Quasi partial equilibrium algebra and a graphical analysis
Where w is the log real wage, and y is log income; log income is a function of the real low income wage bill. Labor demand determines the amount of labor employed.
Take the partial differential with respect to the minimum wage, wmin, and rearrange:
There is kind of a “multiplier” here, in that changes in the real wage bill spur income changes that affect the demand for labor and in turn affect the real wage bill. Note that each component can be signed; however, the overall responsiveness of the real wage bill to the minimum wage cannot unless other assumptions are made. For instance, if one eliminates spillover effects:
Then the impact of the real minimum wage on the real wage bill is given by:
Finally, if the elasticity of the wage with respect to the minimum wage is unity, then the real wage bill rises as long as the (absolute value) of the labor demand elasticity is less than one.
This analysis can be depicted graphically, as in Figure 1 below.
Figure 1: Low wage labor market, initial equilibrium and after imposition of minimum wage.
Initially, quantity labor supplied equals quantity demanded at N1. With the imposition of a minimum (real) wage, the wage rate rises from W1 to W2. In standard partial equilibrium analysis, there is excess labor supply Ns2 – Nd2. However, labor demand is derived; the income variable shifts out the labor demand curve (plausible if the marginal propensity to consume is high for low income households, and the wage bill increases).
Then, quantity of labor demanded rises to Nd3. While unemployment rises, employment also rises. Hence, the standard Econ 1 prediction that employment necessarily falls in response to the imposition of a (binding) minimum wage is a special case of a more general model.
Notice that the impact on the real wage bill, even when no spillover effect is present, is ambiguous. The original real wage bill is the orange shaded area, while the bill afterwards (once again ignoring spillover effects) is the lined area. Clearly, if demand is inelastic, then the wage bill will increase.
The larger the response of income to the low income wage bill, the larger the ultimate increase in the wage bill (to the dotted area).
It’s important to recall that the proposed $9 minimum wage will be eroded by some inflation by the time it is fully implemented, by end-2015. Figure 2 depicts the time series of nominal and real minimum wage rates.
Figure 2: Nominal minimum wage per hour (blue), and real (in Dec. 2012$) (red). 2015M12 real value assumes 2% inflation from 2012M12 through 2015M12. Source: about.com, BLS and author’s calculations.
More discussion by Wonkblog, [Thoma/MoneyWatch], as well as this 2006 post. Here are 2007 CBO calculations of the impact of a minimum wage increase, assuming net employment impact is zero. For a full information/partial equilibrium approach, see Keith Hennessey (whose logic implies that we should immediately drop the minimum wage to zero).