Abstract
Different trends in the distribution of income across countries and in the same country over time are typically observed. The general question we are interested in is to know where these inequalities come from and what explains their differences. By restricting our attention to an artisan economy with no taxation and where individuals have identical preferences but different productivities, we study the impact on the inequality of labour income of particular changes in the way productivities are allocated. Assuming next that the distribution of productivities is fixed, we look for the modifications of the preferences that lead to a more even distribution of income. Finally, we examine the question of how simultaneous changes in preferences and in the distribution of productivities interact in the shaping the distribution of labour income.
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Notes
The Gini index is known to be more sensitive to income changes at the middle than at the tails of the distribution (see, e.g., Atkinson 1970, Section 4). Since there is evidence that the income gaps have mostly increased among the richest and the poorest households, one may reasonably expect that the figures reported in the OECD report underestimate the changes in inequality during the last three decades.
Although we do not follow this route, it must be emphasised that focusing on the distribution of hourly wages has its own merits from the equality of opportunity point of view. Indeed, because the wage rate essentially captures the size of the budget set from which the agent optimally chooses her bundle, it can be taken as a measure of the size of the agent’s opportunity set (see Fleurbaey and Maniquet 2015).
Presumably, the variance of the logarithms is extensively used by labour economists because of it decomposability properties allowing one to measure the respective contribution to overall earnings inequality of wages and hours worked.
Equivalently, in the labour-consumption space, the slopes of the indifference curves decrease with labour time for any given consumption level.
We note that \({C}(\,\cdot \,;{\tilde{u}}) = {C}(\,\cdot \,;{\hat{u}})\) and \({Z}(\,\cdot \,;{\tilde{u}}) = {Z}(\,\cdot \,;{\hat{u}})\), whenever \({\tilde{u}} = {\varphi } \circ {\hat{u}}\) and \({\varphi }\) is increasing. The utility functions \({\tilde{u}}\) and \({\hat{u}}\) represent distinct preferences if there exists \({\varphi }\) increasing and \((c,{\ell }) \in {\mathbb R}_{++} \!\times \! (0,T]\) such that \({\tilde{u}}(c,{\ell }) \ne {\hat{u}}(c,{\ell })\).
Given a binary relation \(\ge _{J}\) over a set \({\mathscr {S}} \!\subseteq \! {\mathbb R}^{n}\) (\(n \geqslant 2\)), we define in the usual way its asymmetric and symmetric components, which we indicate by \(>_{J}\) and \(\sim _{J}\), respectively.
Note however that, if \({\nu }({\mathbf w}^{*}) \ne {\nu }({\mathbf w}^{\circ })\) and \({\mathbf w}^{*} \ge _{RD} {\mathbf w}^{\circ }\), then it is possible to find a distribution \({\tilde{\mathbf {w}}}\) such that (i) \({\mathbf w}^{*} \ge _{RD} {\tilde{\mathbf {w}}}\) and \({\nu }({\mathbf w}^{*}) = {\nu }({\tilde{\mathbf {w}}})\) and (ii) \({\tilde{\mathbf {w}}} \sim _{RD} {\mathbf w}^{\circ }\). For this, it suffices to choose \({\tilde{\mathbf {w}}} = ({{\nu }({\mathbf w}^{*})}/{{\nu }({\mathbf w}^{\circ })}) \, {\mathbf w}^{\circ }\).
As is well-known, the variance of the logarithms violates the principle of transfers according to which a progressive transfer reduces inequality (see Foster and Ok 1999). In this respect, it is worth noting that the variance of the logarithms is coherent with the relative differentials quasi-ordering provided that the distributions have the same geometric mean. In other words, dispersion as measured by the variance of the logarithms always decreases as the result of a uniform proportional progressive transfer (see Ebert and Moyes 2018, Section 5 for details).
There is no loss of generality to restrict attention to consumption vectors that are non-decreasingly arranged given the Spence–Mirrlees condition and the assumption that individuals are labelled according to their productivities.
We have chosen separable utility functions for making computations easier but similar results can be obtained with non-separable utility functions.
The notation \({C}({\mathbf w}^{*};{u}) \ne _{RL} {C}({\mathbf w}^{\circ };{u})\) in Table 1 is intended to mean that the distributions \({C}({\mathbf w}^{*};{u})\) and \({C}({\mathbf w}^{\circ };{u})\) cannot be ranked by the relative Lorenz criterion.
The only case where consumption increases or decreases in the same proportion as productivity is when \({\eta }(C,w;{u}) = 1\). This implies that consumption is proportional to productivity, which happens for instance in the case of the utility function \({u}^{(4)}(c,{\ell }) = {\ln } \, c - {\ell }\).
