# Monetary Theory and the Platinum Coin

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.)

$E_0 \sum_{t = 0}^{\infty} \beta^t [u(q_t) - x_t]$

where $\beta$ is the discount factor, $q$ is the quantity of goods purchased in the DM, and $x$ 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 $u'>0 ; u''<0$.

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

$\phi_t m' = \phi_t m + x_t$

where $\phi$ denotes the price of money in terms of goods, $m$ 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 $V_t(m)$ and the value function for buyers entering the CM as $W_t(m)$.

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

$W_t(m) = \max_{x,m'} [-x_t + \beta V_{t + 1}(m')]$

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

$W_t(m) = \phi_t m + \max_{m'} [-\phi_t m' + \beta V_{t + 1}(m')]$

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 $\phi_t d = c(q_t)$ where $d \in [0,m]$ represents the quantity of money balances offered for trade and $c(q_t)$ represents the disutility generated by sellers from producing the DM good. The value function for buyers in the DM is given as

$V_t(m) = u(q_t) + W_t(m - d)$

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

$V_t(m) = u(q_t) - c(q_t) + \phi_t m$

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

$max_{m} \bigg[-\bigg({{\phi_t/\phi_{t + 1}}\over{\beta}} - 1\bigg)\phi_{t + 1} m + u(q_{t+1}) - c(q_{t+1}) \bigg]$

[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 $\phi_t \geq \beta \phi_{t + 1}$. 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 $d = m$). This means that the buyers' offer can now be written $\phi_t m = c(q_t)$. 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

$\phi_t = \beta \phi_{t + 1}\bigg[ \bigg(u'(q_{t + 1})/c'(q_{t + 1}) - 1 \bigg) + 1 \bigg]$

Since money is neutral in our framework, we can assume that there is a steady state solution such that $q_t = q \forall t$. Thus, the difference equation can be written:

$\phi_t = \beta \phi_{t + 1}\bigg[ \bigg(u'(q)/c'(q) - 1 \bigg) + 1 \bigg]$

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 $u$ and $c$ have standard functional forms. Specifically, assume that $u(q) = {{q^{1 - \gamma}}\over{1 - \gamma}}$ and $c(q) = q$. [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 $\phi_{ss}$ 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. $\phi = 0$). Put in economic terms, there are multiple equilibria that are decreasing in $\phi$, 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.

### 14 responses to “Monetary Theory and the Platinum Coin”

1. Lord

Anything is possible and an asteroid could hit tomorrow, but we shouldn’t assume monetary policy only affects supply, and in particular it can have a direct effect on beta.

2. You have to consider in your assumptions that we are in a liquidity trap ZIRP environment where money today is equal to money tomorrow and there’s no incentive NOT to hold money. Thus your condition for monetary equilbrium is faulty in the current state.

The debate would be much different if we were in a normal rate environment.

3. acarraro

I am sorry but I don’t understand your argument. You have no analysis of the impact of a change in m on the price level (m doesn’t even enter your final equation).

Given equilibrium condition, a infinite price level would be stable, ergo it should be so now. Given that it isn’t, cannot we simply ignore the zero solution?

I looked at the original paper, and it does model increases in the monetary base. But there is nothing in it that let’s you state that no change in the monetary base would produce a difference in the solution. There is virtually no variable to account for the composition. I am confused on that you are trying to state.

4. cmbjive

“I am sorry but I don’t understand your argument.”

His argument sans math is quite simple: In a market where there are multiple actors, even if the Fed held money supply constant there is no guarantee that those multiple actors would go along with the belief that the Fed has controlled inflation.

Indeed, there is every reason to believe that those multiple actors will not go along with the idea of a trillion dollar coin. But since this is an idea that the Left likes it is always good.

5. acarraro

That makes no sense to me.

You are saying that if the treasury issues the coin and buys bonds from the Fed with the funds and then cancels them (thus decreasing the debt) someone would start believing that there will be inflation?

I guess if you believed they are going to do that on a regular basis so that they exhaust the bonds at the Fed, you’d start worrying. But that’s basically saying that the Fed is not keeping the base constant.

If you assume that minting the coin is equal to revoking central bank independence, I would agree it would increase inflation. It wouldn’t be my expectation though.

6. cmbjive

“You are saying that if the treasury issues the coin and buys bonds from the Fed with the funds and then cancels them (thus decreasing the debt) someone would start believing that there will be inflation?”

That not exactly what I’m saying but I’ll play along. Let’s say that Treasury goes through with this insane idea of minting a coin and then just sitting it in the Treasury. Never mind that an oz of platinum only sells for $1,579 the United States government has now effectively stated that it is worth$1 trillion.

The first question you should ask yourself is how is the government going to exactly convert the coin into actual money that will be used to pay its debts. I assume it would effectively take out a line of credit with The Fed and then use that to pay the bills. However, demand for dollars did not increase at any point but the Fed is telling the world that no there really are no dollars out there because the only thing that is being done with the dollars is paying down debt. No one would believe that and they would start pricing goods valued in dollars accordingly. That’s why the blogger stated that there are multiple equilibria even when the money supply is constant. The Fed may be able to control the money supply in the United States (and that’s a BIG may) but it cannot control the price of dollars anywhere else around the world. That’s what happens when your currency becomes the de facto reserve currency for the world.

