## Wednesday, June 25, 2014

### Monetary Frictions In DSGE Models

In an earlier article, I described a simple "endowment" model economy which was based on it working paper by John Cochrane. In that article, prices were extremely flexible, and the only means to pin down the initial price level is via the governmental budget constraint, which is a concept referred to as the Fiscal Theory of the Price Level. In this article, I explain the concept of monetary frictions, which creates an independent means of determining the initial price level. Once these frictions are introduced, the validity of the government budget constraint is more problematic.

This article and other related ones represents a first draft of ideas that I am going to incorporate into a book on the theory of fiscal policy. I have strong objections to the representation of fiscal policy within Dynamic Stochastic General Equilibrium (DSGE) models. In this article and the previous I present a simplified DSGE model and how it is supposed to be analysed. One difficulty in following the logic is that the model is very unrealistic, which is an inherent property of these models.

Once I have covered the base ideas, I will be easier to explain my criticisms.

### Monetary Frictions Versus Price Stickiness

Discussion of price determination in DSGE models typically revolves around the notion of the price stickiness — how easy is it to adjust the prices as time passes. Remember that these models are based upon finding an optimal trajectory from the current time, which I label time zero here for simplicity. Price stickiness is a property of prices in the future, which is for time greater than zero.

The real challenge in these models is: what is the initial price level? If prices are sticky, as in the so-called New Keynesian models, the initial price level should have been fixed at time t=-1 (in order to remain consistent with the model structure*). Assuming initial price level is fixed eliminates the problems I discuss within this article, but it raises a new question – how was the initial price level fixed at time t=-1? Instead, I want to look at alternative means of fixing initial price level within a model, so-called monetary frictions.

My model economy is described in this article. But a briefer description is that there is a single household in the economy that receives 100 apples a year, and has \$1 as a financial asset. The question is – what is the price of an apple at t=0?

In the previous article, I explained John Cochrane's argument how taxes could pin down the initial price level. Here I explain how monetary frictions can do the same thing.

### Types Of Monetary Frictions

There are three types of monetary frictions (not including price stickiness) that could be used to determine the initial price level in my model economy. The first two are standard, and the third is similar but comes from Stock-Flow Consistent modelling. They are:

1. cash-in-hand or velocity constraints;
2. money in the utility function;
3. stock–flow norms.

### Velocity

The velocity of money is defined by the equation of exchange:

MV=Py,

where M = money, V = velocity, P = price level, y= real production. Note that this is a definition, allowing us to calculate the velocity at a particular time (assuming money is non-zero).

As quantity theorists discovered the hard way during the QE exercise, velocity is not constant for real economies. But for simplicity, I will discuss what happens in our simple model economy if velocity is constant.

If I fix velocity to be a nice round 100, and the household sector holds \$1 in money, the price of an apple is \$1. This is because total production is 100 apples, and the equation of exchange tells us that nominal GDP is \$100 (=100×\$1). If velocity drops to 50, the price an apple drops to \$0.50.

We could make velocity a function of things like interest rates, in which case we would have to solve some equations to determine the price level. I leave that as an exercise to the reader.

An alternative formulation is to create a cash-in-hand constraint. These are constraints which require the household hold money at the "beginning" of the time period for transaction purposes, even though it will receive more money at the "end" of the period. For example, it cannot use the proceeds of Treasury bills that mature at the "end" of the period for transactions within that period.

Such constraints are equivalent to a velocity constraint if the transactions are for purchasing goods. But it would not be the same thing if the cash-in-hand constraint applied to the payment of taxes.

### Money In The Utility Function

Since it is often assumed that households can borrow, a velocity constraint is hard to justify from the points of view of microfoundations — the argument that macroeconomic models should be derivable from microeconomic behaviour. An alternative way of getting model households to hold money is to incorporate money holdings into utility function.

I will only summarize how this works here. The price of apples is such that the household would be indifferent between holding money or buying another apple – that is, the level of utility would be unchanged. (From whom the household would buy the apple from is an important open question.)

For some classes of utility functions, you would get a constant velocity if the economy followed a steady-state growth path. In other cases, velocity would be variable.

### Stock-Flow Norms

A somewhat similar concept to money in the utility function would be stock-flow norms in household behaviour. Households would attempt to keep financial assets near target ratios to nominal incomes. The difference is philosophical, as it is more an empirical argument, and not driven by the assumption of optimizing behaviour. Stock-flow norms are unsurprisingly embedded in stock-flow consistent (SFC) models.

### Why DSGE Models Need Monetary Frictions

Absent monetary frictions, DSGE models offer a very clear prediction about real-world economies: households will never hold non-interest-bearing money, such as coins and dollar bills. Although we are moving in that direction, that prediction is obviously incorrect. Therefore monetary frictions have to be introduced to reproduce observed behaviour. They are also needed to pin down the initial price level in these models.

### Does This Really Matter?

In reality the price level is pinned down by prior contractual obligations and posted prices. The only time that prices appear to be unmoored is during a hyperinflation. But even in that case, prices are typically fixed in foreign currency terms, and local prices are just those foreign currency prices translated by the latest foreign-exchange quotation. Prices are not going to infinity, rather the local currency is going to zero in the foreign-exchange market. This is not a situation that is easily modelled in a closed economy DSGE model.

Monetary frictions are an artefact to needed to eliminate the under-determination of the price level in DSGE models. But as the Fiscal Theory of the Price Level shows, the determination of the initial price level it is tied to the government budget constraint. As a result, any analysis of fiscal policy using DSGE models has to grapple with monetary frictions.

I will discuss in a later article why the existence of monetary frictions calls into question the validity of the governmental budget constraint.

Footnote:

* The price level in a New Keynesian DSGE model with Calvo pricing would be “partially” fixed at t=-1; there is a continuum of goods and only some can be repriced at any point in time. But even having some prices fixed would create a reference (partial) price level for the remaining prices that could be adjusted.

(c) Brian Romanchuk 2014

1. More good stuff, Brian, and I enjoyed your earlier piece on Cochrane's paper as well.

It always surprises me how easily people overlook prior contractual obligations when pondering what pegs the price level. Price stickiness certainly matters, but there is a vast network of nominal claims in a modern economy, which have significant real consequences. The problem is they are using models which abstract away from all the ways in which this matters.

1. Thanks. It is interesting how hard it is to pin down the initial price level in these models, when in the real world, it is something we hardly would think about.

2. I don't have a problem with the concept 'MV' - which is just Turnover in business parlance. What I do have a problem with is the decomposition by M, where M is the entire money stock.

If households hold \$2 of money, but only ever spends \$1 of them and the velocity is 100 then the price is still \$1 given the production.

It's only the part of the money stock that has changed hands within the time period that should be counted.

1. Yes, it is hard to tie transaction needs to money aggregates. These models simplify money holding too much, which is part of the problem.

If we look at notes and coins within the money stock, some are held by people for small transactions, but the bulk are used in illegal transactions. (John Cochrane discusses this in his article, in the non-mathematical sections.) This makes it hard to model that part of money holding for transactions, since the illegal transactions are largely not tracked, and the bills and coins are used to store proceeds.

But if we start including bank money, then holdings are probably often driven by portfolio allocation decision, and so it is unclear that "velocity" makes a lot of sense. But you could justify some sort of velocity relationship based on stock-flow norms, but it may be based on there being a relationship between nominal income and money holdings.

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