The Curious Solid Accounting Of Dsge Models

This article is the bit share (of a planned trilogy) of articles on the accounting issues inside Dynamic Stochastic General Equilibrium (DSGE) models. I take away maintain deliberately chosen 1 of the simplest DSGE models I could find, a deterministic (non-random) Ramsey model from the text Recursive Macroeconomic Theory past times Lars Ljungqvist in addition to Thomas J. Sargent. I take away maintain the 3rd edition; the text is referred to every bit [LS2012] herein. My previous article, "The Curious Profit Accounting of DSGE Models," described the relationships for the concern sector. This model has 3 sectors, in addition to yes, the 3rd article volition probable endure titled "The Curious Government Accounting of DSGE Models." That is, I reckon issues amongst all 3 sectors; the macroeconomic accounting identities tell us if nosotros take away maintain a employment amongst 1 sector, this volition rebound to the other sectors.

I volition skip over the introductory fabric inward this kickoff article: what I am writing this? I only desire to utter over 1 betoken for the non-mathematically minded here. The utilisation of "accounting" inward the championship is somewhat unfortunate, every bit the accounting itself appears reasonable. (One tin fence the determination past times Ljungqvist in addition to Sargent to limited accounting identities inward existent terms; the equations arguably should endure multiplied through past times the toll level. The largest concern is that the stock of authorities debt is a nominal quantity, in addition to so changing the toll bird at the initial fourth dimension betoken changes the existent value of the debt. Since in that place is cypher inside the model to pivot downward the toll level, that is a serious concern. I may utter over that inward the give-and-take of the authorities sector inward the adjacent article.)


Once again, I am basing my comments on the model inward department 16.2 ("A nonstochastic economy" - which is an odd agency of writing "A deterministic economy"); equation numbers follow that inward the text (of the bird 16.2.x). I take away maintain made slight changes to the notation; inward item I take away maintain converted fourth dimension dependence into the functional form. In other words, I interpret $c_t$ inward [LS2012] to $c(t)$ herein. (As was noted elsewhere past times Matt Franko, calling this a "DSGE" model is a misnomer; technically it is a Dynamic General Equilibrium model. The problems amongst this model are repeated inward the stochastic variants.)

I realise that most readers are allergic to mathematical formulae; I avoid using them inward most of my articles for adept reasons. However, the utilisation of mathematics hither cannot endure avoided. I would summarise my primary conclusion every bit follows: if nosotros are next measure mathematical conventions, the theatre budget constraint only collapses to the inverse of the governmental budget constraint. This only tells us that the bird of authorities debt is driven past times the sequence of authorities deficits (including involvement payments). If financial policy is exogenous, in addition to so this constraint has almost no outcome on the optimisation problem. In other words, the optimal policy for the theatre sector is to ignore financial constraints when determining the optimal solution. This has the outcome that the model is largely uninteresting every bit a model of the macroeconomy.

Some of the points raised hither were discussed inward about of my before articles. The exclusively difference is that I am confining my attending to 1 item example.

Household Equations

The "infinitely lived representative household" aims to optimise a utility share (16.2.1):
$$
\sum^{\infty}_{t=0} \beta^t u(c(t), l(y)), \beta \in (0,1),
$$
amongst consumption $c$ in addition to leisure $l$. The theatre has 1 unit of measurement of fourth dimension to allocate betwixt labour ($n(t)$) in addition to leisure (16.2.2),
$$
l(t) + n(t) = 1.
$$

There is a unmarried adept inward the economy, which tin endure used either every bit upper-case missive of the alphabet or for consumption (1:1 conversion ratio betwixt the variables). The equation describing production is given past times (16.2.3):
$$
c(t) + g(t) + k(t+1) = F(t, k(t), n(t)) + (1-\delta)k(t),
$$
where $g(t)$ is authorities consumption, $F$ is the production function, in addition to $\delta$ is the depreciation element for capital. (Note that this implies that non-depreciated upper-case missive of the alphabet could endure literally consumed.)

