# The envelope (theorem) please: Profits, efficiency wages, and monopsony

In a very helpful blog post, Paul Krugman tries to make sense of Wal-Mart’s recent statement that it is already reaping some gains from raising wages via reduced turnover costs. Krugman’s main point is as follows. If worker productivity is a function of the wage (through improved morale, lower turnover, etc.), and Wal-Mart was initially maximizing profits, then a small change in wages will leave profits largely unchanged.

As Krugman points out, this is logic of the “envelope theorem.” What I want to clarify in this post is that the logic behind this argument is more general than the particular efficiency wage model Krugman works through.  Any time firms are choosing wages to balance various concerns—as opposed to simply accepting a “market wage” as a constraint—the logic of the envelope theorem applies.  What’s more, two types of empirically relevant models of the labor market—monopsonistic competition and efficiency wages—look pretty similar in this regard, and can be thought of as special cases of a more general model.

Krugman discusses the efficiency wage case, where worker productivity $e(w)$ is a function of the wage, $w$. Krugman mentions monopsony in passing, but doesn’t analyze it explicitly: this is the idea that higher wages allows firms to employ more workers. In presence of search frictions, a higher wage allows a firm to more easily recruit and retain its workers. Such frictions, therefore, give employers some wage setting power. In the textbook monopsony model, the quantity of labor employed $L(w)$ becomes a function of the offered wage.

Of course, both efficiency wage and the monopsony channels may matter. Conversely, neither may be relevant. So overall, we have four cases.

1. Competitive case: $\Pi(L) = V(L \times e) - w \times L$

2. Pure efficiency wage: $\Pi(w,L) = V(L \times e(w)) - w \times L$

3. Pure monopsony case: $\Pi(w) = V(L(w) \times e) - w \times L(w)$

4. General case: $\Pi(w) = V(L(w) \times e(w)) - w \times L(w)$

In the competitive case, wages are fixed by the market, and worker productivity $e$ is unaffected by the wage; so the firm just chooses employment $L$. In the pure efficiency wage case, productivity is affected by wage, but labor supply isn’t; in the monopsony case, the opposite is true. In the general case, both productivity $e(w)$ and labor supply $L(w)$ depend on wages. So the firm chooses a wage, and this determines both its labor supply and the productivity of the workers it employs. Both the ease of recruitment and the increased morale provide “offsets” to the increase in costs due to the wage change.

Now, at the wage $w^*$ where profits are maximized, by definition, the following condition holds:

$\frac{d\Pi(w^{*})}{dw}=0$

To understand this condition, say Wal-Mart was maximizing profits initially and chose its wage at $w*.$  Now, say due to “exogenous” social pressures, it raises wages slightly. A shift like that will not affect profits. Why? Because the gains (ease of recruitment/retention affecting $L(w)$, improved worker morale affecting $e(w)$, whatever else) will be exactly balanced agains the wage costs. This is the envelope theorem in action, and it is a feature of any model where firms choose wages, not just the ones I lay out here.

Now, this logic applies exactly only “on the margin” … meaning for small changes in wages around $w^*$. In reality, Wal-Mart recently raised its bottom wages amounting to something like 5 percent of its wage bill. But what if, as Krugman hypothesizes, Wal-Mart were to raise wages by 20 percent or 40 percent? Does this logic hold exactly? No, because we are outside of the zone of “small changes;” I would therefore qualify Krugman’s argument about offsets here. The extent to which the offsets are sizable will now depend on the quantitative importance of the monopsonisitc and efficiency wage channels … i.e., the shape of the $e(w)$ and $L(w)$ functions when we move away from $w^*$. But what is true is that the labor supply effect and the morale effect lead to a smaller profit loss than is true in the competitive case.

Of course, we have been imagining Wal-Mart just raised its wages “exogenously.” But in truth, Wal-Mart changed wages because it probably faced some pressures—market, social or both. These costs will affect its profit function changing its profit-maxizing wage $w^*$.  But the logic of labor supply and morale effects implies that there will be some offsetting gains accruing to Wal-Mart from raising its wages. The fact that the company reported savings from lower turnover is consistent with the theory, and with academic evidence from wage increases by other major retailers, as well as from the fast food sector after minimum wage hikes.

## 6 thoughts on “The envelope (theorem) please: Profits, efficiency wages, and monopsony”

1. 3rdMoment (@3rdMoment)

A couple of additional points. (Perhaps you omitted them for brevity, but I think they are important.)

* If workers are heterogeneous, one way a higher wage can increase productivity is by allowing the firm to hire/retain better workers. It need not be the case that the same workers become more productive.

* As Krugman points out, his analysis is for a single firm in partial equilibrium. If all firms raise wages, they won’t get the same benefits that a single firm would get by itself.

* As you point out, the envelope theorem does not apply to firms that face a binding constraint on wages. This includes firms that are constrained by the minimum wage. It’s not clear to me what, if anything, this analysis tells us about minimum wage policy.

2. ezra abrams

did you know that there are free addins, at least for firefox, that will check to see if your font color and your background color produce enough contrast to make the font readable ?

Arin Dube may choose between working 10 hours per day for UMass producing papers and teaching performances that are worth V=$200k, OR working 5 hours per day producing papers and teaching performances for V=$120k. UMass will pay him w=$180k in the first case and w=$100k in the second one. Arin Dube solves his maximisation problem and realizes he’s indifferent between the two scenarios, also UMass is indifferent between the two scenarios given that it will have profits of \Pi = \$20k in both cases.