## Some Thoughts on Liquidity

The quantity theory relates not so much to money as to the whole array of financial assets exogenously supplied by the government. If the government debt is doubled in the absence of a government-determined monetary base the price level doubles just as well as in the case of a doubling of the monetary base in the absence of government debt. — Jurg Niehans, 1982

Seemingly lost in the discussion of monetary policies various QEs is a meaningful resolution of our understanding of the monetary transmission mechanism.  Sure, New Keynesians argue that forward guidance about the time path of the short term nominal interest rate is the mechanism, Bernanke argues that long term interest rates are the mechanism, and skeptics of the effectiveness of QE argue that it is the interest rate on excess reserves that is the mechanism.  I actually think that these are not the correct way to think about monetary policy.  For example, there are an infinite number of paths for the money supply consistent with a zero lower bound on interest rates.  Even in the New Keynesian model, which purportedly recuses money from monetary policy, the rate of inflation is pinned down by the rate of money growth (see Ed Nelson’s paper on this).  It follows that it is the path of the money supply that is more important to the central bank’s intermediate- and long-term goals.  In addition, it must be the case that the time path of the interest rate outlined by the central bank is consistent with expectations about the future time path of interest rates.  The mechanism advocated by Bernanke is also flawed because the empirical evidence suggests that long term interest rates just don’t matter all that much for investment.

The fact that I see the monetary transmission mechanism differently is because you could consider me an Old Monetarist dressed in New Monetarist clothes with Market Monetarist policy leanings (see why labels are hard in macro).  Given my Old Monetarist sympathies it shouldn’t be surprising that I think the aforementioned mechanisms are not very important.  Old Monetarists long favored quantity targets rather than price targets (i.e. the money supply rather than the interest rate).  I remain convinced that the quantity of money is a much better indicators of the stance of monetary policy.  The reason is not based on conjecture, but actual empirical work that I have done.  For example, in my forthcoming paper in Macroeconomic Dynamics, I show that many of the supposed problems with using money as an indicator of the stance of monetary policy are the result of researchers using simple sum aggregates.  I show that if one uses the Divisia monetary aggregates, monetary variables turn out to be a good indicator of policy.  In addition, changes in real money balances are a good predictor of the output gap (interestingly enough, when you use real balances as an indicator variable, the real interest rate — the favored mechanism of New Keynesians — is statistically insignificant).

Where my New Monetarist sympathies arise is from the explicit nature in which New Monetarism discusses and analyzes the role of money, collateral, bonds, and other assets.  This literature asks important macroeconomic questions using rich microfoundations (as an aside, many of the critics of the microfoundations of modern macro are either not reading the correct literature or aren’t reading the literature at all).  Why do people hold money?  Why do people hold money when other assets that are useful in transactions have a higher yield?  Using frameworks that explicitly provide answers to these questions, New Monetarists then ask bigger questions. What is the cost associated with inflation? What is the optimal monetary policy? How do open market operations work?  The importance of the strong microfoundations is that one is able to answer these latter questions by being explicit about the microeconomic assumptions.  Thus, it is possible to make predictions about policy with an explicit understanding of the underlying mechanisms.

An additional insight of the New Monetarist literature is that the way in which we define “money” has changed substantially over time.  A number of assets such as bonds, mortgage-backed securities, and agency securities are effectively money because of the shadow banking system and the corresponding prevalence of repurchase agreements.  As a result, if one cares about quantitative targets, then one must expand the definition of money.  David Beckworth and I have been working on this issue in various projects.  In our paper on transaction assets shortages, we suggest that the definition of transaction assets needs to be expanded to include Treasuries and privately produced assets that serve as collateral in repurchase agreements.  In addition, we show that the haircuts of private assets significantly reduced the supply of transaction assets and that this decline in transaction assets explains a significant portion of the decline in both nominal and real GDP observed over the most recent recession.

