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# Category Archives: Macroeconomic Theory

## The Index Number Problem and Inflation

Nick Rowe asks whether or not housing prices should be included in the inflation rate that the Bank of Canada targets. His discussion focuses on whether or not housing prices are sticky or whether they are flexible. His discussion is a standard story that follows from Woodford’s textbook on monetary theory and policy. The idea is that the price index that the central bank uses to target inflation should consist only of sticky prices. However, I find this viewpoint (while commonly accepted) to be counter to the conclusion of the index number problem discussed by Samuelson, Niehans and many others. In addition, I think that there is something to learn from the latter.

Consider a standard microeconomic story. An individual receives income, $I$, and gets utility from consuming goods $x$ and $y$. Let $p_x$ denote the price of $x$ and $p_y$ denote the price of $y$. Further, suppose the utility function is given as $u(x, y)$ and has the usual properties. Thus, the consumer maximization problem is

$\max\limits_{x, y} u(x, y)$

$s.t. p_x x + p_y y \leq I$

The optimal allocation is therefore given as

${{u_x}\over{u_y}} = {{p_y}\over{p_x}}$

where $u_x$ is the marginal utility of $x$ and correspondingly for $y$.

Now you are probably wondering, what does this have to do with inflation? Well the answer is quite simple. In the problem above, there was no discussion of money. This was a real economy. Suppose instead that we are dealing with a monetary economy. In this case, income is money income (i.e. the number of dollars that you earn). In order to solve the allocation problem, we now need to deflate money income by some price index such that income is expressed in real terms. If a change in the money supply has an equiproportional impact on all prices, the choice of the price index is entirely arbitrary. The price of any individual good will suffice as a price index. In other words, we could re-write the budget constraint as

$x + {{p_y}\over{p_x}} y = {{I}\over{p_x}}$

Solving the consumer’s maximization problem yields the same equilibrium condition as that above. In addition, since changes in the money supply have an equiproportionate effect on all prices, the relative price of good $y$ to good $x$ remains unchanged and doesn’t have any effect on the allocation of goods. Additionally, so long as $p_x$ is held constant, then money income will not have any effect on the allocation either.

However, suppose that changes in the money supply do not have equiproportionate effects on prices. To use Nick’s example, suppose that the price of $x$ is sticky and the price of $y$ is flexible. In this case, a change in the money supply will also affect relative prices. In this case, one cannot simply solve the allocation problem by deflating money income by the price of one of the goods. In this case, changes in the money supply will distort the allocation of goods. In addition, this means that it is not sufficient to simply target the sticky price.

The solution to this problem is to choose a price index to deflate money income such that when that index is held constant, there is not any distortion in the allocation of goods. In other words, the objective to choose $P$ such that the budget constraint can be re-written as

${{p_x}\over{P}} x + {{p_y}\over{P}} y = {{I}\over{P}}$

Given the correct choice of $P$, it is straightforward to show that (1) the allocation of goods is determined by the relative prices of the goods, and (2) when $P$ is constant, money income is constant as one moves along an indifference curve.

So how do we construct $P$? Well, Samuelson gave us a class of examples where there was a specific price index that could solve the problem. And it turns out that the correct price index to use is dependent on the preferences of the representative consumer in the model. In particular, consider the following utility function:

$u = \sqrt{xy}$

It is straightforward to show that the correct price index to use in this case is

$P = \sqrt{p_x p_y}$

Here is a brief sketch of why this is true. In a real economy, when there is an increase in income, the individual moves to a higher indifference curve (i.e. utility increases). Thus, when an individual moves along an indifference curve, it must be true that income is constant. A different way of stating the problem above is that the objective is to choose a price index such that when that index is held constant, money income is constant when an individual moves along an indifference curve. We can now show that this is true for the utility function and price index above.

Consider the budget constraint:

$I = p_x x + p_y y = p\bigg({{p_x}\over{p}} x + {{p_y}\over{p}} y\bigg)$

Suppose the price index is given as $P = \sqrt{p_x p_y}$, then this can be re-written as

$I = p\bigg({{\sqrt{p_x}}\over{\sqrt{p_y}}} x + {{\sqrt{p_y}}\over{\sqrt{p_x}}} y\bigg)$

Given the preferences assumed above, the equilibrium condition for the consumer is

${{p_x}\over{p_y}} = {{y}\over{x}}$

Substituting this into the budget constraint yields

$I = p(2\sqrt{xy}) = p(2U)$

Thus, when $p$ is constant, a movement along an indifference curve is associated with a constant amount of money income.

So what does all of this mean?

What this means is that if changes in the money supply result in changes in the relative price of goods, then the optimal policy is one in which there is no inflation. However, the choice of how to measure inflation is not arbitrary in this case. Rather, there is a precise index number that must be used to calculate inflation.

Nick’s point, and the accepted wisdom of many in the discipline, is that when changes in the money supply distort relative prices due to price stickiness, the best thing to do is to target the sticky prices and let the flexible prices adjust. However, the example above rejects this idea. If, say, $p_x$ was a sticky price and $p_y$ was a flexible price, targeting $p_x$ would be insufficient. Doing so would prevent money income from affecting utility, but it would not prevent an adjustment in the relative prices of $x$ and $y$ and would therefore distort the allocation.

What the index number problem suggests is that the choice of the proper price index does not depend on which price is sticky or the source of the relative price variability. Instead, the index number problem suggests that the proper price index is derived from the preferences of the consumer. Thus, when asked if housing prices should be included in the price index used to calculate inflation, the relevant question is not whether housing prices are sticky, but rather whether housing enters a representative consumer’s utility function.

## On Secular Stagnation and Money

Gauti Eggertsson and Neil Mehrotra have a new paper that seeks to provide a formal model of secular stagnation. The paper is a welcome addition to a debate that, prior to their paper, was mostly muddled thoughts sprinkled throughout speeches and blog posts. The purpose of this post is to express doubts about some of the features of their model and also talk about the role of money (which is absent from the choices made in the model, but somehow prevents policy from going below the zero lower bound).

The basic idea in the Eggertsson and Mehrotra (henceforth EM) paper is that some sort of shock, like a de-leveraging shock, can cause the real interest rate to fall below zero. Since monetary policy is limited by the zero lower bound, the central bank (potentially) cannot equate the real interest rate with the real natural rate of interest. The only solution is for the central bank to increase its inflation target until the real interest rate is equal to the natural rate. In fact, EM argue that there is no equilibrium possible if the inflation rate isn’t raised to minus the real natural rate of interest.

Essentially, my problem with the model is as follows. As I will discuss below, the zero lower bound is only a constraint if individuals can hold currency. However, if individuals are capable of holding currency, when the real interest rate on savings is less than the real rate of return on currency (minus the rate of inflation), then everyone will hold currency. Thus, it is not true that no equilibrium exists when the inflation rate is less than minus the real rate of interest. The inclusion of money has important implications for their model in terms of the welfare effects of the shocks generating the so-called secular stagnation.

The EM model can be summarized as follows. The model is an overlapping generations model in which agents live for 3 periods. Thus, at any one point in time, there are three generations living — young, middle-aged, and old. Agents are assumed to only receive an endowment (or produce) in middle-age. Thus, in order to consume when old, agents have to save some of their endowment for old age. To consume when young, agents have to borrow from middle-aged agents. Middle-aged agents save by lending their endowment to young agents. When young agents become middle-aged, they repay their debt to the now old agents who use the repayment to consume. The model is a pure credit economy in the sense that money serves as a medium of account, but not a medium of exchange. The key feature of the model is that young agents are debt constrained. EM assume that young agents can only borrow an amount less than or equal to $D_t$. They assume that this constraint is binding such that young agents always borrow an amount, $D_t$. The key equation in their framework is the equilibrium condition in the savings market, given by

$1 + r_t = {{1+\beta}\over{\beta}} {{(1+g_t)D_t}\over{Y_t - D_{t -1}}}$

where $r_t$ is the real interest rate, $\beta$ is the discount rate, $g_t$ is the rate of population growth, and $Y_t$ is the size of the endowment. Secular stagnation results when the real interest rate falls below zero and the central bank cannot reduce the nominal interest rate sufficiently to clear the market. One potential cause of this phenomenon is de-leveraging. For example, suppose that $D_t$ permanently declines (i.e. young agents find it harder to borrow). In this case, the real interest falls in period $t$ and falls again in period $t+1$. If the decline is large enough, this can cause the real interest rate to be negative.

