# The Reader's Guide to Optimal Monetary Policy

I've been seeing more discussion of monetary policy recently, I suspect in large part due to the wave of high inflation that's sweeping the globe right now. As such, I thought this is a good time to make a post about the wonderful website [*The Reader's Guide to Optimal Monetary Policy*](https://diercks.shinyapps.io/anthony_diercks_shiny_app/).

This website gives you access to a collection of hundreds of macroeconomics papers that are designed to specifically answer the question "what is the optimal monetary policy?" The results are then presented in a scatter-plot format, along with various properties of the optimal policy recommended in each paper, such as the first and second moments of inflation for the optimal policy.

Now that I've sufficiently advertised the website, I'll first discuss what monetary policy is about and what goals it tries to achieve, and then I'll explain the fundamental questions and tradeoffs that come up when we try to come up with an optimal monetary policy. I'll then link this to the estimates from the hundreds of papers listed in the website and broadly how each paper comes up with the recommendations they end up endorsing.

If you're already familiar with what monetary policy is about then feel free to skip the next section.

# What is monetary policy?

Monetary policy is fundamentally about the issuer of a money using their power of issuing or retiring currency in circulation to achieve various aims they find to be desirable. When a country has its own currency, this currency is (usually) managed by the central bank. The central bank has the power to issue[^1] new currency or to retire existing currency by buying it back and removing it from circulation, and as counterfeit money printing is illegal, they are the only institution to legally have this power.

## Implementation

While central banks fundamentally exercise control over the money they manage, they can do this in ways that are somewhat less transparent. The most popular way to implement monetary policy nowadays is some form of nominal interest rate targeting. The simplified version of this is that the central bank announces a nominal[^2] interest rate target, say 3% per year. The central bank then stands ready to accept unlimited deposits from banks and to make unlimited loans to banks at this interest rate.

Ideally, as the central bank has unlimited capacity to issue or retire currency, they can keep accepting deposits or making loans until [arbitrage](https://en.wikipedia.org/wiki/Arbitrage) by banks brings the borrowing costs *between banks* in line with the central bank's target. This then influences (again by arbitrage) the borrowing rates that consumers or businesses face when they go to take out a loan from their local commercial bank, through what's called the [monetary transmission mechanism](https://en.wikipedia.org/wiki/Monetary_transmission_mechanism). The central bank can, in principle, always achieve their target assuming they are willing to make transactions that are of sufficient size to do so, and with sufficiently many counterparties.

This is not the only way central banks can implement monetary policy, however. Historically, commodity standards were a much more popular way to implement policy, especially the gold standard. In this policy regime, instead of having a nominal interest rate target, the monetary authority instead has a gold price target. They maintain some fraction of the total money stock in gold reserves, and intervene in the market as required to maintain a fixed dollar price of gold. At times, monetary authorities could also devalue the currency by increasing their target dollar price of gold, much like modern central banks can raise or lower their target interest rate. This policy was abandoned throughout the 20th century, as fixing the price of gold was seen to have serious problems as a mechanism for implementing monetary policy.

Many alternative monetary policy implementations have been proposed, such as targeting some measure of the money supply or targeting the price of various other financial assets (such as inflation futures or nominal GDP futures). Ultimately, the decision has to be based on the goals the central bank is trying to achieve with its policy, which brings us to the next subsection.

## Objectives

Historically, monetary policy had a rather circumscribed role in trying to keep the value of the currency, equivalently the overall price level of goods and services in the economy measured in units of currency, stable. This role was often weighed against the benefits of seigniorage, or the profit made by the monetary authority through the "inflation tax" by issuing new money, as this is often one of the most convenient means of taxation available to a government with its own currency.

Seigniorage fell out of favor over time as the capability of governments to collect taxes through other means improved, as seigniorage is a tax with particularly undesirable properties. In contrast, some version of price stability remains one of the primary objectives of monetary policy today. However, over time monetary policy has experienced substantial [mission creep](https://en.wikipedia.org/wiki/Mission_creep) as more and more factors began to influence the decisions of central bankers.

