Friday, May 9, 2014

A Contradiction At The Heart Of DSGE Models

Dynamic Stochastic General Equilibrium (DSGE) models are based around the optimising choices of a representative household. This creates difficulties for the analysis of other sectors. The role of the governmental (other than central banks) is an extremely important example of this. In this article, I show that these models imply having to make a choice between two possibilities - either fiscal policy is completely determined by the representative household, or else the representative household's path of financial asset holdings is completely determined by fiscal policy. Either of these choices imply that these models are not particularly useful.

Government Budget Constraints

There are two main parts of the "intertemporal government budget constraint" within DSGE models. Without using (much) mathematical notation, and using a deterministic framework (for reasons I discuss here), they are:

1. (Accounting Identity) The increase in the value of government liabilities from time t to t+1 is the total fiscal deficit, which is typically decomposed into the "primary fiscal balance" and interest expense.
2. (Transversality Condition) The discounted value of government liabilities as time goes to infinity goes to zero.
(Note for fans of stochastic mathematics: you can just insert "expected" into the above phrasing to get the standard stochastic version. The accounting identity equality turns into an inequality in some treatments, but this contradicts the definition of  an "accounting identity".)

There is no debate about the validity of the Accounting Identity, although there are debates about its interpretation - is there a behavioural constraint behind it? I will ignore that debate here. Instead, I will only look at the transversality condition.

Transversality Condition - Why It Is There

Most texts I have read have skipped over why the transversality condition is alleged to hold. The paper "The sustainability of budgets in a stochastic economy" by Henning Bohn gives a relatively easy derivation of the transversality condition. Within that paper, there are references to earlier (more complex) derivations.

Without worrying about the details, it is shown that in an economy with only a government sector and households*, the portfolio holding decisions of households will bind the amount of government debt. The derivation is somewhat complex, as households may borrow from each other, and households have to avoid financing "Ponzi schemes".  Roughly speaking, infinitely long-lived households** will want to run down their financial assets as time goes to infinity. (More accurately, not let the value of those holdings grow faster than the discount rate.)

The derivation of the transversality condition implies that the household sector has the capacity to force the level of government liabilities to an arbitrary level. Although it is expressed in terms of time going to infinity, this only means that the household can force the net present value of government liabilities arbitrarily close to zero on a long enough horizon.

Conversely, if we look at the accounting identity, the household sector has exactly no ability to influence the level of government liabilities as we transition from time t to t+1.

There are only two ways of resolving this:

1. The determination of the increase of governmental liabilities from time t to t+1 is determined completely by the representative household. There is no such thing as "fiscal and monetary policy" which are determined by external "policy makers".
2. The representative household has no ability to set the level of its financial asset holdings. (It can reallocate amongst government liabilities, such as trading "money" for "Treasury Bills".) The combination of fiscal and monetary policy completely determines the financial asset holdings within the economy.
The first possibility is consistent with the idea of a single representative household doing everything, but it makes the models entirely useless for the analysis of policy decisions. The second possibility appears to be correct, but it casts doubt on the entire optimising framework of DSGE models. If the household has no control over the level of its financial assets, they are irrelevant to the optimisation problem. We end up with a radical Real Business Cycle model, in which financial variables (including prices) have no impact on the trajectory of the economy.

Concluding Remark

One of the problems associated with writing down mathematical problems and philosophising about their solution is that one can go astray. DSGE modellers do not really solve their optimisation problems, rather they pretend that they can linearise an unknown solution and they solve the linearisations. This begs the obvious question - if you do not have the full solution, how can you linearise?

Footnote:

* Why it is believed that this result applies to models that include a business sector (in addition to the household and government sectors) is a mystery to me.
** Although studying infinitely long-lived households looks like a remarkably unrealistic modelling assumption, we can pretend that this is households worrying about the financial conditions of their descendants. One could argue that this is just an approximation.

(c) Brian Romanchuk 2014