The second question you should ask yourself is what exactly prevents other countries around the world from doing the same thing. Why should it be only the United States that can mint a coin and then magically state it is worth $1 trillion? South Africa has the world’s largest platinum reserves; can South Africa suddenly declare that any platinum it mines is now worth$1 trillion per ounce? Everyone would scoff at the notion but right now it is considered serious economic policy because Democrats refuse to cut spending and Republicans refuse to allow taxes to rise on everyone making less than $400,000 single/$450,000 married.

There are a host of other issues that speak against trying to monetize our debt through valuating a coin at a $1 trillion (by the way, why doesn’t the United States simply mint a .999999999 oz gold coin and call that a$1 trillion? Maybe it has something to do with not devaluating its own portfolio) but those two above should probably be more than enough. Perhaps Congress should just do the hard work of cutting spending where it is not needed and raise taxes to cover any gaps instead of playing with an atom bomb to solve its problems

7. acarraro

The Federal reserve has been expanding the monetary base for the last 3 years at an unprecedented pace. They don’t even bother minting a coin or printing paper currency. They just add an entry in their accounts.

They are currently sitting on $3 trillion treasuries which they purchased by simply adding an entry in their accounting system. The dollar has not collapsed as a result. Inflation is still under control. Japan has done this for the last 15 years. If the treasury mints the coin, it’s doing exactly the same thing. The reason they need to do it with platinum is that there is a law that prohibits them from printing money or issuing silver or gold coins with a value different from market value. There is a law that says they can print platinum coins of arbitrary value. So the basic questions are: is the treasury going to continue behaving this way and take control of monetary policy from the Fed? Can the Fed (given the laws currently existing) stop them if they do (given that the law says they should if the treasuries actions create too much inflation)? If you believe in the inflationary effect, you must answer yes to both questions. Obviously seeing the government minting the coin should raise your probability of the debt being monetized. The question is: by how much? My belief is that it should raise those expectation by very little. We already know that the government will likely monetize the debt if it becomes unfundable (as it would generate such a financial crisis that the Fed would be forced to intervene). This doesn’t matter if the Fed can sterilize the Treasury behaviour. Can they? In the short term (next 3-5 years), the Fed can sell$5-6 trillion US treasuries ($3 trillion it has and the expected amount from the QE program). The government has a 1$ trillion deficit which cannot be increased unless congress approves a new budget. So the Fed can easily offset the next 5-6 years of coin minting. I would say that’s plenty of ammunition and I am not even considering the effect of raising rates (which they can do without limit). Ergo they only way you can get inflation is if the Fed wants it to go up. Which is the exact same situation in which we are now.

If you are assuming that the President is going to wake up tomorrow morning an order the Treasury to mint a coin a day, then yes the situation is going to get out of control… I think that’s as likely as him waking up tomorrow and deciding to bomb Washington.

8. cmbjive

http://online.wsj.com/article/SB10001424127887323936804578227931992302150.html?mod=WSJ_Opinion_LEFTTopOpinion

“If the treasury mints the coin, it’s doing exactly the same thing. The reason they need to do it with platinum is that there is a law that prohibits them from printing money or issuing silver or gold coins with a value different from market value. There is a law that says they can print platinum coins of arbitrary value.”

In other words they cannot devalue their own assets. Like I said above, the US sits atop the world’s largest gold reserve; simply minting a near-pure gold coin and stating it’s worth $1 trillion would instantly devalue its gold supply (not to mention everybody else’s in that list I gave). Better to mint a stone that it has little of than to mint a stone that it has massive control over. Read the above article and then re-examine your statement above. You place too much faith in the Fed to believe that it has awesome power to control the fate of the dollar by holding steady the money supply. 9. Desolation Jones “The second question you should ask yourself is what exactly prevents other countries around the world from doing the same thing. Why should it be only the United States that can mint a coin and then magically state it is worth$1 trillion? South Africa has the world’s largest platinum reserves; can South Africa suddenly declare that any platinum it mines is now worth \$1 trillion per ounce? ”

There’s a different between a country being to declare what an an ounce of a metal is worth and being able to declare what a metal coin is worth. The US already does the later. The metal value of the zinc in a penny is worth about half a cent. The copper value of a dime is worth around 2 cents. How is minting a platinum coin and saying it’s worth a trillion any different?

10. I find it a little odd that when an increase in the monetary base base is being talked about this blog chooses to minimize inflation risks, saying “increases in the monetary base are not sufficient to cause inflation” but when the monetary base is NOT being increased we’re told that that “might not be sufficient to prevent a significant inflation.”

The context here is the Fed already have said last month it was going to add another trillion to its balance sheet in 2013.

• This post was a follow-up to an earlier post. In that previous post I addressed the risk of inflation through normal channels (i.e. the increase in the monetary base) and I pointed out that the platinum coin might undermine Federal Reserve independence. This post was a response to those who said that my claims regarding inflation were unfounded because the monetary base could remain constant.

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