The reason that I bring this up is because this framework allows us not only to suggest a mechanism through which transaction assets shortages emerge and to examine the role of these shortages in the context of the most recent recession, but also because the theoretical framework can provide some insight into how monetary policy works.  So briefly I’d like to explain how monetary policy would work in our model and then discuss how my view of this mechanism is beginning to evolve and what the implications are for policy.

A standard New Monetarist model employs the monetary search framework of Lagos and Wright (2005).  In this framework, economic agents interact in two different markets — a decentralized market and a centralized market.  The terms of trade negotiated in the decentralized market can illustrate the effect of monetary policy on the price level. (I am going to focus my analysis on nominal variables for the time being.  If you want to imagine these policy changes having real effects, just imagine that there is market segmentation between the decentralized market and centralized market such that there are real balance effects from changes in policy.)  In particular the equilibrium condition can be written quite generally as:

P = (M+B)/z(q)

where P is the price level, M is the money supply, B is the supply of bonds, and z is money demand as a function of consumption q.  I am abstracting from the existence of private assets, but the implications are similar to those of bonds.  There are a couple of important things to note here.  First, it is the interaction of the supply and demand for money that determines the price level.  Second, it is the total supply of transaction assets that determines the price level.  This is true regardless of how money is defined.  Third, note that as this equation is presented it is only the total supply of transaction assets that determine the price level and not the composition of those assets.  In other words, as presented above, an exchange of money for bonds does not change the price level.  Open market operations are irrelevant.  However, this point deserves further comment.  While I am not going to derive the conditions in a blog post, the equilibrium terms of trade in the decentralized market will only include the total stock of bonds in the event that all bonds are held for transaction purposes.  In other words, if someone is holding bonds, they are only doing so to finance a transaction.  In this case, money and bonds are perfect substitutes for liquidity.  This implication, however, implies that bonds cannot yield interest.  If bonds yield interest and are just as liquid as money, why would anyone hold money? New Monetarists have a variety of reasons why this might not be the case.  For example, it is possible that bonds are imperfectly recognizable (i.e. they could be counterfeit at a low cost). Alternatively, there might simply be legal restrictions that prevent bonds from being used in particular transactions or since bonds are book-entry items, they might not as easily circulate.  And there are many other explanations as well.  Any of these reasons will suffice for our purposes, so let’s assume that that is a fixed fraction v of bonds that can be used in transactions.  The equilibrium condition from the terms of trade can now be re-written:

P = (M + vB)/z(q)

It now remains true that the total stock of transaction assets (holding money demand constant) determines the price level.  It is now also true that open market operations are effective in influencing the price level.  To summarize, in order for money to circulate alongside interest-bearing government debt (or any other asset for that matter) that can be used in transactions, it must be the case that money yields more liquidity services than bonds.  The difference in the liquidity of the two assets, however, make them imperfect substitutes and imply that open market operations are effective.  It is similarly important to note that nothing has been said about the role of the interest rate.  Money and bonds are not necessarily perfect substitutes even when the nominal interest on bonds is close to zero. Thus, open market operations can be effective for the central bank even if the short term interest rate is arbitrarily close to zero.  In addition, this doesn’t require any assumption about expectations.

The ability of the central bank to hit its nominal target is an important point, but it is also important to examine the implications of alternative nominal targets.  Old Monetarists wanted to target the money supply.  While I’m not opposed to the central bank using money as an intermediate target, I think that there are much better policy targets.  Most central banks target the inflation rate.  Recently, some have advocated targeting the price level and, of course, advocacy for nominal income targeting has similarly been growing.  As I indicated above, my policy leanings are more in line with the Market Monetarist approach, which is to target nominal GDP (preferable the level rather than the growth rate).  The reason that I advocate nominal income targeting, however, differs from some of the traditional arguments.

