The labor market is a key driver for business cycles. Within standard economic models where production is mainly a function of capital and labor (and productivity which determines the multiplier from these inputs), given that productivity is normally fairly stable, we can only generate a recession via a drop in employment. (The recession of 2020 provides an example of "productivity" dropping as a result of governments shuttering activity, but even then, the mechanism for lower production was stopping workers from going to work. This could be captured in any number of ways within a model.)

My comments here are fairly generic, but I am using the paper "Confidence, Crashes and Animal Spirits" by Roger E. A. Farmer* as an example one could look at. I am certainly not in a position to authoritatively survey the neoclassical literature, but my comments here are based on a sampling of benchmark models.

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Utility Maximization

The basic premise of neoclassical dynamic stochastic general equilibrium (DSGE) macro models is that we somehow find the solution to multiple optimization problems. Typically, there is at least a household sector and business sector that are attempting to generate an optimal outcome. In this article, I am largely ignoring the business sector optimization problem, on the grounds that maximizing profits is a relatively easily understood idea.

For the household sector, there are representative households (some model structures can allow heterogeneous agents) that attempt to maximize their utility. The utility function is based on consumption over the model horizon, typically an infinite horizon, or possibly in terms of a generation's lifetime (in an overlapping generations model). Although heterodox critics mock the infinitely long-lived representative agent, I find it less problematic than overlapping generations models. (The paper I from Roger Farmer noted earlier is based on a representative agent, while he has co-authored a more recent article that provides an overlapping generations version.)

The decisions to be optimized by the household is to choose how many hours to work, and how many units of produced goods (there is a single representative good) to consume. Working provides wages to allow goods purchases, but the household will also have financial assets to use to finance consumption. These financial assets are typically either money, bonds (typically one period Treasury bills that pay the nominal interest rate that the central bank sets via a reaction function), and possibly claims on capital. 

Labor Income and Disutility

I will focus first on labor income. Since goods consumed has to be produced, the household sector has to work. (We rule out inventories or imports.)

The simplest possible utility function would only be an increasing function of consumption. This would tend towards a solution where the household sector works the "maximum" hours to allow it to maximize consumption. The relative price between wages and goods prices could drop, so long as doing so would allow for greater consumption. This arrangement is not particularly useful, since the production would always tend to be "maximized," which is not a very convincing fit to the real world.

The easiest way to get a model with more interesting trade-offs is to add a term to the utility function that reduces the utility based on hours worked (disutility of labor). In the Roger Farmer article, this is referred to as the "Neoclassical Model," and is discussed in Section 2. This framework was extremely standard for workhorse DSGE models before the Financial Crisis hit.

The labor disutility tweak gives a model where there is a trade-off that can lead to less than "maximum" employment, which makes it a viable candidate for modeling business cycles. The problem is if we buy into the micro-foundations hype: does the optimization decisions made in the model correspond to how entities in the real world make decisions? In this case, the interpretation of the model is that recessions are the result of households deciding to go on vacation. It is necessary to get indoctrinated in a doctoral program in a freshwater Ivy League university in order to take that story seriously.

(Although the previous statements appear extreme, they are how one would have to interpret most recessions, particularly 2008. In 2008, there were extreme movements in hours worked, while the changes in wages in prices were small. Since the household sector always makes the optimal hours worked/consumption allocation in each period, the change in hours worked can only be generated by a change in preference parameters in the utility function.)

Search Models

One way to avoid the "depressions are vacation outcome" is the use of search models. A portion of the employed labor force is needed to hire other employees (I refer to them as human resources here). The trick is that finding candidates is harder as the unemployment rate drops so that more workers need to be employed in human resources (who do not contribute to production in the production function). 

This creates a new trade-off dynamic within the labor market. For firms, their human resources workers will be increasingly non-productive as the unemployment rate drops, and so they will not wish to increase employment without limitation. This creates a limitation on the number of employees hired, even without disutility of labor in the household utility function.

The implausibility of Search Models

Although the search models are less laughable than labor disutility, there are still some awkward interpretation issues. 

As Roger Farmer notes in his paper, he models having the entire labor force being re-hired each period. Since the firm starts each period with zero employees, there would be no way to lift itself up by the bootstraps to hire anyone. Instead, we need to interpret this as being a steady-state cost, but this glosses over the reality that the concern for hiring is new hires, which is a small portion of the total labor force. (Farmer notes that the model can be extended to being in terms of new hires, but this adds state variable to the model, without offering interesting extensions.)

The danger with this class of models is taking them seriously -- unemployment is the result of matching problems, so all we need to do is offer job training, and we have an optimal outcome. That was the neoliberal story in the 1990s, and it coincided with a low-pressure economy with rampant underemployment. 

However, if we are just interested in a macro model, the search model creates more trade-offs to the labor market. The assumption of decreasing returns to scale in the production function provides one trade-off (which post-Keynesians argue is dubious). The search function provides another, and it might be possible to calibrate it against unemployment data. If we treat it as an added term to give more realistic behavior, it is defensible.

Multiple Equilibria

The addition of more trade-offs to the macro problem results in the possibility of more equilibria. Roger Farmer notes two ways of dealing with this (more are presumably possible).

1. The traditional solution is model bargaining, which is discussed in Section 11 of his paper. The firm has a bargaining weight, and this added bargaining dynamics pins down a unique solution.
2. Farmer's preference is to close the model via beliefs -- the equilibrium is determined by the market capitalization of the capital stock. Since consumption is partly driven by the market capitalization of the capital stock (since it is owned by the household sector, and is thus part of its budget constraint), circularity is present. There is a belief function added to the model, which pins down the value of capital. This is described in Section 12 of the paper.

My feeling is that if we want to get a handle on the business cycle while sticking to neoclassical mathematical foundations, the approach based on belief functions is one of the few plausible paths forward. This is why I am focusing on that body of work, but it is possible to link it back to other approaches. Those other approaches need to eliminate multiple equilibria somehow, so they have to add other dynamics to the model to pin down a single solution.

Concluding Remarks

There is an ongoing deluge of papers in the DSGE macro tradition, all of which are aimed at improving certain aspects of the core models. From the perspective of the business cycle, the core question is the level of employment, and the challenge for DSGE macro is have a model that does not imply permanent full employment. Search models are one popular way to allow for unemployment. However, the added trade-offs imply equilibria, and dynamics added to pin down a solution.

Roger Farmer's work is the closest approach to the spirit of Hyman Minsky (who followed on from Keynes), which is why I am tracing through the logic of his models. This analysis will be continued in later articles, which will eventually be turned into a chapter in the next volume of my text on recessions.

Footnotes:

* Farmer, Roger EA. "Confidence, crashes and animal spirits." The Economic Journal 122.559 (2012): 155-172.