The Curious Governmental Accounting Of Dsge Macro
Readers who are allergic to mathematics may take away hold a hard fourth dimension amongst this article. Given that I am complaining almost inadequate mathematical notation, I am forced to live to a greater extent than formal than what I prefer. The concepts could live expressed inwards evidently English, but it would live much longer.
The previous 2 articles of this serial are here too here (for the trace of piece of work organisation too theater sector accounting). Note that this article volition utilization the model discussed inwards those articles as an example; the equations are constitute therein. It should live noted that those 2 articles were written before I realised that I was mis-interpreting the mathematical notation inside DSGE macro papers. Another article I wrote discusses the curious notation used; this article re-does that analysis. The get-go 2 articles on the theater too trace of piece of work organisation sector were based on a criterion mathematical convention; they offering an explanation of what goes incorrect if 1 assumes that DSGE macro papers are doing criterion mathematics.
Introduction
This get-go component subdivision of article is a reformulation of an before article: "The Curious Notation of DSGE Models." Until really recently, I was unable to slice together the proofs inwards DSGE models, as at that topographic point were logical leaps taken that did non appear reasonable to me. The fundamental is that DSGE macro texts utilization the same symbols for what are dissimilar mathematical objects. This agency that at that topographic point are hidden "rules of the game" for what are legitimate mathematical operations. As volition live discussed, it is unclear whether the rules invoked past times DSGE researchers tin live codified as a serial of formal statements on sets. Influenza A virus subtype H5N1 reader that dislikes mathematics may want to jump to the afterward parts of the article, which returns to verbalise over how this applies to the analysis of the governmental sector.I take away hold noted inwards the past times the Bourbaki persuasion of mathematics -- that mathematics is properly the report of sets. Although applied mathematics is frequently written inwards a less formal manner, nosotros should live able to relate whatever verbal arguing to a arguing almost sets. Otherwise, the logic becomes vague too maybe inconsistent.
DSGE Macro Conventions
If the DSGE macros modelers are working amongst a model inwards which a unmarried sector appears, they take away hold a mathematical work that either comes from optimal command theory, or a variation of one. Although 1 powerfulness observe issues amongst details (in what laid do fourth dimension serial lie?), at that topographic point is no argue to believe that at that topographic point are serious problems. Difficulties arise from multi-sector models, which take away hold no analogy inside optimal control.Influenza A virus subtype H5N1 DSGE multi-sector model is laid upward inwards a serial of steps.
We get-go define a nation vector $x$ that is laid of fourth dimension serial (such as consumption, hours worked, etc.), all of which are real-valued, too are defined on the fourth dimension axis $t = 0,1,2,...$.
Since it is non clear what infinite fourth dimension serial prevarication in, I volition utilization the laid $T$ as a placeholder as the laid of fourth dimension series.*
The Household Problem is so defined past times partitioning the nation vector $x$ into 2 sets of variables: the determination variables $u_h$, too exogenous variables $\xi_h$. For clarity, nosotros volition announce the nation vector as $x_h$, where $x_h = (u_h^T, \xi_h^T)^T$ ($T$ hither denotes the vector transpose).
(If nosotros await at the Ramsey Problem I referred to inwards previous articles, the theater determination variables are the hours worked, consumption, goods held as capital, too bond holdings. The theater budget constraint implies that at that topographic point are fewer degrees of liberty than that listing implies. Everything else, such as the wage rate, taxation rates, too regime spending are exogenous.)
If nosotros ready the exogenous variables ($\xi_h$), nosotros so 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 business office $U_h$ is the household utility function, too the operator $G_h$ is the household constraint. It is a laid of constraints that applies to the nation variables. Note that this is a vector of constraints, too it is somewhat hard to characterise. For example, if initial bond ($B$) holdings equals 1, so $G_h$ volition include a term:
$$
B(0) - 1 = 0.
$$
Things similar the theater budget constraint imply a sequence of constraints that relate the variables betwixt times 0 too 1, times 1 too 2, etc. In other words, $G_h$ has an infinite number of terms. (Note that if at that topographic point are inequalities (such as $\geq$), nosotros would demand to dissever the constraints into an equality too inequality vector constraints.)
We volition follow inwards the footsteps of DSGE modelers, too largely seat aside the query of existence too uniqueness of an optimal solution.
It must live kept inwards heed that the optimal solution tin only live referred to 1 time nosotros ready the exogenous variables. Otherwise, the work is indeterminate, since nosotros are missing variables that appear inwards the objective business office and/or theater constraint ($G_h$).