We live in a world of imperfect information and imperfect markets. As a result, some people face borrowing constraints.  Often these borrowing constraints mean that individuals have to have collateral.  In addition, lending is often constrained by expected income over the course of the loan.  The fact that we have imperfect information, imperfect markets, and subjective preferences means that these debt contracts are often in nominal terms and that the relevant measure of income used in screening for loans is nominal income.  A monetary policy that targets nominal income can potentially play an important role in two ways.  First, a significant decline in nominal income can be potentially harmful in the aggregate.  While there are often claims that households have “too much debt” a collapse in nominal income can actually cause a significant increase and defaults and household deleveraging that reduces output in the short run.  Second, because banks have a dual role in intermediation and money creation, default and deleveraging can reduce the stock of transaction assets.  This is especially problematic in the event of a financial crisis in which the demand for such assets is rising.  Targeting nominal income would therefore potentially prevent widespread default and develeraging (holding other factors constant) as well as allow for the corresponding stability in the stock of privately-produced transaction assets.

Postscript:  Overall, this represents my view on money and monetary policy.  However, recently I have begun to think about the role and the effectiveness of monetary policy more deeply, particularly with regards to the recent recession.  In the example given above, it is assumed that the people using money and bonds for transactions are the same people.  In reality, this isn’t strictly the case.  Bonds are predominantly used in transactions by banks and other firms whereas money is used to some extent by firms, but its use is more prevalent among households.  David Beckworth and I have shown in some of our work together that significant recessions associated with declines in nominal income can be largely explained through monetary factors.  However, in our most recent work, it seems that this particular recession is unique.  Previous monetary explanations can largely be thought of as currency shortages in which households seek to turn deposits into currency and banks seek to build reserves.  The most recent recession seems to be better characterized as a collateral shortage, in particular with respect to privately produced assets.  If that is the case, this calls into question the use of traditional open market operations.  While I don’t doubt the usefulness of these traditional measures, the effects of such operations might be reduced in the present environment since OMOs effectively remove collateral from the system.  It would seem to me that the policy implications are potentially different.  Regardless, I think this is an important point and one worth thinking about.

## How Much Capital?

Recently, it has become very popular to argue that the best means of financial reform is to require banks to hold more capital. Put differently, banks should finance using more equity relative to debt. This idea is certainly not without merit. In a Modigliani-Miller world, banks should be indifferent between debt and equity. I would like to take a step back from the policy response and ask why banks overwhelmingly finance their activities with debt. It is my hope that the answer to this question will provide some way to focus the debate.

It is clear that when banks finance primarily using equity, adverse shocks to the asset side of a bank’s balance sheet primarily affect shareholders. This seems at least to be socially desirable if not privately desirable. The imposition of capital requirements would therefore seem to imply that there is some market failure (i.e. the private benefit from holding more capital is less than the social benefit). Even if this is true, however, one needs to consider what makes it so.

One hypothesis for why banks hold too little capital is because they don’t internalize the total cost of a bank failure. For example, banks are limited liability corporations and covered by federal deposit insurance. Thus, if the bank takes on too much risk and becomes insolvent, shareholders lose their initial investment. Depositors are made whole through deposit insurance. It is this latter characteristic that is key. If bank shareholders were responsible not only for their initial level of investment, but also for the losses to depositors, banks would have different incentives. In fact, this was the case under the U.S. system of double liability that lasted from just after the Civil War until the Banking Act of 1933. (I have written about this previous here.) Under that system bank shareholders had a stronger incentive to finance using equity. In fact, evidence shows that banks with double liability took on less leverage and less risk than their limited liability counterparts.

Along similar lines the existence of Too Big Too Fail similarly creates greater incentives toward risk-taking and leverage because in the event that the bank becomes insolvent, it will be rescued by the government. Finally, the U.S. tax system treats debt finance more favorable than equity finance.

Of course, a first-best policy solution to these incentive problems would be to eliminate deposit insurance, Too Big to Fail, and the favorable tax treatment of debt finance. However, such reform is either politically infeasible or, in the case of eliminating Too Big to Fail, relies on a strong commitment mechanism by the government. Thus, a second-best policy prescription is to impose higher capital requirements.

