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Sunday, March 25, 2018

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).

\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


  1. Brian: I skimmed this post, and your previous post.
    As you probably know, I don't do math. But here's a couple of comments that might help with the intuition:

    Suppose households own all the machines ("capital"). Firms own nothing. Firms hire labour and machines from households, and pay them by the hour. If individual firms maximise profits, taking prices, wages, and the machine rental rate as given, wage and rental rate equal marginal products of labour and machines. And given constant returns to scale, those payments to households for renting labour and machines must add up to total output, so profits must be zero. (And the number of firms is irrelevant, as long as it's large enough to ensure none has market power over prices and wages and rentals.)

    If you instead assume decreasing returns to scale (like your square root example), then firms would earn positive profits, and you would need to add those profits into the household budget constraint. And the number of firms would matter (with free entry the number of firms would explode to infinity). And it would be a very different model.

    If taxes were lump sum, the stock of government bonds would be irrelevant in this model. Standard Ricardian Equivalence result for an infinitely-lived household. "We owe it to ourselves". The only reason the stock of bonds matters in this model is that taxes are distorting, so will affect the household's consumption/leisure and intertemporal consumption/saving choices.

    Dunno if this helps.

    1. Actually, you could pretty much delete firms from this model, assume the representative household produces output using its own labour and machines, and it wouldn't make any difference. It's just a way to get distorting taxes into the model, because if would be hard for the government to observe wages and machine rentals, to tax them, in a self-employed economy.

    2. OK, will have to think about your comments. They jibe with the bulk of my analysis. The issue I see is that the taxes only matter if we insist that government debt cannot be negative; the households can say, “sure, we’ll pay your taxes - just keep lending us the money to do so.” Mathematically, that is what the discussion of the optimal solution should revolve around.

      The case with positive profits would perhaps be the more interesting case; it is what I was used to. This article was a first draft, and I discovered my conclusions were weaker for this model than the other case (the one in the text by Gali) only after I finished my write up.

      There’s presumably hidden mathematical assumptions (perhaps discussed in the previous 614 pages) that I am missing, that precludes my solution technique. My academic-style complaint is that applied mathematics can’t bury critical assumptions like that.

    3. Brian: I haven't read L&S. It's not really my cup of tea. I'm impressed you are having a crack at it. I think you are mostly getting the intuition.

      There's always hidden (or non-obvious) assumptions in math models. The trick is to figure out which ones might matter.

      I don't see anything that could prevent government debt being negative in this model. Instead of the government owing the households, the households owe the government. And with lump sum taxes and infinitely-lived representative household, it wouldn't matter anyway, whether the right pocket owes the left pocket or vice versa. Lump sum taxes could be negative too (i.e. transfer payments).

      But there's no money in this model (yet). We have to think of bonds being promises to pay goods. Wages and capital rentals and taxes are also paid in goods.

      If firms own the capital, and pay the rents to themselves, then those rents would be "profits" in the accounting sense of the word, and those "profits" would appear in the household budget instead of rents. But since the two are equal (constant returns to scale), it wouldn't make any difference to the model. Households own the firms, just like they "own" the government (they owe its debt to themselves). The whole institutional structure is irrelevant in this model, except for: government spending (which is purely wasteful, since it does not appear in the household's utility function); and the distorting taxes.

    4. The lack of money appearing is an issue (and raised the ire of a certain commenter...), and the zero "accounting profits" made my points less interesting. I was going to get to the governmental side, and discuss some other models as well, in which case the results are more interesting.

      I like L&S since they start off close to the optimal control (and my Ph.D. is in control systems). Other treatments buried the optimal control roots, and the mathematics just mystified me. (Gali's text in particular.)

      Unfortunately, my consulting duties call, so I would have to get back to any other points later...

  2. I remain frustrated with the math presented here. I am focusing on equation 16.2.3. \[c(t) + g(t) + k(t+1) = F(t, k(t), n(t)) + (1-\delta)k(t)\]

    I am not well versed in function notation but, based on what I can learn from Wikipedia and other sources, funtion notation is likened to a black box from which we expect to see emerge a prescribed transformation of variable inputs.

    If the 'black box' analogy is correct, the actual production equation does not appear until later following substitutions into eq 16.2.3. The result, which would be the 'production equation', is \[c(t) + g(t) + k(t+1) = r(t)k(t) + w(t)n(t) + (1-\delta) k(t).\]

    Terms r and delta are both related to changes in value of capital, but presented in different contexts. Hence, we have a single variable, inflation, influencing the production equation in two different but related ways.

    So what is the final usefulness of this equation? The combined effect of double entry seems to mostly offset. The equation remains very confusing to me.

    1. Roger: no.

      The first equation in your comment is an engineering relationship. It tells us that the level of output depends on the level of employment of labour and kapital (machines), and output can be either consumed by households, or government, or used to increase the stock of machines.

      The production function F(t,k(t),n(t)) is not specified. It does not need to be specified. But y=k^0.3xn^0.7 would be a numerical example that fits the model.

      The second equation adds that all output (income) gets paid out as wages and machine rentals, and gets used as per above.

      r is rental paid per machine per period, just like the wage paid for labour. delta is the fraction of machines that explode per period. There is no inflation in this model, because there is no money (yet). Think of it as a barter economy.