In gild to create clean upward our notation, nosotros define the household work operator ${\cal P}_h$ as an operator whose domain is the laid of possible exogenous variables, too for whatever $\xi_h$ inwards that set, ${\cal P}_h = {u_h}$, where ${u_h}$ is the laid of all maximising solutions to the theater work (and of course, ${\cal P}_h \xi_h = \emptyset$ is possible). In words, ${\cal P}_h$ maps the exogenous variable to the solution, too replaces writing downwards the entire theater problem.
We tin so repeat this do for the trace of piece of work organisation sector. We create a novel sectionalization of the nation variables into the trace of piece of work organisation sector determination variables ($u_b$) too exogenous variables ($\xi_b$). The trace of piece of work organisation sector volition likewise take away hold an objective business office ($O_b$) too laid of constraints ($G_b$). Note that determination variables betwixt the trace of piece of work organisation sector do non necessarily overlap amongst the theater sector determination variables. That is, nosotros tin only assume that at that topographic point is the same number of variables inwards the entire nation vector for those sectors ($x_h, x_b$). We tin so define the business sector work operator ${\cal P}_b$ inwards an analogous fashion to ${\cal P}_h$.
For example, consumption (denoted $c$ inwards the Ramsey problem) is non a determination variable for the trace of piece of work organisation sector. As far as the trace of piece of work organisation sector optimisation work is concerned, $c$ is an exogenous variable (and thus a fellow member of $\xi_b$), whereas it is a determination variable for the theater sector (and thus a fellow member of $u_h$).
Finally, at that topographic point is a governmental sector, which may or may non take away hold a objective function. (It has a budget constraint, which volition live a constituent of the regime constraint $G_g$, which is related to the other 2 sectors' budget constraint past times an accounting identity.)
Where my notation hither differs from the DSGE macro papers I take away hold seen is that I take away hold kept the notation for variables deliberately distinct. In the published papers, the variables corresponding to the theater work too trace of piece of work organisation work part the same variable name, too some operations appear to mix the 2 versions of the variables (the validity of which is problematic).
Stitching the Problems Together
We directly take away hold 2 completely independent optimisation problems. The query so arises: how do nosotros create a unmarried model for the economy?Attempting to contrary engineer the descriptions of these DSGE models from diverse texts, I believe that the Definition that DSGE modelers want to use is as follows.
Let $X$ live the laid of equilibrium solutions. This laid is defined as follows: ready whatever $x^* \in X$ (this vector of fourth dimension serial is known as an equilibrium solution). Then, nosotros await at the theater (and business) optimisation work separately. The equilibrium $x^*$ is partitioned into the theater determination variables $u_h^*$ too $\xi_h^*$. We so must take away hold the status that $u_h^*$ is the optimal solution to the theater work amongst 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 nosotros require is that the chosen variables is a possible maximising choice.
The same essay on $x^*$ so must apply for the trace of piece of work organisation sector problem: $u_b^* \in {\cal P}_b \xi_b^*.$ (If the regime has an optimisation work to solve, repeat the procedure again.)
The reasoning for this construction presumably results from similar piece of work on one-sector models inwards microeconomics; I take away hold never seen a DSGE macro newspaper write out this Definition formally. (Even the Ljungqvist too Sargent text reverts to verbal shortcuts for fundamental parts of the equilibrium definition.) Apparently, if the micro-structure changed, the way inwards which the equilibrium is defined could live dissimilar (for example, if the trace of piece of work organisation sector were a monopoly). I believe that some definitions are based on get-go gild status logic, as discussed below.
The sting inwards the tail of this Definition is that at that topographic point is no formal way of finding the laid of equilibria. We are searching over a large number of infinite sequences, too so no numerical technique could create a solution inwards the life bridge of the universe. We volition seat that objection to the side, too continue.
The equilibrium Definition defies a casual agreement because of a not-obvious holding behind it: it is phrased inwards such a fashion that nosotros are finding the optimal determination variables amongst the supposition that exogenous variables are fixed. That is, no affair what alternative the theater (or firm) makes amongst determination variables, all else literally remains equal. This agency that nosotros cannot apply criterion macroeconomic logic to the tidings of the optimality of a solution such as accounting identities holding.
The fundamental representative is theater consumption. The ease of this paragraph may brand really fiddling sense to most readers; the number described hither volition live illustrated inwards length inwards afterward sections. This paragraph hither is the verbal summary. The construction of the theater constraint implies that a theater could take away its hours worked too consumption independently. This powerfulness live truthful for a unmarried sector (household only) model, but it implies a violation of the production business office -- which conveniently only appears inwards the trace of piece of work organisation sector problem. However, since the overall work states that the trace of piece of work organisation sector likewise solves its optimisation problem, nosotros know that the equilibrium solution has to honour the production business office constraint. However, if nosotros incorporated the production business office constraint into the theater problem, it turns into a rather trivial path-planning problem. So nosotros hope that such solutions are rejected. They are rejected because it is hoped that 1 time nosotros ready the equilibrium vector, it is possible that the theater (or firm) could brand a superior determination -- fifty-fifty though the superior alternative to its optimisation work implies a violation of the constraints of the other sector.