Now in a pure credit economy, this shouldn’t be a problem. The market rate of interest should just become negative. However, EM assume two things. First, they assume that the central bank determines the nominal interest rate. Second, they assume that “the existence of money precludes the possibility of a negative nominal rate.” The assumption they seem to be making is that nobody holds currency, but the threat that people could hold currency prevents the nominal rate from going below zero. The reason that this is important is because they make the following statement: “…it should be clear that if the real rate of interest is permanently negative, there is no equilibrium consistent with stable prices.” This argument follows directly from the Fisher equation. If prices are constant, the Fisher equation implies that

$i_t = r_t < 0$

which is a contradiction since we've assumed that $i \geq 0$.

However, if we are to take currency seriously, we should consider the conditions under which people would hold currency. To do so, consider a simple modification to their model. Assume that we endow the initial old agents with currency and assume that the supply of currency is constant such that the price level is constant. Now, middle-aged agents face a portfolio allocation decision. They can lend to young agents at the real rate of interest or they can sell their endowment to old agents for money.

In this modified environment there are three possible equilibria. For both debt and currency to be used in equilibrium, it must be true that the rate of return on debt and the rate of return on currency is equal (it is straightforward to show this by adding money to the EM choice problem and solving out the Kuhn-Tucker conditions). If the rate of return on debt is higher than currency, then nobody holds currency and everybody issues debt. If the rate of return on currency (technically, in an OLG model with a constant supply of currency, this is equal to the rate of population growth) is greater than the rate of return on debt, then everybody holds currency.

This point is not a mere formality. The reason is because EM argue that the world blow up with price stability (actually when they say there is no equilibrium, I think they actually mean that autarky is the equilibrium result). However, the simple addition of currency to the model implies that if the real interest rate ever became negative, all middle-aged agents would simply sell their endowment to old agents in exchange for currency rather than lend to young agents. Thus, if young agents become sufficiently debt constrained, nobody lends to young agents and young agents do not consume. Nonetheless, there is an equilibrium consistent with stable prices.

The importance of the explicit inclusion of currency is as follows. The central bank therefore faces a trade-off. If the central bank increased the growth rate of currency and thereby the inflation rate, they could increase the inflation rate sufficiently such that the inflation rate was equal to minus the real rate of interest. In this case, individuals would be indifferent between debt and currency and the debt market would clear at the desired negative real interest rate. This allows young agents to borrow, which given the assumption of diminishing marginal utility of consumption, means that welfare increases. However, this increase in welfare comes at the expense of a reduction in welfare via inflation. It is well-known that in OLG models, the optimal policy is a constant money supply.

This point might seem subtle, but I think it is important. The reason that I think it is important is because by arguing that the zero lower bound causes autarky when the real rate of interest is sufficiently negative, this overstates the welfare losses from the so-called secular stagnation. Introducing a constant supply of currency in this environment, significantly improves welfare relative to autarky. In fact, in standard OLG models, a constant supply of currency produces an optimal allocation.

## On Pegging the Interest Rate

Back in 2010, Narayana Kocherlakota was subjected to a great deal of criticism about his comment that the FOMCs decision to peg the interest rate at zero for an extended period of time would ultimately lead to deflation. Economists, especially those in the blogosphere, became near apoplectic that Kocherlakota, a distinguished scholar and voting member of the FOMC, would say something so seemingly egregious. Criticism largely came from two camps. The first camp, mostly filled with New Keynesians, argued that this was wrong because what mattered was the market interest rate relative to the natural rate. The second camp, mostly those whose views are consistent with Monetarism, suggested that this was preposterous because Friedman’s 1968 speech to the American Economic Association taught us that attempts to peg the interest rate would lead to inflation, not deflation.

Steve Williamson has taken a lot of flak from fellow economics bloggers because he has relentlessly defended this statement over the years. So last week I did a radical thing, I talked to Steve about this very controversy. In doing so, Steve confirmed what I believed to be his view all along and one that I think is correct: Kocherlakota was correct in his statement AND so was Milton Friedman. To understand why, we need to think about the Fisher equation and the behavior of monetary aggregates.

The entire debate centers around the Fisher equation:

$i_t = r_t + E_t \pi_{t+1}$

where $i$ is the nominal interest rate, $r$ is the real interest rate, and $E_t \pi_{t+1}$ is expected inflation. In the long run, we tend to think of the real interest rate as being determined by real factors, like productivity and preferences. The reason that the Fisher equation is important is because to understand Friedman’s point, we need to understand the effects of monetary policy in the short run and the long run. Thus, we need to understand both the liquidity effect of monetary policy and the Fisher effect.

Suppose, for example, that the central bank increased the rate of money growth. The liquidity effect implies that the short term interest rate would initially decline. However, over time, the Fisher effect implies that expectations of future inflation would increase and push the nominal interest rate higher. Thus, Friedman’s point was as follows. Suppose the current nominal interest rate was 3% and the central bank wanted to peg the nominal interest rate at 2%, in the short run they could achieve this by increasing money growth (i.e. through the liquidity effect). However, the Fisher effect would ultimately push the nominal interest rate higher. Thus, the only way that the central bank could achieve this interest rate peg would be through continuously increasing the rate of money growth to produce lower rates through the liquidity effect. The result of this sort of policy would then be one of ever-increasing rates of money growth and inflation. This result was largely viewed as unsustainable and therefore central banks could not peg the interest rate in this manner.

Kocherlakota’s point was that pegging the nominal interest rate would ultimately lead to deflation. This sounds contradictory to Friedman because he arrives at the opposite conclusion. However, the two points are actually two sides of the same coin. What Kocherlakota is effectively (or perhaps not so effectively, given the outcry) saying is that if the central bank wanted to peg the interest rate at a particular level, then this would require that the central bank start reducing money growth. In fact, for the case in which the central bank kept the nominal interest rate pegged at zero, a positive real interest rate would imply that the central bank would have to pursue a negative rate of money growth to maintain the target.

Thus, Kocherlakota is taking the objective of holding the interest rate constant as given and asking how it is that the central bank can maintain this policy. Friedman, on the other hand, is thinking about how the central bank is likely to try to pursue this policy.

What both Kocherlakota’s argument and Friedman’s argument have in common are what is important. The common element of the argument is that if the Federal Reserve adopts an interest rate peg over a long period of time, the ability to maintain that peg is dependent on the rate of money growth. The central bank can either adopt the rate of money growth consistent with the interest rate peg or they cannot maintain the interest rate peg.

What seemed to confuse people about this entire issue is that their understanding of Federal Reserve policy clearly contradicted any prediction of deflation. Nonetheless, this is the wrong way to interpret Kocherlakota’s argument. The argument he was making was essentially that if the Federal Reserve chose to leave the interest rate at zero for an extended period of time, this would imply that eventually they would have to pursue a policy that was deflationary. If they didn’t pursue that type of policy, they couldn’t maintain their peg of the nominal interest rate. In addition, since we don’t expect the Federal Reserve to pursue a policy consistent with negative rates of money growth, the statement should be seen as a criticism of the Federal Reserve’s original attempt at forward guidance which suggested that the FOMC would keep the interest rate at zero for an extended period of time.

There is an important lesson to be learned from this controversy and debate. The lesson is that even though policy is conducted with the federal funds rate as an intermediate target or with interest on reserves as an instrument, it is still necessary to know the underlying path of the money supply associated with this interest rate policy.

## What is Fair?

I recently read Thomas Piketty’s Capital in the 21st Century (my review of which will soon be published by National Review, for those interested). In reading the book, an implicit theme is that of fairness. Throughout the text, Piketty argues that his evidence on inequality suggests that there is a growing importance of inheritance in the determination of income and that this trend is likely to continue. It seems that Piketty sees this as problematic because it undermines meritocracy and even democracy. Nonetheless, when we start talking about there being too much inequality or too great of an importance of inheritance, this necessarily begs the question: How much is too much?

Economists have common ways of dealing with that question. There are vast literatures on optimal policies of all different types. The literature on optimal policy has a very consistent theme. First, the economist writes down a set of assumptions. Second, the economist solves for the efficient allocation given those assumptions. Third, the economist considers whether a decentralized system can produce the efficient allocation. If the decentralized allocation is inefficient, then there is a role for policy. The optimal policy is the one that produces the efficient allocation.

