Readers who are allergic to mathematics may have a difficult time with this article. Given that I am complaining about inadequate mathematical notation, I am forced to be more formal than what I prefer. The concepts could be expressed in plain English, but it would be much longer.

The previous two articles of this series are here and here (for the business and household sector accounting). Note that this article will use the model discussed in those articles as an example; the equations are found therein. It should be noted that those two articles were written before I realised that I was mis-interpreting the mathematical notation within DSGE macro papers. Another article I wrote discusses the curious notation used; this article re-does that analysis. The first two articles on the household and business sector were based on a standard mathematical convention; they offer an explanation of what goes wrong if one assumes that DSGE macro papers are doing standard mathematics.

## Introduction

This first part of article is a reformulation of an earlier article: "The Curious Notation of DSGE Models." Until very recently, I was unable to piece together the proofs in DSGE models, as there were logical leaps taken that did not appear reasonable to me. The key is that DSGE macro texts use the same symbols for what are different mathematical objects. This means that there are hidden "rules of the game" for what are legitimate mathematical operations. As will be discussed, it is unclear whether the rules invoked by DSGE researchers can be codified as a series of formal statements on sets. A reader that dislikes mathematics may wish to jump to the later parts of the article, which returns to discuss how this applies to the analysis of the governmental sector.I have noted in the past the Bourbaki view of mathematics -- that mathematics is properly the study of sets. Although applied mathematics is often written in a less formal manner, we should be able to relate any verbal statement to a statement about sets. Otherwise, the logic becomes vague and possibly inconsistent.

## DSGE Macro Conventions

If the DSGE macros modelers are working with a model in which a single sector appears, they have a mathematical problem that either comes from optimal control theory, or a variation of one. Although one might find issues with details (in what set do time series lie?), there is no reason to believe that there are serious problems. Difficulties arise from multi-sector models, which have no analogy within optimal control.A DSGE multi-sector model is set up in a series of steps.

We first define a state vector $x$ that is set of time series (such as consumption, hours worked, etc.), all of which are real-valued, and are defined on the time axis $t = 0,1,2,...$.

Since it is not clear what space time series lie in, I will use the set $T$ as a placeholder as the set of time series.*

The Household Problem is then defined by partitioning the state vector $x$ into two sets of variables: the decision variables $u_h$, and exogenous variables $\xi_h$. For clarity, we will denote the state vector as $x_h$, where $x_h = (u_h^T, \xi_h^T)^T$ ($T$ here denotes the vector transpose).

(If we look at the Ramsey Problem I referred to in previous articles, the household decision variables are the hours worked, consumption, goods held as capital, and bond holdings. The household budget constraint implies that there are fewer degrees of freedom than that list implies. Everything else, such as the wage rate, tax rates, and government spending are exogenous.)

If we fix the exogenous variables ($\xi_h$), we then define the optimal input $u_h(\xi_h)$ as:

$$

u_h(\xi_h) = \arg \max_{u \in T} U_h(u, \xi_h),

$$

such that

$$

G_h(u, \xi_h) = 0.

$$

The function $U_h$ is the

*household utility function*, and the operator $G_h$ is the

*household constraint*. It is a set of constraints that applies to the state variables. Note that this is a vector of constraints, and it is somewhat hard to characterise. For example, if initial bond ($B$) holdings equals 1, then $G_h$ will include a term:

$$

B(0) - 1 = 0.

$$

Things like the household budget constraint imply a sequence of constraints that relate the variables between times 0 and 1, times 1 and 2, etc. In other words, $G_h$ has an infinite number of terms. (Note that if there are inequalities (such as $\geq$), we would need to split the constraints into an equality and inequality vector constraints.)

We will follow in the footsteps of DSGE modelers, and largely put aside the question of existence and uniqueness of an optimal solution.

It must be kept in mind that the optimal solution can only be referred to once we fix the exogenous variables. Otherwise, the problem is indeterminate, since we are missing variables that appear in the objective function and/or household constraint ($G_h$).

