# RDP 2005-08: Declining Output Volatility: What Role for Structural Change? Appendix A: Factor Mobility and Output Volatility – A Simple Model

This appendix outlines a simple model that illustrates how output volatility could fall in response to reforms that allow greater mobility of productive resources in response to differential shocks across sectors. The model has two sectors, labelled 1 and 2, and labour is the only factor of production. There are two (divisible) units worth of labour available. Production functions are identical for each sector:

where yi is output of sector i, li is labour employed in sector i and 0<α≤1. Productivity shocks are embodied in Ai, which takes one of three possible values depending on three (equally likely) states of the world. In the steady state, Ai is assumed to be unity for both sectors, and demand is such that it is optimal to allocate one unit of labour to each sector, resulting in aggregate output, Y = 2. In the bad state of the world, sector 1 is assumed to suffer a negative productivity shock (with sector 2 unaffected), while in the good state of the world, sector 1 is assumed to benefit from a positive productivity shock (again with sector 2 unaffected). For the purposes of illustration, two parameterisations are considered, one with constant returns to labour (α = 1), and one with diminishing returns to labour (α = 0.7), broadly consistent with the labour share of income.

Consider two extreme cases of labour mobility. In one, regulations impede any transfer of labour across sectors and the allocation remains fixed according to steady state levels. In the other, these impediments are removed allowing labour to move freely so as to equate the marginal product of labour across sectors, which in competitive markets is equal to the economy-wide wage.[21] Results are summarised in Table A1.

Productivity   Labour allocation Total output Average output Variance of output State of the world 2.00 0.67 2 1 1 1 3.00 1 1 1 1 2.00 0 1 1 1 1.00 2.67 0.89 2 1 2 0 4.00 1 1 1 1 2.00 0 1 0 2 2.00 2.00 0.67 2 1 1 1 3.00 1 1 1 1 2.00 0 1 1 1 1.00 2.32 0.54 2 1 1.82 0.18 3.34 1 1 1 1 2.00 0 1 0 2 1.62 Source: authors' calculations

The main results are as follows. With flexible labour, output is higher under both the bad and good states of the world (average output is higher for the case of both constant and decreasing returns to labour). However, the comparison of the variance of aggregate output across inflexible and flexible labour markets depends on the nature of the production function. Under constant returns to scale, the flexible labour market case results in a higher variance of output than in the inflexible labour market case. In contrast, under decreasing returns to labour, the variance of output is less under the flexible labour market case. The variance of output in the inflexible and flexible labour market regimes is equivalent at reasonably high levels of α (that is, α equal to about 0.86). For α less than this, the gains in output during the bad state arising from the ability to reallocate labour are larger than the gains in the good state of the world.

The magnitude of the decline in the variance of output implied by the model described in Table A1 under the case of α = 0.7 appears relatively modest, especially considering that it compares the extreme cases of no flexibility and complete flexibility to reallocate resources across sectors. There are, however, likely to be other features of the real world that could act to amplify the impact of reforms that lead to more flexible and efficient reallocation of productive resources. For example, in reality, extended periods of unemployment can lead to a loss of human capital, thereby accentuating the impact of adverse shocks in a world where the unemployed are not as readily absorbed by those sectors faring relatively better during a downturn.

## Footnote

For simplicity, prices of outputs are assumed to be fixed to unity, which can occur if for example both outputs are tradable and the country in question is a small open economy. [21]