### Wallace Neutrality: It all depends on why, exactly, money is valuable

8/11/2014 10:30:00 AM
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Noah Smith has raised an interesting question: Can the Fed set interest rates? The answer may surprise you.

It is typically assumed in practice that a currency's monetary authority has the ability to set interest rates, and that it does so primarily by manipulating the supply of money in open-ended operations to achieve it's interest target--a process known as "open market operations" (OMO). Most monetary DSGE models include a Taylor Rule that actually skip over OMO and assume that the monetary authority sets the interest rate outright, without ever bothering to compute the time-path of money supply required. However, Smith points to a paper by Neil Wallace called "A Modigliani-Miller Theorem for Open-Market Operations" which defines a principle of "Wallace neutrality" which says that the standard treatment is all wrong--OMO cannot affect either inflation or interest rates at all! That's quite a claim--no matter how much money the fed "prints" (via OMO), it will never cause inflation. As for the well-established empirical relationship between inflation and money supply increases, Wallace says this:
"Most economists are aware of considerable evidence showing that the price level and the amount of money are closely related. That evidence, though, does not imply that the irrelevance proposition [Wallace neutrality] is inapplicable to actual economies. The irrelevance proposition applies to asset exchanges [OMO] under some conditions. Most of the historical variation in money supplies has not come about by way of asset exchanges; gold discoveries, banking panics, and government deficits and surpluses account for much of it."
Basically, he's arguing that the observed correlation between prices and money may not apply to the measured changes in money supply brought about by OMO. It's quite a claim.

Let me offer a model. There's an infinitely lived household that derives utility from consumption $C_t$ and real money balances $\frac{M_t^h}{p_t}$ where $M_t^h$ is the households nominal bank account balance at the beginning of period $t$, and $p_t$ is the price of a unit of consumption in period $t$. The utility function is given by $$\sum_{t=0}^\infty\beta^t\left[ln\left(C_t\right)+ln\left(\frac{M_t^h}{p_t}\right)\right].$$ Further, the household is endowed in each period with an income of $y_t$ units of the consumption good, and must pay $\tau_t$ units in taxes to the fiscal authority. In addition to saving money $M_{t+1}^h$ for next period the household can invest in bonds $B_{t+1}$ that will yield gross real interest of $R_{t+1}$ in period $t+1$ giving us the budget constraint $$C_t+B_{t+1}+\frac{M_{t+1}}{p_t}\leq y_t+R_tB_t-\tau_t.$$ We assume (as Wallace does) that households have perfect foresight of all other agent's actions. Solving the household's constrained maximization problem yields $$C_{t+1}=\beta R_{t+1} C_t$$ which describes the inter-temporal consumption tradeoff for a given interest rate, as well as $$\frac{1}{\beta}\left(\frac{p_{t+1}}{p_t}+\frac{1}{R_{t+1}}\right)\frac{M_{t+1}^h}{p_{t+1}}+B_{t+1}^h+\frac{M_{t+1}^h-M_t^h}{p_t}-R_tB_t^h +\tau_t=y_t$$ which can be thought of as describing the conditions under which the household is indifferent between money, bonds, and consumption, a necessary condition for an optimum.

In addition to the household there are two other agents: a monetary authority (aka the Federal Reserve) and a fiscal authority (aka Congress). Congress sets government consumption levels $G_t$, measured in units of the consumption good, issues bonds $B_{t+1}$ that must be repaid at gross real interest $R_{t+1}$, all financed by lump-sum taxation of $\tau_t$ units of the household's income. Thus the Congress is bound by the budget constraint $$G_t+B_{t+1}=R_tB_t+\tau_t.$$ The Fed engages in OMO by buying bonds financed by increasing the money supply, according to the budget constraint $$B_{t+1}^f-R_tB_t^f=\frac{M_{t+1}-M_t}{p_t}.$$

We define an equilibrium as a sequence of prices $\left\{p_t,R_t\right\}_{t=0}^\infty$ such that the fiscal authority's budget constraint is satisfied, the monetary authority's constraint is satisfied, household utility is maximized, and $B_{t+1}^h+B_{t+1}^f=B_{t+1},$ (bond market clearing condition) $M_{t+1}^h=M_{t+1},$ (money market clearing condition) and $C_t+G_t=y_t$ (aggregate resource constraint) for all periods. Plugging in the fiscal and monetary authority's budget constraints, along with the bond and money market clearing conditions into the solution to the household problem yields $$\frac{1}{\beta}\left(\frac{p_{t+1}}{p_t}+\frac{1}{R_{t+1}}\right)\frac{M_{t+1}}{p_{t+1}}=y_t-G_t.$$ This is Robert Barro's famous "Riccardian Equivalence" result--holding expenditures constant (and assuming perfect foresight and only lump-sum taxation), government debt has absolutely no effects whatsoever on equilibrium--it is as if government spending entered directly into households' budget constraints. Recall that $$C_{t+1}=\beta R_{t+1} C_t$$ which combined with the aggregate resource constraint implies $$R_{t+1}=\frac{y_{t+1}-G_{t+1}}{\beta\left(y_t-G_t\right)}$$ and further combining that with the above result yields $$\pi_{t+1}=\frac{\frac{y_t-g_t}{y_{t+1}-G_{t+1}}\left(Q-\frac{M_t}{p_t}\right)}{y_t-G_t-\frac{1}{\beta}\left(Q+\frac{M_t}{p_t}\right)}$$ where $Q\equiv \frac{M_{t+1}-M_t}{p_t}$ was the Fed's bond purchases, or OMO, and the left hand side, $\pi_{t+1}\equiv\frac{P_{t+1}-p_t}{p_t}$ is the inflation rate. I think there's a way to make this result a little prettier, but no matter, in this form it tells us what we wanted to find out:the Fed's bond purchases--$Q$ in the equation--clearly and unambiguously cause inflation, a direct contradiction of Wallace neutrality.

