Interpreting Dsge Mathematics
This article uses some mathematical notation, rendered yesteryear MathJax. Some browsers volition create got difficulties rendering these equations. For non-mathematicians, they may hold upward able to follow my logic, but I am sticking fairly around the mathematics. What the arguments hateful inwards champaign English linguistic communication is opened upward to interpretation, but inwards my view, nosotros involve to convey that the behavior of published models are arbitrary, together with non drawn from the constraints of optimisation.
Update: I added a modest department discussing 1 potential mainstream answer to this article. The construction I discsuss is only of to a greater extent than involvement if it is extended to the sticky-price models, assuming that tin give the axe hold upward done. Also, I got some interesting feedback from Brian Albrecht (@BrianCAlbrecht) together with Roger E. A. Farmer (@farmerff) on Twitter. I involve to digest the information, but every bit I suspected, my approach towards DSGE macro mathematics is somewhat literalist. There is a large "back story" behind the diverse equations that appear inwards published papers together with books. This is roughly what I argued; I may reconsider whether or non my assessment at the bottom of the article is likewise harsh later.
What the Mathematical Problem Looks Like
(Please depository financial establishment notation that this article is mainly aimed at representative family DSGE models. Other DSGE models may hold upward less problematic.)If nosotros expect at a DSGE model newspaper together with drive to interpret the contents into a well-posed mathematical problem, it has the impression of looking something similar the next model, labelled M1.
Model M1. Define a set of nation variables x, which are fourth dimension serial on fourth dimension axis T. That is, for each chemical factor $x_i$ of x, $x_i(t) \in {\mathbb R}$ for all $t \in T$.
Let the set of all possible $x$ hold upward denoted A (the set of whatever conceivable nation vector).
Impose a set of constraints C on the nation variables x. We define the set of viable solutions ${\cal F}$, which is a subset of A, for which C(x) is true.
Define a utility business office U that is a business office of x.
The solution to the model is the vector $x^*$ which maximises $U$, that is:
\[
x^* ={\mathrm argmax}_{x \in {\cal F}} U(x).
\]
(What happens inwards the instance of $x^*$ non existing or non-unique is some other question.)
The measure practise inwards other fields of mathematics would hold upward initiative of all to lay out the Definition of M1, together with so plough to method of solution.
DSGE Macro Papers
The layout of published DSGE macro papers diverges from measure mathematical practice. Each economical sector is set out separately, together with so the writer starts working out diverse optimality weather condition -- called first-order conditions. Once all sectors are defined, the global model solution is approached. I flora this confusing; the solution method is mixed upward alongside the job statement, together with so it was unclear whether equations were portion of the constraints of the problem, or portion of the solution.If nosotros rewrite the job declaration into an organised fashion, nosotros larn the next construction (Model M2).
Let the set of all possible $x$ hold upward denoted A (the set of whatever conceivable nation vector).
Impose a set of constraints C on the nation variables x.
We impose an add-on set of constraints O on the nation variable x ("first-order constraints"). The set of x for which C(x) together with O(x) are truthful is the set of equilibrium solutions, E.
Define a utility business office U that is a business office of x.
The solution to the model is the vector $x^*$ which maximises $U$, that is:
\[
x^* = {\mathop{\mathrm argmax}}_{x \in E} U(x).
\]
However, when nosotros dig farther into the proof, the writer only seems to uncovering the solution based on the first-order weather condition together with the constraints. We are truly at a novel problem, M3.
Model M3. Define a set of nation variables x, which are fourth dimension serial on fourth dimension axis T. That is, for each chemical factor $x_i$ of x, $x_i(t) \in {\mathbb R}$ for all $t \in T$.
Let the set of all possible $x$ hold upward denoted A (the set of whatever conceivable nation vector).
Impose a set of constraints C on the nation variables x.
We impose an add-on set of constraints O on the nation variable x (first-order constraints). The set of x for which C(x) and O(x) are truthful is the set of equilibrium solutions, E.
The solution is a vector inwards E.
That is, nosotros only uncovering a nation vector for which the first-order weather condition together with constraints hold. To what extent the solution is optimal (as inwards models M1 or M2), that has to hold upward the outcome of the first-order weather condition O.
When pose this way, what is happening becomes obvious: the writer of the newspaper believed that the initiative of all companionship weather condition imply optimality when applied to M1. That is, if $O(x^*)$ is true, so $x^*$ is the solution to M1. (The fact that "first-order conditions" comes from optimisation theory should create got been a clue.)
The ground why I flora this hard to follow is that it was really clear that O did non imply optimality when nosotros expect at the job M1. I assumed that I was missing added constraints.