One may banking concern complaint that the tidings "hope" appeared inwards the tidings of equilibrium determination. From the Bourbaki perspective, "hope" is non a well-defined functioning on sets.
The Attempted Solution Technique
Discussing problems for which nosotros take away hold no way to determine a solution leads to a theoretical dead inwards applied mathematics.
I take away hold personal sense amongst this. In my corner of command theory, every work ended upward beingness 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, too so 1 ended upward having to brand conservative assumptions. It was clear that I had hitting a dead cease for research; it was either observe a novel academic field, or larn a undertaking inwards finance. Guess what I chose?
In gild to maintain along this path, DSGE macro researchers decided that fifty-fifty if they cannot observe the solution, they could maybe determine some of its properties. Constraints are constitute past times the researcher that stand upward for relationships betwixt the nation variables if a solution to the optimisation work is indeed optimal. They may live constitute past times applying a Bellman equation, Lagrange multipliers, whatever. The listing of constraints to live included is exclusively determined past times the researcher. If the researcher inwards query is denoted $R$, so the theater first gild weather ${\cal F}^R_h$ is a laid of constraints on the vector on the nation vector such that:
$$
{\cal F}^R_h x^*_h(\xi_h) = 0,
$$
if $x^*_h(\xi_h)$ is the optimising solution to the theater work ($x^*_h(\xi_h)$ represents the optimal determination vectors given a laid of exogenous variables).
Note that it powerfulness take away hold a lot of piece of work to shoehorn some get-go gild weather into this notation -- such as the transversality status -- so I am leaving deliberately vague what laid ${\cal F}^R_h$ resides in.
The Definition of "equilibrium" inside the DSGE literature is surprisingly vague. Some definitions of equilibrium may live based on these get-go gild conditions, which is somewhat awkward mathematically.
Example. I volition directly illustrate what such a "first gild condition" operator looks like, using a unproblematic non-economic optimisation problem. Assume that nosotros take away hold to variables: the determination variable $w$ too an exogenous variable $a$. The nation variable is the vector $(w, a)^T$. We desire to observe 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 work becomes:
$$
w^* = \arg \max \hat{U}(w) = -(w -2)^2.
$$
I volition move out the reader to validate that the choice:
$$
w = 2 a
$$
will accomplish the maximum for $U$. Therefore, the "first gild condition" operator associated amongst this is ${\cal F} x = w - 2a$, too so if ${\cal F} x = 0$, nosotros take away hold an optimal solution. However, banking concern complaint that so long as $a$ is a gratuitous parameter, nosotros do non take away hold a well-posed optimisation problem: nosotros take away hold no way of choosing the optimal $w$ inwards the absence of noesis of $a$. Therefore the "first gild condition" operator is somewhat unusual; it is putting a constraint on variables that holds over a laid of optimisation problems.This is an approach that I am quite unfamiliar with, hence I am non exclusively certain almost the formal description of this object.
Similarly, the researcher $R$ determines the get-go gild weather for the trace of piece of work organisation sector -- ${\cal F}^R_b$.
The solution to the macro work is the laid of nation variables $X$ such that:
$$
X = \{ x: {\cal F}^R_h x = 0, {\cal F}^R_b x = 0\}.
$$
It is hoped that this laid matches the laid of equilibria.
The work is that these get-go gild weather are generated past times researchers using solution techniques that ignores the added constraints that prevarication exterior the optimisation problem. The get-go gild weather may only apply for solution vector $x$ such that $x$ violates the constraints for the other sector. For example, the get-go gild weather that apply to the theater sector powerfulness violate the constraints on some other sector.
As volition live discussed next, the way inwards which conflicts are resolved lies largely at the discretion of the researcher.
Fiscal Policy inwards Macro Models
In the tidings of the Ramsey Problem inwards department 16.2 of Recursive Macroeconomic Theory past times Lars Ljungqvist too Thomas J. Sargent (denoted [LS2012] here), nosotros run into the transversality status (which is related to the so-called intertemporal governmental budget constraint). I do non want to reproduce all of the equations inwards that model; the mass of the of import ones appeared inwards the get-go 2 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 take away hold taken the liberty of denoting a fourth dimension serial $y$ as $y(t)$, non $y_t$ as inwards [LS2012]; their fourth dimension serial notation was erratic.) To refresh the important of the notation, $b$ is the stock of regime debt held past times the theater sector, too $R$ is the existent charge per unit of measurement of interest. (This model does non characteristic money; all variables are expressed inwards existent terms.)
