long period of time to enable analysts to isolate the impact of downturns separately from other factors affecting inequality trends. One exception is the study by Aaberge etal. (2000) of the distributional impact of the early-1990s recession in Sweden— this is discussed in Section 1.4. More common is informal description. For example, Krueger etal. (2010) usefully review the association between inequality among working-age households and the business cycle for a number of countries. They observe that 'in all countries earnings inequality at the bottom increases during recessions' (2010: 8), but '[t]he general pattern is that, in all countries and in all recessions, inequality in disposable income during the recession rises less than inequality in earnings, reflecting the significant role played by automatic stabilizers. Quantitatively this role appears to be larger in some countries (i.e. Canada, Sweden, Germany) and smaller in others (US, Italy)' (2010: 9).

In sum, analysis of the distributional impact of recessions using decomposition approaches (whether by population subgroup or income source) shows that it is possible for a macroeconomic downturn to lead to either a decrease or an increase in overall income inequality. There are multiple elements that may change in offsetting directions, so the net effect is unclear in principle. There is some descriptive evidence that, in practice, recessions are associated with greater inequality, but that evidence mainly refers to working- age households rather than the whole population, and what happens to other groups in the population can affect the distributional outcome for the population as a whole.

In Chapter 2, we draw on the decomposition by income source framework to guide our analysis. In the rest of this chapter, we consider other approaches and tools, and review empirical work about the distributional impacts of recessions.