Effective measures tackle the root of inequality
To lessen inequality a progressive income tax is ineffective, and there are good alternatives. As employees look at their net income, the market will rebound by correcting for progressive taxation: a progressive income tax will lead to even bigger differences in (gross) salaries. Education, intrinsic motivation, and inheritance tax are more effective to lessen inequality.
Progressive income tax
A progressive income tax means higher income is taxed at a higher rate: the marginal tax rate is higher than the average tax rate. In other words, when you get a pay raise, a higher proportion of that raise is taxed than the proportion of your total income. A method for this is to create tax brackets: additional income over a certain threshold is taxed at a higher rate.
A progressive income tax has several (presumed) benefits: (1) a low tax on (low) income decreases the wedge between before and after tax income, and hence makes hiring (low income) employees more attractive; (2) progressive taxation of income optimizes the total tax take; (3) progressive taxation fosters a more equal income distribution. I will question whether progressive taxation indeed lessens inequality, and then offer alternatives.
A rebound effect reverts us back to inequality
Employers and employees will respond to progressive taxation in a way that will restore (part of) inequality. Employees look at their net earnings when accepting a job. When an employee ends up in a low tax bracket, this employee will accept a lower gross income. Likewise, when an employee ends up in a high tax bracket, this employee will demand more gross compensation.
After the introduction of a (more) progressive income tax, you would expect modest raises for those in the lower brackets and strong raises for those in higher brackets. From a gross income perspective inequality then even worsens. This rebound effect restores the “sufficient” level of net earnings, and inequality is back where it was.
Accounting for other effects makes little difference
A price effect counters this rebound effect. Employers would realize their best paid workers become pricier in gross salary terms. In a competitive market, these workers become less attractive to hire. Supply/demand dynamics will drive down high gross salaries. This price effect of progressive taxation will probably narrow the income gap slightly.
The price effect is again countered by a constraint effect: scarcity of skilled employees. Skilled employees must be motivated to take on more responsibilities. When these skills are essential to the business, that might mean companies will have to spend more on their high earners. This leaves less of the cake for low earners.
Counter effect on counter effect, it seems that progressive taxation might not dent inequality as much as hoped.
Less inequality: alternatives for progressive taxation
Progressive taxation of income is ineffective because taxes do not determine the initial income distribution. It is a blunt instrument to lessen inequality. It does not address the causes of an initial unequal income distribution, but only for the symptoms.
Alternative measures, that solve for the root causes, can lower inequality:
Accessible education (socially, financially) is where less inequality starts. The more society can supply the economy with skilled people, the less inequality will flow back. The less scarce skills are, the smaller income differences.
More attractive work demands lower compensation. Public sector workers (education, healthcare, …) tend to have lower salaries compared with private sector workers. These public sector workers seem more intrinsically motivated: part of their compensation is the value of contributing to society. The lesson is that when jobs are more attractive, they demand less compensation. If we can unlock more intrinsic motivation in high earners, through social change perhaps, that might lessen income inequality.
Inheritance tax targets the destination of earnings and not its source, contrary to an income tax, and therefore the rebound effect is less strong. Bequests entrench existing inequality, which obviously hampers efforts to lessen inequality. It seems better to incentivize high earners to spend their money in their own lifetime.
These alternatives will contribute more to future equality, than income tax ever will.