When Piketty and others talk about inequality and policy, however, they aren’t really talking about efficiency. Meritocracy-type arguments are about fairness. Economists, however, often shy away from discussing fairness. The reason is often simple. Who defines what is fair? Let’s consider an example. Suppose there are two workers, Adam and Steve, who are identical in every possible way and to this point have had the exact same economic outcomes. In addition, assume that we only observe what happens to these individuals at annual frequencies. Now suppose that this year, Adam receives an entirely random increase in pay, where random simply refers to something that was completely unanticipated by everyone. However, this year Steve loses his job for an entirely random reason (e.g. a clerical error removed Steve from the payroll and it cannot be fixed until next year). After this year, Adam and Steve go back to being identical (the clerical error is fixed!) and continue to experience the same outcomes the rest of their lives.

This is clearly a ridiculously stylized example. However, we can use this example to illustrate the difference between how economists evaluate policies. For someone concerned with a meritocratic view of fairness, the ideal policy in the ridiculous example above is quite clear. Adam, through no actions of his own, has received a windfall in income. Steve, through no fault of his own, has lost his income for an entire year. Someone only concerned with meritocracy would argue that the ideal policy is therefore to tax the extra income of Adam and give it to Steve.

Most economists, armed with the same example would not necessarily agree that the meritocratic policy is ideal. The most frequently used method of welfare analysis is the idea of Pareto optimality. According to Pareto optimality, a welfare improvement occurs when at least one person can be made better off without making another person worse off. In our example above, Pareto optimality implies that the optimal policy is to do nothing because taxing Adam and giving the money to Steve makes Adam worse off.

Advocates of meritocracy, however, are unlikely to be convinced by such an argument. And there is reason to believe that they shouldn’t be convinced. For example, if Adam and Steve both knew that there was some random probability of unemployment ex ante, they might have chosen to behave differently. For example, suppose that Adam and Steve each knew in advance that there was some probability that one of them would lose their job. They might have each purchased insurance against this risk. If we assume the third party insurer can costly issue insurance and earns zero economic profit, then when Steve became unemployed, he would receive his premium back plus what is effectively a transfer from Adam.

Of course, in this example, there still isn’t any role for policy. Private insurance, rather than policy, can solve the problem. Nonetheless, as I detail below, this does give us a potentially better starting place for discussing fairness, efficiency, and inequality.

Suppose that inequality is entirely driven by random idiosyncratic shocks to individuals and that these events are uninsurable (e.g. one cannot insure themselves against being born to poor parents, for example). There is a potential role for policy here that is both fair and efficient. In particular, the policy would correspond to what economists traditionally think of as ex ante efficiency. In other words, a fair policy would be the policy that individuals would choose before they knew the realization of these random shocks.

As it turns out there is a sizable literature in economics that examines these very issues and derives optimal policy. The conclusions of this literature are important because (1) they take the meritocratic view seriously, and (2) they arrive at policy conclusions that are often at odds with those proposed by advocates of meritocracy.

It is easy to make an argument for meritocracy. If people make deliberate decisions that improve their well-being, then it is easy to make the case that they are “deserving” of the spoils. However, if people’s well-being is entirely determined by sheer luck, then those who are worse off than others are simply worse off due to bad luck and a case can be made that this is unfair. Unfortunately, for advocates of meritocracy, all we observe in reality are equilibrium outcomes. In addition, individual success is often determined by both deliberate decision-making and luck. (No amount of anecdotes about Paris Hilton can prove otherwise.) I say this is unfortunate for advocates of meritocracy because it makes it difficult to determine what amount of success is due to luck and what is due to deliberate actions. (Of course, this is further muddled by the fact that when I say luck, I am referring to entirely random events, not the definition of the person who once told me that “luck is when preparation meets opportunity.”)

Nevertheless, our economic definition of fairness allows us to discuss issues of inequality and policy without having to disentangle the complex empirical relationships between luck, deliberate action, and success. Chris Phelan, for example, has made a number of contributions to this literature. One of his papers examines the equality of opportunity and the equality of outcome using a definition of fairness consistent with that described above. Rather than examining policy, he examines the equality of opportunity and outcome within contracting framework. What he shows is that inequality of both opportunity and outcome are both consistent with this notion of fairness in a dynamic context. In addition, even extreme inequality of result is consistent with this definition of fairness (such extreme inequality of opportunity, however, are not supported so long as people care about future generations).

Now, of course, this argument is not in any way the definitive word on the subject. However, the main point is that a high degree of inequality is not prima facie evidence of unfairness. In other words, it is not only difficult to disentangle the effects of luck and deliberate action in determining an individuals income and/or wealth, it is actually quite difficult to figure out whether a particular society is fair simply by looking at aggregate statistics on inequality.

This point is especially important when one thinks about what types of policies should be pursued. Advocates of a meritocracy, for example, often promote punitive policies — especially policies pertaining to wealth and inheritance. Piketty, for example, advocates a global, progressive tax on wealth. The idea behind the tax is to forestall the importance of inheritance in the determination of income and wealth. While this policy might be logically consistent with that aim, but it completely ignores the types of things that we care about when thinking about optimal policy.

For example, consider the Mirrlees approach to optimal taxation. The basic starting point in this type of analysis is to assume that skills are stochastic and the government levies taxes on income. The government therefore faces a trade-off. They could tax income highly and redistribute that income to those with lower skill realizations. This represents a type of insurance against having low skills. On the other hand, high taxes on income would discourage high skill workers from producing. The optimal policy is one that best balances this trade-off. As I note in my review of Piketty in National Review, this literature also considers optimal taxation with regards to inheritance. The trade-off here is that high taxes on inheritance discourage wealth accumulation, but provide insurance to those who are born to poor parents. The optimal policy is the one that best balances these incentives. As Farhi and Werning point out in their work on inheritance, it turns out that the optimal tax system for inheritance is a progressive system. However, the tax rates in the progressive system are negative (i.e. we subsidize inheritance with the subsidization getting smaller as the size of the inheritance gets larger). The intuition behind this is simple. This system provides insurance without reducing incentives regarding wealth accumulation.

Economists are often criticized as being unconcerned with fairness. This is at least partially untrue. Economists are typically accustomed to thinking about optimality in the context of Pareto efficiency. As a result, economists looking at two different outcomes will be hesitant to suggest that a particular policy might be better than another if neither represents a Pareto improvement. Nonetheless, this doesn’t mean that economists are unconcerned with the issue of fairness nor does it suggest that economists are incapable of thinking about fairness. In fact, economists are capable of producing a definition of fairness and the policy implications thereof. The problem for those most concerned with fairness is the economic outcomes and policy conclusions consistent with this definition might not reinforce their ideological priors.

## Exchange Rate Volatility and Alternative Money: The Case of Bitcoin

David Andolfatto has written a very good post on Bitcoin and why it might have positive value. In particular, he provides an excellent overview of what Bitcoin actually is (an electronic record of transactions) and how this relates to the insight that “money is memory.” (On this point, see also, William Luther’s paper, “Bitcoin is Memory.”) Nevertheless, I have some questions about the post regarding David’s discussion of the volatility of Bitcoin and how this impacts the choice of what to use as money. In this post, I hope to address this point and perhaps add some additional substance to the issue.

David ends his post talking about whether Bitcoin would make for a good form of money. This is an interesting question and one that often isn’t given sufficient thought. In David’s own research, however, he has emphasized that the characteristic that determines whether an asset is useful as money is whether that asset is information-sensitive (this claim is often prevalent in Gary Gorton’s work as well). The basic idea is that if the value of an asset is sensitive in the short-run to news that has private value, but no social value, then short run fluctuations in the price of the asset will preclude its use as a medium of exchange. Thus, David suggests looking at how the value of Bitcoin has changed over time. In his graph, he plots the price of bitcoins in terms of dollars. As you can see if you read his post (or if you know anything about Bitcoin), the price of bitcoins relative to dollars is quite volatile — especially over the last year.

However, I wonder whether looking at the volatility of the exchange rate between Bitcoin and the dollar is the best way to determine whether Bitcoin would be a good form of money. There are two reasons why I say this.

First, as David points out, this volatility could be the result of the fact that people view the supply of Bitcoin as being fixed (the supply of Bitcoin will eventually be fixed), but the demand for Bitcoin is fluctuating. David notes that this is consistent with the type of behavior we observe under commodity standards. When there is a change in the demand for gold, the purchasing power of gold varies (at times considerably) even though the long run purchasing power is constant.

I have heard others make this argument as well and this seems very plausible. Nevertheless, it is useful to recall the way in which free banking systems operated. For example, in a free banking system in which banks issued notes that were backed by gold, the supply of bank notes fluctuated with demand. Increases in the demand for money caused an increase in the supply of notes. These changes in the supply of notes, however, needn’t imply any change in the purchasing power of gold. Issuing bank notes redeemable in gold was thereby capable of reducing the volatility of the purchasing power of gold. Similarly, a financial intermediary today could issue bank notes redeemable in bitcoins and reduce the volatility of the purchasing power of bitcoins.

