The Great Vacation: Recessions In DSGE Models (Part I)

Neoclassical models are built around optimising behaviour. The logic for this is somewhat reasonable: one should expect the private sector to look out after its own interests, and not be tricked by policymakers into self-defeating behaviour. The aspiration is hard to argue against, the problem is the implementation. When it comes to recession analysis, the most blatant problems are in the modelling of household sector behaviour. Since working is voluntary, the hours worked in a period is allegedly a decision variable that can be controlled unilaterally by the household in order to optimise its utility. However, since employment is voluntary — so is unemployment. The result is that recessions can be seen as the optimal decision of households to stop working. As wags have described it, The Great Depression was actually The Great Vacation.

The Great Vacation Effect is a pathological result that few people take seriously. The implication is that it is easy to avoid recessions — tell people to stop cutting back their work hours! However, no matter what one's opinions of politicians are, it is safe to say that no politician would make such a silly suggestion. Even if we put aside the quite obvious criticisms one can make about the silliness of voluntary unemployment, we are stuck with pathological dynamics embedded in the model, hampering recession analysis.

(Note that this topic was broken into two parts. This part sets up the background of The Great Vacation Effect. For many people, this may be the most interesting part — since it makes DSGE models look silly. However, even if we want to put aside the question of silliness, practical issues remain, which are deferred to the second part.)

Not Easily Avoided

Most verbal descriptions of dynamic stochastic general equilibrium models give the mistaken impression that The Great Vacation Effect is just a feature of Real Business Cycle (RBC) models. These models were the first generation of dynamic stochastic general equilibrium models. My guess that RBC models are associated with The Great Vacation Effect because RBC model supporters tended to be free market absolutists who take voluntary unemployment literally.  Robert Chernomas and Ian Hudson give some statements to this effect in Chapter 6 of their book "The Profit Doctrine: Economists of the Neoliberal Era."*

However, adding innovations like price stickiness (which converts an RBC model into a so-called New Keynesian model) generally does not eliminate the vacation effect. However, the New Keynesians realise that The Great Vacation Effect is embarrassing to the credibility of the DSGE project, so they change the subject to other features of this model. Nevertheless, so long as hours worked is a voluntary choice in the household optimisation problem, the effect remains. I discuss means the implications of trying to eliminate this effect later.

Two Optimisations Mashed Together

There are a wide variety of DSGE macro models, but my concern are the class of models where there are at least two optimisation problems which are meant to be joined via general equilibrium: a household problem, and a firm problem. In the context of recessions, the firm's problem seems more plausible, so that it was adherents of DSGE macro wish to discuss. I will turn to it in a later article. For now, I will focus on the household problem, but offer a minimal discussion of its linkage to the business sector.

An optimisation problem normally consists of two mathematical structures.

1. There is an objective function, which maps the possible choices of decision variables to a numerical (real) value. We want to find the choice of decision variables that gives the maximum (or minimum) of this function. A choice of the decision variables that results in the maximum (minimum) objective function is termed the optimising solution.
2. Constraints on the decision variables are then specified. This specification can be done in a multitude of ways. Note that the decision variables can be quite complex -- a set of time series variables (a variable defined on either a discrete or continuous time axis). In economic problems, the constraints are the mathematical relations that define the "laws of motion" of the model economy. For example, the production function will map the hours worked, capital, and productivity variables to the level of production. The set of all possible choices of variables are termed feasible solutions. For example, economic trajectories that violate accounting identities within the model are not feasible, and so it does not matter if they generate a greater utility.

When we look at the mathematics, there are a lot more details to be concerned with. One key concern is the existence and uniqueness of optimal solutions. If we want an economic model to be an optimal solution, to be useful, it needs to exist and be unique. Existence is straightforward — if there is no solution that meets the constraints, the model says literally nothing. The less obvious concern is uniqueness. If there are multiple solutions, we have a straightforward problem: which one is the one we use to look at and compare to the real world? Furthermore, we run into the practical problem that if there are multiple solutions, algorithms will either not converge (thus no solution can be discussed), or may converge to a sub-optimal solution.

Digression: Equilibria in Economics

(I expect that I will write this up in a separate article. I have written on this subject in the past, but I have run across a reference that confirmed my worst fears about how neoclassicals use the concept of equilibrium.)

For historical reasons, mainstream economists refer to solutions of their models as "equilibria." This is how the concept was first visualised, and mainstream economists stuck to the terminology despite it making very little sense when compared to how the term is used in other fields of applied mathematics or physics.

The DSGE macro model is defined as two sets of mathematical entities.

1. The "equations of motion" of the economy: accounting constraints, production functions, etc.
2. A set of constraints upon variables defined in the first part, mainly in the form of relationships in terms of the derivatives of variables. These are referred to as "first order conditions."

The hope is to find a solution of the economic time series variables that meets the two sets of constraints. A solution is called an "equilibrium," and the risk is that the solution is not unique — "multiple equilibria."

The first order conditions are chosen by looking at properties of independent optimisation problems (the Household Problem, the Firm Problem). One of the better-known relationships is that the wage rate is assumed to equal the marginal productivity of labour (the derivative of the production function with respect to the labour variable). 

The hope is that the solution found will correspond to the optimal solutions of the independent optimisation problems.

