The Curious Household Accounting Of DSGE Models

This article is the second part (of a planned trilogy) of articles on the accounting issues within Dynamic Stochastic General Equilibrium (DSGE) models. I have deliberately chosen one of the simplest DSGE models I could find, a deterministic (non-random) Ramsey model from the text Recursive Macroeconomic Theory by Lars Ljungqvist and Thomas J. Sargent. I have the third edition; the text is referred to as [LS2012] herein. My previous article, "The Curious Profit Accounting of DSGE Models," described the relationships for the business sector. This model has three sectors, and yes, the third article will likely be titled "The Curious Government Accounting of DSGE Models." That is, I see issues with all three sectors; the macroeconomic accounting identities tell us if we have a problem with one sector, this will rebound to the other sectors.

I will skip over the introductory material in this first article: what I am writing this? I just want to discuss one point for the non-mathematically minded here. The use of "accounting" in the title is somewhat unfortunate, as the accounting itself appears reasonable. (One can debate the decision by Ljungqvist and Sargent to express accounting identities in real terms; the equations arguably should be multiplied through by the price level. The largest concern is that the stock of government debt is a nominal quantity, and so changing the price level at the initial time point changes the real value of the debt. Since there is nothing within the model to pin down the price level, that is a serious concern. I may discuss that in the discussion of the government sector in the next article.)


Once again, I am basing my comments on the model in section 16.2 ("A nonstochastic economy" - which is an unusual way of writing "A deterministic economy"); equation numbers follow that in the text (of the form 16.2.x). I have made slight changes to the notation; in particular I have converted time dependence into the functional form. In other words, I translate $c_t$ in [LS2012] to $c(t)$ herein. (As was noted elsewhere by Matt Franko, calling this a "DSGE" model is a misnomer; technically it is a Dynamic General Equilibrium model. The problems with this model are repeated in the stochastic variants.)

I realise that most readers are allergic to mathematical formulae; I avoid using them in most of my articles for good reasons. However, the use of mathematics here cannot be avoided. I would summarise my main conclusion as follows: if we are following standard mathematical conventions, the household budget constraint just collapses to the inverse of the governmental budget constraint. This just tells us that the level of government debt is driven by the sequence of government deficits (including interest payments). If fiscal policy is exogenous, then this constraint has almost no effect on the optimisation problem. In other words, the optimal policy for the household sector is to ignore financial constraints when determining the optimal solution. This has the effect that the model is largely uninteresting as a model of the macroeconomy.

Some of the points raised here were discussed in some of my earlier articles. The only difference is that I am confining my attention to one particular example.

Household Equations

The "infinitely lived representative household" aims to optimise a utility function (16.2.1):
$$
\sum^{\infty}_{t=0} \beta^t u(c(t), l(y)), \beta \in (0,1),
$$
with consumption $c$ and leisure $l$. The household has 1 unit of time to allocate between labour ($n(t)$) and leisure (16.2.2),
$$
l(t) + n(t) = 1.
$$

There is a single good in the economy, which can be used either as capital or for consumption (1:1 conversion ratio between the variables). The equation describing production is given by (16.2.3):
$$
c(t) + g(t) + k(t+1) = F(t, k(t), n(t)) + (1-\delta)k(t),
$$
where $g(t)$ is government consumption, $F$ is the production function, and $\delta$ is the depreciation factor for capital. (Note that this implies that non-depreciated capital could be literally consumed.)

The government issues one-period bills, that pay a real rate of interest $R(t)$. The household budget constraint (16.2.6) is:
$$
c(t) + k(t+1) + \frac{b(t+1)}{R(t)} = (1 - \tau_n(t)) w(t)n(t) + (1 - \tau_k(t)) r(t)k(t) + (1-\delta) k(t) + b(t).
$$
The variable $w$ is the (real) wage rate, $r$ is the real return on "borrowed capital" (see previous article) and $\tau_n, \tau_k$ are the tax rates on wage and capital income respectively. I am assuming here that all capital is borrowed during a time period, although that is a problematic assumption (as discussed in the previous article).

