# Can the cyclicality of the real wage be used as a measure for the degree of competiveness in the labour market?

When reading the answers to this question of mine, I'm left wondering if cyclicality of real wages can be seen as an indicator for the nature and degree of competition in the labour market.

In perfect competition, we would have high procyclical wages, under monopsony acyclicality.

What would be the cyclicality, if the labour market could resemble a oligopoly(a highly unionized market could resemble an oligopoly, right?), or if the labour market worked as if in monopolistic competition?

### Formal model: Wage determination under bargaining

Let's be a bit formal. We are interested in modeling wage determination. For sake of brevity, I will skip some parts, also I will say nothing about how applicants and firms actually match. We want to see how their wages get formed, conditional on having matched.

Let $p$ denote productivity of workers, and hence also the state of the business cycle. Hence, we will index everything by $p$. $w(p)$ is the wage level of hires negotiated given current productivity of $p$. If we normalize labor supply to 1, $p$ will also be TFP.

Workers have some outside option to working for a specific firm. That might be informal work, unemployment benefits or similar - denote that $b$. Also, the worker has some chance at meeting some other firm. Denote this aggregate outside-option as $U(w(p), p)$. The firm's outside option is to wait for another applicant and hire him instead. Denote that by $V(w(p), p)$.

When the workers and the firm meet, we assume them to negotiate the wage. Bayesian bargaining will need one additional parameter, the bargaining power $\beta$, which is not to be confused with the outside option. Think of it as "how good someone is at bargaining". Here, $\beta$ will denote how good workers are at bargaining, compared to the firms.

Since employment is not an instantaneous contract, but will hold for some time, we need two variables that contain the present-discounted value (PDV) of the relationship for both workers and the firm. Let $E(w(p), p)$ denote the value of employment to the worker. It will be related to the wage rate ($w(p)$), and on how long we expect the worker to be employed. Let $J(w(p), p)$ denote the value of the employed to the firm. It will be related to the per-period profit ($p - w(p)$) that it gets from the worker in every period, and the number of periods that the worker will be working for that firm.

Then, the Nash bargaining solution is given by

$$w(p) = \max_w [E(w, p) - U(w, p)]^\beta [J(w, p) - V(w, p)]^{1-\beta}$$

subject to the constraint that neither firms nor the worker "can make losses" from the wage. That is $w(p) \geq b$, and $w(p) \leq p$.

$\beta = 0$ means that workers have no bargaining power. In that case, Wages are given by the workers fundamental outside option $b$. $\beta = 1$ means that the firms have no bargaining power. They will make zero profits always, and workers getting the total surplus, $w(p) = p$.

You can already intuitively see that as we changed $\beta$ from 0 to 1, the cyclicality of the wages increased (from a variable that is more or less constant, to a very cyclical variable). Trust me that it is also the case for the whole unit interval domain of $\beta$.

Or even better, do it yourself. This is a part of the Diamond-Mortsensen-Pissarides model. A great reference is Pissarides' book Equilibrium Unemployment Theory.

### Conclusion

The model ignores many things and says ceteris paribus, yes. There are many pitfalls to this, so the c.p. is a very strong one.

In particular, what are some mechanisms that would interfere?

• In some sectors, matches between workers and firms hold longer. This is irrelevant in the standard model with risk-neutral workers, but will affect the cyclicality of the wage rate under risk aversion (i.e. under the same $\beta$, and fundamentals $p, b$ you will get a different rate of procyclicality ).
• Workers with different levels of risk aversion might sort themselves into different sectors. To the extent that firms insure the workers partially against wage changes over the business cycle, varying wage cyclicality might stem from risk-aversion rather competition
• Some sectors might be under stronger government control. As a simple case, wages might be "stuck" at the minimum wage for low wage employees. This wouldn't necessarily lead to the conclusion that restaurants are monopsonists.

So, in order to use wage cyclicality as a metric for competition on the labor side, you need to be able to control for all these mechanisms, which is somewhat difficult.

In general, no. The procyclicality of wages can also It could also be a measure of labor market frictions instead of the level of competition.

For an example that highlights the difference, consider an economy where only union electricians can do electrical work and where the union controls the members and their hours. If the union can frequently and costlessly adjust their total labor supply, then they don't have a lot of friction in their supply but they (by construction) have limited competition through their monopoly labor supply. On the other hand, laws mandated costly separation processes for employees such as 2 years severance or, more extremely, lifetime employment make it very difficult to adjust the quantity of labor supplied but this isn't exactly a competition issue. There could be many employees, many employers, and no licensing restrictions and yet this could still significantly dampen the cyclicality of the real wage by making employment not adjust to cyclical productivity shocks.

This might be somewhat splitting hairs because both restrict the adjustment of labor demanded but conceptually they are quite different.

Additionally, there may be reasons related to other features of the economy or the way that the data is gathered for wages to be less procyclical in one place than another. For example, measurement error attenuates statistical relationships towards zero so in two otherwise identical countries, if one has wage measured with more error then the correlation between wage and your favorite business cycle measure will be shrunk towards zero in that country relative to the truth.