The Curious Annotation Of Dsge Models

This article is an appendix to my before articles on dynamic stochastic full general equilibrium (DSGE) model accounting (here too here). The work amongst my agreement of DSGE appears to hold upwards that I assumed that the DSGE modellers were using mathematical notation inward a measure fashion. I directly realise that the hugger-mugger sauce to DSGE modelling is a blatant disregard to mathematical notation. I had pointed out that I was missing something; my supposition that at that topographic point were to a greater extent than or less super of import theorems from microeconomics that everyone was invoking, but non formally specifying. Instead, the respond is much simpler: the notation used was misleading, if non outright incorrect.

(I postulate maintain been hitting amongst a divulge of projects of late -- including a MMT presentation at Concordia University terminal night. As a result, this article is relatively brief too tentative. I volition furnish to a to a greater extent than meatier conclusions when I verbalise over the handling of the governmental sector. Note that my before articles hinted at what I verbalise over here, but I had non gone to far inward formalising this. The formalisation is a fundamental step; if the notation is wrong, the entirely way to bargain amongst the work is to ready the notation.)

The Proper Notation

This is my outset endeavour at sketching out how this could hold upwards formalised. As a result, it is an informal endeavour at formalisation. There's a few parts that are sketchy, but that is reflective of the truthful Definition of DSGE models.

We volition pace dorsum too expect at a generic macro DSGE problem. Let $x$ hold upwards the vector of all fourth dimension serial variables inside the model. (That is, at whatever given fourth dimension $t$, $x(t) \in R^n$.) This volition include cost variables, activeness variables, etc. Let $u$ hold upwards the laid of "decision variables" inside $x$, too $p$ hold upwards the other variables (generally prices).

We volition define the identify outset social club atmospheric condition denoted ${\cal F}_h$ as

Given an objective role $O_h$, too a constraint operator (see banking concern complaint below) $C_h$, ${\cal F}_h$ is a laid of atmospheric condition chosen past times a researcher R  past times the utilisation of Lagrange multiplier technique such that at that topographic point is a laid $X_h$ where:
$$
C_h x_h = 0, \forall x_h \in X_h,
$$
too ${\cal F}_h x_h$ holds $\forall x \in X_h$, too $u_h$ is the optimising alternative of $O_h$ fixing the $p_h$.

In words, ${\cal F}_h$ holds for optimising choices of determination variables for the identify problem.

We similarly define the outset social club atmospheric condition for the concern sector ${\cal F}_b$, replacing the $h$ subscripts amongst $b$ subscripts (there is a concern sector objective role $O_b$, too constraint $C_b$.

The DSGE macro work is: Find the laid $X$ such that both $F_h x$ too $F_b x$ hold, too $C_h x = 0$ too $C_b x = 0$, for all $x \in X$.

The disconcerting part of this work Definition is the curious appearance of the phrase  "set of atmospheric condition chosen past times a researcher R." I accept a somewhat difficult draw of piece of work opinion towards mathematics: mathematics is the report of sets, too operations on sets (and the laid of mathematical logic that defined the laid of allowed mathematical manipulations). In this case, nosotros postulate maintain a ghost inward the car - nosotros postulate maintain a human existence popping upwards too picking atmospheric condition that define the laid of allowed operations. Once the constraints are fixed, nosotros tin forcefulness out larn ahead using measure mathematics, but at that topographic point is otherwise no obvious fixed algorithm to bargain amongst these models. Different researchers could conceivably pick dissimilar constraint sets, too halt upwards amongst dissimilar optimising solutions.

This is hardly satisfactory, but it describes what DSGE modellers are trying to do.

The Wrong Notation

What tripped me upwards was that DSGE modellers wrote downwards their problems, too denoted the same variables across the dissimilar problems (the identify problem, the theater problem, too the overall solution) using the same variable. Using my notation, they only used $x$ at all appearances of the variables.

The work is that if $x_h = x_b$, nosotros tin forcefulness out therefore convert the identify work to maximise $O_h$ dependent land to the constraints:
$$
C_h x = 0,
$$
and
$$
C_b x = 0.
$$
The concern sector constraint $C_b$ contains the production function. This agency that nosotros could wreak havoc on the optimisation work past times substituting the production role into the identify budget constraint. The optimisation work therefore collapses to a picayune problem: what is the degree of production that optimises identify utility? Financial constraints disappear.

(As a specific example, they utilisation the same variable parent to announce hours worked inward both the identify too firm's optimisation problem, likewise equally identify consumption. The production role too firm's profits human relationship are expressed equally constraints on those variables. This implies that nosotros tin forcefulness out force those constraints dorsum into the identify optimisation work equally additional constraints. This creates a picayune solution. My previous articles laissez passer on examples of the confusion generated past times a measure DSGE textbook treatment, using their master copy notation.)

This approach violated the rules of the game. Why? The DSGE researchers wrote downwards the bloody mathematical specification inward a horribly wrong fashion.

The serial volition conclude amongst an explanation how this messes upwards the news of financial policy inward these models.


(c) Brian Romanchuk 2018