The firm's (real) "pure" profits $\Pi$ are given by (16.2.17):
$$
\Pi(t) = F(t, k(t), n(t)) - r(t)k(t) - w(t)n(t)
$$
(this was discussed in the previous article).

We will then apply the result that pure profits are zero (as discussed in Section 16.2.3); $\Pi(t) = 0 \forall t$. One of the properties of mathematical models is that if we have a relationship between variables in one part of the system, we can then apply that relationship to other equations. In this case, if pure profits are zero, then:
$$
F(t, k(t), n(t)) = r(t)k(t) + w(t)n(t).
$$

We can then substitute this into (16.2.3):
$$
c(t) + g(t) + k(t+1) = r(t)k(t) + w(t)n(t) + (1-\delta) k(t).
$$

We then subtract this equation from the household budget constraint (16.2.6) to get:
$$
\frac{b(t+1)}{R(t)} - g(t) = -\tau_n w(t)n(t) - \tau_k r(t)k(t) + b(t).
$$

Or,
$$
\frac{b(t+1)}{R(t)} - b(t) = g(t)   -\tau_n w(t)n(t) - \tau_k r(t)k(t) = g(t) - T(t).
$$

This is just the governmental accounting constraint (16.2.5), although I have defined $T$ to be the total tax take. In word, the increase in government debt is equal to the primary deficit (and interest costs derived from $R(t)$).

This accounting identity does not add any information to the optimisation problem. In particular, it does not appear to have an effect on consumption or work decisions.

This can be seen as follows. We can define a new variable $h$ that is the total output received by the household in a period:
$$
h(t) = c(t) + k(t+1),
$$
and $\alpha$ is the fraction of output that is saved ($k(t+1) = \alpha(t) h(t)$, $c(t) = (1-\alpha(t))h(t)$. Under the reasonable assumption that capital and consumption cannot be negative, we see that $\alpha(t) \in [0,1] \forall t$.

We can then recast the optimisation problem as a path-planning problem in terms of variables $n(t)$ and $\alpha(t)$, both variables being confined to the unit interval $[0,1]$. These are decision variables that represent real economy variables; there is no financial aspect to them. I will then assert that the utility function as a function of $n$ and $\alpha$ is a continuous function. (This assertion is perhaps not obvious to validate, but I would need to drag in all the assumptions about continuity that are specified in the text, but not reproduced here. Conversely, if the utility function is not continuous in this fashion, the discussions of the solution in [LS2012] faces even greater mathematical issues.)

We can then look at the optimisation problem on any finite interval $t= 0,1, ..., T$. In this case, the set of possible $n$ and $\alpha$ forms a closed, compact set. Unless I am suffering a brain freeze, we should be able to apply Theorem 3 of Section 19 of Elements of the Theory of Functions and Functional Analysis by A.N. Kolmogorov, and S.V. Fomin, to conclude that the utility function achieves its maximum value for at least one choice of $n, \alpha$ (under my assumption of continuity).

The original problem is an infinite horizon. However, under the continuity assumption -- and if we assume that the supremum of the infinite horizon problem exists -- we can approach within any arbitrary $\epsilon$ of the supremum using the $n, \alpha$ formulation over a long enough forecast horizon. (I will denote the supremum of the infinite horizon utility function as $U$.)

As long as there are no constraints that limit the choice of $\alpha, n$, the optimising solution is independent of other variables. The next section discusses the possibility that taxes would limit the set of feasible solutions.

Imagine that the initial household debt holdings was a factor in determining the value of the supremum. In this case, if we have two different levels of household debt ($b(t)$ and $\tilde{b}(t)$, with $b(t) \neq \tilde{b}(t)$, the supremum values for the two optimisation problems would differ. Without loss of generality, assume that $\tilde{U} > U$. However, the $\alpha, n$ construction method for the solution is independent of the initial debt levels, and would therefore it is possible to approximate $\tilde{U}$ arbitrarily closely for both optimisation problems. This contradicts the assumption that $U$ was the optimal value for the problem.