In the US, the Federal Reserve (hereafter referred to as "the Fed") didn't even exist until the early 20th century, and the role of managing monetary policy in a gold standard regime was left to the Treasury Department. The Fed was created by the Wilson administration in order to fight financial panics that had become commonplace in late 19th and early 20th century US.  The Fed would lend money to banks that were facing bank runs and therefore coming under pressure to liquidate their long-run investments at big losses. If the bank was not fundamentally insolvent, the Fed would accept its assets as collateral and provide the bank with short-term financing to meet its obligations to depositors.[^3] The hope was with such a commitment bank runs themselves would become less frequent, as depositors would know that the Fed would step in to provide a lifeline to a solvent bank that was threatened with a bank run. This is  now known as the Fed's role as "lender of last resort", and has become part of the Fed's broader objective of promoting financial stability.

The Phillips curve, an inverse relationship between unemployment and inflation proposed by William Phillips, led to a further expansion of the role of monetary policy. (More on the motivation behind expecting such a relationship to exist later.) Now, the goal was not to simply have price stability, but to actively trade off inflation against unemployment. When inflation was judged to be too high, the central bank would push it down at the expense of creating some unemployment; and when unemployment was judged to be too high they would do the reverse. Over time, this would evolve into the macroeconomic stabilization objective of monetary policy: while monetary policy can't by itself make a country wealthy, it was thought that it could be used to lower the variance of economic growth.

As monetary policymakers got access to increasing amounts of higher frequency information, it became apparent that demand for money fluctuates a lot over time. Some of the fluctuations are predictable: people go shopping in Christmas week, so in this week demand for money spikes in a seasonal pattern. Other changes in demand are less predictable; and occur either in acute spikes due to financial distress in some sector of the economy, or due to long-run technological changes such as the use of credit cards for transactions becoming commonplace. Faced with this situation, central bankers decided the best high-frequency tool for mitigating this volatility in money demand was to target the *opportunity cost* of money[^4] instead of its *supply*, which is how a brief experiment with money supply targeting in the 1980s came to an end and how nominal interest rate targeting became dominant.

Even this list does not fully cover the wide range of goals modern central banks aim to pursue, but it will be enough for this post. In short, we're looking at the following goals:

- Price stability
- Financial stability
- Macroeconomic stability

Before moving on to discussion of optimal monetary policy, there are a few important points to understand about how monetary policy works.

## Important points

### The zero lower bound

The zero lower bound (ZLB) on nominal interest rates is a consequence of the fact that most monetary systems offer people the ability to hold cash. Cash pays 0% interest, so in principle if interest rates ever fell below 0%, lenders and depositors would hold cash paying 0% interest instead of loans or deposits that pay negative interest. Ideally this puts a lower bound of 0% on nominal interest rates.

In practice this doesn't quite work because holding cash has a cost, but these costs are usually not much larger than 1%/year, so while the strict zero lower bound is not true in the real world some weaker version of it still poses a serious constraint to how low monetary policy can push nominal interest rates to be. 

### The Fisher relation

The Fisher relation is an equation that ties nominal interest rates, expected inflation and real interest rates together. It's a consequence of arbitrage between nominal and real investments.

Suppose you have $1. You can either lend this for one year at a nominal interest rate of \\( i \\), or you can lend it on an inflation-indexed basis at a real interest rate of \\( r \\). An inflation-indexed loan is one such that the borrower pays back an amount that's proportional to a price level index, such as the CPI, when the loan is due instead of a fixed dollar amount.

The Fisher relation says that in expectation these investments should have the same rate of return. The first investment nets you \\( 1+i \\) dollars at the end of the year, while the second nets you \\( (1+r)(1 + \pi) \\) where \\( \pi \\) is the inflation rate over the period. So the Fisher relation says

\\[ (1+i) = (1+r) (1 + \mathbb E[\pi]) \\]

and on a first order approximation this is roughly \\( i \approx r + \mathbb E[\pi] \\). As an example, if normal US Treasuries are trading at a yield of 5%/year but inflation-indexed TIPS bonds are trading at a yield of 2%/year, that means[^5] the market expects roughly 3%/year inflation over the period of the maturity of the bonds.

