Yesterday I argued that the platinum coin is a bad idea. In doing so I received a substantial amount of pushback. Some have argued that while the platinum coin might be a dumb idea, it is preferable to being held hostage by recalcitrant Republicans. Others argued that my claims about the potential inflationary effect of the platinum coin were overblown. With regards to the first claim, I have very little to add other than the fact that I don’t subscribe to the “two wrongs make a right” theory of public policy. The second claim, however, is more substantive. It is also something about which economic theory has something to say.

In many contemporary models, money is either excluded completely or is introduced using a reduced form approach, such as including real money balances in the utility function. These models are ill-equipped to tackle the effects of the introduction of the platinum coin because they either assume that money always has value (it generates utility) or that it has no value whatsoever. An analysis of the effects of the platinum coin should be backed by an understanding of what gives money value in a world of fiat money and the conditions necessary to insure a unique equilibrium in which money has value. In doing so, one can show that having the Fed conduct open market sales to offset the increase in the monetary base from the minting of the platinum coin (i.e. holding the money supply constant) might not be sufficient to prevent a significant inflation.

To illustrate the properties of money, I am going to employ the monetary search model of Lagos and Wright. (If you’re allergic to math, scroll down a bit.) The reason that I am employing this approach is because it is built on first principles, its explicit about the conditions under which a monetary equilibrium exists, and can be used to derive a dynamic equilibrium condition that can shed light on the value of money.

The basic setup is as follows. Time is discrete and continues forever. There are two types of agents, buyers and sellers. Each time period is divided into two subperiods. In the first subperiod, buyers and sellers are matched pairwise and anonymously to trade (we will call this the decentralized market, or DM). In the second subperiod, buyers and sellers all meet in a centralized (Walrasian) market (we will call this the centralized market, or CM). What makes buyers and sellers different are their preferences. Buyers want to purchase goods in the DM, but cannot produce in that subperiod. Sellers want to purchase goods in the CM, but cannot produce in that subperiod. Thus, there is a basic absence of double-coincidence of wants problem. The anonymity of buyers and sellers in the DM means that money is essential for trade. Given this basic setup, we can examine the conditions under which money has value and this will allow us to discuss the implications of the platinum coin. (Note that we can confine our analysis to buyers since sellers will never carry money into the DM since they never consume in the DM.)

Suppose that buyers have preferences:

where is the discount factor, is the quantity of goods purchased in the DM, and is the quantity of goods produced by the buyer in the CM. Consumption of the DM good provides utility to the buyer and production of the CM good generates disutility of production. Here, the utility function satisfies .

The evolution of money balances for the buyer is given by:

where denotes the price of money in terms of goods, denotes money balances, and the apostrophe denotes an end of period value. Now let's denote the value function for buyers in the DM as and the value function for buyers entering the CM as .

Thus, entering the CM, the buyer's value function satisfies:

Using the evolution of money balances equation, we can re-write this as

In the DM, buyers and sellers are matched pairwise. Once matched, the buyers offer money in exchange for goods. For simplicity, we assume that buyers make take-it-or-leave-it offers to sellers such that where represents the quantity of money balances offered for trade and represents the disutility generated by sellers from producing the DM good. The value function for buyers in the DM is given as

Using the linearity of and the conditions of the buyers' offer, this can be re-written as:

Iterating this expression forward and substituting into $W$, we can then write the buyer's problem as:

[If you're trying to skip the math, pick things up here.]

From this last expression, we can now place conditions on whether anyone will actually hold fiat money. It follows from the maximization problem above that the necessary condition for a monetary equilibrium is that . Intuitively, this means that the value of holding fiat money today is greater than or equal to the discounted value of holding money tomorrow. If this condition is violated, everyone would be better off holding their money until tomorrow indefinitely. No monetary equilibrium could exist.

Thus, let's suppose that this condition is satisfied. If so, this also means that money is costly to hold (i.e. there is an opportunity cost of holding money). As a result, buyers will only hold an amount of money necessary to finance consumption (in mathematical terms, this means ). This means that the buyers' offer can now be written . This gives us the necessary envelope conditions to solve the maximization problem above. Doing so, yields our equilibrium difference equation that will allow us to talk about the effects of the platinum coin. The difference equation is given as

Since money is neutral in our framework, we can assume that there is a steady state solution such that . Thus, the difference equation can be written:

This difference equation now governs the dynamics of the price of money. We can now use this assess claims that the platinum coin would not have any inflationary effect.

Suppose that and have standard functional forms. Specifically, assume that and . [I should note that the conclusions here are robust to more general functional forms as well.] If this is the case, then the difference equation is a convex function up to a certain point at which the difference equation becomes linear. The convex portion is what is important for our purposes. The fact that the difference equation is convex implies that the difference equation intersects the 45-degree line used to plot the steady-state equilibrium in two different places. This means that there are multiple equilibria. One equilibrium, which we will call is the equilibrium that is assumed to be the case by advocates of the platinum coin. They assume that if we begin in this equilibrium, the Federal Reserve can simply hold the money supply constant through open market operations and in so doing prevent the price of money (i.e. the inverse of the price level) from fluctuating.

However, what this suggestion ignores is that the difference equation also intersects the 45-degree line at the origin. Coupled with the range of convexity of the difference equation, this implies that there are multiple equilibria that converge to an equilibrium in which money does not have value (i.e. ). Put in economic terms, there are multiple equilibria that are decreasing in , which means that they increasing in the price level. It is therefore possible to have inflation even with a constant money supply. The beliefs of economic agents are self-fulfilling.

In terms of the platinum coin, this implies that the explicit monetization of the debt by minting the platinum coin can potentially have disastrous effects even if the president states that the infusion is temporary and even if the Federal Reserve conducts open market operations to offset the increase in the monetary base caused by the deposit of the coin by the Treasury. In short, if the debt monetization were to have a significant impact on inflation expectations, it is possible that the United States could experience significant inflation even if the Federal Reserve tried to hold the money supply constant. The very idea that this represents a possible outcome should render the platinum coin to be a bad idea.