Back to Partial Equilibrium
It turns out what was happening had a uncomplicated explanation:the first-order weather condition were determined sector yesteryear sector, without whatever reference to the global model.If nosotros desire to utilisation economical jargon, the first-order weather condition are based on "partial equilibrium" conditions: nosotros expect at equilibrium inwards a unmarried sector or marketplace at a time, without reference to what is happening inwards other sectors or markets. This is every bit opposed to "general equilibrium", where all markets (including futurity markets) plough over equilibrium simultaneously. Since the "GE" inwards DSGE explicitly refers to full general equilibrium, did non the solution method create got to expect at the global optimisation problem?
The respond appears to hold upward no (for at to the lowest degree most of the representative family models I struggled with).
Simple Example
Influenza A virus subtype H5N1 total DSGE model is complex; I volition only expect at a cut-down 1 menstruum model. The handling is roughly based on the model developed inwards Chapter 2 of Jordi GalÃ's Monetary Policy, Inflation, together with the Business Cycle: An Introduction to the New Keynesian Framework. I am next the construction used inwards that text.The family sector starts alongside an initial coin residuum M. It provides labour to the line concern sector, together with uses its reward to buy output. The line concern sector aims to maximise profits.
The next variables are defined:
- P = Price of goods (nominal).
- W = Wage charge per unit of measurement (nominal).
- N = Number of hours worked.
- C = Amount of goods consumed (real units).
The family sector wants to maximise its utility business office U(C,N). (We assume that increased consumption improves utility; increased piece of work subtracts from uutility,)
The family sector has a budget constraint:
$$\begin{equation}
PC \leq WN + M. \label{eq:budget}
\end{equation}$$
That is, the amount of spending on goods (PC) is less than or equal to the wage pecker (WN) summation the initial stock of money.
We so outflow to the line concern sector. The declaration is that the line concern sector wants to maximise profits. I label profits F, and it is defined by:
\[
F = PC - WN.
\]
Total output is given yesteryear a production function:
\[
C = Influenza A virus subtype H5N1 N^{1-\alpha}, Influenza A virus subtype H5N1 \in {\mathbb R_+}.
\]
The commons logic is: maximum profits occurs when $\frac{dF}{dN} = 0$, together with nosotros tin give the axe apply to the previous 2 equations to get:
$$\begin{equation}
\frac{W}{P} = (1-\alpha) Influenza A virus subtype H5N1 N^{-\alpha}. \label{eq:marginal}
\end{equation}$$
This appears to pivot downward the human relationship betwixt reward together with prices; nosotros so apply this human relationship to the first-order weather condition for the family sector. (This pace is really standard; the textbook applies it to the model inwards Chapter 2, although the expressions are slightly to a greater extent than complex.)
However, this logic assumes that at that spot is no human relationship betwixt the wage pecker together with line concern revenue. Unfortunately, such a constraint exists: equation ($\ref{eq:budget}$).
If nosotros apply ($\ref{eq:budget}$) to the Definition of F. nosotros regard that:
\[
F \leq M.
\]
It is straightforward to regard that maximum profits is equal to M. The values of P together with W are essentially gratis to convey whatever value that they wish, so long every bit they brand it at F=M.
We tin give the axe so apply the human relationship $F=M$ to ($\ref{eq:budget}$), together with nosotros get:
\[
PC - WN = M.
\]
That is, the family budget constraint becomes an equality.
We tin give the axe build an optimising solution every bit follows.
- Apply $C = Influenza A virus subtype H5N1 N^{1-\alpha}$, together with insert into U. That is, cast $\hat{U}(N) = U(A N^{1-\alpha}, N).$
- The measure constraints on the cast of U ensure that $\hat{U}$ is differentiable alongside honour to N. Furthermore, at that spot is a unique $N^* \in {\mathbb R}_+$ such that the derivative of $\hat{U}$ alongside honour to $N$ is nada at $N^*$, together with $\hat{U}(N^*)$ is indeed the maximum of $\hat{U}$.
- Set $C^* = Influenza A virus subtype H5N1 (N^*)^{1-\alpha}.$
- Fix the wage $W \in {\mathbb R}_+$.
- Set $P = \frac{WN^* + M}{C^*} = \frac{N^*}{C^*} W + \frac{M}{C^*}$. That is, a unit of measurement wage cost summation a markup.The constraint $F = M$ trivially holds.
In champaign English, the optimal solution is that prices together with reward are set so that:
- the family sector industrial plant the issue of hours that it feels is optimal ("full employment");
- the line concern sector's profits stand upward for the entire starting stock of coin held yesteryear the family sector; and
- wage rates are indeterminate, but prices are set every bit a markup over reward to attain the fixed target turn a profit level.