In order to clean up our notation, we define the

*household problem operator*${\cal P}_h$ as an operator whose domain is the set of possible exogenous variables, and for any $\xi_h$ in that set, ${\cal P}_h = {u_h}$, where ${u_h}$ is the set of all maximising solutions to the household problem (and of course, ${\cal P}_h \xi_h = \emptyset$ is possible). In words, ${\cal P}_h$ maps the exogenous variable to the solution, and replaces writing down the entire household problem.

We can then repeat this exercise for the business sector. We create a new partition of the state variables into the business sector decision variables ($u_b$) and exogenous variables ($\xi_b$). The business sector will also have an objective function ($O_b$) and set of constraints ($G_b$). Note that decision variables between the business sector do not necessarily overlap with the household sector decision variables. That is, we can only assume that there is the same number of variables in the entire state vector for those sectors ($x_h, x_b$). We can then define the

*business sector problem operator*${\cal P}_b$ in an analogous fashion to ${\cal P}_h$.

For example, consumption (denoted $c$ in the Ramsey problem) is not a decision variable for the business sector. As far as the business sector optimisation problem is concerned, $c$ is an exogenous variable (and thus a member of $\xi_b$), whereas it is a decision variable for the household sector (and thus a member of $u_h$).

Finally, there is a governmental sector, which may or may not have a objective function. (It has a budget constraint, which will be a component of the government constraint $G_g$, which is related to the other two sectors' budget constraint by an accounting identity.)

Where my notation here differs from the DSGE macro papers I have seen is that I have kept the notation for variables deliberately distinct. In the published papers, the variables corresponding to the household problem and business problem share the same variable name, and some operations appear to mix the two versions of the variables (the validity of which is problematic).

## Stitching the Problems Together

We now have two*completely independent*optimisation problems. The question then arises: how do we create a single model for the economy?

Attempting to reverse engineer the descriptions of these DSGE models from various texts, I believe that the definition that DSGE modelers

*want to use*is as follows.

Let $X$ be the set of equilibrium solutions. This set is defined as follows: fix any $x^* \in X$ (this vector of time series is known as an equilibrium solution). Then, we look at the household (and business) optimisation problem separately. The equilibrium $x^*$ is partitioned into the household decision variables $u_h^*$ and $\xi_h^*$. We then must have the condition that $u_h^*$ is the optimal solution to the household problem with the exogenous variables fixed at $\xi_h^*$. That is, $u_h^* \in {\cal P_h}\xi^*_h.$ This notation covers the possibility of multiple optimal solutions; all we require is that the chosen variables is a possible maximising choice.

The same test on $x^*$ then must apply for the business sector problem: $u_b^* \in {\cal P}_b \xi_b^*.$ (If the government has an optimisation problem to solve, repeat the process again.)

The reasoning for this structure presumably results from similar work on one-sector models in microeconomics; I have never seen a DSGE macro paper write out this definition formally. (Even the Ljungqvist and Sargent text reverts to verbal shortcuts for key parts of the equilibrium definition.) Apparently, if the micro-structure changed, the way in which the equilibrium is defined could be different (for example, if the business sector were a monopoly). I believe that some definitions are based on first order condition logic, as discussed below.

The sting in the tail of this definition is that there is no formal way of finding the set of equilibria. We are searching over a large number of infinite sequences, and so no numerical technique could construct a solution in the life span of the universe. We will put that objection to the side, and continue.

The equilibrium definition defies a casual understanding because of a not-obvious property behind it: it is phrased in such a fashion that we are finding the optimal decision variables with the assumption that exogenous variables are fixed. That is, no matter what choice the household (or firm) makes with decision variables, all else literally remains equal. This means that we cannot apply standard macroeconomic logic to the discussion of the optimality of a solution such as accounting identities holding.

The key example is household consumption. The rest of this paragraph may make very little sense to most readers; the issue described here will be illustrated in length in later sections. This paragraph here is the verbal summary. The structure of the household constraint implies that a household could choose its hours worked and consumption independently. This might be true for a single sector (household only) model, but it implies a violation of the production function -- which conveniently only appears in the business sector problem. However, since the overall problem states that the business sector also solves its optimisation problem, we know that the equilibrium solution has to respect the production function constraint. However, if we incorporated the production function constraint into the household problem, it turns into a rather trivial path-planning problem. So we

*hope*that such solutions are rejected. They are rejected because it is

*hoped*that once we fix the equilibrium vector, it is possible that the household (or firm) could make a superior decision -- even though the superior choice to its optimisation problem implies a violation of the constraints of the other sector.