[A quick note: It is perhaps true that the U.S. government might decide that they don't want to allow financial intermediaries to issue bank notes, in which case my proposed solution to this source of volatility would not be operable. I would add though that it is not operable because of a legal restriction and not the volatility of the asset price.]

Second, and perhaps much more importantly, in models with competing money supplies the exchange rate does not factor in to the choice of allocation or welfare analysis. This is true even of David’s own research. For example, in quoting the price of bitcoins in terms of dollars, David is citing an exchange rate. However, in his research, the price volatility that matters for an asset is the own rate of return volatility. I think that this distinction matters.

To illustrate why I think that this distinction matters, let’s consider a simple overlapping generations model. There are two types of agents, young and old. Each lives for two periods. At any point in time, there is a generation of young and a generation of old. The population is assumed to be constant. There is one good to trade and it is non-storable. The young receive an endowment, y, of the consumption good. The old do not receive an endowment of goods. Thus, money is essential. There are two assets that can be used as a possible medium of exchange. The first is fiat currency. The second is bitcoins. The initial old carry both currency and bitcoins into the first period. The aggregate supply of bitcoins is fixed. The aggregate supply of currency, $N_t$, is assumed to grow at the gross rate $x$ (i.e. $N_{t + 1} = x N_t$).

Let’s consider the first and second period budget constraints for future generations in our model (i.e. everybody except the initial old). In the first period, future generations can use their endowment for consumption or they can sell some of this endowment for money and/or bitcoins. Thus, the first-period budget constraint is:

$c_{1,t} + m_t + b_t \leq y$

where $c_{1,t}$ denotes the consumption when young in period t, $m$ is real currency balances, and $b$ denotes real balances of bitcoins.

Denote $v_t$ as the price of currency in terms of goods at time t. Similarly, denote the price of bitcoins in terms of goods as $u_t$. Thus, the rate of return on currency is $v_{t + 1} / v_t$. Now let’s assume that there is some cost, $\tau$ that individuals have to pay when they use bitcoin to make a purchase. The rate of return on bitcoins is then given as $(1-\tau){{u_{t+1}}\over{u_t}}$. Thus, the second-period budget constraint can be written as

$c_{2,t+1} = {{v_{t+1}}\over{v_t}} m_t + (1 - \tau){{u_{t+1}}\over{u_t}} b_t$

But we can derive a precise definition of the rate of return on money. It follows from our first period budget constraint that we have:

$m_t = v_t n_t = (y - b_t - c_{1,t})$

where $n_t$ denotes nominal currency balances. Define the total nominal currency stock as $N_t$ and the size of the population, which as assumed to be constant as $P$. This implies an aggregate demand function for currency:

$v_t N_t = P(y - b_t - c_{1,t})$

Thus, the rate of return on money is

${{v_{t+1}}\over{v_t}} = {{P(y - b_{t+1} - c_{1,t+1})}\over{P(y - b_t - c_{1,t})}}{{N_t}\over{N_{t + 1}}}$

From above, we know that the currency supply grows at a gross rate $x$. This implies that in a stationary allocation (i.e. where consumption paths a constant across generations), the rate of return on currency is

${{v_{t+1}}\over{v_t}} = {{1}\over{x}}$

By similar logic, it is straightforward to show that in a stationary allocation ${{u_{t+1}}\over{u_t}} = 1$ because the supply of bitcoins was assumed to be fixed.

Thus, our stationary allocation budget constraints are:

$c_1 + m + b \leq y$

$c_2 = {{1}\over{x}}m + (1 - \tau) b$

In the present model, money and bitcoins are perfect substitutes (i.e. there only purpose is to serve as proof of a previous transaction when trading with future young generations). Thus, the real rates of return on money and bitcoins must be equal for both to exist in equilibrium. In other words, it must be true that ${{1}\over{x}} = (1 - \tau)$. We can re-write the second-period constraint as

$c_2 = {{1}\over{x}}(m + b)$

Combining these budget constraints, we have a lifetime budget constraint:

$c_1 + x c_2 \leq y$

Now let’s consider the basic implications of the model. First, the conditions under which both currency and bitcoins would be held in equilibrium is dependent on their relative rates of return. If these rates of return are equal, then both assets are held. This condition is independent of the exchange rate. Second, lifetime budget constraint outlines the feasible set of allocations available given the agents’ budget. Assume that utility is a function of consumption in both periods. The allocation decision in this case is dependent on the rate of return on currency, which is the same as the rate of return on bitcoins. The allocation decision is therefore contingent on the equilibrium rate of return. The exchange rate between currency and bitcoins plays no role in the allocation decision. In addition, one can show that this is the identical lifetime budget constraint that would exist in a currency economy (i.e. one in which Bitcoin doesn’t circulate). This last characteristic implies that neither the existence of bitcoins nor the exchange rate between bitcoins and currency have any effect on welfare.

So what does this mean? Basically what all of this means is that the exchange rate between currency and bitcoins is irrelevant to the decision to hold bitcoins, to the allocation decision, and to welfare. [Note: This isn't new, Neil Wallace taught us this type of thing over 30 years ago.]

The analysis above is a bit unfair to David for two reasons. First, the framework above skirts David’s main point, which is that Bitcoin is information sensitive whereas currency is not. Second, David doesn’t necessarily mean that the exchange rate between the dollar and bitcoins is relevant for the type of analysis above. Rather, what he means is that since the dollar is relatively stable, the exchange rate serves as a proxy for the own price of bitcoins in terms of goods.

What I would like to do now is to amend the framework above to make bitcoins information sensitive. The results regarding the exchange rate of bitcoins into dollars remain. However, an interesting result emerges. In particular, one can show that it is the risk premium associated with bitcoins that can help us to understand the choice of whether to hold bitcoins or dollar-denominated currency as well as factor into the allocation decision. I will then speculate as to whether this risk premium is sufficient to explain the differences in rates of return between bitcoins and dollar-denominated currency.

For simplicity, let’s now denote the rate of return on bitcoins as $r$. In addition, we will assume that $r$ is stochastic. In particular the assumption is that the rate of return is entirely determined by random news events. Thus, we can re-write our first- and second-period budget constraints, respectively, as

$c_{1,t} + m_t + b_t \leq y$

$c_{2,t+1} = {{1}\over{x}} m_t + r_{t+1} b_t$

The objective of future generations is to maximize $u(c_{1,t}) + v(c_{2,t+1})$ subject to the two constraints above. Combining the first-order conditions with respect to $m$ and $b$ and using the definition of covariance, we have the following equilibrium condition:

$E_t r_{t+1} = {{1}\over{x}} - {{cov[r_{t+1},v'(c_{2,t+1})]}\over{v'(c_{2,t+1})}}$

where the second term on the right-hand side measures the risk premium associated with bitcoins (note that this is in fact a risk premium since the covariance between the rate of return on bitcoins and the marginal utility of consumption is negative). If both assets are to be held in equilibrium, then the equibrium condition must hold. If the risk premium is too high, it is possible that nobody would hold bitcoins and they would only hold currency. This confirms David’s view that information sensitivity could affect the decision to hold bitcoins. However, this does not show up in the exchange rate, but rather in the relative rates of return. The risk premium similarly affect the allocation decision. Consider, for example, that the lifetime budget constraint can now be written as

$c_{1,t} + x c_{2,t+1} + x \omega \leq y$

where $\omega$ is used to simplify notation and denote the risk premium and the aggregate supply of bitcoins has been normalized to one. It is straightforward to see that when the risk premium is zero (i.e. bitcoins are not information sensitive) then the lifetime budget constraint is the same as that outlined above. The existence of a positive risk premium alters the budget set.

So what does all of this mean?

Essentially what it means is that looking at the exchange rate between bitcoins and the dollar is not a useful indicator about whether or not bitcoins would actually make for a good money. Even if we view the exchange rate between bitcoins and dollars as a useful proxy of the price of bitcoins in terms of goods, the exchange rate is not the correct measure for analysis. Rather, to evaluate whether bitcoins are a viable alternative/substitute for dollars, we need to know the relative rates of return on bitcoins and dollars and the risk premium associated with the fact that bitcoins are information sensitive.