# Optimal monetary policy

Now that we have some grasp of which goals monetary policy aims to achieve, let's think about which policies would be optimal for each goal individually. I'll make a separate section for each goal, as some of them require more in-depth discussion.

## Price stability

Well, this one is the easiest, in principle. If you want price stability, just pick some measure of the price level, such as the CPI or the PCE index, and target it at its current level forever. This will produce, on average, a 0%/year inflation rate as inflation is just the rate of growth in such an aggregate price index.

This does have a problem though, and it comes from a combination of the Fisher relation and the zero lower bound. Note that if you put these two together, we get the constraint that \\( r + \mathbb E[\pi] \geq 0 \\) or \\( \mathbb E[\pi] \geq -r \\): on average, inflation should be at least as large as the negative of the real interest rate. Historically this was not a common phenomenon, but nowadays it's actually not unusual to see real interest rates being negative. In such situations it will be impossible for the monetary authority to have a monetary policy that produces, on average, 0% inflation.

This problem is only minor, though, because it's possible to adjust the policy such that the central bank targets the price level in year \\( n \\) to be \\( P(n) = P(0) (1 + \pi)^n \\) for a fixed average inflation rate \\( \pi \\), say 2%/year. This has the benefit that it still makes price levels far in the future predictable, so such a policy can still serve essentially the same purpose with a sufficiently large \\( \pi \\) to not violate the zero lower bound. So optimal policies trying to address price stability concerns in general have this form, where \\( \pi \\) is chosen to optimize other goals of policy.

Note that this is not *quite* an inflation target, because in this policy the central bank commits to offsetting above target inflation in year \\( n \\) with below target inflation in year \\( n+1 \\) and vice versa. An inflation target would instead try to have an expected inflation rate of \\( \pi \\) each year regardless of whether the central bank overshot or undershot its inflation target in previous years. The benefit of targeting the price level instead of inflation is that it makes long-term contracts indexed to the currency much more predictable, ensuring greater price stability.

## Financial stability

Financial problems are generally caused by the maturity mismatch on the portfolio of banks or bank-like institutions[^6], which nowadays extend far beyond ordinary commercial banks.

These institutions borrow money short-term and lend it out long-term, and try to profit from the spread they charge between the two interest rates. This is beneficial because depositors or lenders to a bank often don't know when they will need to take out their money, but when this risk is averaged over many such depositors it becomes much more manageable from the bank's point of view. Therefore the bank is able to use short-term financing to support long-term investment projects and eliminate a source of inefficiency in the capital markets.

The main problem is that this makes banks particularly exposed to interest rate risk. The value of the long-term loans they own are often much more sensitive to interest rates than the short-term debts they owe to lenders or depositors, and this means a sudden increase in interest rates can leave a bank insolvent if it didn't sufficiently hedge against this risk. Likewise, a sudden decrease in interest rates can leave long-term borrowers in a bad position, as it makes it much more difficult to service long-run debts. For all these reasons, trying to stabilize the financial system usually pushes monetary policy towards stabilizing the nominal interest rate at some level \\( i \\).

In fact, there's a nice theoretical argument that the correct target value for \\( i \\) is just zero. This is because \\( i \\) is essentially the opportunity cost of holding money, as explained here[^4]. So the fact that \\( i \\) is not zero means people will economize on money holdings as it costs them some return to hold money instead of e.g. bank deposits or bonds. This is not desirable, because money is actually very cheap to produce, so it makes no sense for people to have to economize on it as if it were scarce. Therefore the optimal monetary policy from the point of view of minimizing the inefficiency caused by the necessity to hold money for transactions is to target the nominal interest rate at zero: [the Friedman rule](https://en.wikipedia.org/wiki/Friedman_rule).