The authorities issues one-period bills, that pay a existent charge per unit of measurement of involvement $R(t)$. The theatre 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 existent render on "borrowed capital" (see previous article) in addition to $\tau_n, \tau_k$ are the taxation rates on wage in addition to upper-case missive of the alphabet income respectively. I am assuming hither that all upper-case missive of the alphabet is borrowed during a fourth dimension period, although that is a problematic supposition (as discussed inward the previous article).

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

We volition in addition to so apply the outcome that pure profits are null (as discussed inward Section 16.2.3); $\Pi(t) = 0 \forall t$. One of the properties of mathematical models is that if nosotros take away maintain a human relationship betwixt variables inward 1 share of the system, nosotros tin in addition to so apply that human 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 tin in addition to so 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 in addition to so subtract this equation from the theatre 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 only the governmental accounting constraint (16.2.5), although I take away maintain defined $T$ to endure the full taxation take. In word, the growth inward authorities debt is equal to the primary deficit (and involvement costs derived from $R(t)$).

This accounting identity does non add together whatsoever information to the optimisation problem. In particular, it does non look to take away maintain an outcome on consumption or piece of work decisions.

This tin endure seen every bit follows. We tin define a novel variable $h$ that is the full output received past times the theatre inward a period:
$$
h(t) = c(t) + k(t+1),
$$
in addition to $\alpha$ 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 supposition that upper-case missive of the alphabet in addition to consumption cannot endure negative, nosotros reckon that $\alpha(t) \in [0,1] \forall t$.

We tin in addition to so recast the optimisation employment every bit a path-planning employment inward terms of variables $n(t)$ in addition to $\alpha(t)$, both variables existence confined to the unit of measurement interval $[0,1]$. These are determination variables that stand upwards for existent economic scheme variables; in that place is no financial aspect to them. I volition in addition to so assert that the utility share every bit a share of $n$ in addition to $\alpha$ is a continuous function. (This assertion is perchance non obvious to validate, but I would quest to drag inward all the assumptions virtually continuity that are specified inward the text, but non reproduced here. Conversely, if the utility share is non continuous inward this fashion, the discussions of the solution inward [LS2012] faces fifty-fifty greater mathematical issues.)

We tin in addition to so aspect at the optimisation employment on whatsoever finite interval $t= 0,1, ..., T$. In this case, the laid of possible $n$ in addition to $\alpha$ forms a closed, compact set. Unless I am suffering a encephalon freeze, nosotros should endure able to apply Theorem 3 of Section nineteen of Elements of the Theory of Functions in addition to Functional Analysis past times A.N. Kolmogorov, in addition to S.V. Fomin, to conclude that the utility share achieves its maximum value for at to the lowest degree 1 pick of $n, \alpha$ (under my supposition of continuity).

The master employment is an interplanetary space horizon. However, nether the continuity supposition -- in addition to if nosotros assume that the supremum of the interplanetary space horizon employment exists -- nosotros tin approach inside whatsoever arbitrary $\epsilon$ of the supremum using the $n, \alpha$ formulation over a long plenty forecast horizon. (I volition announce the supremum of the interplanetary space horizon utility share every bit $U$.)

As long every bit in that place are no constraints that boundary the pick of $\alpha, n$, the optimising solution is independent of other variables. The adjacent department discusses the possibility that taxes would boundary the laid of viable solutions.

Imagine that the initial theatre debt holdings was a element inward determining the value of the supremum. In this case, if nosotros take away maintain ii unlike levels of theatre debt ($b(t)$ in addition to $\tilde{b}(t)$, amongst $b(t) \neq \tilde{b}(t)$, the supremum values for the ii 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, in addition to would thence it is possible to gauge $\tilde{U}$ arbitrarily closely for both optimisation problems. This contradicts the supposition that $U$ was the optimal value for the problem.

This calls into interrogation the entire give-and-take of the solution inward [LS2012]. Influenza A virus subtype H5N1 whole host of weather that allegedly thing for the theatre optimisation exercise non look inward truthful optimal solution. The employment is only a path-planning employment for existent variables: what is the optimal bird of production in addition to saving given the bird of existent authorities purchases ($g$)? The concern sector, taxes, in addition to debt balances are superfluous to the mathematical problem.

