### The Model Economy

This model economy is a simplified version of the flexible price model discussed in the appendix of the working paper. This model is a Real Business Cycle (RBC) DSGE model, with a “pure endowment” economy (there is no labour required to create output).

- You are the head of the “representative household” (which is assumed to have heirs for all time), and your decisions somehow represent the whole household sector.
- You have an apple tree in your back yard that produces 100 apples per year, and that is the total economic output for the economy.
- You hold $1 in money at present, which we label as time zero.
- You want to maximise your household utility over from t=0 to infinity. Your utility is based only upon the number of apples you eat in each period. Future consumption also gives utility, but you apply a discount rate on future consumption.

So the question is: what is the trajectory of economic variables that maximises your utility?

### Price Level Indeterminacy

The problem with this situation is that it is unclear what the price of apples should be. You get no utility from holding money, only by eating apples. It is very clear that the optimal solution is that you consume all 100 apples every year. (In other words, the economy fully utilises all the real resources of the economy, which is a characteristic of Real Business Cycle models.)

Although there is no debate about the trajectory of the real variables, the question is what is happening in the monetary economy. Since the optimal strategy for the household is to consume all of its apples and there are no other apples to buy, the household has no use for its money in transactions. You could interpret this in two ways:

- Money is worthless, and the price of apples are infinite.
- The value of money is unknown (indeterminate), but the number of apples that will be supplied at any price is zero.

In order to create a demand for money, the government has to impose taxes. (This is the key argument behind Chartalism, which is the theory of state money.) The household wants to hold money to pay future taxes, as the government will not loan it money in order to allow it to meet tax obligations. For some unknown reason it is assumed that the household gets no utility from meeting tax obligations (by avoiding the “disutility” of being thrown in jail for not paying taxes), rather this obligation is enforced as a constraint on the optimal solution.* This constraint leads to what he refers to as the Government Bond Valuation formula:

(This formula is described in my previous post.)

For simplicity, I will short-circuit the infinite sum in the formula and reduce it to a single-period condition.

What is assumed that the government pays off the debt in the next period, and runs a balanced budget for all time thereafter. In this case, the Bond Valuation Formula collapses to:

*The real (face) value of government debt = the present value of the real primary surplus in period 1, discounted at the real rate of return.*

The argument is that the value of the debt must be this value because:

- If the surplus is more than that, the household sector would end the period with a negative amount of money, and we assumed that the government will not lend money to households.
- The surplus is less than that (or there is a deficit), the household will end up with positive financial assets at the end of the period, which will never be taxed away. He argues that the household would want to trade away the excess financial asset holdings (which are government liabilities – debt and money).

(I believe there is a leap in logic in the second case, but I will discuss that elsewhere.)

One important thing to note that in this simplified framework, the government undertakes no operations in the real economy – it does not supply or demand goods (apples, in the current example). This simplifies the analysis, but it is also what helps drive the indeterminacy of the price level. The operations in financial assets are completely decoupled from the transactions in apples (since the household is both buying and selling the same amount of apples, the net cash flows due to apples are always zero).

The interesting thing is how the surplus has to be defined – it has to be defined in real terms. For my example, assume that the household discount rate and the interest rate is zero to keep the numbers simple. In this case, the price level would be expected to be constant. Since the initial amount of government liabilities is $1, the surplus would have to $1 at time 1 in order eliminate the outstanding government liabilities. But if that fiscal strategy was specified in nominal terms, there is nothing to pin down the price level.

Instead, Cochrane’s paper defines the amount of tax as a percentage of nominal income. If the tax rate was set at 1% of nominal income, the household would set the price of 1 apple at $1. This means that the household nominal income is $100 (100 apples times $1), and so it has a tax liability of $1. If the tax rate were 2%, the price of an apple would be $0.50, as 2% of $50 gives the required $1 in taxes. Since the amount of taxes is the same percentage of GDP regardless of the price level, the tax is effectively defined in real terms.

As he notes, having taxes being specified in real terms makes sense. This matches practice for most modern taxes, as we rarely see head taxes that are fixed nominal amounts.

### Monetary Policy – Pah!

Looking at the Bond Valuation Formula carefully shows an interesting property of the price level – its path will be determined by the path of surpluses. The only role of monetary policy is to affect the level of interest payments, which influences the amount of government debt outstanding.

In the working paper, John Cochrane discusses this in detail, and he gives examples how monetary policy “works” by creating a fiscal response. I do not have a lot to add, other than agreeing with the point that fiscal policy and monetary policy need to be coordinated. This should not be a surprise to most economists, but it may be news to some members of the ECB.

### Budget Constraint Or Valuation Rule?

There is a small literature discussing the Fiscal Theory of the Price Level, and the main debate is whether the Bond Valuation Formation/Governmental Budget Constraint is a constraint on behaviour, or a valuation formula. The standard view is that it is a constraint on behaviour that must hold (largely by assumption); the alternative view is that the valuation of government debt and/or the price level adjusts to the expected fiscal path.

If the initial price level for the economy has not been determined, such as in flexible price models, and there are no “monetary frictions”** the valuation interpretation seems to be the most tenable. (UPDATE: I discuss monetary frictions further in this article.) There is no other way of fixing the initial price level. The central bank can determine the path of inflation, but that is the rate of change of the price level relative to an indeterminate initial level.