One may note that the word "hope" appeared in the discussion of equilibrium determination. From the Bourbaki perspective, "hope" is not a well-defined operation on sets.

## The Attempted Solution Technique

Discussing problems for which we have no way to determine a solution leads to a theoretical dead in applied mathematics.

*I have personal experience with this. In my corner of control theory, every problem ended up being equivalent to solving a Hamilton-Jacobi-Bellman equation or inequality. (Yes, that includes the Bellman Equation that some economists are all excited about.) However, numerical solution was out-of-reach, and so one ended up having to make conservative assumptions. It was clear that I had hit a dead end for research; it was either find a new academic field, or get a job in finance. Guess what I chose?*

In order to continue along this path, DSGE macro researchers decided that even if they cannot find the solution, they could possibly determine some of its properties.

*Constraints are found by the researcher*that represent relationships between the state variables if a solution to the optimisation problem is indeed optimal. They may be found by applying a Bellman equation, Lagrange multipliers, whatever. The list of constraints to be included is entirely determined by the researcher. If the researcher in question is denoted $R$, then the household

*first order conditions*${\cal F}^R_h$ is a set of constraints on the vector on the state vector such that:

$$

{\cal F}^R_h x^*_h(\xi_h) = 0,

$$

if $x^*_h(\xi_h)$ is the optimising solution to the household problem ($x^*_h(\xi_h)$ represents the optimal decision vectors given a set of exogenous variables).

Note that it might take a lot of work to shoehorn some first order conditions into this notation -- such as the transversality condition -- so I am leaving deliberately vague what set ${\cal F}^R_h$ resides in.

The definition of "equilibrium" within the DSGE literature is surprisingly vague. Some definitions of equilibrium may be based on these first order conditions, which is somewhat awkward mathematically.

**Example.**

*I will now illustrate what such a "first order condition" operator looks like, using a simple non-economic optimisation problem. Assume that we have to variables: the decision variable $w$ and an exogenous variable $a$. The state variable is the vector $(w, a)^T$. We want to find the variable $w$ to maximise $U$ where:*

$$

U(w, a) = - (w -2 a)^2,

$$

*where $w, a \in R$. For example, if $a = 1$, the problem becomes:*

$$

w^* = \arg \max \hat{U}(w) = -(w -2)^2.

$$

*I will leave the reader to validate that the choice:*

$$

w = 2 a

$$

*will achieve the maximum for $U$. Therefore, the "first order condition" operator associated with this is ${\cal F} x = w - 2a$, and so if ${\cal F} x = 0$, we have an optimal solution. However, note that so long as $a$ is a free parameter, we do not have a well-posed optimisation problem: we have no way of choosing the optimal $w$ in the absence of knowledge of $a$. Therefore the "first order condition" operator is somewhat unusual; it is putting a constraint on variables that holds over a set of optimisation problems.This is an approach that I am quite unfamiliar with, hence I am not entirely sure about the formal description of this object.*

Similarly, the researcher $R$ determines the first order conditions for the business sector -- ${\cal F}^R_b$.

The solution to the macro problem is the set of state variables $X$ such that:

$$

X = \{ x: {\cal F}^R_h x = 0, {\cal F}^R_b x = 0\}.

$$

It is hoped that this set matches the set of equilibria.

The problem is that these first order conditions are generated by researchers using solution techniques that ignores the added constraints that lie outside the optimisation problem. The first order conditions may only apply for solution vector $x$ such that $x$ violates the constraints for the other sector. For example, the first order conditions that apply to the household sector might violate the constraints on another sector.

As will be discussed next, the way in which conflicts are resolved lies largely at the discretion of the researcher.