This might all seem like semantics, after all, if we think the exchange rate is a good proxy of the price of bitcoins in terms of goods, then the rate of return could just be measured as the rate of change in the exchange rate. Nonetheless, this distinction seems especially important given the nature of the exchange rate between bitcoins and dollars. In particular, just looking at David’s graph of the exchange rate, it is plausible that the time series follows a random walk with a drift (I had trouble acquire actual data rather than ready-made graphs on this so if anybody has the data please send it along). This is important because if this is correct, the variance of the exchange rate is time-dependent. However, in terms of rates of change, the data would be stationary and therefore have some constant, finite variance. Thus, in this hypothetical example, looking at the exchange rate using David’s criteria about information sensitivity would indicate that bitcoin is a very bad money because the variance of the exchange rate is time dependent. In contrast, if the rate of return on bitcoin is stationary, then it is not immediately clear from the data whether or not bitcoin is a good money. This is why we need the model as it helps us to understand what properties the rate of return must possess to make a good money.

## Yellen, Optimal Control, and Dynamic Inconsistency

For much of his career, Milton Friedman advocated a constant rate of money growth — the so-called k-percent rule. According to this rule, the central bank would increase the money supply at a constant rate, k, every year. In this case, there would be no need for an FOMC. A computer could conduct monetary policy.

The k-percent rule has often been derided as a sub-optimal policy. Suppose, for example, that there was an increase in money demand. Without a corresponding increase in the money supply, there would be excess money demand that even Friedman believed would cause a reduction in both nominal income and real economic activity. So why would Friedman advocate such a policy?

The reason Friedman advocated the k-percent rule was not because he believed that it was the optimal policy in the modern sense of phrase, but rather that it limited the damage done by activist monetary policy. In Friedman’s view, shaped by his empirical work on monetary history, central banks tended to be a greater source of business cycle fluctuations than they were a source of stability. Thus, the k-percent rule would eliminate recessions caused by bad monetary policy.

The purpose of this discussion is not to take a position on the k-percent rule, but rather to point out the fundamental nature of discretionary monetary policy. A central bank that uses discretion has the ability to deviate from its traditional approach or pre-announced policy if it believes that doing so would be in the best interest of the economy. In other words, the central bank can respond to unique events with unique policy actions. There are certainly desirable characteristics of this approach. However, Friedman’s point was that there are very undesirable characteristics of discretion. Just because a central bank has discretion doesn’t necessarily mean that the central bank will use it wisely. This is true even of central banks that have the best intentions (more on this point later).

The economic literature on rules versus discretion is now quite extensive. In fact, a substantial amount of research within the New Keynesian paradigm is dedicated to identifying the optimal monetary policy rule and examining the robustness of this rule to different assumptions about the economy. In addition, there has been a substantial amount of work on credible commitment on the part of policymakers.

Much of the modern emphasis on rules versus discretion traces back to the work of Kydland and Prescott and the idea of dynamic inconsistency. The basic idea is that when the central bank cannot perfectly commit to future plans, we end up with suboptimal outcomes. The idea is important because Kydland and Prescott’s work was largely a response to those who viewed optimal control theory as a proper way to determine the stance of monetary policy. The optimal control approach can be summarized as follows:

The Federal Open Market Committee (FOMC) targets an annual inflation rate of 2% over the long run and an unemployment rate of 6% (the latter number an estimate of the economy’s “natural” unemployment rate).

Under the optimal control approach, the central bank would then use a model to calculate the optimal path of short-term interest rates in order to hit these targets.

In short optimal control theory seems to have a lot of desirable characteristics in that policy is based on the explicit dynamics of a particular economic framework. In addition, it is possible for one to consider what the path of policy should look like given different paths for the models state variables. Given these characteristics, the story describing optimal control linked above is somewhat favorable to this approach and notes that the optimal control approach to monetary policy is favored by incoming Fed chair Janet Yellen. Thus, it is particularly useful to understand the criticisms of optimal control levied by Kydland and Prescott.

As noted above, the basic conclusion that Kydland and Prescott reached was the when the central bank has discretionary power and use optimal control theory to determine policy, this will often result in suboptimal policy. Their critique of optimal control theory rests on the belief that economic agents form expectations about the future and those expectations influence their current decision-making. In addition, since these expectations will be formed based in part on their expectations of future policy, this results in a breakdown of the optimal control framework. The reason that this is true is based on the way in which optimal control theory is used. In particular, optimal control theory chooses the current policy (or the expected future path of policy, if you prefer) based on the current state variables and the history of policy. If expectations about future policy affect current outcomes, then this violates the assumptions of optimal control theory.

Put differently, optimal control theory generates a path for the policy instrument for the present policy decision and the future path of policy. This expected future path of the monetary policy instrument is calculated taking all information available today as given — including past expectations. However, this means that the value of the policy instrument tomorrow is based, in part, on the decisions made today, which are based, in part, on the expectations about policy tomorrow.

There are two problems here. First, if the central bank could perfectly commit to future actions, then this wouldn’t necessarily be a problem. The central bank could, for example, announce some state-contingent policy and perfectly commit to that policy. If the central bank’s commitment was seen as credible, this would help to anchor expectations thereby reinforcing the policy commitment and allowing the central bank to remain on its stated policy path. However, central banks cannot perfectly commit (this is why Friedman not only wanted a k-percent rule, but also sometimes advocated that it be administered by a computer). Thus, when a central bank has some degree of discretion, using optimal control theory to guide policy will result in suboptimal outcomes.

In addition, discretion creates additional problems if there is some uncertainty about the structure of the economy. If the central bank has imperfect information about the structure of the macroeconomy or an inability to foresee all possible future states of the world, then optimal control theory will not be a useful guide for policy. (To see an illustration of this, see this post by Marcus Nunes.) But note that while this assertion casts further doubt on the ability of optimal control theory to be a useful guide for policy, it is not a necessary condition for suboptimal policy.

In short Kydland and Prescott expanded and bolstered Friedman’s argument. Whereas Friedman had argued that rules were necessary to prevent central banks from making errors that were due to timing and ignorance of the lag in effect of policy, Kydland and Prescott showed that even when the central bank knows the model of the economy and tries to maximize an explicit social welfare function known to everyone, using optimal control theory to guide policy can still be suboptimal. This is a remarkable insight and an important factor in Kydland and Prescott receiving the Nobel Prize. Most importantly, it should give one pause about the favored approach to policy by the incoming chair of the Fed.

## My Two Cents on QE and Deflation

Steve Williamson has caused quite the controversy in the blogosphere regarding his argument that quantitative easing is reducing inflation. Unfortunately, I think that much of the debate surrounding this claim can be summarized as: “Steve, of course you’re wrong. Haven’t you read an undergraduate macro text?” I think that this is unfair. Steve is a good economist. He is curious about the world and he likes to think about problems within the context of frameworks that he is familiar with. Sometimes this gives him fairly standard conclusions. Sometimes it doesn’t. Nonetheless, this is what we should all do. And we should evaluate claims based on their merit rather than whether they reinforce our prior beliefs. Thus, I would much rather try to figure out what Steve is saying and then evaluate what he has to say based on its merits.

My commentary on this is going to be somewhat short because I have identified the point at which I think is the source of disagreement. If I am wrong, hopefully Steve or someone else will point out the error in my understanding.

The crux of Steve’s argument seems to be that there is a distinct equilibrium relationship between the rate of inflation and the liquidity premium on money. For example, he writes:

Similarly, for money to be held,

(2) 1 – L(t) = B[u'(c(t+1))/u'(c(t))][p(t)/p(t+1)],

where L(t) is the liquidity premium on money. For example, L(t) is associated with a binding cash-in-advance constraint in a cash-in-advance model, or with some inefficiency of exchange in a deeper model of money.

He then explains why QE might cause a reduction in inflation using this equation:

…the effect of QE is to lower the liquidity premium (collateral constraints are relaxed) which … will lower inflation and increase the real interest rate.

Like Steve, I agree that such a relationship between inflation and the liquidity premium exists. However, where I differ with Steve seems to be in the interpretation of causation. Steve seems to be arguing that causation runs from the liquidity premium to inflation. In addition, since the liquidity premium is determined by the relative supplies of alternative transaction assets, monetary policy controls inflation by controlling the liquidity premium. My thinking is distinct from this. I tend to think of the supply of public transaction assets determining the price level (and thereby the rate of inflation) with the liquidity premium determined given the relative supply of assets and the rate of inflation. Thus, we both seem to think that there is this important equilibrium relationship between the rate of inflation and the liquidity premium, but I tend to see causation running in the opposite direction.