This calls into question the entire discussion of the solution in [LS2012]. A whole host of conditions that allegedly matter for the household optimisation do not appear in true optimal solution. The problem is just a path-planning problem for real variables: what is the optimal level of production and saving given the level of real government purchases ($g$)? The business sector, taxes, and debt balances are superfluous to the mathematical problem.

Furthermore, if household sector financial balances do not matter, then it does not matter what the prices are. The imposition of sticky prices or other effects will have no effect on the optimal solution.

Do Taxes Matter?

It is possible that taxes could prevent the household from achieving the global optimising solution, if we do not allow the household to have negative debt holdings (either at all times, or in the limit). Since taxes are levied as a percentage of labour/capital income, too high an income (output) could result in a negative financial asset balance. The set of feasible solutions is more complex.

In this case, there is an obvious dependence upon the initial household financial balance; a larger balance implies that the household sector can have greater consumption before its financial asset balance hits its constraint.

It is unclear how interesting this financial constraint is. All it suggests is that the government should avoid driving its debt levels to zero, as that results in a sub-optimal outcome relative to the case where government debt levels remain strictly positive for all time.

In any event, it is unclear how this solution relates to the discussion in the text. In fact, the text suggests that negative government debt levels are optimal (Section 16.4), and so the constraint I suggest here does not exist in their version of the model.

How to Interpret This?

There is a general aversion in the DSGE macro literature to present worked examples. Extremely general models are written down, and then properties of the model are then derived. These rules are then expected to tell us something about the functioning of the macro-economy. However, it is very much unclear whether we get a well-formed optimisation problem if we fix the model structure, and try to find the optimal solution. The lack of numerical examples helps shroud the entire field in an unnecessary level of mystery -- and also makes it very difficult to see whether mathematical operations are legitimate.

It is clear that they wish that the problem is for households to take a given trajectory for future prices, and then find an optimal consumption path that obeys a financial constraint. This resembles a financial planning optimisation problem. The key is that we pretend that the decisions of the household have no effect on prices; it is too small to matter.

The problem is that there are circular flows of income in the economy, and so the spending returns to the household (less the financial drain of taxes, and the addition provided by government spending). Since financial assets will just return to the household, they largely drop out of the determination of the optimal solution.

If the solution discussed by DSGE modellers differs from the optimal solution technique I discussed above, their solution is diverging from the optimal utility level. However, they assert that the solution in fact optimises utility (definition of the Ramsey Problem in Section 16.3, on page 620). So why are they wasting time on the various first order conditions they identify in the text?

The only way of explaining this divergence is that we need to relabel all of the variables in the model, and we need to distinguish between the actual variables, and each sector's forecast of that variable. Each sector runs a separate optimisation on its version of the variables, which can somehow differ from the true levels.

For example, the household could have forecasts for its wages and capital income. It can then attempt to optimise relative to those forecasts, and thus we get the financial planner optimisation problem. However, we then need to ask -- how can each sector have its own forecasts, and what happens forecasts diverge between sectors? All of these forecasts would have to be identified as separate variables, and then we would need a separate set of equations or constraints that relates those variables, and how the economic variables relate to those expectations. Needless to say, none of the required mathematical steps are spelled out, and readers have to guess what is going on.

Furthermore, we run into an interesting problem. If we interpret the widespread use of Lagrange multipliers as attempting an optimisation on forecast variables, if the "optimal" solution differs from the true optimum, households are no longer doing model-consistent optimisations. They are instead following a heuristic, that leads to sub-optimal outcomes. Why are the DSGE behavioural heuristics privileged versus the ones in stock-flow consistent models? Was not the whole point of DSGE macro that it described optimising behaviour? If the behaviour is suboptimal, we might as well look at the data and see how sectors actually behave in the real world.

Concluding Remarks

To say that published DSGE treatments of multi-sector models are muddled is an understatement. There is no clear mathematical statement on how the optimisations that are done on a sector-by-sector basis relate to the global solution.

(c) Brian Romanchuk 2018