Interpreting DSGE Mathematics

To describe the mathematics of Dynamic Stochastic General Equilibrium (DSGE) as confusing is an understatement. Although trained in applied mathematics, I always had difficulties following the logic used. I now realise that the economists were not solving the global optimisation problem set out at the beginning of the paper. In fact, they are solving a different mathematical model. Importantly, this new formulation no longer solves the original optimisation problem. As a result, it is incorrect to assume that model behaviour reflects optimisation by households.

This article uses some mathematical notation, rendered by MathJax. Some browsers will have difficulties rendering these equations. For non-mathematicians, they may be able to follow my logic, but I am sticking fairly close to the mathematics. What the arguments mean in plain English is open to interpretation, but in my view, we need to accept that the behaviour of published models are arbitrary, and not drawn from the constraints of optimisation.

Update: I added a small section discussing one potential mainstream response to this article. The construction I discsuss is only of more interest if it is extended to the sticky-price models, assuming that can be done. Also, I got some interesting feedback from Brian Albrecht (@BrianCAlbrecht) and Roger E. A. Farmer (@farmerff) on Twitter. I need to digest the information, but as I suspected, my approach towards DSGE macro mathematics is somewhat literalist. There is a large "back story" behind the various equations that appear in published papers and books. This is roughly what I argued; I may reconsider whether or not my assessment at the bottom of the article is too harsh later. 

What the Mathematical Problem Looks Like

(Please note that this article is mainly aimed at representative household DSGE models. Other DSGE models may be less problematic.)

If we look at a DSGE model paper and attempt to translate the contents into a well-posed mathematical problem, it has the impression of looking something like the following model, labelled M1.

Model M1.  Define a set of state variables x, which are time series on time axis T. That is, for each element $x_i$ of x, $x_i(t) \in {\mathbb R}$ for all $t \in T$.

Let the set of all possible $x$ be denoted A (the set of any conceivable state vector).

Impose a set of constraints C on the state variables x. We define the set of feasible solutions ${\cal F}$, which is a subset of A, for which C(x) is true.

Define a utility function U that is a function of x.

The solution to the model is the vector $x^*$ which maximises $U$, that is:
\[
x^* ={\mathrm argmax}_{x \in {\cal F}} U(x).
\]

(What happens in the case of $x^*$ not existing or non-unique is another question.)

The standard practice in other fields of mathematics would be first to lay out the definition of M1, and then turn to method of solution.

DSGE Macro Papers

The layout of published DSGE macro papers diverges from standard mathematical practice. Each economic sector is laid out separately, and then the author starts working out various optimality conditions -- called first-order conditions. Once all sectors are defined, the global model solution is approached. I found this confusing; the solution method is mixed up with the problem statement, and so it was unclear whether equations were part of the constraints of the problem, or part of the solution.

If we rewrite the problem statement into an organised fashion, we get the following structure (Model M2).

Model M2. Define a set of state variables x, which are time series on time axis T. That is, for each element $x_i$ of x, $x_i(t) \in {\mathbb R}$ for all $t \in T$.

Let the set of all possible $x$ be denoted A (the set of any conceivable state vector).

Impose a set of constraints C on the state variables x.

We impose an addition set of constraints O on the state variable x ("first-order constraints"). The set of x for which C(x) and O(x) are true is the set of equilibrium solutions, E.

Define a utility function U that is a function of x.

The solution to the model is the vector $x^*$ which maximises $U$, that is:
\[
x^* = {\mathop{\mathrm argmax}}_{x \in E} U(x).
\]

This seems reasonable, but it is troubling that there does not appear to be a systematic way of determining what the first order conditions are. My assumption was that they result from some un-specified theorems taken from microeconomics.

However, when we dig further into the proof, the author just seems to find the solution based on the first-order conditions and the constraints. We are actually at a new problem, M3.

Model M3. Define a set of state variables x, which are time series on time axis T. That is, for each element $x_i$ of x, $x_i(t) \in {\mathbb R}$ for all $t \in T$.

Let the set of all possible $x$ be denoted A (the set of any conceivable state vector).

Impose a set of constraints C on the state variables x.

We impose an addition set of constraints O on the state variable x (first-order constraints). The set of x for which C(x) and O(x) are true is the set of equilibrium solutions, E.

The solution is a vector in E.

That is, we just find a state vector for which the first-order conditions and constraints hold. To what extent the solution is optimal (as in models M1 or M2), that has to be the result of the first-order conditions O.