    2. Thanks Nick. The perspective you present is considerably different from my beginning perspective.

      One thing: I think the time notation \((t)\) represents a data accumulation that occurs over a time period. In contrast, the time notation \((t+1)\) would be the single point reading at time point one.

      If that would be the correct notation assignment, I might still quible with the equation over the apparent useage of time intervals.

    3. Time notation: all series are defined for time=0, 1, 2, 3, ...
      x(t) refers to the variable at the current time; for example x(0) refers to the value of x at time zero.
      x(t+1) refers to the value at the next time point, So if t=0, x(t+1) is the value at time 1. This is used to tie capital levels and debt levels to the previous period.

    4. Thanks Brian. I think I have it now.

      The production equation assumes no beginning capital. As a result of that assumption, at end of any time period, \(k(t+1) = k(t).\)

      If there was a beginning capital quantity, we would have at end of any time period (t) \(k(t + 1) = k(t + 0) + k(t).\)

      I think that is the correct interpretation.

  3. The phrase "we owe it to ourselves" is true only if the household sector is NOT divided into three segments: wealthy, working class, and poor. In that case the wealthy own equity and debt instruments which represent future cash flow obligations of the working class. The government can be seen as another type of firm which intermediates between the wealthy and working class households. So government policy has much to do with the evolving structure of the financial float in a context where bailouts may occur, debts may be discharged in bankruptcy, financial assets may be zombie items or written off, and government may run a surplus or deficit.

  4. "To say that published DSGE treatments of multi-sector models are muddled is an understatement."

    I am but a mere lowly student of this area but it seems to me that DSGE modelling is a complete contrivance with questionable logic.

    Firstly, the model is specified and defined. It is said to be based on GE but this hardly appears to be the case - it would seem it is more like a partial equilibrium approach. As Brian says, the major mathematical operation is the use of optimization algorithms. The model is linearized to remove cyclical elements. Parameters are estimated (guesstimated) and then the model equations are calibrated and massaged to fit the data.

    All this seems a patchwork of steroidal techniques and hardly elegant.

    Henry Rech

  5. The curious non-existence of profit in economics
    Comment on Brian Romanchuk on “The Curious Household Accounting Of DSGE Models”

    Krugman once put it in a nutshell: “most of what I and many others do is sorta-kinda neoclassical because it takes the maximization-and-equilibrium world as a starting point.”

    Krugman, of course, is an idiot. But he is not the only one, just the opposite, he is the mouthpiece of the stupid majority. To this basket of deplorables belong also Lars Ljungqvist, Thomas Sargent, the rest of DSGEers, Brian Romanchuk, Roger Sparks, Nick Rowe, and the rest of tireless nonsense bloggers.

    What unites these folks is scientific incompetence, more specifically, the inability to realize that maximization-and-equilibrium has always been and will always be a methodologically inadmissible starting point. To recall, these are the verbalized neo-Walrasian axioms:

    HC1. There exist economic agents.
    HC2. Agents have preferences over outcomes.
    HC3. Agents independently optimize subject to constraints.
    HC4. Choices are made in interrelated markets.
    HC5. Agents have full relevant knowledge.
    HC6. Observable economic outcomes are coordinated, so they must be discussed with reference to equilibrium states. (Weintraub)

    These premises contain three plain NONENTITIES (constrained optimization, rational expectations, equilibrium) and therefore are forever unacceptable. Being incompetent scientists, though, most economists swallowed this inane stuff hook, line and sinker from Jevons/Walras/Menger onward to DSGE.

    Every theory/model that contains just one NONENTITY is scientifically worthless.

    What has to be realized in addition is that false premises require a pigtail of false auxiliary assumptions. So, in order to be applicable, constrained optimization requires the auxiliary assumption of a production functions with decreasing returns. Thus, a silly behavioral assumption determines the physical properties of production in the upside-down world of standard economics.

    One auxiliary assumption that is clearly at odds with reality is zero profits.

    “The zero profit attributed to perfect competition does not arise from perfect competition at all, but merely from the assumption that aggregate profit is zero.” (Murad)

    “… since it is impossible to have an economy where everyone is making profits. Aggregate profit for an entire (closed) economy must be zero, hence if any firm is making profits, some other firm must be making losses.” (Boland)

    “Wherever entrepreneurs make profits … they expand production; wherever they incur losses, production is contracted. In equilibrium therefore, there are neither profits nor losses. Walras thus created the abstraction of the zero-profit entrepreneur under perfect competition.” (Niehans)

    “Profit theory has long been regarded as one of the more unsatisfactory branches of economics. . . . One reason for this is that economists have not asked the right questions about profit.” (Murad)

    In fact, economists do not understand since 200+ years what profit is.#1 This includes Walrasians, Keynesians, Marxians, Austrians, Pluralists, DSGEers, MMTers and, last but not least, Brian Romanchuk.

    Lars Ljungqvist’s and Thomas Sargent’s DSGE is proto-scientific dreck. Nobody with more than two brain cells needs three lengthy posts to arrive at this conclusion.

    Egmont Kakarot-Handtke

    #1 For details see cross-references Profit


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