[Readers of this blog might be familiar with my friend and philosopher of science who introduced me to the Van Fraassen paper that I talked about in my post on the nature of reality. He has agreed to publish the following thesis on The Policy Tensor. I will reserve my comments for the comments section. You are welcome to join the conversation.]
One thing that gets said in textbooks on macroeconomics is that it follows from the assumption that firms maximize profits that workers will be paid their marginal product — that is, that their compensation will match their contribution to the firm’s output. This is a rather incredible thesis. After all, one might think that a wage which is proportional to the laborer’s contribution is a fair wage. But it is surprising, to say the least, that the firm’s fairness could follow from the firm’s greed.
Here’s how the argument goes: For simplicity’s sake, assume that a particular firm produces its good using only labor, and no capital (no machines, tools, or intermediate goods). Suppose, for instance, that the firm owns a large stock of land for which it pays no rent, and that it employs laborers to work the land with their bare hands. The firm then takes the good — yams, let’s say — to market and sells them at the price of 1 dollar per yam. (These assumptions just make the math a bit cleaner — everything I’m about to say would go through just as well without them.)
Let’s denote the total number of yams produced in year with ‘,’ the total number of laborers that the firm hires in year with ‘,’ and the wage it pays those laborers with ‘.’ Since the number of yams the firm produces is determined by the number of laborers it hires (more laborers means more yams and fewer laborers means fewer yams), we can write:
That is, yams produced in year is a function, , of laborers hired in year . Let me make two assumption about the functional form of . (These assumptions, unlike the ones above, do matter. The argument won’t go through without them.) The first assumption is that is an increasing function of — more laborers means more yams. The second assumption is that is a concave function. That is: it looks something like the function shown in figure 1.
This assumption amounts to the claim that laborers bring the firm diminishing marginal returns — if hiring its first laborers brings the firm yams, then hiring extra laborers brings the firm fewer than extra yams. This could be due to the fact that there is only so much land to be worked, so that, with each extra laborer, there’s less extra work for that laborer to do than there was extra work for the previous laborer to do.
Now, let’s consider the firm’s profits in year , which I’ll denote ‘.’ The firm earns a dollar for every yam sold at market, so it takes home from the market. However, it doesn’t get to keep all of this profit, because it has to pay each of its workers the wage . Since there are workers, the firm makes a total payment of to its workers. Therefore, total profit in year is given by:
We assume that the firm is greedy — that is wants to maximize . Then, the firm faces the following optimization problem:
And the first-order condition of this maximization problem is
But this just says that the wage is equal to the marginal product of labor, . If we think that this is a fair wage, then it follows from profit maximization that laborers are paid a fair wage. Incredible! Perhaps Ayn Rand was right all along!
Unfortunately, even if we accept the dubious premise that the marginal product of labor is a fair wage, this argument does not show that firm’s greed will induce them to pay this fair wage. What the argument shows is that the marginal product of labor will be set to the wage — and not that the wage will be set to the marginal product. That is, profit maximization explains why the marginal product is what it is, rather than explaining why the wage is what it is. If you think about it, the optimization problem I set up above couldn’t possibly explain why the wage is what it is, because that maximization problem takes the wage as given. That maximization problem,
asked the following question: ‘Given that the wage is set to , what is the value of that maximizes the expression ?’ That is: we are asking how many laborers the firm will hire, taking the wage as a given. We are decidedly not asking how much the firms will pay its laborers, taking the dispositions of the laborers as given. In this optimization problem, the firm is choosing how many workers to employ; it is not choosing how much to pay them.
The assumption in play here is that firms are wage-takers. They don’t have any control over the wage that they pay their employees. That wage is set by the conditions in the labor market. And since the firm is a small participant in the labor market, it can’t exercise any control over the wage. If it were to offer a slightly lower wage, then it would not be able to find any workers at all, since there would be other firms who would be willing to pay those workers more, and the workers would prefer working for those other firms at that higher wage.
Suppose that we relax this assumption. Suppose that we modify the model so that the labored supplied, ‘,’ is a function of the wage the firm offers,
and the firm must now maximize profits by deciding not how many workers to employ, but rather the wage at which it is going to employ them. That is: the firm picks the wage, and then laborers get to make a decision about whether or not they want to work at that wage. The number of laborers who say yes depends upon what wage the firm offers. Of course, the firm doesn’t have to hire all the workers who say yes, but they can’t hire any more workers than the number that say yes. In a situation like this, the firm faces the following maximization problem:
Since the firm gets to choose the wage, however, it will surely lower the wage if labor supplied, , is in excess of the labor it demands, . So, the constraint will be an equality , and we can reduce the above optimization problem to this:
where is a function of .
Interestingly, the first-order condition for this optimization problem is this:
That is, the firm will choose to pay its laborers their marginal products minus . This term depends (roughly) on how many workers are willing to supply their labor for wages slightly higher and slightly lower than the offered wage .
From this perspective, we can now see that, in the first maximization problem, our assumption that the firm was a wage-taker really amounted to the assumption that was infinite. That is, we assumed that the function looked like this:
(where is the equilibrium wage.) This assumption amounts to the claim that there are no laborers in the work force who are willing to work for any less than the equilibrium wage .
But, of course, this assumption is absurd. There are many reasons to supply your labor for less than the equilibrium wage. For one, there are search and wait costs associated with finding other firms who will employ you at a higher wage. However, if we relax this assumption, and assume that the labor supply function the firm faces looks something more like this:
Then it will turn out that is everywhere finite, and that, therefore, the wage a profit-maximizing firm will choose to pay its workers will be strictly less than their marginal product.
Moreover, there was nothing special about the firm I just considered. So, what goes for it should go for every profit-maximizing firm in the economy. But if every firm in the economy is offering wages lower than the marginal product, then that just means that the equilibrium wage is going to be lower than the marginal product of labor. So, as long as we assume that firms are the ones who get to set wages – and that, for instance, employees don’t get to negotiate with their employers – profit maximization doesn’t, as a matter of fact, imply that workers are paid the putatively fair wage of . What it implies is that workers are paid systematically less than their marginal products.