In my previous post, I noted how the Fed has recently doubled the supply of base money. At the same time, the price-level (however measured) has shown no correspondingly dramatic change. This led me to question a simple version of a popular theory of the price-level; the so-called Quantity Theory of Money (QTM).
Maybe I was a little too hasty. After receiving several thought-provoking comments on that post, I am led to re-think what I was trying to say. The basic idea is that new money used to purchase assets (a la the Fed's MBS purchase program) need not have any price-level effect -- even if the increase in the money supply is permanent. On the other hand, it seems that it is also possible to square the observed phenomenon with a conventional QTM. Below, I discuss both interpretations.
A portfolio choice model
To sharpen the discussion, let me write down a simple model. The technology governing the return to capital investment works as follows. Let k denote capital investment at date t. Then the return to this investment is realized at date t+1 and is given by z+ f(k), where f(.) is strictly increasing and concave (diminishing returns to investment) and z+ denotes the realization of a productivity shock.
Let ω denote information available at date t that is useful in forecasting future productivity z+. I like to refer to ω as news. Let E[z+|ω] denote the conditional forecast over future productivity. Hence, for a given level of domestic investment k, the expected return to an additional unit of investment is given by the expected marginal product of capital, E [z+|ω]f ' (k).
Assume that there is one other asset in this economy: government money/debt (I make no distinction here). Let R denote the gross nominal interest rate paid on government money (think of the interest currently being paid on reserves by the Fed). Assume that R is a policy parameter and that it is held fixed over time; net interest on maturing debt is financed with lump-sum taxes.
Let p(ω) denote the price-level at date t (I write it this way because I anticipate that the equilibrium price-level may depend on news). The (gross) rate of inflation is defined by
Π(ω+,ω) = p(ω+) / p(ω).
If agents are risk-neutral (assume this is so), then for both capital and money to be willingly held in the wealth portfolios of individuals, the following "no-arbitrage-condition" must hold:
 E[ z+ | ω]f ' (kd ) = E[ R / Π(ω+,ω) | ω]
The LHS of  is the expected marginal product of capital and the RHS is the expected real return on government money. Condition  implies that investment demand kd is decreasing in R and increasing in (expected) inflation.
Let y denote aggregate saving which, in what follows, I view as a parameter (this allows me to focus on the forces that determine the composition of saving, rather than its level). Consequently, the demand for real money balances is determined residually by qd = y−kd.
Market Clearing. If M is the nominal supply of base money at date t, then market-clearing implies that M = p(ω)qd(ω) must hold at every date. At date t+1, this implies M+ = p(ω+)qd(ω+). Assume that the money supply grows at a constant rate μ, so that M+ =μM. (Let the new money is injected by way of helicopter drops). It follows that the equilibrium inflation rate must satisfy
Π(ω+,ω) = p(ω+) /p(ω) = μ[qd(ω) / qd(ω+) ]. Now, combine this expression with  to form:
 E[ z+ | ω]f ' ( y −qd(ω) )qd(ω) = (R / μ)E[ qd(ω+ ) | ω]
Given a stochastic process for news, condition  can be used to solve for the unknown function qd(ω).
Simplification: Actually, most of what I want to say here can be gleaned from the simplified case where we just shut down the aggregate news shock (nevertheless, I think it is useful to keep the underlying stochastic structure in mind). In this case, condition  reduces to
 z f ' ( y −qd ) = (R / μ)
Interpretation. I like to think of this model as capturing an element of banking (or of intermediation, in general). The bank (consolidated banking sector) must decide how to allocate its deposits y between cash q and loans that finance private sector capital spending k.
Under this interpretation, the nominal value of "bank money" is given by pk. From this, we may then construct a broad money aggregate as the sum of government money and private money, M1 = M + pk. Exploiting the market-clearing condition p = M / q, we may rewrite this latter expression as:
 M1 = [ 1 + (k / q)] M
The term in the square brackets in  is the so-called money multiplier (M1 / M). Note that the money multiplier here is endogenous; it depends on the loan-to-cash ratio (k / q), which is determined by the banking sector in relation to prevailing macroeconomic conditions. [You can verify that p = M1 / y as well].
1. An asset purchase program
O.K., so here is what I had in mind vis-a-vis my earlier post. Consider equation  and hold all parameters fixed. Then the equilibrium price-level is determined by pt = Mt / qd (The demand for real money balances remains constant in a stationary equilibrium -- all nominal quantities expand at the same rate as base money...a very QTM result, by the way).
What is important to realize here is that the price-level is pinned down by the ratio of the supply and demand for fiat money. While it is also true here that p = M1 / y, the price-level depends on M1 only to the extent that the supply and demand for fiat money influences M1 (i.e., via equation ).