## Macroeconomic stability

Initially it may be tempting to think that monetary policy has macroeconomic effects primarily because money is used as a *medium of exchange*: people pay for goods and services using money, so it seems intuitive that monetary policy can have an effect on the real economy. This is true, but the problem is that this effect usually turns out to be second-order in the nominal interest rate, which as discussed above is a measure of how scarce money is in the economy. This is because \\( i = 0 \\) is optimal, so you can imagine welfare as a function of \\( i \\) having a global maximum at \\( i = 0 \\). This means the first derivative vanishes at \\( i = 0 \\) and consequently deviations from \\( i = 0 \\) only have second order effects on welfare. So unless we have a hyperinflation and therefore \\( i \\) gets very large as a consequence of the Fisher effect, it's unlikely that deviations of \\( i \\) from \\( 0 \\) will have much explanatory power when it comes to macroeconomic volatility.

The most commonly accepted explanation for why monetary policy has substantial macroeconomic effects comes from money's role in the economy as a *unit of account*. If prices and wages are quoted in dollars, and for whatever reason they are adjusted at less than the optimal frequency, then monetary policy can affect relative prices in the economy and therefore have an impact on real production in the economy. 

For instance, if workers resist nominal wage cuts and the monetary authority implements a deflationary monetary policy that drives the overall price level down, then in real or purchasing power terms the wages of workers will be going up. As employers can't afford to hire workers at this higher real wage, and as workers are unwilling to take nominal pay cuts, they end up having to fire workers instead. This continues until the marginal product of labor rises sufficiently to compensate for the higher real wage businesses must pay to workers, and therefore the deflation results in an increased rate of unemployment and a lower amount of total production. There are other similar frictions that have the same effect, but the essential ingredient is always that nominal prices are slow to adjust for one or another reason and this allows monetary policy to have real effects.

"Wait", you might say. "You just said that the deviations of the nominal interest rate from zero can't have big macroeconomic effects because \\( i = 0 \\) is optimal. So if relative prices on the market are set optimally, then shouldn't first order deviations from those prices still only have second order effects on welfare?"

You would be right: in ideal market conditions this argument would fail for essentially the same reason that the \\( i = 0 \\) argument fails. The way macroeconomists get around this is to assume that relative prices in the economy are *not* set optimally, because the economy has a lot of agents trading with each other with various amounts of market power, and this leads relative market prices to be different from the frictionless optimal values they ideally would take. If this is true, then inflation messing up these prices and wages further can have *first order* effects.[^7]

In this case, the way for monetary policy to remedy the inefficiency is to create inflation when real wages need to fall (for example due to a recession), so employers don't actually have to cut nominal wages. In addition, having some average level of inflation that's somewhat above 0%/year can also help because not raising wages may be somewhat easier than cutting wages on nominal terms even in ordinary economic circumstances, so some amount of inflation can be generally useful to achieve the necessary adjustment in real wages without nominal wages needing to be cut.

Working out the optimal policy for achieving this usually gives a result that looks like a nominal GDP level target, with the same caveats as I've already discussed in the context of price level targets. The idea is that as nominal GDP growth is the sum of real economic growth and inflation, at a time when real growth slows down a nominal GDP target pushes up inflation to compensate for this. This higher inflation then enables real wages throughout the economy to adjust to the economic shock without anyone needing to get nominal wage cuts or a slower pace of nominal wage raises.

If you think the decisive friction is the difficulty of other prices to adjust and not of wages, the optimal policy can end up looking somewhat different, e.g. it could end up being a price level target.

# What does the literature say?

As you can check [on the website](https://diercks.shinyapps.io/anthony_diercks_shiny_app/), averaging the recommendations of 124 papers published about optimal monetary policy from 2010 to 2020 gives an average expected inflation of 0.55% and an average inflation volatility of 1%.

This might seem quite low, but it shouldn't be surprising in light of the discussion in the previous section. If we took all goals into consideration, we could end up with a nominal GDP level target that aims for a slightly positive inflation rate on average, as this compromise is not far from the optimal policy on all fronts. It would also end up with similar numbers to this: inflation would be slightly above zero and the volatility of inflation would be roughly the volatility of real growth, which in the US is indeed on the order of 1%/year. In addition, if you look at the specific papers in detail, you'll see that they indeed get their results by optimizing for one or more of the goals I've discussed earlier.

