### Profit-sharing is not the solution to inequality

8/22/2014 04:43:00 PM
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In Fortune, sociologist Joseph Blasi thinks he has a better solution to inequality than Piketty's proposed global wealth tax. His solution is this: instead of taxing wealth as a way to give the non-rich an advantage in capital markets, we should simply promote "profit-sharing" between firms and their employees, by having firms compensate employees in stocks and dividends. But here's the problem: don't bring a sociologist to an economics fight profit-sharing is bad for workers. Especially the poorest ones.

You don't need a mathematical model to see why. Individual corporations are extremely risky. They may make a mistake--a flaw in the computer chip design, or a contaminate in their food products--that will put them out of business tomorrow, or at least cause their stocks to plunge shortly before it's time for you to retire. And even when they do everything right, corporations are still extremely risky--a mad cow scare in Virginia can easily put your beef packing plant in California out of business, or a new technology can come along and make your paper company obsolete. Merely by working at a company, workers are subjected to huge amounts of risk because they could loose their future wages at any moment if they are laid off. But, at least in that eventuality their savings will not also be lost when they loose their future wages.

Except when they have a profit-sharing agreement with their company. These agreements require that part of worker's wages will be paid not in cash but in stocks, with strings attached about when and how these stocks can be traded. The result is that profit-sharing arrangements require workers to invest not just their future wage incomes in their extremely risky employer, but also a large share of their savings too. Now when their employer goes bankrupt, they loose both their source of income and their life savings. Profit-sharing magnifies risks to workers, and no one is harmed more by this than the poor.

I'm not saying that we shouldn't encourage the poor to invest in equities. I'm saying that they are much better off being able to diversify that investment to insure themselves against the extremely high amount of risk that any single firm represents.

We can examine this with math. Households value consumption according to $U=E\left[u\left(C\right)\right]$ where $u$ is a continuous strictly concave increasing function--that is, households want to consume more, but experience diminishing returns. The household's wage income is $y$, and there are two identical firms that earn $\pi_i$ profits. Each firm faces a probability $p$ that it will have a disaster (say, a recall) that will reduce profits by $D$ so that $\pi_i=\pi$ with probability $1-p$ and $\pi_i=\pi-D$ with probability $p$. The two firms are identical, but their disaster risks are independent. A household works for just one of these firms.

Under profit sharing, household utility is $$U=pu\left(\pi-D+y\right)+\left(1-p\right)u\left(\pi+y\right)$$ because the individual is forced, by the profit sharing agreement, to own the firm's equity and therefore bears the firm's disaster risk. Without profit sharing, total worker's compensation is the same, but the worker is no longer required to invest in his own firm's equity. He therefore buys a diversified portfolio of both firm's equity, such that his portfolio earns $\pi$ with probability $\left(1-p\right)^2$, $\pi-\frac{1}{2}D$ with probability $2\left(1-p\right)p$, and $\pi-D$ with probability $p^2.$ The expected returns on this portfolio are identical to the expected returns in the profit-sharing case, but with a much smaller variance--he is less likely to earn $\pi$, but a lot more likely to earn more than $\pi-D$. The utility without profit sharing is $$U=\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)pu\left(\pi-\frac{1}{2}D+y\right)+p^2u\left(\pi-D+y\right).$$ The only assumption needed to show that $U$ is higher (and therefore the household is better off) without profit-sharing than with is that $u$ is strictly concave, which is equivalent to saying that people are risk averse which is equivalent to saying people experience diminishing returns.

In fact, most economists would just say that my result follows directly from the definition of concavity, though this may be less obvious to non-math folks. Indeed, what constitutes a mathematical "proof" depends entirely on the knowledge level of the reader. If you are aware, for example, that a continuous function $f$ is strictly concave on a set $C$ if and only if for any $x,y\in C$ $$f\left(\frac{x+y}{2}\right)\gt\frac{f\left(x\right)+f\left(y\right)}{2}$$ I could simply start there. But if you don't know that, then a proof that assumes that doesn't really prove anything, does it? I had planned on simply linking you to the wikipedia page to assert that this is true--you can bug them if you remain unconvinced--but it turns out wikipedia only asserts the weak form of concavity ($\geq$) and not the strict form I'm using here ($\gt$). So I'm just going to tell you that this result is easily shown by taking wikipedia's definition of strict concavity and setting $t=\frac{1}{2}$. That let's us jot down the following proof:

Proof:
Recall that utility in the non-profit-sharing case was given by $$U=\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)pu\left(\pi-\frac{1}{2}D+y\right)+p^2u\left(\pi-D+Y\right).$$ Therefore,
\begin{align}
U&=\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)pu\left(\pi-\frac{1}{2}D+y\right)+p^2u\left(\pi-D+Y\right)\\
&\gt\left(1-p\right)^2u\left(\pi+y\right)+2\left(1-p\right)p\frac{u\left(\pi+y\right)+u\left(\pi-D+y\right)}{2})+p^2u\left(\pi-D+Y\right)\\
&=\left(1-p\right)u\left(\pi+y\right)+pu\left(\pi-D+Y\right)
\end{align}
which was the utility in the profit sharing case. So, profit sharing is worse.