# The minimum wage and elasticity of labor demand

Submitted as homework for the Microeconomics Principles MOOC.

A couple weeks ago I argued that an increase in minimum wage could decrease unemployment, because workers’ increased spending power would lead to increased consumption and therefore would require firms to hire more workers to keep up with demand for products.

Now, I will argue that this only works if the price elasticity of demand for labor (ED) is much less in absolute terms than the price elasticity of the labor supply. In other words, unemployment will decrease only if the higher minimum wage results in relatively few workers being let go.

Why is demand for labor price inelastic? Simply put, it’s easy to raise wages by, say, 25%, but it’s hard to raise each worker’s productivity by the same amount. So a firm that has to increase its workers’ pay won’t be able to maintain the same level of productivity unless it doesn’t fire very many workers after the pay hike. The firm’s labor costs are just going to have to go up, if it wants to keep productivity at the same level. It therefore makes sense that demand for labor would be price inelastic.

But how inelastic does it have to be, in order for a minimum wage increase to lead to decreased unemployment? Let’s work it out mathematically. For simplicity’s sake, I’m going to make a few assumptions:

1. The price elasticity of the labor supply is 1. (Because I only care about relative elasticity between supply and demand, I think this is a fair assumption.)
2. The current minimum wage P0 is the equilibrium price for labor. (I do not believe that this is true for the US right now, but that’s an argument for another time.)
3. A minimum wage increase (Δ%P) of 25% is being debated. (This is approximately Pres. Obama’s current proposal.)
4. There is no unemployment insurance, disability insurance, etc., to give workers an income if they are not working.

We can represent all workers’ total wages (and thus, their spending power) after the wage increase in terms of their initial total wages like this:

WR [wage ratio] = TW1 / TW0 = (1 + EDΔ%P)(1 + Δ%P),

where (1 + EDΔ%P) represents the amount of labor (or number of workers) demanded after the wage increase and (1 + Δ%P) represents their new wage as a percentage of P0. Since in this case price elasticity of demand is defined as the percent change in quantity of labor demanded divided by the percent change in the price of labor, (1 + EDΔ%P) equals the percentage of the original demand for labor that remains after the wage increase.

In order for unemployment to decrease while the minimum wage increases, the increase in total wages will have to offset the decreased demand for labor–workers’ total spending power must be big enough to require firms to hire more workers to keep up with increased consumption. In mathematical terms, (WR)(1 + EDΔ%P) must be greater than 1. In fact, it probably has to be much greater than one, because only a fraction of low-wage workers’ new spending power is going to go toward the labor costs of the firms where they spend their additional wages.

Let’s fill in these formulas with some numbers and see what we find.

Ex1. First, what if the demand for labor is only slightly inelastic (ED = -0.8)?

WR = (1 – [0.8 * 0.25])(1 + 0.25) = 1

(WR)(1 + EDΔ%P) = 1 * 0.8 = 0.8 < 1

So in this case, fewer workers (80% of the original total) end up sharing the same sized pie. The workers as a group do not have more purchasing power than they did before, so their increased consumption will not translate into greater demand for labor.

Ex2. What if demand for labor is moderately inelastic (ED = -0.4)?

WR = (1 – [0.4 * 0.25])(1 + 0.25) = 0.9 * 1.25 = 1.125

(WR)(1 + EDΔ%P) = 1.125 * 0.9 = 1.0125 ≈ 1

In this case, only 10% of the workers lose their jobs, but even so, total purchasing power increases by 12.5%. Will 12.5% more total consumption be enough to offset the loss of 10% of the original jobs? Probably not, but that effect is beginning to seem technically possible.

Ex3. What is demand for labor is very inelastic (ED = -0.2)?

WR = (1 – [0.2 * 0.25])(1 + 0.25) = 0.95 * 1.25 = 1.19

(WR)(1 + EDΔ%P) = 1.19 * 0.95 = 1.13 > 1

Here, only 5% of the workers lose their jobs, but total spending power increases by (nearly) 19%. It is conceivable that that this new spending power will generate enough additional demand to cause firms to hire back the workers who lost their jobs, and then some.

In general, my argument agrees with an argument made by the economist Leif Danziger, that “for each minimum wage rate there exists a critical value of the elasticity of labor demand … such that an increase in the minimum wage rate makes workers better off if labor demand is less elastic than the critical value, but worse off if labor demand is more elastic than the critical value.” Given my assumptions, the critical price elasticity of labor demand for a minimum wage increase of 25% is around -0.2.

But I had assumed that there is no unemployment insurance. Danziger does not make that assumption. In fact, he shows that unemployment benefits are critical in locating the equilibrium value of price elasticity of labor demand. The more generous unemployment benefits currently are, the more it makes sense to increase the minimum wage.

Assignments, Econ MOOC