Furthermore, if theatre sector financial balances exercise non matter, in addition to so it does non thing what the prices are. The imposition of mucilaginous prices or other effects volition take away maintain no outcome on the optimal solution.

Do Taxes Matter?

It is possible that taxes could foreclose the theatre from achieving the global optimising solution, if nosotros exercise non allow the theatre to take away maintain negative debt holdings (either at all times, or inward the limit). Since taxes are levied every bit a percent of labour/capital income, likewise high an income (output) could outcome inward a negative financial property balance. The laid of viable solutions is to a greater extent than complex.

In this case, in that place is an obvious dependence upon the initial theatre financial balance; a larger remainder implies that the theatre sector tin take away maintain greater consumption before its financial property remainder hits its constraint.

It is unclear how interesting this financial constraint is. All it suggests is that the authorities should avoid driving its debt levels to zero, every bit that results inward a sub-optimal outcome relative to the illustration where authorities debt levels stay strictly positive for all time.

In whatsoever event, it is unclear how this solution relates to the give-and-take inward the text. In fact, the text suggests that negative authorities debt levels are optimal (Section 16.4), in addition to so the constraint I propose hither does non be inward their version of the model.

How to Interpret This?

There is a full general aversion inward the DSGE macro literature to acquaint worked examples. Extremely full general models are written down, in addition to and so properties of the model are in addition to so derived. These rules are in addition to so expected to tell us something virtually the functioning of the macro-economy. However, it is rattling much unclear whether nosotros larn a well-formed optimisation employment if nosotros laid the model structure, in addition to attempt to uncovering the optimal solution. The lack of numerical examples helps hide the entire land inward an unnecessary bird of mystery -- in addition to also makes it rattling hard to reckon whether mathematical operations are legitimate.

It is clear that they wishing that the employment is for households to accept a given trajectory for futurity prices, in addition to and so uncovering an optimal consumption path that obeys a financial constraint. This resembles a financial planning optimisation problem. The primal is that nosotros pretend that the decisions of the theatre take away maintain no outcome on prices; it is likewise modest to matter.

The employment is that in that place are round down flows of income inward the economy, in addition to so the spending returns to the theatre (less the financial drain of taxes, in addition to the improver provided past times authorities spending). Since financial assets volition only render to the household, they largely drib out of the determination of the optimal solution.

If the solution discussed past times 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 inward fact optimises utility (definition of the Ramsey Problem inward Section 16.3, on page 620). So why are they wasting fourth dimension on the diverse kickoff lodge weather they seat inward the text?

The exclusively agency of explaining this departure is that nosotros quest to relabel all of the variables inward the model, in addition to nosotros quest to distinguish betwixt the actual variables, in addition to each sector's forecast of that variable. Each sector runs a assort optimisation on its version of the variables, which tin somehow differ from the truthful levels.

For example, the theatre could take away maintain forecasts for its reward in addition to upper-case missive of the alphabet income. It tin in addition to so endeavor to optimise relative to those forecasts, in addition to thus nosotros larn the financial planner optimisation problem. However, nosotros in addition to so quest to enquire -- how tin each sector take away maintain its ain forecasts, in addition to what happens forecasts diverge betwixt sectors? All of these forecasts would take away maintain to endure identified every bit assort variables, in addition to and so nosotros would quest a assort laid of equations or constraints that relates those variables, in addition to how the economical variables relate to those expectations. Needless to say, none of the required mathematical steps are spelled out, in addition to readers take away maintain to guess what is going on.

Furthermore, nosotros come across an interesting problem. If nosotros interpret the widespread utilisation of Lagrange multipliers every bit attempting an optimisation on forecast variables, if the "optimal" solution differs from the truthful optimum, households are no longer doing model-consistent optimisations. They are instead next a heuristic, that leads to sub-optimal outcomes. Why are the DSGE behavioural heuristics privileged versus the ones inward stock-flow consistent models? Was non the whole betoken of DSGE macro that it described optimising behaviour? If the demeanour is suboptimal, nosotros mightiness every bit good aspect at the information in addition to reckon how sectors genuinely conduct inward the existent world.

Concluding Remarks

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

(c) Brian Romanchuk 2018