Comparison to Existing Results
It should hold upward noted that this illustration assumes flexible prices, together with so it is equivalent to the Real Business Cycle (RBC) models. The RBC solution also features the same optimal output grade every bit my outcome here, but alongside the W/P ratio pinned downward yesteryear the equation given inwards GalÃ's text. So long every bit the solution stays away from the budget constraint, that comprises some other set of optimal solutions (since the cost grade is also unconstrained, until something changes to strength the cost grade to a detail level). (That is, Galà is right inwards arguing that the ratio is an optimal choice, but alongside the supposition that nosotros are non hitting budget constraints, which was non specified.)As a result, the construction hither could easily hold upward interpreted every bit a pathological corner case. To hold upward interesting, it needs to hold upward extended to where prices appear to affair for the solution (such every bit New Keynesian models alongside Calvo pricing). I am having off-line discussions alongside Alexander Douglas nearly that analysis.
Stock-Flow Inconsistency
Things larn fifty-fifty uglier if nosotros start looking at inter-temporal optimisations. We regard that the optimal strategy is for the line concern sector to absorb all of the coin from the family sector. What happens thereafter?The commons declaration yesteryear mainstream economists is that family bond/money holdings jibe authorities issuance; that is, family bond holdings determine the amount of debt outstanding. This runs into the obvious problem: they forgot nearly the line concern sector holdings of authorities liabilities.
Unless the line concern sector is to create got an ever-growing pile of financial assets, it has to homecoming profits to the family sector every bit dividends.
If nosotros permit dividends to hold upward returned inwards the same period, the outcome is indeterminate: profits would hold upward arbitrarily large. Let D hold upward dividends.
\[
PC = WN + M + D.
\]
\[
F = M + D.
\]
Any $D \in {\mathbb R}$ is a solution, together with thence may hold upward arbitrarily large. Formally, no solution would be to the maximisation problem. The only agency of getting a finite solution is to impose a status on dividend payments. Very few DSGE papers discussed dividends, together with the lawsuit on the family budget constraint was minimised. That is, the fact that at that spot is a feedback loop was by together with large non discussed.
No Longer an Optimisation Problem
If nosotros follow the spirit of DSGE macro, nosotros only impose the partial equilibrium first-order conditions, together with assay to uncovering a solution (model M3). However, the solution to this novel job no longer is an inter-temporal optimisation, together with nosotros cannot conclude whatever properties nearly its solution from optimisation theory.For example, there is no ground for the Transversality Condition to hold, since the model is sub-optimal.
The attractiveness of DSGE models is also clearer. All you lot involve to do is impose arbitrary first-order weather condition to constrain the arrangement to follow some sub-optimal solution trajectory; all you lot involve is some justification from microeconomics to impose the condition. However, instead of recognising that you lot are forcing sectors to conduct inwards a sub-optimal fashion (which evidently would brand the model dependent acre to the Lucas Critique), you lot tin give the axe instead give the impression that the sub-optimal outcome is the outcome of optimising behaviour!
The conclusions drawn from DSGE models tin give the axe also hold upward seen every bit beingness the outcome of the only arbitrary nature of the "first-order conditions" chosen. If the writer decides that financial policy has no lawsuit on the economy, it only gets dropped from the "first-order conditions." Since the model solution is sub-optimal anyway, what departure does it make? Furthermore, it becomes clear why controversies nearly the nature of DSGE model solutions be (for example, the Fiscal Theory of the Price Level, together with the "neo-Fisherian" debate). Since the model solutions are essentially the outcome of arbitrary choices, whatever cast of behavior tin give the axe hold upward achieved for what is allegedly the same optimisation problem.
Concluding Remarks
My concern alongside DSGE models (at to the lowest degree those of the representative family variety) is straightforward: they are non well-posed mathematical models. We create got no thought what the properties of these objects are, together with nosotros create got no ground to believe that they refer to whatever optimisation job solution.More constructively, the advantages of the stock-flow consistent (SFC) approach to modelling are much to a greater extent than apparent. Unlike DSGE models that are defined yesteryear heuristic (and largely arbitrary) partial equilibrium "first-order conditions," SFC models are stock-flow consistent. (Accounting identities ever hold.) Furthermore, the alleged disadvantage of SFC models -- that they do non stand upward for optimisation problems -- is also shared yesteryear DSGE models inwards practice. Influenza A virus subtype H5N1 researcher tin give the axe impose heuristic behavior weather condition -- so-called "first-order conditions" -- on SFC model behavior inwards just the same agency every bit tin give the axe hold upward done alongside a DSGE model.
(c) Brian Romanchuk 2017
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