A little welfare math

In an earlier post, I mentioned an argument that welfare decreases wages. The idea is that employers don't need to pay as much, since the wages they provide will be supplemented by welfare. However, in a simple analysis, welfare clearly increases wages. It could be that the simple analysis is wrong (as I showed is the case for a simple analysis of minimum wage), but the simple analysis should still be a beginning for understanding.

Since this is only a personal blog, I did just the bare minimum of research into US welfare systems. I immediately learned that US welfare is shit. I can't believe how little safety net there is in this country. Here are some basic facts:

• The major programs are Social Security, Medicaid, TANF, and SNAP. I ignored the first two because those are more complicated and conditional. 
• TANF (Temporary Assistance for Needy Families) is for extremely poor families with children. For example, a household of four people in Indiana must have income under 36% of the poverty level, and the maximum payment is 17% of the poverty level (and that's assuming your income is zero). 
• Additionally, TANF has a lifetime limit of 5 years, and only 2 years without a job. 
• TANF websites are horrible and most of them won't give the relevant information in an accessible manner. 
• SNAP (Supplemental Nutrition Assistance Program, or food stamps) applies to a much broader class of people, and pays more. For a household of four, you must have gross income under $2552 a month to qualify (or 130% of the poverty level), and you are allotted $632 (with 30% of your "net income" deducted). 
• People in the SNAP program must get a job within three months.

For the sake of analysis, I will take a simplified SNAP as a model. Let's just say that we have a four-person household, and the payments are $630 a month minus 25% of gross income.

The first thing you should notice is that SNAP effectively creates a 25% tax on the first $2520 of your household's income. Already we can see why welfare should increase wages. Workers will demand more money for the same amount of labor, because they're only seeing 75% of that money. Of course, there's a tradeoff between wages and employment. If workers only see 75% of their wages, more of these workers may choose not to get a job at all. Or employers may choose not to hire them in light of the higher wages.

This is all complicated by SNAP's requirement that people get a job. Now that may decrease wages, if it motivates people to look for some job, any job, in order to maintain eligibility.

There's a second way that welfare can increase wages: the decreasing marginal utility of money. I'm making these numbers up, but say you need $100 a month to eat, and $200 a month to eat well. The first $100 prevents you from starving, whereas the next $100 only prevents you from hating your food. If there is no welfare, you might be willing to work a shit job for low pay because it's still better than starving. If you're already getting $100 a month from welfare, you would demand more money for your labor.

------------------------------------------

The rest of this post will do a simple mathematical analysis of welfare. You must have javascript enabled in order for MathJax to render the *\LaTeX* equations.

In a simple analysis, the number of low-wage workers employed and the amount of wage they earned is determined by the intersection of supply and demand curves. The demand curve is the number of workers that employers are willing to hire. The supply curve is the number of people willing to work for a certain wage.

 

In panel a I show typical supply and demand curves. However, there are arguments that the demand curve for low-wage workers is unusual in that it has "zero elasticity" (panel b) or "negative elasticity" (panel c). These arguments are complicated and controversial, so I'm just going to ignore the demand curve for now.

Welfare affects the utility function of workers. Normally, we'd just say

**U = L w - c(L)**

 where U is the utility function, L is the fraction of laborers employed, w is the wage, and c(L) is the total cost of all those people working. c(L) is basically a measure of people's subjective preferences. But now the total earnings of each worker is the wage plus welfare, which we'll call t(w). So we have

**U = L t(w) + (1-L) t(0) - c(L)**

 A further complication is the decreasing marginal utility of money discussed previously. If u(w) is the utility of wage w, the the total utility function is

**U = L u(t(w)) + (1-L) u(t(0)) - c(L)**

Here I plot the t(w), u(w), and u(t(w)) that I used. t(w) is wages plus the simplified SNAP payment described previously. We know very little about u(w), so I just used *u(w) = Log(\frac{w + $500}{$500})*.

The supply curve is defined by the point where the marginal worker no longer benefits from working. In other words, it's the point where U no longer increases with L. The derived supply curve is

**u(t(w)) = u(t(0)) + \frac{dc}{dL}(L)**

 Below I show the supply curves before and after applying welfare (assuming *\frac{dc}{dL}(L)* is just proportional to L).

So as far as I can tell, the idea that welfare is "subsidizing" employers makes no sense. Welfare appears to increase wages.  The number of workers may increase or decrease, depending on the elasticity of the demand.

There are a million caveats and complications here, and you should not trust the basic analysis.  For instance, note that the wages payed by the employer are not the same as the profit earned by the employees, since employees miss out on the welfare they could have gotten by being unemployed.  This is just the effective 25% tax from welfare.

One way to get rid of the effective tax is to guarantee everyone a base income. That is, give everyone $630 a month, regardless of their other earnings. This would still raise wages, because of decreasing marginal utility. But then we'd probably have to worry about inflation. I'm not even gonna go there.

I couldn't find any studies (that I understood) on the impact of welfare on wages, so I have no way of knowing whether any of my conclusions are correct or incorrect. I think, however, it was good to learn about our welfare system. If it took a math problem to motivate me to do a bit of research, it was worth it.