This second-best policy solution, however, is contingent upon the characteristics above being the only source of the socially inefficient level of capital. I would argue that even in the absence of these characteristics banks might still be biased toward debt finance and that imposing capital requirements could actually result in a loss in efficiency along a different dimension of welfare.

The reason that capital requirements could be welfare-reducing has to do with the unique nature of bank liabilities. Banks issue debt in the form of deposits (and, historically, bank notes), which circulate as a medium of exchange. Thus, bank debt serves a social purpose over and above the private purpose of debt finance. This social function is important. In a world that consists entirely of base money, for example, individuals will economize on money balances because money does not earn a pecuniary yield. As a result, the equilibrium quantity of consumption and production will not equal the socially optimum quantity. Bank money, or inside money, has the potential to be welfare improving. In fact, the main result of Cavalcanti and Wallace was that feasible allocations with outside (or base) money are a strict subset of those with inside money. Imposing strict capital requirements would reduce the set of feasible allocations and thereby reduce welfare along this dimension.

Now some might be quick to dismiss this particular welfare criteria. After all, greater stability of the financial system would seem to be more important than whether the equilibrium quantity of production is the socially optimum quantity. However, this ignores the potential interaction between the two. Caballero, for example, has argued that there is a shortage of safe assets. This claim is consistent with what I argued above. If the supply of media of exchange is not sufficient to allow for the socially optimum quantity of output then there is a transaction asset shortage. As a result, there is a strong incentive for banks to create more transaction assets. This can explain while interest rates were low in early part of the decade and can similarly explain the expansion in the use of highly-rated tranches of MBS in repurchase agreements prior to the financial crisis.

In other words, the shortage of transaction assets described above creates an incentive for banks to create new such assets in the form of new debt finance. Thus, it is possible that banks have a bias toward debt finance that would exist even independent of Too Big To Fail, deposit insurance, limited liability, and the tax system. In addition, one could argue that the desire to create such transaction assets played an important role in the subsequent financial crisis as some of the assets that were previously considered safe become information-sensitive and thereby less useful in this role.

To the extent that one believes that the transaction asset shortage is significant, policymakers face a difficult decision with respect to capital requirements. While imposing stronger capital requirements might lead to greater financial stability by imposing greater losses on shareholders, this requirement can also exacerbate the shortage of transaction assets. Banks and other financial institutions will then have a strong incentive to attempt to mitigate this shortage and will likely try to do so through off-balance sheet activities.

This is not meant to be a critique of capital requirements in general. However, in my view, it is not obvious that they are sufficient to produce the desired result. One must be mindful of the role that banks play in the creation of transaction assets. It would be nice to have an explicit framework in which to examine these issues more carefully. In the meantime, hopefully this provides some food for thought.

P.S. Miles Kimball has suggested to me that capital requirements coupled with a sovereign wealth fund could assist in financial stability and fill the gap in transaction assets. I am still thinking this over. I hope to have some thoughts on this soon.

## Armen Alchian, 1914 – 2013

Armen Alchian passed away today at the age of 98. Others have chimed in with their thoughts on Alchian and his work (see here and here, for example) and I thought that I would as well. As many others have noted, Alchian was insightful and clever. He had a unique ability to communicate clever, unique, and insightful ideas in a way that suggested that these ideas were obvious. In fact, these ideas were often profound in both clarifying topics and in inspiring the work of fellow economists.

The aspect of Alchian’s work that I have found most insightful and most challenging is thinking about the economy as a coordination problem. There are few, if any, economists who have done as much in terms of thinking about economics in this light as Alchian did and the profession is much better for his insights.

