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I'm interested in the following quote that came up in this earlier answer.

Second, when labor supply responses are along the extensive margin only, which is the empirically relevant case, the co-existence of a minimum wage with a positive tax rate on low-skilled work is always (second-best) Pareto inefficient.A Pareto improving policy consists of reducing the pre-tax minimum wage while keeping constant the post-tax minimum wage by increasing transfers to low-skilled workers, and financing this reform by increasing taxes on higher paid workers. Importantly, this result is true whether or not rationing induced by the minimum wage is efficient or not.

How are higher paid workers not worse off, given the increased taxes they have to pay?

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(Note that this answer implicitly makes reference to the specific model in Lee and Saez.)

Short answer: the increased taxes on high-skilled workers exactly offset the higher real wages they obtain from a decline in the minimum wage for low-skilled workers.

Longer answer: Suppose that I'm the government, and I decide to lower the minimum wage $\bar{w}$. The direct effect will make low-skilled workers currently earning the minimum wage worse off, while making high-skilled workers better off. (The fall in low-skilled wages means an increase in high-skilled wages.)

Furthermore, a lower minimum wage means that fewer low-skilled workers will be rationed out of the labor market. These workers will be better off.

As the government, I want to turn this into a Pareto improvement - which it currently isn't, because workers who are already earning the minimum wage are hurt by the decrease. I try the simplest possible offset: I adjust taxes so that everyone's after-tax wage rate is exactly the same as before. The combination of this tax rate change and the minimum wage decrease means that everyone's welfare is unchanged, except for some low-skilled workers who were previously unemployed and now can work. These workers are better off - hence overall we have a Pareto improvement. Great!

There's only one catch: I didn't verify that this policy was feasible for the government. Maybe the proposed change in tax rates would violate the government's budget constraint.

This is where some slightly more involved logic comes in, and it's useful to think about the infinitesimal case for simplicity. Before the policy change, output is produced by a mix of low- and high-skilled workers, who I'll call the "old" workers. The minimum wage increase leads to additional low-skilled workers entering the mix; I'll call them the "new" workers. These new workers produce additional output and earn income. At the margin, though, (pretax) wage equals marginal product - so when an infinitesimal number of new workers is added, they increase output by exactly as much as they draw away in earnings, and the total earnings of the old workers are unchanged. There's a redistribution in earnings among the old workers, toward the high-skilled and away from the low-skilled, but this is just a zero-sum redistribution, and the government can use taxes to reverse a zero-sum redistribution while maintaining a balanced budget - so this part is fine.

All we need to worry about now is the budgetary impact of the new low-skilled workers. But this is simple: we're initially taxing low-skilled work at some rate $\tau_1$, and the net budgetary effect of more low-skilled entry will be positive (even after the infinitesimal decrease in $\tau_1$ that's part of the reform) as long as $\tau_1>0$. New workers are good for the budget as long as their tax rate is positive.

So that wraps it up: we see that there is a potential Pareto improvement involving a decrease in the minimum wage and an offsetting change in tax rates, and this policy change is feasible as long as the initial tax rate on minimum wage work is positive: $\tau_1>0$. Lee and Saez emphasize this point (in conjunction with their other results) to demonstrate that in their model, the minimum wage can only be rationalized as a policy that's complementary to a labor subsidy $\tau_1\leq 0$. They really like this purported complementarity, and it's one of the most popular current arguments in favor of minimum wages.

(I happen to think that the way they prove this complementarity, which relies on the assume-a-can-opener assumption of efficient labor market rationing, is incredibly silly. But that's a different part of the paper, not directly related to your question here.)

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The reason is that at the same time the wage of the high-skilled increases.

By reducing the minimum wage, the number of people working in the low-skilled sector increases (involuntary unemployment is reduced) which leads to an increase in the wage of the high-skilled.

The corresponding proposition in the paper is Proposition 3, the argument of which is

" Suppose that the government reduces the minimum wage ($d\bar{w}<0$) while keeping $c_0$, $c_1$,$c_2$ constant. Reducing the minimum wage leads to a positive employment effect $dh_1 > 0$ as involuntary unemployment is reduced, improving the welfare of the newly employed workers and increasing tax revenue as $τ_1>0$. The increase $dh_1>0$ also leads to a change $dw_2 > 0$. However, because $h_1 d\bar{w} + h_2 dw_2 = 0$ through the no-profit condition (6), the mechanical fiscal effect of $d\bar{w}$ and $dw_2$, keeping $c_1$ and $c_2$ constant, is zero. Because $c_0$,$c_1$,$c_2$ remain constant, nobody's welfare is reduced. The increase in welfare due to the reduction in unemployment remains a-fortiori true if rationing is not efficient. Therefore, this reform is a second-best Pareto improvement." (Lee and Saez 2012)

If you can access the paper, useful complements on the general equilibrium dynamics determining wages are in Appendix A.

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  • $\begingroup$ "which leads to an increase in the wage of the low skills." Did you mean to say an increase in the wage of the high skills workers? $\endgroup$ – jmbejara Dec 15 '14 at 0:54

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