Now, imagine that the Fed surprises the market by printing up some new money, using it to purchase some private sector capital (MBS). To my way of thinking of things, the new money created in this manner does not alter the supply of fiat money--even if it does increase the supply of government money. Since the supply/demand conditions for fiat money remain unaltered, the price-level remains unchanged.
Note that in this operation, the supply of private money falls by the amount of the Fed purchase, so that M1 remains unchanged. All that has happened here is a relabelling of the currency (the Fed is simply acting like a private intermediary). Prior to the experiment, the total supply of private (backed) money was pk. Imagine now dividing pk into two components: private money and government backed money. An econometrician will observe an increase in base money supply accompanied by disintermediation in the banking sector. But a rose by any other name is still a rose. [rose = backed money].
2. A flight to quality
In the view of some, a recession is caused by a sudden "excess demand for the medium of exchange." I interpret this to mean an exogenous increase in the demand for government money/debt.
Of course, as someone cast in the "New Monetarist" mold, I am not inclined to believe in exogenous money demand shocks. People do not, in my view, simply wake up one morning en masse wanting more cash or government treasuries. Well, maybe they do. On the other hand, I do not think it far-fetched to imagine that any change in money demand might not be driven by something more fundamental.
The fundamental shock I want to consider here is "bad news." That is, assume that E[ z+ | ω] is increasing in ω, so that a high realization of ω corresponds to "good news" and a low realization of ω corresponds to "bad news."
What typically happens near a cyclical peak is the arrival of information that leads private sector actors to revise downward their forecasts on the return to additional capital investments (e.g., new residential structures). A part of the information flow might take the form of rising foreclosure rates, for example. In any case, the point is that the sudden arrival of bad news leads to a form of rational pessimism.
What are the implications of a bad news shock in the context of the model developed here? Properly, we should study . But it turns out that the basic intuition can be gleaned equally well from  and interpreting an exogenous decline in z as a bad news shock.
Mechanically, condition  asserts that the real demand for government debt qd is decreasing in z. This follows from the strict concavity of f(.). So a decrease in z leads to an increase in qd; an econometrician is likely to identify this event as a positive money demand shock.
Intuitively, bad news leads banks to rebalance the composition of their assets--they want to contract their lending activities (the forecasted returns are no longer as attractive) and they want to substitute into higher grade government money/debt--there is a "flight to quality." The consequent increase in money demand is deflationary (in the model, there is a sudden drop in the price-level). Recall: pt = Mt / qd . The contraction in bank-lending leads to a subsequent drop in (expected) GDP, owing to reduced levels of investment today.
Note that the decrease in (k / q) here results in a decrease in M1 and the money multiplier, even for fixed base money supply M; see condition . Needless to say, this effect (disintermediation) seems present in the data.
Now, according to this interpretation, the decline in the price-level was avoided by recent Fed actions. In particular, the Fed increased the supply of (interest-bearing) fiat money to accommodate the apparent increase in demand for these government liabilities. Note that, according to this view, the Fed had better stand ready at some point in the future to reverse its money injection, if a huge increase in the price-level is to be avoided (once good news returns, and private investment demand picks up).
I'm not sure which interpretation I like better. It is possible that both interpretations contain an element of truth. You tell me (maybe you think both models are lousy).
Let me address what I think will be one sure criticism. The model I use above makes no distinction between government money and debt. To put things another way, money is treated simply like any other asset. In fact, money--in the form of cash--is special; it is a medium of exchange. The model here simply views money as a store of value.
Well, the first thing I would say in response to this is that at least my model has money! (unlike, say, conventional "cashless" NK models). And anyway, the money/debt object in my model does circulate as a medium of exchange. In the OLG model, for example, the asset passes from one group of agents to another in a sequence of spot exchanges. I might further add that the existence of government money/debt is welfare-improving in the model (of course, I have not modeled the government as potentially exploiting its monopoly on fiat for the purpose of extracting seigniorage -- such an extension is possible, but beside the point here). But ultimately, my response is this: what phenomenon might be better understood, in the context of the present discussion, by distinguishing money from debt? You tell me how what I have done here is in any way misleading because of my failure to take the medium of exchange role of money more seriously.
It is interesting to note that in making no distinction between money and debt here, the Fed finds itself in a perpetual liquidity trap. That is, any swap of money and bonds is completely irrelevant for anything real or nominal. The two objects are perfect substitutes in the model. In fact, I think this describes the current situation along one dimension rather well. In particular, interest-bearing reserves are virtually indistinguishable from government treasuries as far as banks are concerned (so swaps of treasuries for interest-bearing reserves should have little, if any, consequences).
Along the same vein, if you want, think of the case in which R = 1 (zero nominal interest rate). This is the scenario that most people associate with a liquidity trap, but as the preceding discussion makes clear, one does not need to hit the zero lower bound to speak of liquidity traps.