## Fiscal Policy in Macro Models

In the discussion of the Ramsey Problem in section 16.2 of*Recursive Macroeconomic Theory*by Lars Ljungqvist and Thomas J. Sargent (denoted [LS2012] here), we run into the transversality condition (which is related to the so-called intertemporal governmental budget constraint). I do not wish to reproduce all of the equations in that model; the bulk of the important ones appeared in the first two articles of this series. The constraint is as follows (equation 16.2.16):

$$

\lim_{T \rightarrow \infty} \left( \prod_{i=0}^{T-1} R^{-1}(i) \right) \frac{b(t+1)}{R(t)} = 0.

$$

(I have taken the liberty of denoting a time series $y$ as $y(t)$, not $y_t$ as in [LS2012]; their time series notation was erratic.) To refresh the meaning of the notation, $b$ is the stock of government debt held by the household sector, and $R$ is the real rate of interest. (This model does not feature money; all variables are expressed in real terms.)

This relationship appeared after an iteration on an expression that was derived from a Lagrange multiplier.

Ljungqvist and Sargent then state:

As discussed in chapter 13, the household would not like to violate these transversality conditions by choosing $k_{t+1}$[Note: $k$ is capital; I have ignored the capital transversality condition, which is equation 16.2.15 in LS2012]or $b_{t+1}$ to be larger, because alternative feasible allocations with higher consumption in finite time would lead higher lifetime utility.

*(They go on to note that the limit cannot be negative as a result of borrowing constraints, a constraint which they of course omitted from the problem specification.)*

I would like to note that [LS2012] is one of the first references that attempted to give a formal justification of the transversality condition within the many DSGE macro papers I have read. One randomly chosen example is from the working paper "Bubbles and the Intertemporal Government Budget Constraint" by Stephen LeRoy (my version is from October 10, 2004), one reads:

Letting $n$ go to infinity and "applying the usual transversality condition," ...(At least he had scare quotes...)

Returning to the quote in [LS2012], in my view the text is misleading relative to the mathematics. The fact that they appeal to anthropomorphism in their discussion of what are supposed to be sets ("the household would not like...") is a signal that we are on shaky ground.

The first thing to keep in mind is that [LS2012] derived that "pure profits" are zero in equilibrium (discussed in my earlier article). The implication is that bond holdings of the business sector are always equal to zero, and so the governmental and household sector bond holdings are the negative of the other. (The bond holdings are better thought of as a pure debt security, which could be issued by either sector.) As a result, if the transversality condition applies, it applies equally to both the government and household sector.

*As an aside, the implications of the transversality condition are often mis-stated. They do not imply that the debt-to-GDP ratio will remain bounded, it is purely defined by the discount rate. If the discount rate is less than the long-term GDP growth rate, the debt-to-GDP ratio will in fact go to zero. On the other hand, if the discount rate is greater than the long-term growth rate, the debt-to-GDP ratio is allowed to become arbitrarily large. (Neither possibility appears to make much sense.)*

For simplicity, I will replace the "distortionary tax" in this model (which is a tax that depends upon output) with a lump sum tax (as seen in the model in [Gali2008]. (If this leap is distressing, we can just set tax rates to zero.) If the business sector has zero bond holdings, we know that the evolution of government debt is equal to the previous debt level plus the primary deficit plus interest costs. (Since we are using a discount bond convention, that formula is slightly more complex.) The formula is (an algebraic rearrangement of 16.2.5):

$$

\frac{b(t+1)}{R(t)} = g(t) + \tau(t) + b(t),

$$

where $\tau$ is the lump sum tax variable. We see that the trajectory of debt is completely determined by fiscal policy. The only reason the transversality condition holds is because fiscal policy is set to allow it to be true. There is nothing forcing the government to set policy in this fashion, other than the fact that the definition of equilibrium chosen by DSGE researchers will not exist. If there is no equilibrium, the model has no solution -- and so it offers no guide as to what might happen.**

We can now return to the statement by Ljungqvist and Sargent. The text "alternative feasible allocations with higher consumption in finite time would lead higher lifetime utility" is questionable. They seem to ignore the definition of feasibility found within the Ramsey problem (page 619 of the third edition): that equation (16.2.3) holds:

$$

c(t) + gt) + k(t+1) = F(t, k(t), n(t)) + (1 - \delta) k(t).

$$

This is the "accounting identity" for supply and demand for the single (real) good in this model, and includes the production function. This equation specifically bars the household from increasing its consumption solely by selling bonds: in order for the equation to balance,

*the number of hours worked would have to increase*. The rising disutility of work will tend to cancel out the increased consumption, and so we cannot draw any conclusions about utility.