When put this way, what is happening becomes obvious: the author of the paper believed that the first order conditions imply optimality when applied to M1. That is, if $O(x^*)$ is true, then $x^*$ is the solution to M1. (The fact that "first-order conditions" comes from optimisation theory should have been a clue.)

The reason why I found this difficult to follow is that it was very clear that O did not imply optimality when we look at the problem M1. I assumed that I was missing added constraints.

Back to Partial Equilibrium

It turns out what was happening had a simple explanation:the first-order conditions were determined sector by sector, without any reference to the global model.

If we want to use economic jargon, the first-order conditions are based on "partial equilibrium" conditions: we look at equilibrium in a single sector or market at a time, without reference to what is happening in other sectors or markets. This is as opposed to "general equilibrium", where all markets (including future markets) reach equilibrium simultaneously. Since the "GE" in DSGE explicitly refers to general equilibrium, did not the solution method have to look at the global optimisation problem?

The answer appears to be no (for at least most of the representative household models I struggled with).

Simple Example

A full DSGE model is complex; I will just look at a cut-down one period model. The treatment is roughly based on the model developed in Chapter 2 of Jordi Galí's Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework. I am following the structure used in that text.

The household sector starts with an initial money balance M. It provides labour to the business sector, and uses its wages to purchase output. The business sector aims to maximise profits.

The following variables are defined:

• P = Price of goods (nominal).
• W = Wage rate (nominal).
• N = Number of hours worked.
• C = Amount of goods consumed (real units).

We assume that all output is purchased (Y=C, in economist jargon).

The household sector wants to maximise its utility function U(C,N). (We assume that increased consumption improves utility; increased work subtracts from uutility,)

The household sector has a budget constraint:

$$\begin{equation}
PC \leq WN + M. \label{eq:budget}
\end{equation}$$

That is, the amount of spending on goods (PC)  is less than or equal to the wage bill (WN) plus the initial stock of money.

We then jump to the business sector. The argument is that the business sector wants to maximise profits. I label profits F, and it is defined by: 
\[
F = PC - WN.
\]

Total output is given by a production function:
\[
C = A N^{1-\alpha}, A \in {\mathbb R_+}.
\]
The usual logic is: maximum profits occurs when $\frac{dF}{dN} = 0$, and we can apply to the previous two equations to get:
$$\begin{equation}
\frac{W}{P} = (1-\alpha) A N^{-\alpha}. \label{eq:marginal}
\end{equation}$$
This appears to pin down the relationship between wages and prices; we then apply this relationship to the first-order conditions for the household sector. (This step is very standard; the textbook applies it to the model in Chapter 2, although the expressions are slightly more complex.)

However, this logic assumes that there is no relationship between the wage bill and business revenue. Unfortunately, such a constraint exists: equation ($\ref{eq:budget}$).

If we apply ($\ref{eq:budget}$) to the definition of F. we see that:
\[
F \leq M.
\]

It is straightforward to see that maximum profits is equal to M. The values of P and W are essentially free to take any value that they wish, so long as they arrive at F=M.

We can then apply the relationship $F=M$ to ($\ref{eq:budget}$), and we get:
\[
PC - WN = M.
\]
That is, the household budget constraint becomes an equality.

We can construct an optimising solution as follows.

• Apply $C = A N^{1-\alpha}$, and insert into U. That is, form $\hat{U}(N) = U(A N^{1-\alpha}, N).$ 
• The standard constraints on the form of U ensure that $\hat{U}$ is differentiable with respect to N. Furthermore, there is a unique $N^* \in {\mathbb R}_+$ such that the derivative of $\hat{U}$ with respect to $N$ is zero at $N^*$, and $\hat{U}(N^*)$ is indeed the maximum of $\hat{U}$.
• Set $C^* = A (N^*)^{1-\alpha}.$
• Fix the wage $W \in {\mathbb R}_+$.
• Set $P = \frac{WN^* + M}{C^*} = \frac{N^*}{C^*} W + \frac{M}{C^*}$. That is, a unit wage cost plus a markup.The constraint $F = M$ trivially holds.

In other words, the set of solutions is infinite; real variables are pinned down, but nominal prices are indeterminate. There is no reason to believe that ($\ref{eq:marginal}$) holds, but it may be possible to adjust wages until it is satisfied. In other words, it was not really a constraint on the optimal solution; it was an arbitrary condition that might prevent us from determining the true set of optimal solutions.