No doubt, in the next couple of days, there will be excellent discussions of his great works as a scholar and a teacher, but what is perhaps the best biographical information about Alchian is found in his “Principles of Professional Advancement”, in which he provides a light-hearted guide to success in academia and also discusses some of his important papers. I think that this excerpt in which Alchian details an experience during his time as a consultant at RAND does a lot to summarize the inquisitive nature of Alchian’s mind and his astute ability to use the economic way of thinking to analyze the problems at hand:

I like to brag that I did the first “event study” in corporate finance, bank in the 1950s and 1960s. The year before the H-bomb was successfully created, we in the economics division at RAND were curious as to what the essential metal was — lithium, beryllium, thorium, or some other. The engineers and physicists wouldn’t tell us economists, quite properly, given the security restrictions. So I told them I would find out. I read the U.S. Department of Commerce Year Book to see which firms made which of the possible ingredients. For the last six months of the year prior to the successful test of the bomb, I traced the stock prices of those firms. I used no inside information. Lo and behold! One firm’s stock prices rose, as best I can recall, from about $2 or$3 per share in August to about $13 per share in December. It was the Lithium Corp. of America. In January, I wrote and circulated a memorandum titled “The Stock Market Speaks.” Two days later I was told to withdraw it. The bomb was tested successfully in February, and thereafter the stock price stabilized. That entire speech is filled with similar anecdotes that demonstrate the way in which Alchian thought and how that influenced his research. Today is a day to sit down with Economic Forces at Work and appreciate the brilliance of Armen Alchian. ## Re-Thinking Financial Reform Over at National Review Online I advocate reviving double liability for banks. Here is an excerpt: The banking system in the U.S. hasn’t always been like this. Between the Civil War and the Great Depression, banks did not have limited liability. Instead, they had double liability. When a bank became insolvent, shareholders lost their initial investment (just as they do under limited liability today). But in addition, a receiver would assess the value of the asset holdings of the bank to determine the par value of the outstanding shares. Shareholders had to pay an amount that could be as high as the current value of their shares in compensation to depositors and creditors. Shareholders and bank managers (who were often shareholders themselves) thus had a stronger incentive than they do today to assess the risk of investments accurately, because they were risking not just their initial investment but the total value of the banks’ assets. Shareholders also had an incentive to better monitor bank managers and the bank balance sheet. ## What I’m Reading 1. The New Dynamic Public Finance by Narayana Kocherlakota 2. The Redistribution Recession by Casey Mulligan 3. The Bretton Woods Transcripts, edited by Kurt Schuler and Andrew Rosenberg 4. Misunderstanding Financial Crises by Gary Gorton ## Let’s Talk About Interest on Reserves Recently, there has been a great deal of discussion about paying interest on excess reserves and the corresponding implications for money and monetary policy. While much of this discussion has been interesting, it might be useful to consider the impact of the influence of paying interest on reserves in the context of an explicit macroeconomic model so that we might better understand the dynamics of the effects of such a policy. In addition, a model allows us to be explicit about the assumptions that we are making and also to keep are logic consistent. Fortunately, we do not need to start from scratch on this topic as Peter Ireland has written an excellent paper entitled, “The Macroeconomic Effects of Interest on Reserves.” Before we discuss the impact of paying interest on reserves, it might be beneficial to talk about how this impacts the market for reserves using a straightforward supply and demand analysis, as Ireland does. Consider a simple supply and demand graph with the interest rate on the vertical axis and the quantity of reserves on the horizontal axis. Typically, in the market for reserves, the demand for reserves is a standard downward sloping demand curve. This is because a higher federal funds rate means that there is a higher opportunity cost of holding reserves rather than lending them to banks. If the Federal Reserve sets a target for the federal funds rate, the supply curve is horizontal at that interest rate. Where the supply curve intersects the demand curve is where one gets the unique quantity of reserves necessary to clear the market. One can therefore think of the Fed as providing the quantity of reserves necessary to maintain its interest rate target. Now let’s suppose that the Fed starts paying interest on reserves. In this case, the supply curve remains the same (horizontal at the federal funds rate), but the demand curve changes. In particular, with demand curve for reserves is now