Thinking

Why the Equilibrium Wage is Lower than the Marginal Product of Labor

[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 t with ‘Y_t,’ the total number of laborers that the firm hires in year t with ‘L_t,’ and the wage it pays those laborers with ‘w_t.’  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:

{\large Y_t = \phi( L_t ) }

That is, yams produced in year t is a function, \phi, of laborers hired in year t.  Let me make two assumption about the functional form of \phi.  (These assumptions, unlike the ones above, do matter.  The argument won’t go through without them.)  The first assumption is that \phi is an increasing function of L_t — more laborers means more yams.  The second assumption is that \phi is a concave function.  That is: it looks something like the function shown in figure 1.

Figure 1: The Production Function

This assumption amounts to the claim that laborers bring the firm diminishing marginal returns — if hiring its first n laborers brings the firm x yams, then hiring n extra laborers brings the firm fewer than x 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 t, which I’ll denote ‘\pi_t.’  The firm earns a dollar for every yam sold at market, so it takes home \$ Y_t 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 w_t.  Since there are L_t workers, the firm makes a total payment of w_t L_t to its workers.  Therefore, total profit in year t is given by:

\pi_t = Y_t - w_tL_t

We assume that the firm is greedy — that is wants to maximize \pi_t.  Then, the firm faces the following optimization problem:

\max_{L_t} \Big\{ Y_t - w_tL_t \Big\}

And the first-order condition of this maximization problem is

w_t = \frac{\partial Y_t}{\partial L_t}

But this just says that the wage is equal to the marginal product of labor, \partial Y / \partial L.  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 givenThat maximization problem,

\max_{L_t} \Big\{ Y_t - w_tL_t \Big\}


asked the following question: ‘Given that the wage is set to w_t, what is the value of L_t that maximizes the expression Y_t - w_tL_t?’  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, ‘L_t^s,’ is a function of the wage the firm offers,

L^s_t = \lambda(w_t)

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:

\max_{w_t,~ L^d_t} \Big\{ Y_t - w_t L^d_t \Big\}\quad subject to  \quad L^d_t \leq L^s_t(w_t)

Since the firm gets to choose the wage, however, it will surely lower the wage if labor supplied, L^s_t, is in excess of the labor it demands, L^d_t.  So, the constraint L^d_t \leq L^s_t(w_t) will be an equality L^d_t = L^s_t(w_t), and we can reduce the above optimization problem to this:

\max_{w_t} \Big\{ Y_t - w_t L_t \Big\}

where L_t is a function of w_t.

Interestingly, the first-order condition for this optimization problem is this:

w_t = \frac{\partial Y_t }{ \partial L_t } - \frac{ L_t }{ \partial L_t / \partial w_t }

That is, the firm will choose to pay its laborers their marginal products minus L_t \partial w_t / \partial L_t.  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 w_t.

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 \partial L_t / \partial w_t was infinite.  That is, we assumed that the function \lambda looked like this:

Figure 2

(where \overline{w} 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 \overline{w}.

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:

Figure 3

Then it will turn out that \partial L_t / \partial w_t 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 \partial Y_t / \partial L_t.  What it implies is that workers are paid systematically less than their marginal products.

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2 thoughts on “Why the Equilibrium Wage is Lower than the Marginal Product of Labor

  1. JJ says:

    Clearly written. I would think this argument resembles the situation faced by unskilled labour the most.

    The more unique a worker is, the less homogeneous L_t is (another crucial assumption to the model).

    I would argue that even when labor is completely ‘unskilled’ it is still not accurate to view it as homogeneous. For example, dependability and friendliness are prized traits.

    Another point that shoudl be considered is the cost to the employer of finding/replacing/training workers. Perhaps you find this neglible in comparison to the costs of the seeker of employment, but it bears some mention. Also the relative size of the costs between employer/employee is not the only important factor, one should also try to understand the interaction between the cost to the employer of constantly rehiring versus the discount the employer recieves to the marginal rate from increasing job insecurity.

  2. The part of the picture that is missing is the competition between firms which is supposed to drive their wage offers up to the competitive equilibrium. So one sets up a game theoretic model with an infinite number of firms and solves for the unique Nash equilibrium wherein the equilibrium wage equals the marginal product of labor. This is just a variant of the first fundamental theorem of welfare economics.

    But of course it is a highly stylized model and any relaxation of this assumption of perfect competition immediately implies deviation from the equilibrium. In reality, firms routinely have market power–in this case, monopsonistic power in the labor market–which implies they will corner a nonzero surplus and wages will be lower than the marginal product of labor.

    The “transaction costs” are what makes the labor market less than perfectly flexible. They are real and have similar effects on the wage level and even employment. Its unclear how far we end up from the Pareto optimal equilibrium. One suspects that the differential is substantial and thereby feeds into the labor theory of value and its connection to capitalist accumulation in Marx.

    These are nontrivial questions and one would think that Libertarians and economists of the Austrian school would pay more attention to this than neoclassical economists who just assume it away by claiming that the competitive model is an excellent first approximation. The Austrians acknowledge that significant deviations persist and that it is precisely these “supernormal profits” that provide the incentive for the process of creative destruction.

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