This human relationship appeared after an iteration on an aspect that was derived from a Lagrange multiplier.
Ljungqvist too Sargent so state:
As discussed inwards chapter 13, the theater would non similar to violate these transversality weather past times choosing $k_{t+1}$ [Note: $k$ is capital; I take away hold ignored the working capital alphabetic character transversality condition, which is equation 16.2.15 inwards LS2012] or $b_{t+1}$ to live larger, because alternative viable allocations amongst higher consumption inwards finite fourth dimension would atomic number 82 higher lifetime utility.(They cash inwards one's chips on to banking concern complaint that the limit cannot live negative as a termination of borrowing constraints, a constraint which they of class omitted from the work specification.)
I would similar to banking concern complaint that [LS2012] is 1 of the get-go references that attempted to give a formal justification of the transversality status inside the many DSGE macro papers I take away hold read. One randomly chosen representative is from the working newspaper "Bubbles too the Intertemporal Government Budget Constraint" past times Stephen LeRoy (my version is from Oct 10, 2004), 1 reads:
Letting $n$ cash inwards one's chips to infinity too "applying the park transversality condition," ...(At to the lowest degree he had scare quotes...)
Returning to the quote inwards [LS2012], inwards my persuasion the text is misleading relative to the mathematics. The fact that they appeal to anthropomorphism inwards their tidings of what are supposed to live sets ("the theater would non like...") is a dot that nosotros are on shaky ground.
The get-go thing to maintain inwards heed is that [LS2012] derived that "pure profits" are null inwards equilibrium (discussed inwards my before article). The implication is that bond holdings of the trace of piece of work organisation sector are ever equal to zero, too so the governmental too theater sector bond holdings are the negative of the other. (The bond holdings are improve idea of as a pure debt security, which could live issued past times either sector.) As a result, if the transversality status applies, it applies every bit to both the regime too theater sector.
As an aside, the implications of the transversality status are frequently mis-stated. They do non imply that the debt-to-GDP ratio volition rest bounded, it is purely defined past times the discount rate. If the discount charge per unit of measurement is less than the long-term gross domestic product growth rate, the debt-to-GDP ratio volition inwards fact cash inwards one's chips to zero. On the other hand, if the discount charge per unit of measurement is greater than the long-term growth rate, the debt-to-GDP ratio is allowed to cash inwards one's chips arbitrarily large. (Neither possibility appears to brand much sense.)
For simplicity, I volition supplant the "distortionary tax" inwards this model (which is a taxation that depends upon output) amongst a lump total taxation (as seen inwards the model inwards [Gali2008]. (If this jump is distressing, nosotros tin simply laid taxation rates to zero.) If the trace of piece of work organisation sector has null bond holdings, nosotros know that the development of regime debt is equal to the previous debt bird summation the primary deficit summation involvement costs. (Since nosotros are using a discount bond convention, that formula is slightly to a greater extent than 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 total taxation variable. We reckon that the trajectory of debt is completely determined past times financial policy. The only argue the transversality status holds is because financial policy is laid to allow it to live true. There is zippo forcing the regime to laid policy inwards this fashion, other than the fact that the Definition of equilibrium chosen past times DSGE researchers volition non exist. If at that topographic point is no equilibrium, the model has no solution -- too so it offers no guide as to what powerfulness happen.**
We tin directly render to the arguing past times Ljungqvist too Sargent. The text "alternative viable allocations amongst higher consumption inwards finite fourth dimension would atomic number 82 higher lifetime utility" is questionable. They seem to ignore the Definition of feasibility constitute inside the Ramsey work (page 619 of the tertiary 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 render too demand for the unmarried (real) expert inwards this model, too includes the production function. This equation specifically bars the theater from increasing its consumption solely past times selling bonds: inwards gild for the equation to balance, the number of hours worked would take away hold to increase. The rising disutility of piece of work volition tend to cancel out the increased consumption, too so nosotros cannot pull whatever conclusions almost utility.
What may take away hold happened is that the quoted text refers to an alternative Definition of feasibility, which refers to the theater work only. The fact that they take away hold 2 definitions of "feasibility" inside the same mathematical work is curious. If 1 were attempting to do the mathematics carefully, nosotros would demand to distinguish 2 types of feasibility: unmarried sector feasibility, too macro feasibility (which is the status that 16.2.3) holds. The idea experiment of increasing consumption without changing hours worked is unmarried sector feasible, but non macro feasible. Once again, beingness careful amongst the Definition of objects clears upward needless ambiguity.