downward-sloping for all rates above the interest rate on reserves. At the interest rate on reserves, the demand curve is horizontal. Why? Suppose that the federal funds rate is above the interest rate on reserves. In this case, an increase in the federal funds rate, holding the interest rate on reserves constant, causes a reduction in the demand for reserves. In other words, when the federal funds rate is above the interest rate on reserves, the opportunity cost of holding reserves is now the spread between the federal funds rate and the interest rate paid on reserves. So why do people think that money doesn’t matter in this context? They think that money doesn’t matter because when the federal funds rate target is equal to the interest rate on reserves, the supply curve is horizontal at the same interest rate at which the demand curve is horizontal. This implies that there is a continuum of values for reserves that can be an equilibrium. Unfortunately, this is where most of the debate stops in the blogosphere. Those who think that money is irrelevant point to this latter result and conclude that any quantity of base money is consistent with equilibrium and therefore the actual quantity doesn’t matter. However, as Ireland notes, this leaves many questions unanswered: [The preceding analysis ignores] the effects that changes in output, including those brought about in the short run by monetary policy actions themselves, may have on the demand for reserves. And to the extent that changes in the interest rate paid on reserves get passed along to consumers through changes in retail deposit rates, and to the extent that those changes in deposit rates then set off portfolio rebalancing by households, additional effects that feed back into banks’ demand for reserves get ignored as well. One cannot tell from these graphs whether changes in the federal funds rate, holding the interest rate on reserves fixed either at zero or some positive rate, have different effects on output and inflation than changes in the federal funds rate that occur when the interest rate on reserves is moved in lockstep to maintain a constant spread between the two; if that spread between the federal funds rate and the interest rate on reserves acts as a tax on banking activity, those differences may be important too. The point of developing a corresponding macroeconomic model is to fundamentally assess whether or not these hypothesized effects are of any significance. To do so, Ireland extends a New Keynesian model to have a banking system and a shopping time specification to motivate the use of a medium of exchange. Since this is a large scale model and this is a blog post, I will spare further details of the model and refer interested readers to the paper. However, I would like to discuss what Ireland finds as it relates to the discussion among econobloggers (there are more results that are of interest as well). First, and perhaps most importantly for the blogosphere discussion, Ireland’s model demonstrates that even if they pay interest on reserves, the Fed still has to use open market operations to adjust the supply of bank reserves in order to change the price level. In other words, not only does the monetary base remain important, it is still necessary to pin down the price level. Second, there are important implications for how the Fed conducts open market operations. Specifically, in a world without interest on reserves, when the Fed raises its target for the federal funds rate it correspondingly reduces the supply of reserves. However, in Ireland’s model, the impulse response function for reserves following a change in monetary policy is just the opposite. In his model the central bank would have to increase bank reserves in response to a tightening of monetary policy as a result of an increase in the demand for reserves from banks, which in turn are caused by the portfolio reallocations of households. This is because a contractionary monetary policy causes a reduction in the user cost of deposits, which raises the demand for deposits and thereby the demand for reserves. As noted above, there are other results of interest and I would encourage anyone who wants to have a serious discussion about interest on reserves to read the paper in its entirety. Nevertheless, just to summarize, the importance of Ireland’s paper is to present an explicit macroeconomic model that allows us to talk about the short-run and long-run behavior of the monetary base when the Fed pays interest on reserves. The implications of his model is that the monetary base is important in both the long- and short-run. In the short run, the Fed has to adjust the supply of bank reserves in accordance with their desired interest rate target. This response differs depending on whether interest is paid on reserves, but in either case, this behavior is necessary. In addition, and most importantly, the nominal stock of reserves is essential for influencing the price level in the long run. In other words, the monetary base is not irrelevant. ## 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.) Suppose that buyers have preferences: $E_0 \sum_{t = 0}^{\infty} \beta^t [u(q_t) - x_t]$ 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.