Becker's comprehensive analysis leaves me with little to add, especially as I am not permitted to comment publicly on the constitutionality of the "big box" ordinance because (if it does go into effect) its constitutionality is likely to be challenged, and in my court to boot.
The first-order economic analysis of minimum wage laws shows that they reduce employment by raising the price of labor; the Law of Demand teaches that an increase in the price of a good reduces the quantity of it that is demanded. A second-order analysis complicates the picture. Price affects supply as well as demand. An increase in the price of labor might attract into the labor force individuals who, at the existing price, prefer to go to school, engage in crime, work part time, or subsist on welfare. If, moreover, there is a large sector exempt from the law, the law's main effect may be to shift workers to the exempt sector rather than to reduce overall employment. The higher wages in the covered sector, by driving up employers' costs in that sector, will tend to reduce the demand for the products and services produced by those employers and to increase the demand for substitute products and services produced in the exempt sector, which in turn will increase the demand for labor in that sector.
What seems relatively clear, however, is that the brunt of the disemployment effect of the minimum wage will be felt by marginal workers. For example, some teenagers whose marginal product (that is, their contribution to the employer's profits) was just at or only slightly above the minimum wage will if the minimum wage is raised be replaced by slightly more productive teenagers from affluent households who were not attracted to working when the wage was lower.
The smaller the sector covered by the minimum wage law (and the coverage of the "big box" ordinance is very limited), the more dramatic the disemployment effects of the law are likely to be. The demand for labor as a whole is inelastic, but the demand for labor by an individual company or a small group of companies is likely to be quite elastic. Not because the company can easily substitute capital for labor, but because it cannot pass on increased costs to its customers if it has many competitors who have lower labor costs by virtue of being exempt from the minimum wage. Such a company, assuming it faces an upward-sloping average-cost curve (meaning that its average cost rises with its output--the normal assumption about a firm's cost structure in a market with many firms, because if its costs were invariant to its output it could expand indefinitely), can control its labor costs only by reducing its output and thus laying off workers. One especially draconian way of doing this is by relocating the firm's plants or other facilities from the jurisdiction imposing the high minimum wage to a jurisdiction that has a lower minimum wage. Becker points out that this may be a consequence of the Chicago ordinance because it does not reach Chicago's suburbs. It is a reason for believing that state minimum wages are likely to have fewer disemployment effects that local minimum wages, and the federal minimum wage fewer disemployment effects than state minimum wages.
At the current minimum wage in Illinois of $7.75 an hour, an employee who works 2000 hours a year (a 40-hour week with two weeks of annual vacation) and is paid the minimum wage earns only $15,500 a year. This is a pittance, though if the minimum-wage employee's spouse is employed at a significantly higher wage, the family's income may not be at a hardship level. Similarly, the minimum-wage employee may be an elderly person who receives social security and Medicare and may have a company pension in addition. These possibilities show that minimum wage laws, even if they had no disemployment effects, would be a clumsy instrument for combating poverty. A better approach than raising the mininum wage would be increasing the earned-income tax credit (negative income tax), which is a method of increasing the earnings of marginal workers without confronting their employer with a higher cost of labor and thus inducing the employer to discharge those workers whose marginal product is lower than the minimum wage. But this would be difficult for an individual city or even state to do; it would require federal action.
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