But rather than simply conclude here, let me outline what I am saying within the context of a simple model. Consider the equilibrium condition for money in a monetary search model:

$E_t{{p_{t+1}}\over{\beta p_t}} = \sigma E_t[{{u'(q_{t+1})}\over{c'(q_{t+1})}} - 1] + 1$

where $p_t$ is the price level, $\beta$ is the discount factor, $q_t$ is consumption, and $\sigma$ is the probability that a buyer and seller is matched. Thus, the term in brackets measures the value of spending money balances and $\sigma$ the probability that those balances are spent. The product of these two terms we will refer to as the liquidity premium, $\ell$. Thus, the equation can be written:

$E_t{{p_{t+1}}\over{\beta p_t}} = 1 + \ell$

So here we have the same relationship between the liquidity premium and the inflation rate that we have in Williamson’s framework. In fact, I think that it is through this equation that I can explain our differences on policy.

For example, let’s use our equilibrium expression to illustrate the Friedman rule. The Friedman rule is designed to eliminate a friction. Namely the friction that arises because currency pays zero interest. As a result, individuals economize on money balances and this is inefficient. Milton Friedman recommended maintaining a market interest rate of zero to eliminate the inefficiency. Doing so would also eliminate the liquidity premium on money. In terms of the equation above, it is important to note that the left-hand side can be re-written as:

${{p_{t+1}}\over{\beta p_t}} = (1 + E_t \pi_{t + 1})(1 + r) = 1 + i$

where $\pi$ is the inflation rate and $r$ is the rate of time preference. Thus, it is clear that by setting $i = 0$, it follows from the expression above that $\ell = 0$ as well.

Steve seems to be thinking about policy within this context. The Fed is pushing the federal funds rate down toward the zero lower bound. Thus, in the context of our discussion above, this should result in a reduction in inflation. If the nominal interest rate is zero, this reduces the liquidity premium on money. From the expression above, if the liquidity premium falls, then the inflation rate must fall to maintain equilibrium.

HOWEVER, there seems to be one thing that is missing. That one thing is how the policy is implemented. Friedman argued that to maintain a zero percent market interest rate the central bank would have to conduct policy such that the inflation rate was negative. In particular, in the context of our basic framework, the central bank would reduce the interest rate to zero by setting

$\pi_t = \beta$

Since $0 < \beta < 1$, this implies deflation. More specifically, Friedman argued that the way in which the central bank could produce deflation was by shrinking the money supply. In other words, Friedman argued that the way to produce a zero percent interest rate was by reducing the money supply and producing deflation.

In practice, the current Federal Reserve policy has been to conduct large scale asset purchases, which have substantially increased the monetary base and have more modestly increased broader measures of the money supply.

In Williamson's framework, it doesn't seem to matter how we get to the zero lower bound on nominal interest rates. All that matters is that we are there, which reduces the liquidity premium on money and therefore must reduce inflation to satisfy our equilibrium condition.

In my view, it is the rate of money growth that determines the rate of inflation and the liquidity premium on money then adjusts. Of course, my view requires a bit more explanation of why we are at the zero lower bound despite LSAPs and positive rates of inflation. The lazy answer is that $\beta$ changed. However, if one allows for the non-neutrality of money, then it is possible that the liquidity premium not only adjusts to the relative supplies of different assets, but also to changes in real economic activity (i.e. $q_t$ above). In particular, if LSAPs increase real economic activity, this could reduce the liquidity premium (given standard assumptions about the shape and slope of the functions $u$ and $c$).

This is I think the fundamental area of disagreement between Williamson and his critics — whether his critics even know it or not. If you tend to think that non-neutralities are important and persistent then you are likely to think that Williamson is wrong. If you think that non-neutralities are relatively unimportant or that they aren't very persistent, then you are likely to think Williamson might be on to something.

In any event, the blogosphere could stand to spend more time trying to identify the source of disagreement and less time bickering over prior beliefs.

## Monetarism, Debt, and Observational Equivalence

I have heard a number of people say over the years that one of the best things about reading Adam Smith and Henry Thornton and other classical economists is that they argued their points fairly. In particular, Smith and Thornton argued in favor of their own views and against opposing views while taking these opposing views at face value. They did not attack straw men. They did not caricature their intellectual adversaries (in fact, Thornton and Smith were intellectual adversaries to some extent in their views on the role of bank notes, bills of exchange, and the operation of the monetary system).

This characteristic is, at times, missing from contemporary discourse. This doesn’t mean that modern disagreements are fraught with malice. However, sometimes ideas are not given the proper understanding sufficient for critique. Franco Modigliani, for example, once joked that what we would now call real business cycle theory blamed recessions on mass outbreaks of laziness. Similarly, when Casey Mulligan published his most recent book on the recession in which he argued that expansions of the social safety net can explain a significant fraction of the increase in unemployment, others shrugged this off by saying that this was akin to saying that soup lines caused the Great Depression.

My point is not to defend Casey Mulligan or the real business cycle theorists. It is perfectly reasonable to view real business cycle theory as unconvincing without referencing mass outbreaks of laziness. Rather my point is that more care needs to be taken to understand opposing theories and views of business cycles, growth, etc. so that one can adequately articulate criticisms and rebuttals to such views.

The fact that there is little understanding of (or perhaps just little credit given to) opposing viewpoints is never more apparent than when predictions of two different theories are observationally equivalent. To give an example, consider two explanations of the cause of the most recent recession. Please note that these are not the only two explanations and that the explanations that I give are sufficiently broad to encapsulate a number of more nuanced views.

The first explanation of the recession is what I will refer to as the Debt Theory. According to this view, the expansion that preceded the recession was fueled by an unsustainable accumulation of debt. There are many varieties of this theory that emphasize different factors that caused the run-up of debt, such as monetary policy, policies that subsidize housing, etc. Regardless of the reason that “too much” debt was accumulated, the debt eventually reached a point (most often argued as the beginning of the collapse in housing prices) that was unsustainable and hence the beginning of a recession. The recession is largely the result of de-leveraging.

The second explanation is what I will refer to as the Money Theory. According to this view, it is a deviation between the supply and demand of money (broadly defined) that ultimately results in reduced spending and, as a result, a lower level of real economic activity. As a result, when the large haircuts became apparent in the market for mortgage-backed securities, this reduced the supply of transaction assets thereby causing a deviation between the supply and demand for money. The Federal Reserve, in its failure to provide a sufficient quantity of transactions assets, thereby allowed this deviation to persist and resulted in decline in nominal, and ultimately, real spending.

As these brief descriptions imply, there doesn’t appear to be much overlap between the two views. However, they actually produce a number of observationally equivalent implications. For example, advocates of the Money Theory point to the negative rates of money growth in broad measures of the money supply as evidence that the Federal Reserve failed to provide adequate liquidity. Nonetheless, this observation is consistent with the Debt Theory. According to this view, de-leveraging reduces the demand for credit and therefore reduces the need of financial intermediaries to create new debt instruments that are used as transaction assets. Thus, we would expect a decline in money growth in both cases.

On the other hand, advocates of the Debt Theory point out that there is a strong relationship between counties that had higher levels of debt prior to the recession and the reductions in consumption during the recession. Nonetheless, this observation is also consistent with the Money Theory. Most advocates of the Money Theory are intellectual descendants of Milton Friedman. In Friedman’s theory of money demand, money is considered similar to a durable good in that individuals hold a stock of money to get the flow of services that come from holding money. Thus, contra the transactions view of money demand, individuals do not draw down money balances during a recession. Instead individuals make adjustments to different parts of their portfolio, most notably consumer debt. In other words, we would observe de-leveraging under both frameworks.

To distinguish between the two views it is not sufficient to point to characteristics that they have in common (although those observations are still important). It is also necessary to find areas in which the theories differ so that one is able to develop an empirical approach to assess each framework’s validity.

The examples given above are obviously simplifications, but this is what makes being an economist difficult. It is not enough to use inductive reasoning to support one’s theory. One must be able to differentiate between other theories that would produce observationally equivalent results. Admittedly, this is a problem that exists to a greater extent in the blogosphere than it does in academic journals. The reason is obvious. If one submits a paper to an academic journal, a good reviewer is able to spot the ambiguities between testing the predictions of a particular theory and contrasting the predictions of theories with observationally equivalent predictions. In the blogosphere, the “reviewers” are commenters and colleagues. However, the differences don’t often get resolved. Perhaps this is because there is no gatekeeper that prevents the blog post from being published. (Ironically, the lack of a gatekeeper is perhaps the best quality of the blogosphere because it allows discourse to take place in public view.) Nonetheless, given the degree to which blog posts and debates in the blogosphere ultimately spill over into the popular financial press and public debate, it is important to be careful and considerate regarding opposing views.

