### Why Doesn't the Minimum Wage Cause Unemployment?

2/18/2013 11:08:00 PM
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Mark Thoma and others are puzzled about why the minimum wage doesn't seem to affect employment the way theory predicts for perfectly competitive markets. One answer is that the markets are not perfectly competitive. Maybe, but there isn't a whole lot of empirical evidence of monopsony either. So, what else could it be?

The conventional thinking looks like this:
We set a wage above the market clearing wage. At that wage, employers want to hire fewer workers, but workers want to work more hours--the minimum wage causes unemployment.

But that analysis violates even the most cursory look at real world markets: in the chart, everyone earns exactly the same wage. You could argue that this graph is for a specific subset of the labor market, but even that makes very little sense--in the real world we observe people of drastically different wages collaborating to produce the same products. That suggests that different-wage labor are to a very strong degree complements in the labor market. So let me postulate a production function in which we have workers getting paid different wages while collaborating to produce the same good: $$Y=AL_p^\alpha L_r^{1-\alpha}$$ where $L_p$ refers to labor supplied by "poor" workers and $L_r$ refers to labor supplied by "rich" workers. We can think of type $r$ and type $p$ workers as differing in their skill sets. Maybe $r$ types have college degrees and work as managers, while $p$ types lack college degrees. We can establish the inequality of their pay by analyzing the firm's profit maximizing conditions: $$profit=A L_p^\alpha L_r^{1-\alpha}-w_pL_p-w_rL_r$$ which is maximized whenever $$\frac{w_p}{w_r}=\frac{\alpha}{1-\alpha} \frac{L_r}{L_p}$$ Therefore, we can prove that $r$ types will be paid a higher wage than $p$ types whenever $$L_r<\frac{1-\alpha }{\alpha}L_p$$

My point is this: A minimum wage law aims to increase the ratio $\frac{w_p}{w_r}$. We know from profit maximization that this ratio depends two things: the relative factor intensity $\alpha$ as well as the ratio of the quantities of the two types of labor. We assume that $\alpha$ is fixed, so profit maximization means that the minimum wage law increases the ratio $\frac{L_r}{L_p}$. There are two ways to do this. One way is to decrease $L_p$. The other way is to increase $L_r$.  The empirical literature says that there is no change in total employment $L_r+L_p$, which suggest that, infact, we are doing both--the minimum wage is causing workers to switch from being $p$ types to $r$ types.

Now, we have to be careful here, because whether this logic makes sense depends a lot on the household side of the problem. For example, I could easily write down a utility function that leads to $L_r<L_p$ simply because the households prefer $p$ type work to $r$ type work. But, if that were the case, then while a minimum wage law might cause some workers to switch from $p$ to $r$, we should also see an overall decrease in employment.

Which is why I'm sticking with a story along the lines of college versus no college workers. We can easily imagine that there would be heterogeneity in college education since some people are born in households that endow them with the resources to go to college, while other aren't. And it is logically plausible that people without college would all prefer to go to college rather than become unemployed in the event of a minimum wage hike. Of course, this is more extreme than reality--don't take the "college" versus "no-college" example literally, it is just meant to represent that people with $p$ type skills can retrain themselves, at a cost, with $r$ type skills. In fact, it needn't be the case that the one is any more or less educated than the other.

This kind of model could explain why we aren't seeing any employment effects of minimum wage hikes in the data, without needing to resort to the argument that labor markets are highly monopsonistic (for which there is little empirical evidence).

It is also worth adding that this model has its own policy implications. If the minimum wage law is prompting poor people to pay a fixed cost to retrain themselves, then technically we aren't doing them any favor--the fact that they were unwilling to spare this expense before the minimum wage hike is proof that it was not utility maximizing to do so. As an alternative, the government could directly finance these retraining costs (though we need to keep in mind the inefficiency that arises from distortionary taxation).

My hope is not to assert that this is the way the world is. I just want to try to expand the discussion beyond the narrow Econ 101 vs Monopsony discussion. Income Insurance 9/23/2013 12:11:00 AM
Lack of economic reform can also cause unemployment.