In fact, the only argue that this work is non-trivial is because the Definition of equilibrium relies on at that topographic point beingness superior non-feasible solutions relative to the trivial solution. (The trivial solution is what happens if the trace of piece of work organisation sector does non exist: you lot insert the production business office constraint into the theater optimisation problem, too it becomes a pure path-planning work amongst no financial components. I take away hold discussed this inwards before articles.) The only way of stopping the trivial solution from dominating is that the status that all exogenous variables are assumed to live constant regardless of the alternative of the determination variables. This is what allows the theater sector to pass to a greater extent than now: the production business office constraint that exists betwixt hours worked too output is specifically ignored when testing whether a solution meets the Definition of "equilibrium."
There is no way to relate the status to the existent world, since it is logically incoherent if nosotros believe that the variables inwards the model correspond to existent earth variables. If a theater increases consumption past times spending out of bond holdings, either the holdings migrate to the trace of piece of work organisation sector, or else it is returned via "capital rental" or wages. In the get-go case, the trace of piece of work organisation sector bond holdings are non-zero, which agency that the regime seat inwards bonds is no longer the mirror of the theater sector. If the trace of piece of work organisation sector bond holdings rest at zero, so the extra spending all the same results inwards the same terminal bond holdings. This is why the transversality status does non appear inwards stock-flow consistent models: it violates accounting identities if you lot essay to model the effect.
In models amongst monetary policy, similar effects would presumably limit the choices for monetary policy. However, the get-go gild weather which are examined appear to live largely at the discretion of the DSGE modeller -- since nosotros cannot characterise whatever solutions that powerfulness exist. Researchers are interesting inwards demonstrating the efficacy of monetary policy, so the get-go gild constraints on policy appear to live less explored.
Concluding Remarks
Although DSGE modellers pride themselves on their mathematical skills, 1 is struck past times how the center Definition of equilibrium is largely glossed over inwards the handling of multi-sector models. One potential explanation is that the Definition relies on invalidating potential solutions past times comparing them to trajectories that specifically violate hard constraints of the multi-sector model. This is a holding that is somewhat less than satisfying, too powerfulness enhance the eyebrows of skeptics.In whatever event, statements that DSGE macro models are truthful macro models inwards which accounting identities agree do non stand upward for the spirit of the mathematics. In fact, analysis is confined to "first gild conditions" that are defined for the optimisation work of a unmarried sector, too these get-go gild weather explicitly ignore the constraints imposed past times other sectors.
Appendix: Zero Measure Households
There may live a mathematical dorsum even out behind these models that appears to offering a justification for the violation of accounting identities. (From the Bourbaki perspective, having a dorsum even out for a arguing almost sets is somewhat hard to grasp.) The even out is that nosotros take away hold an uncountable infinity of households, too so a theater is a laid of mensurate zero. It tin take away its determination variables inwards whatever affair it chooses, too at that topographic point is no number on aggregate variables. (Aggregates are generating past times taking a Lebesgue integral over the laid of households; a laid of null mensurate tin do whatever it wants, too non number the Lebesgue integral -- putting aside monstrosities similar the Dirac delta "function.")In which case, only the toll variables volition matter; anything that is integrated tin live ignored past times a laid of null measure. This agency that nosotros tin lop out the inconvenient hard constraints.
The even out is so that the global solution so has to align amongst the optimal alternative for the laid of null measure.
Although this is cute, this agency that the theater nosotros are looking at is exactly non representative; it is anti-representative. There is no laid of households amongst non-zero mensurate that could ignore the hard constraints inwards their choices, since their decisions volition take away hold a non-zero number on integrated variables. We are all the same stuck amongst invalidating solutions based on comparisons that are non viable for non-zero mensurate sets of households.
Footnotes:
* Specifying the laid inwards which fourth dimension serial appear inwards is a special that could live quite of import for infinite horizon problems. Just specifying that a fourth dimension serial is finite for all $t \in R_+$ may non live enough. For example, whatever variable that looks similar $w(t) = 10^{10^t}$ is technically finite for all time, but it volition do a really expert undertaking of causing whatever infinite total it appears inwards to non converge. It is unclear whether diverse solution techniques used to determine optimal solutions volition hold upward the inclusion of such fourth dimension serial as possible solutions.
** The vagueness of the implications of an "unsustainable financial policy" inwards articles written past times mainstream economists is thus non a surprise.
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