What may have happened is that the quoted text refers to an alternative definition of feasibility, which refers to the household problem only. The fact that they have two definitions of "feasibility" within the same mathematical problem is curious. If one were attempting to do the mathematics carefully, we would need to distinguish two types of feasibility: single sector feasibility, and macro feasibility (which is the condition that 16.2.3) holds. The thought experiment of increasing consumption without changing hours worked is single sector feasible, but not macro feasible. Once again, being careful with the definition of objects clears up needless ambiguity.

In fact, the only reason that this problem is non-trivial is because the definition of equilibrium relies on there being superior non-feasible solutions relative to the trivial solution.

*(The trivial solution is what happens if the business sector does not exist: you insert the production function constraint into the household optimisation problem, and it becomes a pure path-planning problem with no financial components. I have discussed this in earlier articles.)*The only way of stopping the trivial solution from dominating is that the condition that all exogenous variables are assumed to be constant regardless of the choice of the decision variables. This is what allows the household sector to spend more now: the production function constraint that exists between hours worked and output is specifically ignored when testing whether a solution meets the definition of "equilibrium."

There is no way to relate the condition to the real world, since it is logically incoherent if we believe that the variables in the model correspond to real world variables. If a household increases consumption by spending out of bond holdings, either the holdings migrate to the business sector, or else it is returned via "capital rental" or wages. In the first case, the business sector bond holdings are non-zero, which means that the government position in bonds is no longer the mirror of the household sector. If the business sector bond holdings remain at zero, then the extra spending still results in the same final bond holdings. This is why the transversality condition does not appear in stock-flow consistent models: it violates accounting identities if you try to model the effect.

In models with monetary policy, similar effects would presumably limit the choices for monetary policy. However, the first order conditions which are examined appear to be largely at the discretion of the DSGE modeller -- since we cannot characterise any solutions that might exist. Researchers are interesting in demonstrating the efficacy of monetary policy, so the first order constraints on policy appear to be less explored.

## Concluding Remarks

Although DSGE modellers pride themselves on their mathematical skills, one is struck by how the core definition of equilibrium is largely glossed over in the treatment of multi-sector models. One potential explanation is that the definition relies on invalidating potential solutions by comparing them to trajectories that specifically violate hard constraints of the multi-sector model. This is a property that is somewhat less than satisfying, and might raise the eyebrows of skeptics.In any event, statements that DSGE macro models are true macro models in which accounting identities hold do not represent the spirit of the mathematics. In fact, analysis is confined to "first order conditions" that are defined for the optimisation problem of a single sector, and these first order conditions explicitly ignore the constraints imposed by other sectors.

## Appendix: Zero Measure Households

There may be a mathematical back story behind these models that appears to offer a justification for the violation of accounting identities. (From the Bourbaki perspective, having a back story for a statement about sets is somewhat difficult to grasp.) The story is that we have an uncountable infinity of households, and so a household is a set of measure zero. It can choose its decision variables in any matter it chooses, and there is no effect on aggregate variables. (Aggregates are generating by taking a Lebesgue integral over the set of households; a set of zero measure can do whatever it wants, and not effect the Lebesgue integral -- putting aside monstrosities like the Dirac delta "function.")In which case, only the price variables will matter; anything that is integrated can be ignored by a set of zero measure. This means that we can lop out the inconvenient hard constraints.

The story is then that the global solution then has to align with the optimal choice for the set of zero measure.

