A Contradiction At The Midpoint Of Dsge Models

Dynamic Stochastic General Equilibrium (DSGE) models are based some the optimising choices of a representative household. This creates difficulties for the analysis of other sectors. The move of the governmental (other than cardinal banks) is an extremely of import illustration of this. In this article, I demonstrate that these models imply having to brand a pick betwixt ii possibilities - either financial policy is completely determined past times the representative household, or else the representative household's path of financial property holdings is completely determined past times financial policy. Either of these choices imply that these models are non peculiarly useful.

Government Budget Constraints

There are ii primary parts of the "intertemporal authorities budget constraint" inside DSGE models. Without using (much) mathematical notation, as well as using a deterministic framework (for reasons I utter over here), they are:

1. (Accounting Identity) The increase inwards the value of authorities liabilities from fourth dimension t to t+1 is the total financial deficit, which is typically decomposed into the "primary financial balance" as well as involvement expense.
2. (Transversality Condition) The discounted value of authorities liabilities every bit fourth dimension goes to infinity goes to zero.

(Note for fans of stochastic mathematics: you lot tin merely insert "expected" into the higher upwards phrasing to instruct the measure stochastic version. The accounting identity equality turns into an inequality inwards some treatments, but this contradicts the Definition of  an "accounting identity".)

There is no debate nearly the validity of the Accounting Identity, although in that place are debates nearly its interpretation - is in that place a behavioural constraint behind it? I volition ignore that debate here. Instead, I volition only human face at the transversality condition.

Transversality Condition - Why It Is There

Most texts I bring read bring skipped over why the transversality status is alleged to hold. The newspaper "The sustainability of budgets inwards a stochastic economy" past times Henning Bohn gives a relatively slow derivation of the transversality condition. Within that paper, in that place are references to before (more complex) derivations.

Without worrying nearly the details, it is shown that inwards an economic scheme with only a authorities sector as well as households*, the portfolio belongings decisions of households volition bind the amount of authorities debt. The derivation is somewhat complex, every bit households may borrow from each other, as well as households bring to avoid financing "Ponzi schemes".  Roughly speaking, infinitely long-lived households** volition desire to run downward their financial assets every bit fourth dimension goes to infinity. (More accurately, non allow the value of those holdings grow faster than the discount rate.)

The Contradiction

The derivation of the transversality status implies that the identify sector has the capacity to strength the degree of authorities liabilities to an arbitrary level. Although it is expressed inwards terms of fourth dimension going to infinity, this only agency that the identify tin strength the cyberspace acquaint value of authorities liabilities arbitrarily closed to zip on a long plenty horizon.

Conversely, if nosotros human face at the accounting identity, the identify sector has precisely no might to influence the degree of authorities liabilities every bit nosotros transition from fourth dimension t to t+1.

There are only ii ways of resolving this:

1. The determination of the increase of governmental liabilities from fourth dimension t to t+1 is determined completely past times the representative household. There is no such matter every bit "fiscal as well as monetary policy" which are determined past times external "policy makers".
2. The representative identify has no might to ready the degree of its financial property holdings. (It tin reallocate alongside authorities liabilities, such every bit trading "money" for "Treasury Bills".) The combination of financial as well as monetary policy completely determines the financial property holdings inside the economy.

The kickoff possibility is consistent with the thought of a unmarried representative identify doing everything, but it makes the models only useless for the analysis of policy decisions. The instant possibility appears to live on correct, but it casts incertitude on the entire optimising framework of DSGE models. If the identify has no command over the degree of its financial assets, they are irrelevant to the optimisation problem. We cease upwards with a radical Real Business Cycle model, inwards which financial variables (including prices) bring no acquit on on the trajectory of the economy.

Concluding Remark

One of the problems associated with writing downward mathematical problems as well as philosophising nearly their solution is that ane tin instruct astray. DSGE modellers practice non actually solve their optimisation problems, rather they pretend that they tin linearise an unknown solution as well as they solve the linearisations. This begs the obvious interrogation - if you lot practice non bring the total solution, how tin you lot linearise?

Footnote:

* Why it is believed that this lawsuit applies to models that include a employment organisation sector (in add-on to the identify as well as authorities sectors) is a mystery to me.
** Although studying infinitely long-lived households looks similar a remarkably unrealistic modelling assumption, nosotros tin pretend that this is households worrying nearly the financial weather of their descendants. One could debate that this is merely an approximation.

See Also:

• The summary of my complaints nearly DSGE models.

(c) Brian Romanchuk 2014