In plain English, the optimal solution is that prices and wages are set so that:

• the household sector works the number of hours that it feels is optimal ("full employment");
• the business sector's profits represent the entire starting stock of money held by the household sector; and
• wage rates are indeterminate, but prices are set as a markup over wages to achieve the fixed target profit level.

This outcome meets the optimisation problem conditions as stated in the text; but the solution bears no resemblance to what the text says it is.

Comparison to Existing Results

It should be noted that this example assumes flexible prices, and so it is equivalent to the Real Business Cycle (RBC) models. The RBC solution also features the same optimal output level as my result here, but with the W/P ratio pinned down by the equation given in Galí's text. So long as the solution stays away from the budget constraint, that comprises another set of optimal solutions (since the price level is also unconstrained, until something changes to force the price level to a particular level). (That is, Galí is correct in arguing that the ratio is an optimal choice, but with the assumption that we are not hitting budget constraints, which was not specified.)

As a result, the construction here could easily be interpreted as a pathological corner case. To be interesting, it needs to be extended to where prices appear to matter for the solution (such as New Keynesian models with Calvo pricing). I am having off-line discussions with Alexander Douglas about that analysis.

Stock-Flow Inconsistency

Things get even uglier if we start looking at inter-temporal optimisations. We see that the optimal strategy is for the business sector to absorb all of the money from the household sector. What happens thereafter?

The usual argument by mainstream economists is that household bond/money holdings match government issuance; that is, household bond holdings determine the amount of debt outstanding. This runs into the obvious problem: they forgot about the business sector holdings of government liabilities.

Unless the business sector is to have an ever-growing pile of financial assets, it has to return profits to the household sector as dividends.

If we allow dividends to be returned in the same period, the result is indeterminate: profits would be arbitrarily large. Let D be dividends.
\[
PC = WN + M + D.
\]
\[
F = M + D.
\]
Any $D \in {\mathbb R}$ is a solution, and thus may be arbitrarily large. Formally, no solution would exist to the maximisation problem. The only way of getting a finite solution is to impose a condition on dividend payments. Very few DSGE papers discussed dividends, and the effect on the household budget constraint was minimised. That is, the fact that there is a feedback loop was generally not discussed.

No Longer an Optimisation Problem

If we follow the spirit of DSGE macro, we just impose the partial equilibrium first-order conditions, and try to find a solution (model M3). However, the solution to this new problem no longer is an inter-temporal optimisation, and we cannot conclude any properties about its solution from optimisation theory.

For example, there is no reason for the Transversality Condition to hold, since the model is sub-optimal.

The attractiveness of DSGE models is also clearer. All you need to do is impose arbitrary first-order conditions to constrain the system to follow some sub-optimal solution trajectory; all you need is some justification from microeconomics to impose the condition. However, instead of recognising that you are forcing sectors to behave in a sub-optimal fashion (which apparently would make the model subject to the Lucas Critique), you can instead give the impression that the sub-optimal outcome is the result of optimising behaviour!

The conclusions drawn from DSGE models can also be seen as being the result of the entirely arbitrary nature of the "first-order conditions" chosen. If the author decides that fiscal policy has no effect on the economy, it just gets dropped from the "first-order conditions." Since the model solution is sub-optimal anyway, what difference does it make? Furthermore, it becomes clear why controversies about the nature of DSGE model solutions exist (for example, the Fiscal Theory of the Price Level, and the "neo-Fisherian" debate). Since the model solutions are essentially the result of arbitrary choices, any form of behaviour can be achieved for what is allegedly the same optimisation problem.

Concluding Remarks

My concern with DSGE models (at least those of the representative household variety) is straightforward: they are not well-posed mathematical models. We have no idea what the properties of these objects are, and we have no reason to believe that they refer to any optimisation problem solution.

More constructively, the advantages of the stock-flow consistent (SFC) approach to modelling are much more apparent. Unlike DSGE models that are defined by heuristic (and largely arbitrary) partial equilibrium "first-order conditions," SFC models are stock-flow consistent. (Accounting identities always hold.) Furthermore, the alleged disadvantage of SFC models -- that they do not represent optimisation problems -- is also shared by DSGE models in practice. A researcher can impose heuristic behaviour conditions -- so-called "first-order conditions" -- on SFC model behaviour in exactly the same way as can be done with a DSGE model.

(c) Brian Romanchuk 2017