For instance, if one imposes the further restriction that \(0 \leqslant {\eta }(C,w;{u}) < 1\), then condition (4.6), in conjunction with the Spence–Mirrlees condition, implies that the consumption function is concave. On the other hand, \({u}^{(3)} (c,{\ell }) = \ln \, c - ({1}/{c}) - {\ell }\) is an example of a utility function that generates a consumption function that is concave but whose elasticity is increasing everywhere.
Leaving aside the fact that productivities are difficult to observe, the major problem is that the consumption patterns one observes in practice are determined jointly by the distribution of productivities and by different institutional arrangements. This makes it difficult to separate the changes in the distribution of consumption that stem from modifications in the allocation of productivities from those that result, for instance, from modifications of the tax system, all other things being the same.
None of the utility functions that we have retained for illustration have the property to exhibit decreasing or constant consumption derivatives in the logarithm.
In fact, it can been shown that the conditions identified in Sect. 5 are necessary and sufficient in order for a uniform proportional progressive transfer of productivities to give rise to a generalised Lorenz improvement in consumption.
Another possibility would be to compare directly the (bi)dimensional distributions of consumption and leisure by means of criteria like those suggested by Atkinson and Bourguignon (1982) as was done, for instance by McCaig and Yatchew (2007). Then, the problem would amount to identifying the properties of the consumption function—or, equivalently, the restrictions to be placed on the preferences—that ensure that less dispersed productivities result in less unequally distributed consumption-leisure bundles.
Consider the utility function \({u}^{(6)}(c,{\ell }) := -e^{-c} - {\ell }\) and the distributions of productivities \({\mathbf w}^{\circ } = (1.50,1.55,2.15,2.20)\) and \({\mathbf w}^{*} \simeq (1.725,1.782,1.869,1.913)\). Distribution \({\mathbf w}^{*}\) is obtained from \({\mathbf w}^{\circ }\) by means of a uniform proportional progressive transfer plus a small increment to the benefit of the most productive agent, hence \({\mathbf w}^{*} >_{MERD} {\mathbf w}^{\circ }\). Because the elasticity of \({C}(w,{u}^{(6)})\) is decreasing, we deduce from Proposition 4.3 that \({C}({\mathbf w}^{*},{u}^{(6)}) >_{RL} {C}({\mathbf w}^{\circ },{u}^{(6)})\). However, the fact that income inequality decreases unambiguously does not prevent inequality of utilities to increase. Indeed, we get \({V}({\mathbf w}^{\circ },{u}^{(6)}) >_{RL} {V}({\mathbf w}^{*},{u}^{(6)})\), where \({V}({\mathbf w}^{\circ },{u}^{(6)})\) and \({V}({\mathbf w}^{*},{u}^{(6)})\) represent the distributions of utilities at the equilibrium when the allocations of productivities are respectively \({\mathbf w}^{\circ }\) and \({\mathbf w}^{*}\). Now, if we substitute for \({u}^{(6)}\) the utility function \({\tilde{u}}^{(6)} := \bigl (e^{({u}^{(6)} + 5)}\bigr )^{2}\), which is an equally valid representation of the preference ordering, then we obtain the converse ranking, namely \({V}({\mathbf w}^{*},{\tilde{u}}^{(6)}) >_{RL} {V}({\mathbf w}^{\circ },{\tilde{u}}^{(6)})\).
For linear-in-labour preferences, consumption is independent of exogenous income, which implies that the way the latter is allocated among the population in the artisan economy has no impact on income inequality. When preferences are linear-in-consumption, this is no longer true: depending on the amount of non-labour income, the elasticity of consumption with respect to productivity may be increasing, decreasing or even non-monotonic. Choose the utility function \({u}^{(2)}(c,{\ell }) = c - e^{{\ell }}\) and let \({C}(w,m;{u}^{(2)})\) be the corresponding consumption function that depends now on productivity w and exogenous income m. After some algebra, we get \({C}(w,m;{u}^{(2)}) = m + w \, {\ln } \, w\) and it can be checked that \({\eta }(C,w,m;{u}^{(2)}) = {w \, (1 + {\ln } w)}/{(m + w \, {\ln } \, w)}\) is increasing on the interval \((1, + \infty )\) when \(m = 0\) and decreasing otherwise.
It is still an open question to know whether the constancy of the consumption elasticity is also necessary for inequality to decrease as the result of a uniform proportional progressive transfer of productivities.
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Acknowledgements
This paper forms part of the research project Heterogeneity and Well-Being Inequality (Contract No. HEWI/ANR-07-FRAL-020) of the ANR-DFG programme whose financial support is gratefully acknowledged. We are indebted to Stephen Bazen, the associate editor and two anonymous referees for very useful comments and suggestions when preparing this version. Needless to say, none of the persons mentioned above should be held responsible for remaining deficiencies.
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Ebert, U., Moyes, P. Talents, preferences and income inequality. Soc Choice Welf 51, 13–50 (2018). https://doi.org/10.1007/s00355-017-1105-1
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DOI: https://doi.org/10.1007/s00355-017-1105-1