[Note: For an example of someone who tries to disentangle the issues surrounding the Debt View and the Money View, see Robert Hetzel's The Great Recession: Market Failure or Policy Failure?]

## Are Capital Requirements Meaningless?

Yes, essentially.

The push for more strict capital requirements has become very popular among economists and policy pundits. To understand the calls for stricter capital requirements, consider a basic textbook analysis of a consolidated bank balance sheet. On the asset side, banks have things like loans, securities, reserves, etc. On the liability side a traditional commercial bank has deposits and something called equity capital. Given our example, equity capital is defined as the difference between the bank assets and deposits (note that banks don’t actually “hold” capital).

So why do we care about capital?

Suppose that assets were exactly equal to deposits. In this case the equity capital of bank would be non-existent. As a result, any loss on the asset side of the balance sheet of the bank would leave the bank with insufficient assets to cover outstanding liabilities. The bank would be insolvent.

Now suppose instead that bank’s assets exceed their deposits and the bank experiences the same loss. If this loss is less than the bank’s equity capital then the bank remains solvent and there is no loss to depositors.

The call for capital requirements is driven by examples like those above coupled with the institutional environment in which banks operate. For example, banks have limited liability. This means that shareholders are subjected only to losses to their initial investment in the event that a bank becomes insolvent. Put differently, bank shareholders are not assessed for the losses to depositors. Since the private cost of insolvency is less than the public cost, shareholders have an incentive to favor riskier assets than they would otherwise. Conceivably, this notion is well-understood since the the shift to limited liability for banks in the early 1930s was coupled with the creation of government deposit insurance. However, while deposit insurance insulates deposits from losses due to insolvency, it also has the effect of encouraging banks to take on more risk since depositors have little incentive to monitor the bank balance sheet.

Within this environment capital requirements are thought to reduce the risk of insolvency. By requiring that banks have equity capital greater than or equal to some percentage of their assets, this should make banks less likely to become insolvent. This is because, all else equal, a greater amount of capital means that a bank can withstand larger losses on the asset side of their balance sheet without becoming insolvent.

There is nothing logically wrong with the call for greater capital requirements. In fact, calls for greater capital requirements represent a seemingly simple and intuitive solution to the risk of financial instability. So why then does the title of this post ask if capital requirements are meaningless? The answer is that calls for higher capital requirements ignore some of the realities of how banks actually operate.

The first problem with capital requirements is that they impose a substantial information cost on bank regulators. How should value be reported? Should assets be marked-to-market or considered at book value? Some assets are opaque. More importantly, this practice shifts the responsibility of risk management from the bank to the regulator. This makes the actual regulation of banks difficult.

Second, and more importantly, is that capital requirements provide banks with an incentive to circumvent the intentions of the regulation while appearing compliant. In particular, a great deal of banking over the last couple of decades has been pushed off the bank balance sheet. Capital requirements provide banks with an incentive to move more assets off their balance sheet. Similarly, banks can always start doing bank-like things without actually being considered a bank thereby avoiding capital requirements altogether. In addition, this provides an opportunity for non-banks to enter the market to provide bank-like services. In other words, the effect of capital requirements is to make the official banking sector smaller without necessarily changing what we would consider banking activity. Put simply, capital requirements become less meaningful when there are substitutes for bank loans.

Advocates of capital requirements certainly have arguments against these criticisms. They would be perhaps correct to conclude that the costs and imperfections associated with the actual regulation would be work it if capital requirements brought greater stability to the system. However, the second point that I made above would seemingly render this point moot.

For example, advocates of higher capital requirements seem to think that redefining what we call a bank and adopt better general practices for accounting reporting would eliminate the second and most important problem that I highlighted above. I remain doubtful that such actions would have any meaningful impact. First, redefining what a bank is to ensure that banks and non-banks in the current sense remain on equal footing regarding capital requirements is at best a static solution. Over time, firms that would like to be non-banks will figure out how to avoid being considered a bank. Changing the definition of a bank only gives them the incentive to change the definition of their firm. In addition, I remain unconvinced that banks will be unable to circumvent changes to general accounting practices. Banks are already quite adept at circumventing accounting practices and hiding loans off of their balance sheets.

Those who advocate capital requirements are likely to find this criticism wanting. If so, I am happy to have that debate. In addition, I would like to point out that my skepticism about capital requirements should not be seen as advocacy of the status quo. In reality, I favor a different change to the banking system that would provide banks with better incentives. I have written about this alternative here and I will be writing another post on this topic soon.

## 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.


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• •

### 具有混合时变时滞的2-D离散时间非线性切换系统的H∞控制问题研究

1. 燕山大学理学院, 秦皇岛 066004
• 收稿日期:2022-04-11 修回日期:2022-05-16 发布日期:2022-11-04
• 通讯作者: 彭丹,Email:dpeng1219@163.com.
• 基金资助:
河北省省级科技计划资助(F2020203037),国家自然科学基金杰青项目(61825304),河北省自然科学基金创新研究群体项目(F2020203013)资助课题.

LIU Mengjie, PENG Dan. Study of H Control Problem for 2-D Discrete-Time Nonlinear Switched Systems with Mixed Time-Varying Delays[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(10): 2774-2793.

### Study of H∞ Control Problem for 2-D Discrete-Time Nonlinear Switched Systems with Mixed Time-Varying Delays

LIU Mengjie, PENG Dan

1. School of Science, Yanshan University, Qinhuangdao 066004
• Received:2022-04-11 Revised:2022-05-16 Published:2022-11-04

In this paper,we study the H control problem of two-dimensional (2-D) discrete-time nonlinear switched systems with mixed time-varying delays based on the Roesser model.Firstly,an improved Lyapunov function is implemented by introducing some sum of state vectors containing both single and double forms,so as to strive to obtain additional information related to the time delays.Secondly,a 2-D admissible edge-dependent average dwell time (2AED-ADT) method for 2-D discrete-time nonlinear switched systems is introduced.Using this method,sufficient conditions are given by the linear matrix inequalities (LMIs) to guarantee the exponential stability and H performance of the 2-D discrete-time nonlinear switched systems.Further,based on the above stability results,a dynamic output feedback (DOF) controller is designed to stabilize the nonlinear closed-loop switched systems with H performance index γ.Finally,the superiority and validity of the obtained results are verified by two numerical calculation examples.

MR(2010)主题分类:

