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Wednesday, March 28, 2018

The Curious Notation Of DSGE Models

This article is an appendix to my earlier articles on dynamic stochastic general equilibrium (DSGE) model accounting (here and here). The problem with my understanding of DSGE appears to be that I assumed that the DSGE modellers were using mathematical notation in a standard fashion. I now realise that the secret sauce to DSGE modelling is a blatant disregard to mathematical notation. I had pointed out that I was missing something; my assumption that there were some super important theorems from microeconomics that everyone was invoking, but not formally specifying. Instead, the answer is much simpler: the notation used was misleading, if not outright incorrect.

(I have been hit with a number of projects recently -- including a MMT presentation at Concordia University last night. As a result, this article is relatively brief and tentative. I will return to a more meatier conclusions when I discuss the treatment of the governmental sector. Note that my earlier articles hinted at what I discuss here, but I had not gone to far in formalising this. The formalisation is a key step; if the notation is wrong, the only way to deal with the problem is to fix the notation.)

The Proper Notation

This is my first attempt at sketching out how this could be formalised. As a result, it is an informal attempt at formalisation. There's a few parts that are sketchy, but that is reflective of the true definition of DSGE models.

We will step back and look at a generic macro DSGE problem. Let $x$ be the vector of all time series variables within the model. (That is, at any given time $t$, $x(t) \in R^n$.) This will include price variables, activity variables, etc. Let $u$ be the set of "decision variables" within $x$, and $p$ be the other variables (generally prices).

We will define the household first order conditions denoted ${\cal F}_h$ as

Given an objective function $O_h$, and a constraint operator (see note below) $C_h$, ${\cal F}_h$ is a set of conditions chosen by a researcher R  by the use of Lagrange multiplier technique such that there is a set $X_h$ where:
$$
C_h x_h = 0, \forall x_h \in X_h,
$$
and ${\cal F}_h x_h$ holds $\forall x \in X_h$, and $u_h$ is the optimising choice of $O_h$ fixing the $p_h$.

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

We similarly define the first order conditions for the business sector ${\cal F}_b$, replacing the $h$ subscripts with $b$ subscripts (there is a business sector objective function $O_b$, and constraint $C_b$.

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

The disconcerting part of this problem definition is the curious appearance of the phrase  "set of conditions chosen by a researcher R." I take a somewhat hard line stance towards mathematics: mathematics is the study of sets, and operations on sets (and the set of mathematical logic that defined the set of allowed mathematical manipulations). In this case, we have a ghost in the machine - we have a human being popping up and picking conditions that define the set of allowed operations. Once the constraints are fixed, we can go ahead using standard mathematics, but there is otherwise no obvious fixed algorithm to deal with these models. Different researchers could conceivably pick different constraint sets, and end up with different optimising solutions.

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

The Wrong Notation

What tripped me up was that DSGE modellers wrote down their problems, and denoted the same variables across the different problems (the household problem, the firm problem, and the overall solution) using the same variable. Using my notation, they just used $x$ at all appearances of the variables.

The problem is that if $x_h = x_b$, we can then convert the household problem to maximise $O_h$ subject to the constraints:
$$
C_h x = 0,
$$
and
$$
C_b x = 0.
$$
The business sector constraint $C_b$ contains the production function. This means that we could wreak havoc on the optimisation problem by substituting the production function into the household budget constraint. The optimisation problem then collapses to a trivial problem: what is the level of production that optimises household utility? Financial constraints disappear.

(As a specific example, they use the same variable name to denote hours worked in both the household and firm's optimisation problem, as well as household consumption. The production function and firm's profit relationship are expressed as constraints on those variables. This implies that we can push those constraints back into the household optimisation problem as additional constraints. This creates a trivial solution. My previous articles give examples of the confusion generated by a standard DSGE textbook treatment, using their original notation.)

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

The series will conclude with an explanation how this messes up the discussion of fiscal policy in these models.


(c) Brian Romanchuk 2018

4 comments:

  1. Brian,

    I am still working on eq. 16.2.3. I am convinced that either the book is wrong or your interpretation is incorrect. Here is my thinking of how it should be:

    The productive firm is represented by a stylized income statement with income = expenses. Income is represented by consumption translated to a form of price.

    Income in the real world is never equal to expenses except by accident. Hence, we need a balancing term which in this model is profit or loss. I will call it \(k_p(t).\)

    The capital conversion rate for labor is \(k_w(t) .\) The rest of the terms are labeled as you labeled them.

    We assume that all capital is borrowed and is borrowed at the same rate that labor is expended. Hence, the last capital is borrowed when the last hour is worked. This borrowed capital is rented at an interest rate of \(r(t) .\) Interest will be treated as an expense. Important--borrowed capital is all converted to labor cost in my formula.

    [This model is not the model of Nick where production produces both capital and salable product. Here capital substitutes for product and labor.]

    Here is the accounting: \[c(t) + g(t) = r(t) [k_b(t) = k_w(t)n(t)] + k_w(t)n(t) + k_p(t).\]

    Simplify to read \[c(t) + g(t) = k_w(t)n(t)[ r(t) + 1 ] + k_p(t). \]

    This equation should work for prices converted to goods and for prices in green dollars. Just be sure that every term is in the same unit.

    Changing the focus, I think we should define our sectors by decision-ownership categories. Each sector can make a buy-sell-continue decision. By that standard, we have here sectors of consumer, government, firm, capital lender, and worker. Each sector shares in the decisions and gives-gets part of the economic pie.

    The accounting in this equation has capital flowing between five sectors. We can think of a firm as being an amalgamator who brings this all together and keeps a profit (if any).

    Conclusion: I much appreciate this series of articles. They have prompted some deep thinking on my part. I hope you can agree with my processes, but I have learned whatever your reactions might be. Thanks.

    ReplyDelete
    Replies
    1. You’re free to do whatever model you want, but this is the model they used in the text. There’s more than one mathematical model you can look at.

      Accounting profits are normally non-zero in this model; it’s just that all profits are immediately distributed as “rent” in the same period. And since the model is deterministic, the future is known with certainty, so the rent amount can be forecast perfectly. Since the model is internally consistent, albeit unrealistic.

      These models barely handle a business sector, adding in a financial sector is nearly impossible.

      Delete
  2. Brian Romanchuk

    You announced a tripartite analysis of an elementary DSGE model and after part 2 you suddenly realize that the notation of DSGE models is “curious”. This tells everyone that you have not understood from the very beginning what you are talking about.

    At the end of your second post I summed up: “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.”

    Your analysis and critique of DSGE is a pointless exercise. Everybody knows by now that the microfoundations approach is dead. Standard economics is dead. DSGE is dead.

    Economics needs a Paradigm Shift because the main approaches ─ Walrasianism, Keynesianism, Marxianism, Austrianism ─ are axiomatically false and materially/ formally inconsistent.

    Endless recycling of long dead theories/models is not science.

    Economics is what Feynman called a cargo cult science, economists are failed/fake scientists, the Bank of Sweden Prize in Economic Sciences is a deception of the public.  Economics is one of the worst cases of theory failure in the history of modern science ─ and you are part of it.

    Egmont Kakarot-Handtke

    ReplyDelete
    Replies
    1. I’ll admit that it is somewhat silly, but there was another possibility (that there were deep theorems that “everyone else” knows about, except me. I gave them the benefit of the doubt.

      And yes, I am probably beating a dead horse. However, the neoclassical replacements for DSGE representative agent macro are almost certainly be roughly the same thing, with just a few tweaks here and there. So we need to deal with the simpler cases first.

      Delete

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