Although this is cute, this means that the household we are looking at is precisely not representative; it is anti-representative. There is no set of households with non-zero measure that could ignore the hard constraints in their choices, since their decisions will have a non-zero effect on integrated variables. We are still stuck with invalidating solutions based on comparisons that are not feasible for non-zero measure sets of households.

**Footnotes:**

* Specifying the set in which time series appear in is a detail that could be quite important for infinite horizon problems. Just specifying that a time series is finite for all $t \in R_+$ may not be enough. For example, any variable that looks like $w(t) = 10^{10^t}$ is technically finite for all time, but it will do a very good job of causing any infinite sum it appears in to not converge. It is unclear whether various solution techniques used to determine optimal solutions will survive the inclusion of such time series as possible solutions.

** The vagueness of the implications of an "unsustainable fiscal policy" in articles written by mainstream economists is therefore not a surprise.

Brian,

ReplyDeleteGood news, I think I understand Eq. 16.2.3 when written this way

\(c(t) + g(t) + k(t+1) = F(t, k(t), n(t)) + (1-\delta)k(t).\)

This is an inventory equation. Consumption + carryover = production + last period carryover as depreciated.

Unfortunately, the logic breaks down completely when government (or household) borrows the commodity and promises to pay interest. How does a consumer become a borrower when nothing is ever exchanged?

Could this be the root of the DSGE curious accounting?

Maybe the intent is for government to take control of the carryover inventory. For release of control, government promises to pay an interest bonus. Present owners might agree to this if government promised more interest than the private market annually offered.

If this is the intent, it should effect production very little.

The business sector borrows the capital from the household sector, and runs production. This does not make a whole lot of sense, but that’s how the model is set up. The exchanges are between the business sector and buyers - the government and households.

DeleteAfter I take your comments into consideration, and my own comments into consideration, my thinking tends to evolve. I think improves, with the puzzle pieces fitting tighter together.

DeleteI will delete my last two comments, replacing them with a better(?) line of thinking.

Now I go back to the first post in this series "....curious profit accounting.." to draw some new insight.

"There is a single good produced in the economy, and one unit of this good can be converted to one unit of capital, denoted k(t)."

This is a TWO commodity economy! The produced good is a commodity and capital is a commodity.

"We define household consumption as c(t) and government consumption as g(t). The accounting for real output is given by (16.2.3):"

This is misleading. It is not exactly the accounting for real output. It is the accounting for the real output

converted to terms of the capital commodity and added to the inventory of capital.Unfortunately, Eq. 16.2.3 has a typo, with a misplaced parenthesis. It should read \(c(t) + g(t) + k(t+1) = F(t, k_b, n) + (1-\delta )k(t).\)

16.2.3 now makes sense as an accounting of the capital exchanged during production and sale (of the produced good), added to the capital existing in inventory outside of the production process. Eq. 16.2.3 is an inventory equation expressed in terms of capital.

So what should the production equation look like? Following the constraints given, we could write \(c(t) + g(t) = F(t, k_b, n) + k_p(t)\) where \(k_p(t)\) is the firm's profit. This accounts for all production translated into terms of capital.

When writing the last two comments (now deleted), I was still thinking that you were writing about barter production. Now I think you were writing about a two commodity exchange economy, which much better aligns with my most recent post.

Oh well, something to think about when your other projects are complete.

Thanks for spotting the typo.

DeleteHowever, it is still a single good economy. Imagine that we start with 100 units of capital, then produce 10 units, and there is no depreciation. We can consume 110 units (assuming g=0); that is, we have consumed the initial capital as well as the production. This does not make a lot of sense for a modern economy, but we could imagine it as “eating our seed corn.” If capital is always increasing, this does not show up. In any event, this is what the equation says is happening.

Hmmm. Yes, your example is of consuming capital.

DeleteBut aren't we talking about production of a gizmo? So if we produced 10 gizmos, we could only sell 10. This even if each gizmo is worth one capital unit.

I think this gizmo distinction makes the profit equation (which followed) incorrect. Profit is not a built in part of the cost of production. Profit is better considered as an error between what customers pay and the unavoidable cost of production.

Sometimes I wish I wouldn't think about this stuff so much. I find it to be a fascinating puzzle.