When one reads advanced mathematical texts, one does not see phrases like "we hope that a solution exists." The problem with the DSGE literature is that existence and uniqueness of solutions is blown over. At best, appeals are made to various theorems -- without any attempt of validating that the model in question is in the class of mathematical systems to which the theorem can be applied. This disregard of mathematical niceties explains why the DSGE literature appears opaque to many outsiders. (My reaction to seeing the papers was inevitably: “they cannot do that!” It took an extremely long search to find a text that described the mathematical definition of equilibrium using standard mathematics.) Applied mathematics articles tend to be somewhat sloppy — to avoid confusing readers who may be from different backgrounds — but the DSGE economic literature is well outside the publication norms of other fields that I had contact with.

The Household Problem — What do We Know About It?

The lack of rigour in discussing the solution to DSGE models raises an extremely uncomfortable point in discussing solution properties. Even if we accept the assumption that there is a solution to the mathematical model, is there any guarantee that this solution will correspond to the optimal solution of the true optimisation models from which the first order conditions are taken? For example, can we assume that the solution is the optimal consumption/work path for the household given the time series of prices and wages?

This seems to be the intention, but I have never seen any attempt to validate such a claim in the models described in the recent literature. In particular, first order conditions are given, but there is no attempt to show that the conditions chosen are exhaustive. My guess is that one would need to go back to the literature of the 1960s to find such results (which I have never seen cited in the modern literature). However, even if such a theoretical result exists, does it incorporate the mathematical structure of new DSGE models?

We are faced with a few options.

1. Have faith in the textual representations by authors about the mathematical properties of models. My experience in academia and fixed income has led me to not take authors' representations about their work at face value.
2. Attempt to re-develop the model in a rigorous fashion from scratch. I have doubts about the feasibility of such a project, and it is certainly not something a non-academic can afford to waste time on.
3. Only accept that a model has a unique solution if it can be calculated numerically. We can then look at the properties of this solution. This is an extremely sensible methodology, but there is a hidden catch: we can only examine the properties of one model solution at a time. We are not in the position to make any generalisations about model solution properties.
4. Discuss the model on the basis that we are making an assumption that the solution exists and is unique, and the solution is in fact the optimal solution for the sub-problems. That is the approach I take here, since I need to make generalisations about the solution properties.

I will now turn to discuss the properties of model solutions — conditional on the existence and uniqueness of solutions, and that the solution does meet the optimality condition of the sub-problems.

Basic Neoclassical Model Structure

The class of models that I am interested in are similar to classic RBC models, but I am allowing for wider classes of models, such as standard New Keynesian models.

1. There is a household problem, where one or more "representative" households determine an optimal path of consumption and hours worked over their lifetime (either overlapping generations, or infinitely long-lived). Although the decisions are made for an arbitrarily small household unit, the decision is supposed to scale up to a part or all of the household sector. The household takes (at least) three key price series as given: wages, output goods prices, the rate of interest, then optimises utility while subject to a budget constraint. (Note that goods can be composite goods if we have Calvo pricing.)
2. There is a firm problem, where the firm decides how many workers to hire, as well as possibly "renting" capital from households. (The capital stock is owned by households, then the business sector rents it each period.) Output is determined by a production function. Once again, the representative firm takes prices (wages, output prices) as given.
3. There can be a government sector, quite often with a central bank. The central bank typically has an interest rate reaction function that is conditional on solution variables.

The aggregate DSGE model takes a set of first order conditions from these two problems, and then assumes that markets clear. For example, the number of hours the household sector decides to work has to match the number of hours the business sector wants to hire. Since inventories are typically assumed out of existence, the amount produced must equal the amount consumed.

Household Optimal Solution

Under the twin assumptions of there being a solution to the overall model and that the solution matches the household optimisation problem used to determine the first order conditions, we can then discuss the properties of the solution as being conditional on the given set of prices (wages, goods prices, interest rate).

We need to be careful about this: we have to take wages and prices as given, and we cannot start asking ourselves what would happen if the household sector did something different. (This is "partial equilibrium thinking.") 

Nevertheless, so long as hours worked is a decision variable, we can say that it is the optimal solution for households. If employment falls, that was the optimal decision of households. So yes, The Great Vacation Effect is real for any model with this structure. Given that most advances in DSGE modelling revolves around adding features to the business sector (e.g., price stickiness solely shows up in firm behaviour), they also feature the vacation effect. (The term DSGE is vague, and so covers models that I would not consider macro models. But for the macro models, I cannot remember seeing any that did not have one or more versions of the Household Optimisation Problem embedded within it.)

The more pressing issue is the discussion of why recessions happen. We want to compare these models to observed data. We can then make a major leap of faith: what happens if we could exactly fit a model in this class to a run of historical data?

Coming Next

The next part of this discussion will be to look at what the predictions are about recessions if we assume that observed data were generated by a model of this type. Even if one is not concerned about the silliness of interpreting depressions as mass vacations, recession modelling is still warped by the chosen model structure.

Footnote:

* “The Great Vacation: Rational Expectations and Real Business Cycles.” The Profit Doctrine: Economists of the Neoliberal Era, by Robert Chernomas and Ian Hudson, Pluto Press, London, 2017, pp. 106–124. JSTOR, https://www.jstor.org/stable/j.ctt1jktsbd.11 . Accessed 5 Feb. 2021.

(c) Brian Romanchuk 2021