If the initial price level is fixed (that is, it was assumed to be fixed in the previous time step), then the valuation rule runs into difficulties. The left hand side of the equation is fixed, and there is nothing that forces the right hand side to match the left hand side. For example, the government could just decide to run deficits, and thus the right hand side would never sum to the positive quantity on the left hand side of the equation. That said, if the initial price level is assumed to be fixed, it is unclear that the government budget constraint must hold, for the same reason.

### Equilibrium Versus Off-Equilibrium

Part of the discussion of the Fiscal Theory of the Price Level in the literature revolves around “equilibrium” versus “out of equilibrium” states. To my mind, this discussion is confused, and based on verbal discussions of supply and demand curves in the absence of modelled time. In DSGE models, the trajectory of economic variables (or at least their expected values) are the solution to an optimisation problem – which is implicitly assumed to exist and be unique. The possibility that economic variables (like prices) do not follow this optimal trajectory (or trajectories) is rejected. If we allow the possibility of non-optimal solutions, the models tell us nothing, since any trajectory that meets accounting constraints becomes a valid solution, and there is no way to distinguish which one to use.

In the mathematics of dynamic systems, an equilibrium is a point in the state space where the state variable would remain if it starts at exactly that point. It is not a property of the entire trajectory of the state variable, which is what DSGE model solutions are. Broadly speaking, the discussion of equilibrium appears to be a verbal projection of the properties of the real world onto a system of mathematical equations. Obviously, mathematical proofs can tell us nothing about those verbal formulations.

### Concluding Remarks

The Fiscal Theory of the Price Level is a way of determining the initial price level in a DSGE model that does not have monetary frictions. Given that we generally do not see societies waking up each day and struggling to determine the price level, the real world implications of this are rather remote. I would instead use this as an indication that such DSGE models are underdetermined and thus somewhat questionable.

Once we introduce monetary frictions, the Fiscal Theory of the Price Level starts to face problems. But as I will discuss in upcoming article, this same is true for the entire “governmental budget constraint”.

**Footnotes:**

* The household could conceivably borrow from the government to pay taxes. The DSGE model frameworks may or may not allow this to happen, but the assumption is that household borrowings have to go to zero as time goes to infinity (the “No Ponzi” condition). Realistically, governments do not lend to households to allow them to pay taxes, so it would make sense to enforce the no-borrowing rule at all times. This would require a slight modification to the proofs in the working paper, but that is already known.

** Monetary frictions would include having money in the utility function, or having a cash-in-advance constraint. The simplest such constraint would be adding the constraint MV=Py (Money × Velocity = Price Level × Real Output). The amount of money creates a constraint on the price level, eliminating the under-determination in the model. See this article for a longer discussion.

(c) Brian Romanchuk 2014

To your four model assumptions, you could add one more: Government places a tax of $1 per year on the land used to produce the apples.

ReplyDeleteThis will provide a lower limit on the value of the apples if it is further assumed that the apple producer will not want to subsidize the situation with money from another source.

I do not think there would be an upper limit on the value of the apples in money terms.

If the tax was $1/year (assuming interest rates were zero), government liabilities would be zero after the first year. The household could not pay taxes in the second year (this a pure endowment economy with a single household). If the tax continued in the second year, the household would have to sell apples to the government, which would presumably consume them. This would result in a balanced budget, but it violates a simplification in the John Cochrane paper - that the government does not consume real goods. (I obliquely referenced that assumption, but did not make it explicit.)

DeleteThe model economy begins with $1. The $1 is property that came initially from government because that is what fiat money is.

ReplyDeleteThe head of household is initially wealthy to to extent that $1 constitutes wealth. The head of household is also wealthy to the extent that an annual income of 100 apples per year constitutes wealth. I do not see in this model economy that there is any intersection of apples and money that allows for valuating apples in terms of money.

I think it is safe to allow that government might want back the property that it created, the recovery method to be in the form of taxes of some kind, but where is the link to apples? The wealth represented by $1 seems completely disconnected to the wealth represented by 100 apples per year.

Yes, there is no link between money and apples. That is why these models are fairly strange. The linkage is created by the logic (1) the total value of the tax paid is $1, and (2) the tax is a percentage of nominal income - which is the price of the apples times the number of apples produced.

DeleteAnother way of specifying the tax would be to say it is the value of a certain number of apples. For example, it could be specified as 1 apple. In order to generate the required $1 in tax, the price of an apple has to be $1.

This concept does not make much real world sense. I am trying to explain the logic before I cover where it breaks down. I will be writing that article "shortly"; it represents a fairly big problem with the treatment of fiscal policy in DSGE models.

This comment has been removed by the author.

ReplyDelete1) Yes; issue 1 period debt, pay off next period. Ending activity after one period was my way of simplifying things, not a general feature of these models.

Delete2) Not operating in the real economy was added by myself to simplify things (again). It is unclear to me what price the household would sell the apples to the government (which determines the price level). Why not overcharge the government, since we know that it has to buy? You end up needing to get a handle on what equilibrium represents, and that is pretty slippery. I have read a spectacular amount of deep mathematical treatments, and I have still have not seen a description of general equilibrium that makes sense to me.

3) If I have read the Daniel article, I've forgotten what it says. It would need some extensions beyond what was in Cochrane's paper (and my example has further simplifications), since the government bond valuation formula would be meaningless with respect to the initial price level. Within Cochrane's framework, it just says that discounted surpluses equal zero, and the initial price level could be anything (since there are no initial nominal stocks to scale against). Without looking up the paper, it coulf be that money provides the initial nominal scaling stock, and we then need to incorporate seigneurage into the equations. (My explanation assumes that money holdings are zero, so no seigneurage, which adds terms to the formula.)