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 [1] Kaczorek T. Two-Dimensional Linear Systems. Berlin:Springer-Verlag, 1985.[2] 陈东彦,于浍.状态饱和2-D离散系统的稳定性分析.系统科学与数学, 2014, 34(2):171-178.(Chen D Y, Yu H. Stability analysis of 2-D discrete systems with state saturation. Journal of Systems Science and Mathematical Sciences, 2014, 34(2):171-178.)[3] Roesser R. A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, 1975, 20(1):1-10.[4] Fornasini E, Marchesini G. Doubly-indexed dynamical systems:State-space models and structural properties. Theory of Computing Systems, 1978, 12(1):59-72.[5] Yang R N, Zheng W X, Yu Y. Event-triggered sliding mode control of discrete-time twodimensional systems in Roesser model. Automatica, 2020, 114(5):108813.[6] Ahn C K, Wu L, Shi P. Stochastic stability analysis for 2-D Roesser systems with multiplicative noise. Automatica, 2016, 69:356-363.[7] Dami L, Benhayoun M, Benzaouia A. Admissibility and stabilization of singular continuous 2D systems described by Roesser model. Multidimensional Systems and Signal Processing, 2020, 31(2):673-687.[8] Peng D, Nie H M. Stabilisation for 2-D discrete-time switched nonlinear systems with mixed time-varying delays under all modes unstable. International Journal of Systems Science, 2022, 53(4):757-777.[9] Peng D, Xu H S. A novel approach to delay-variation-dependent stability analysis of 2-D discretetime systems with mixed delays. IEEE Access, 2019, 7:99817-99829.[10] Benzaouia A, Hmamed A, Tadeo F. Stability conditions for discrete 2D switching systems, based on a multiple Lyapunov function. 2009 European Control Conference (ECC), 2009, 2706-2710.[11] Benzaouia A, Hmamed A, Tadeo F, et al. Stabilisation of discrete 2D time switching systems by state feedback control. Journal of the Franklin Institute, 2011, 42(3):479-487.[12] Shi S, Ma Y J R, Ren S Q. Asynchronous filtering for 2-D switched systems with missing measurements. Multidimensional Systems and Signal Processing, 2018, 30(2):543-560.[13] Duan Z X, Xiang Z R. State feedback H∞ control for discrete 2D switched systems. Journal of the Franklin Institute, 2013, 350(6):1513-1530.[14] Duan Z X, Ghous I, Xia Y Q, et al. H∞ control problem of discrete 2-D switched mixed delayed systems using the improved Lyapunov-Krasovskii functional. International Journal of Control Automation and Systems, 2020, 18(8):2075-2087.[15] Duan Z X, Xiang Z R. Output feedback H∞ stabilization of 2D discrete switched systems in FM LSS model. Circuits, Systems, and Signal Processing, 2014, 33(4):1095-1117.[16] Huang S P, Xiang Z R. Delay-dependent stability for discrete 2D switched systems with state delays in the Roesser model. Circuits Systems&Signal Processing, 2013, 32(6):2821-2837.[17] Fei Z Y, Shi S, Zhao C, et al. Asynchronous control for 2-D switched systems with mode-dependent average dwell time. Automatica, 2017, 79:198-206.[18] Gao L J, Cao Z B, Zhang M, et al. Input-to-state stability for hybrid delayed systems with admissible edge-dependent switching signals. Journal of the Franklin Institute, 2020, 357:8823-8850.[19] Liu H Y, Gao L J, Wang Z Y, et al. Asynchronous l2-l∞ filtering of discrete-time impulsive switched systems with admissible edge-dependent average dwell time switching signal. International Journal of Systems Science, 2021, 52(8):1564-1585.[20] Hien L V, Vu L H, Trinh H. Stability of two-dimensional descriptor systems with generalized directional delays. Systems&Control Letters, 2018, 112:42-50.[21] Hien L V, Trinh H. Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model. Nonlinear Analysis:Hybrid Systems, 2017, 24:45-57.[22] Wang J, Liang J, Dobaie A M. Dynamic output-feedback control for positive Roesser system under the switched and TS fuzzy rules. Information Sciences, 2018, 422:1-20.[23] Xu H, Zou Y, Lu J, et al. Robust H∞ control for a class of uncertain nonlinear two-dimensional systems with state delays. Journal of the Franklin Institute, 2005, 342(7):877-891.[24] Hua C C, Liu G P, Guan X P. Switching regulation based stabilisation of discrete-time 2D switched systems with stable and unstable modes. IET Control Theory&Applications, 2018, 12(7):953-960.[25] Duan Z X, Xiang Z R, Karimi H R. Delay-dependent H∞ control for 2-D switched delay systems in the second FM model. Journal of the Franklin Institute, 2013, 350(7):1697-1718.[26] Hien L V, Trinh H. Stability of two-dimensional Roesser systems with time-varying delays via novel 2D finite-sum inequalities. IET Control Theory&Applications, 2016, 10(14):1665-1674.[27] Peng D, Hua C C. Improved approach to delay-dependent stability and stabilisation of twodimensional discrete-time systems with interval time-varying delays. IET Control Theory&Applications, 2015, 9(12):1839-1845.[28] Fridman E, Shaked U. An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, 2002, 47(11):1931-1937.
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Volume 2
, Issue 2

1965

Doubtless the above heading will be to most of our readers as cryptic as the meaning as some of the more outlandish examples of the so-called "Strine language" which have recently appeared in the newspapers.

In the October 1964 issue of Parabola, the article on the Four Color Problem called your attention to the existence of numerous unsolved mathematical problems which can be stated in quite simple non-technical terms.

J31 Prove that, given five consecutive integers, it is always possible to find one which is relatively prime to all the rest.

J21 (i) In a box there is a number of balls $c$ of different colours. What is the smallest number of balls for which we can say that, however the colours are distributed, there is at least one set of $s$ balls of the same colour? Justify your statement.

$X$ is a point inside a polygon and $AB$ is one of the sides of the polygon. Show that the perimeter of the triangle $ABX$ is shorter than the perimeter of the polygon.


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The Origin of the Ket in Quantum Mechanics

Origin of the Ket

The origin of the ket in quantum mechanics arises from the need to describe the concept of superposition as well as a system that 1) changes with respect to time and 2) has so-called states. The ket can symbolize such a state. The concept of superposition is closely related to the mathematical operation of addition.

In the words of Paul Dirac, “the superposition process is a kind of additive process and implies that states can in some way be added to give new states. The states must therefore be connected with mathematical quantities of a kind which can be added together to give other quantities of the same kind. The most obvious of such quantities are vectors” [1]. Dirac then states “it is desirable to have a special name for describing the vectors which are connected with the states of a system in quantum mechanics, whether they are in a space of a finite or an infinite number of dimensions. We shall call them ket vectors, or simply kets, and denote a general one of them by a special symbol $\ket{}$. If we want to specify a particular one of them by a label, $A$ say, we insert it in the middle, thus $\ket{A}$.

Superposition

Apparently, superposition is a “kind of additive process.” What I am unsure about is how superposition implies “that states can…be added to give new states.” Where does this implication come from? There is another section in Dirac’s book [1] that explains superposition, so it is logical to look there. It’s a long story but I’ll attempt to make it short. Dirac elaborates on the “polarization of photons” and the “interference of photons.” He claims that these are examples of superposition. Note that this claim uses the assumption that a photon exists. But this post is not about proving the existence of photons…

A sentence that summarizes the essence of superposition is “the general principle of superposition of quantum mechanics applies to the states…of any one dynamical system.” It is interesting that Dirac tends to focus on one system in contrast to many systems. Anyway, superposition “requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states.” In my opinion, this can be interpreted as a reductionist viewpoint, meaning that a whole system can be decomposed into its parts. But this is just one interpretation. To be clear, “the original state must be regarded as the result of a kind of superposition of the two or more new states…”

Furthermore, “any state may be considered as the result of a superposition of two or more other states…” Viewed in a slightly different way, “any two or more states may be superposed to give a new state.” It appears that the word ‘superposed’ is almost synonymous with the word ‘added.’ I use the words ‘almost’ and ‘appears’ because superposition does have some connotations that the mathematical operation of addition does not. Also, if any state can be formed by adding other states, it follows that quantum mechanics should be able to describe any state for any system that can be described by quantum mechanics. That is a strong implication; it implies that quantum mechanics does not break down. For quantum mechanics to be correct, it should break down when it is expected to, if the theory does break down at all.

It is possible that this idea of superposition originated in part from Fourier analysis, since Dirac states that superposition is “like the procedure of resolving a wave into Fourier components.” Anyway, I’ll run with the idea of superposition, because why not. After all, electrostatics seems like a good theory, and it relies on some type of superposition.

Some Properties of Kets

I am just going to follow Dirac’s book for these properties. The first property is that

1. “Ket vectors may be multiplied by complex numbers and may be added together to give other ket vectors, e.g. from two ket vectors $\ket{A}$ and $\ket{B}$ we can form $c_1 \ket{A} + c_2 \ket{B} = \ket{R}$, say, where $c_1$ and $c_2$ are any two complex numbers.

Since any ket can be written in terms of two or more kets, this process can continue without end, leading to a never-ending sum of kets,

$\Sigma_N \ket{N} = \ket{R}$

in which $\ket{R}$ is another ket and $N$ is a label to distinguish one ket from another. In math, sometimes one can write  $\int dN$ (or something similar) instead of $\sum_N$, in order to move from the discrete to the continuous, so in certain situations it might be reasonable to write

$\int dN \ket{N} = \ket{R}$

but from a Calculus standpoint, this means there exists an area under several kets or vectors, which doesn’t make much sense to me visually. Nevertheless, maybe it is still valid to write this, in some abstract sense.

In short, a ket is a symbol for a vector. By convention, this vector is a column vector. On the other hand, a row vector is symbolized by a bra $\bra{}$. The labelling convention for a bra is analogous to the case of a ket, so it is allowable to label a bra with $B$ by writing $\bra{B}$. Now it is possible to multiply a bra with a ket, to obtain a scalar product, assuming the dimensions of the vectors match in the middle; this multiplication is written as $\langle B| A \rangle$,  a 1×1 matrix or a scalar. On the other hand, $\ket{A}\bra{B}$ is a matrix depending on the dimensions of the ket and the bra. Notice that the inner dimension of $\ket{A}\bra{B}$ is 1×1 regardless of the other two dimensions, so it is possible to produce a non-square matrix with $\ket{A}\bra{B}$.

References

[1] P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford Science Publications. 1958.

Christina Daniel

I am a research assistant in theoretical physics. This website, derive-it.com, serves to organize my ongoing learning and research as well as to provide a resource to other learners around the world.