There is a good reason to target inflation to be higher than the 0.55%/year average recommendation in the real world, and it again has to do with the zero lower bound. Modern central banks operationally implement monetary policy through interest rate targets, even if their high level goal is more like a 2%/year inflation target. This means the zero lower bound on nominal interest rates seriously limits their ability to conduct monetary policy effectively in some situations, so they have a temptation to set a higher inflation target so as to "have room" to reduce nominal interest rates in a recession if necessary.

The literature does not care much about this because using interest rate targets to implement monetary policy when your actual goal is to target inflation is not optimal to begin with. In this case, you're better off implementing policy by buying and selling NGDP futures or CPI futures. However, as these assets don't actually exist in a large enough scale for central banks to use them in operational implementation of policy, they are stuck with using interest rates even though they don't actually want to be targeting interest rates at a high level. This discrepancy means there's some reason for them to push inflation higher than what would be optimal, as it gives them more room to maneuver in operational terms.

# What do I think?

Overall, my considered view is that targeting interest rates operationally doesn't make sense unless you also want to target interest rates as part of your policy regime, as in the Friedman rule for example. Unless you want to do this, you should be implementing monetary policy through buying and selling CPI futures, NGDP futures or other similar assets.

If you can't do this and are stuck with interest rate targeting as an implementation mechanism but you think difficulties of nominal wage and price adjustment are a big problem, then I would recommend a relatively high inflation target, possibly higher than 2%/year. This is because this raises the average level of nominal interest rates by the Fisher effect and thus gives central banks more room to cut interest rates during economic slumps, while the costs of this inflation are likely going to be second order. However, I'd make it clear that this is entirely an artifact of the poor way monetary policy is implemented and executed by central banks right now, and first best is definitely to get them to change the way they do this.

[^1]: It's a misconception that central banks print money, at least in the US. The physical task of printing dollar bills is actually handled by the US Treasury Department. However, it's the Federal Reserve that issues new currency. So they do the economic equivalent of "printing money", which you'll often encounter in colloquial usage as a description of what the Federal Reserve does even though it's, strictly speaking, false.

[^2]: A nominal interest rate is one that's denoted in the money the central bank is managing. The Federal Reserve targets an interest rate in dollars, not in pounds or euros, because dollars are the currency that they manage.

    It's important to note that this is not the *real* interest rate, i.e. the interest rate net of the expected change in the value of money, or in inflation. A money that is expected to lose value will generally have higher borrowing costs to compensate lenders for the additional expected loss they incur when they make a loan. More on this [here](###the-fisher-relation).

[^3]: This continues to be relevant in the modern age. When AIG, an insurance company, experienced a severe run on its short-run debt financing in 2008; the Fed decided that they could accept AIG's portfolio of long-run investments as collateral and provide them with sufficient financing to save them from immediate bankruptcy.

[^4]: If you hold cash instead of investing your money in interest-bearing securities, the return you forgo is the nominal interest rate paid on those securities. So this is the opportunity cost of, or "the price you pay" for, holding money instead of such securities for some period of time.

[^5]: This argument doesn't quite work for two reasons. One is that the Fisher relation doesn't have to hold exactly, as the arbitrage between real and nominal investments is only statistical: it equates the average return of the two investments while these don't really have to be the same *a priori*. Two is that TIPS bonds are not quite inflation-indexed, though they are quite close to being so.

    Still, as a first approximation the calculation should be fine; the deviations from it are going to come from higher order effects.

[^6]: Confusingly, nowadays institutions that are called "banks" often don't take interest rate risk at all, because they sell their loans to other financial institutions instead of keeping them on their balance sheet. However, there are still economic equivalents of banks, sometimes called "shadow banks", that do the same maturity transformation and are similarly vulnerable to interest rate shocks.

[^7]: Of course there are also somewhat exotic monetary economics models in which the optimal value of \\( i \\) is actually *negative*, which can't be achieved due to the zero lower bound, and so deviations of \\( i \\) from zero *also* have first order effects. These models seem even less compelling to me than the market power stories about relative prices, though.