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We provide an example that shows that in the Alesina and Angeletos (2005) model one can obtain multiplicity even if luck plays no role in the economy. Thus, it is not critical that the noise to signal ratio be increasing in taxes, or that desired taxes are increasing in the noise to signal ratio.


Fairness and Redistribution: Comment †
By Rafael Di Tella and Juan Dubra* In an influential paper, Alesina and Angeletos (2005)-henceforth, AA-argued that a preference for fairness could lead two identical societies to choose different economic systems. In particular, two equilibria might arise: one with low taxes and a belief that the income-generating process is "fair" because effort is important (an "American" equilibrium) and another with high taxes and the belief that the process is "unfair" because luck prevails. Piketty (1995) had shown that a similar pattern could arise from standard preferences if initial beliefs about the relative importance of effort and luck in generating income differed across the two societies, while Benabou and Tirole (2006) study this issue using more realistic preferences (Buera, Monge-Naranjo, and Primiceri 2011 discuss the evolution of beliefs about economic systems). A key contribution of AA is to obtain these two equilibria from identical societies assuming agents prefer outcomes that are fair, an important modification because fairness considerations seem central in the demand for redistribution, and because in several settings (as in some ultimatum games) such preferences for fairness can lead to large (material) inefficiencies.
In this note we report a difficulty we encountered when interpreting the results in AA: we find multiplicity (and demand for redistribution) even if luck plays no role. In other words, there is multiplicity even if the equilibrium tax rate is independent of the signal-to-noise ratio (a quantity that expresses how important effort is, relative to luck, in the determination of income). This conflicts with the notion that the signal-to-noise ratio plays a central role in generating multiplicity with AA preferences for fairness.

I. The AA Model
The economy is populated by a measure 1 continuum of individuals i ∈ [0, 1], who live for two periods: in the first period the individuals accumulate capital; in the middle of their lives the taxes are set; in the second period, individuals exert effort (work). Total pretax lifetime income is (1) where A is talent, k is the capital accumulated during the first period, e is effort during the second period, η is "noise" or "luck", and α ∈ (0, 1) is a technological constant. The government imposes a flat tax rate τ on income and redistributes the proceeds in a lump sum fashion, so that the individual's consumption is, for govern- where u i is private utility from own consumption, investment and effort, β i is an impatience parameter, γ is "distaste for unfair outcomes," and Ω is a measure of the social injustice in the economy. AA assume that A, η, and β are i.i.d. across agents, and that for δ = A 2 β, Cov(δ, η) = 0. We let _ δ be the mean of δ, and δ m its median; AA also assume Δ = _ δ − δ m ≥ 0 and normalize δ m = 2. Similarly, _ η is the mean of η and η m its median.

AA define social injustice as
where u i is the actual level of private utility, and  u i is a measure of the "fair" level of utility the individual should have (deserves) on the basis of his talent and effort. They define The individual chooses k when taxes haven't been set, so anticipating a tax rate of τ e (which will be equal to the actual τ in equilibrium), he maximizes with respect to k, and using the actual tax rate in equation (3) maximizes with respect to e to obtain (4) The critical features of the model that generate equilibrium multiplicity are (a) that the optimal tax rate is decreasing in the signal-to-noise ratio and (b) that the equilibrium signal-to-noise ratio is in turn decreasing in the tax rate (p. 970).
We now present an example with no noise, no luck, and therefore a constant noiseto-signal ratio, that still has multiple equilibria.
Set δ m = 2, This means, in principle, but we will now check it, that given an expected tax rate of τ e j for j = l, m, h, the tax rate that maximizes the utility of the voter with the median values of the shocks is τ e j ; that is, there is multiplicity of equilibria, even though luck plays no role.
We now check that given a tax rate of τ e l the tax rate that maximizes the utility of the individual with the median values of the shocks is again τ = τ e l (the cases of τ e m and τ e h are similar and omitted). First note that the optimal tax rates for _ 100,000 we still have U m (1, τ e l ) < U m (0, τ e l ) < U m ( τ e l , τ e l ). Therefore, the tax τ ∈ [0, 1] that maximizes U m (τ, τ e l ) must solve dU(τ, τ e l )/d τ = 0. We know that d U( τ e l , τ e l )/d τ = 0 (by definition of τ e l ), so we only need to check that it is the global maximum among τ ∈ [0, 1] which is ensured by concavity in the domain: d 2 U m (τ, τ e l )/d ( τ 2 ) ≈ −1.296 × 1 0 −5 τ 2 + 1.0331 × 1 0 −2 τ − 1.3781 < 0 (for all τ ∈ [0, 1]).

B. Discussion
Note that with no luck in the model, σ η 2 = 0, for τ e = τ we obtain that for σ  y 2 the variance of "fair" income (the signal in AA), Ω = τ 2 σ  y 2 = σ δ 2 τ 2 ( 1 − τ) 2 , which is nonmonotonic in τ, while one might expect unfairness to increase with taxes. 1 Hence, it is possible that the key insights in AA can be restored if other definitions of what is fair are used. For example, one alternative definition involves keeping taxes in the definition of fair consumption (in AA "fair" consumption involves no taxes and no luck). 2 Alesina and Cozzi (2012) analyze multiplicity using another approach where the AA preferences are normalized by average income. Another possibility would be to insist that the effort imputed in "fair" consumption takes into account that there are no taxes. In other words, it may be more reasonable to modify AA so that the k i and e i used to substitute into  y i in equation (2), are not those associated to the case where taxes may be positive. Finally, one may also insist on preferences for fairness that are consistent with the empirical evidence. For example, Levine (1998) andRotemberg (2008) demonstrate that preferences for "reciprocal altruism" are consistent with the available evidence from the ultimatum games, while Di Tella and Dubra (2010) show that they lead to multiplicity in an economy similar to that presented in AA.
One difficulty for exploring these conjectures in the AA framework is that a counterexample to the main theorem can be produced because AA claim that the individual with the median values of the shocks is the median voter, but in general he is not. In the online Appendix we give an example where the equilibrium tax rate, the one preferred by the median voter, is not the one identified in AA. The tax rate identified as the equilibrium in AA would be defeated in voting by the one preferred by the median voter (which can be shown to be a Condorcet winner, even if the median voter theorem does not apply). This wedge between the prediction of the AA model and what would happen in that economy is relevant, since it is currently not known if in the AA model multiplicity can arise when the equilibrium tax rate is one that, when anticipated, maximizes the utility of the median voter (in one special case, when _ δ = δ m , Di Tella, Dubra, and MacCulloch 2010 show how to analyze the AA model, establishing that the median voter's preferred tax rate is a Condorcet winner. But this case is not very relevant empirically, since it implies mean income equal to median income, and does not allow for a Meltzer-Richard effect.). 3 In brief, we believe that the main point in AA, namely, that a preference for fairness can lead to multiple equilibria, is potentially valid but some aspects of the particular framework they propose need to be revised.