So did I just disprove Neil Wallace and embarrass the American Economic Review with a first-year grad student homework problem?

It all boils down to differing views of why money has value embedded in Wallace's and my models. It's a surprisingly difficult question for monetary economists. We all know, intuitively, why money is valuable--you need it to buy stuff and pay taxes--and microeconomics professors will give you hours-long lectures on the double-coincidence of wants and what a marvelous innovation the idea of liquidity represents. But none of those reasons answer the fundamental question of why people hold money, which is different than using it for transactions. Why not, for example, put all your savings into bonds until you want to buy stuff, then simultaneously sell the bonds and buy the stuff such that your cash account never rises above zero? After all, so long as interest rates are above zero, you actually lose money by holding it!

The model above is a standard money-in-utility (MIU) model at the core of most monetary New Keynesian models. The basic assumption needed to get MIU models is this: people want the flexibility that comes with holding your assets in a highly liquid form. The standard alternative, and the one that Milton Friedman believed in more or less his whole life (but did not fair well in the Great Recession) is the cash-in-advance model, which says that not only do you need to have money to buy things, but for some unstated reason you must actually possess that money in liquid form for a period of time prior to making your purchase. Neither of these models exhibits Wallace neutrality, nor anything close to resembling it.

In Wallace's model, money is neither desired for it's liquidity nor even necessary to make purchases. Money has no actual function in Wallace's economy. So why the heck would agents in his model want to buy and sell money? That's a very good question.

Wallace's model is what's known as an overlapping-generations model (OLG). In these models, unlike the model above, households live brief lives and die. Younger generations want to save up to insure their future incomes, but older generations want to dissave--to sell off whatever assets they have and consume as much as possible before they die. There are two ways to save, in general: either invest in some type of durable good, or lend money to a borrower. If there's no durable good to invest in, you have to find credit-worthy borrowers if you want to save. Unfortunately, the old generation (the only people who might want to borrow) is not credit-worthy--they are about to die soon, and therefore have no future incomes with which to repay. Because of this problem, OLG models are actually quite prone to inefficiency. One way to reduce that inefficiency is to introduce a fictitious durable "good," like money, that can be traded as if it was a storage technology for perishable consumption goods. In this way the young can purchase money from either the old or the Fed, and when they become old, sell that money in exchange for consumption (ie, buy stuff). In this way, money acts, not as a medium of exchange, but as a line of credit to people who would otherwise not be credit-worthy--in an OLG economy, money facilitates loans from the young to the old, benefiting both by increasing the latter's consumption and insuring the former's future consumption. Unlike an actual loan which dies when the borrower dies, money is a durable good and will outlive it's owner. However, in Wallace's model, there actually is an alternative durable good the young could invest in, so that when the Fed engages in OMO, it is actually just buying durable goods that the young would have invested in, and selling a virtually identical durable good--money--in it's stead, in an economy where money itself is neither desirable nor necessary. No wonder this has no effects!

So it is in situations where money exists only to facilitate intergenerational transfers and nearly identical alternative durable goods exist, where the money supply is expanded specifically by swapping between the two, and only in these kinds of situations, that economies exhibit Wallace neutrality. I do not deny that Wallace's model to some extent does resemble reality--we really do use money as a type of intergenerational loan--but this exists along side other reasons to hold money, such as the liquidity preference of MIU models. That alone ensures that the real world is not Wallace neutral. Wallace neutrality is significantly less robust in theory than Riccardian Equivalence or it's older brother, Miller-Modigliani. This is the point that neither Smith nor anyone else seems to have made.

But then, lots of other people have made good points too--basically, all of the same critiques of Riccardian Equivalence also apply to Wallace neutrality, which requires that individuals have perfect information, non-distortionary taxation, etc.
Max 8/12/2014 03:25:00 AM
"we really do use money as a type of intergenerational loan"

Money or government bonds?

The Fed doesn't have any control over the national debt, right?

(Great post, by the way!)