## 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.

## On SNAP Eligibility and Spending

William Galston has an op-ed in the Wall Street Journal that begins as follows:

We are entering a divisive debate on the Supplemental Nutrition Assistance Program (SNAP), popularly known as food stamps. Unless facts drive the debate, it will be destructive as well.

I certainly agree with this statement. Unfortunately, I found the op-ed misleading and vague (a vague op-ed can be somewhat forgiven since word counts are limited).

The basic premise of Galston’s op-ed is that critics of the increased spending on food stamps are misguided in their criticisms. For example, he explains:

The large increase in the program’s cost over the past decade mostly reflects worsening economic conditions rather than looser eligibility standards, increased benefits, or more waste, fraud and abuse.

[...]

The food-stamp program’s costs have soared since 2000, and especially since 2007. Here’s why.

First, there are many more poor people than there were at the end of the Clinton administration. Since 2000, the number of individuals in poverty has risen to 46.5 million from 31.6 million—to 15% of the total population from 11.3%. During the same period, the number of households with annual incomes under \$25,000 rose to 30.2 million (24.7% of total households) from 21.9 million (21.2%).

Critics complain that beneficiaries and costs have continued to rise, even though the Great Recession officially ended in 2009. They’re right, but the number of poor people and low-income households has continued to rise as well.

Thus, according to Galston, we can explain much of the increase in food stamp spending on the rise of poverty over the last 13 years (and especially the last 6 years). If Galston is correct, then we could examine the ratio of households who are receiving SNAP benefits to the number of people below the poverty line. Supposing that he is correct, we would expect that this ratio would be constant (or at least roughly so). In other words, as the number of people below the poverty line increased, the number of households receiving SNAP benefits would increase in direct proportion.

Such a comparison, however, casts doubt on Galston’s claim. Casey Mulligan, in his book The Redistribution Recession, has taken great effort to actually calculate such ratios. What Mulligan found is that from 2007 to 2010, the number of families below 125% of the federal poverty level increased by 16%. That is indeed a large increase. However, the number of households receiving SNAP benefits increased by 58%. This means that the SNAP recipiency ratio, or the ratio of households receiving SNAP to that below 125% of the poverty line (a higher threshold that Galston himself uses), rose by 37%.

So what can explain the fact that recipients are rising so much faster than poverty? One possible explanation are eligibility requirements. Since 2008, there have been several changes to eligibility for food stamps. For example, the Farm Bill passed in 2008 increased the maximum benefit that beneficiaries could receive, it excluded some income from the formula used to determine eligibility, and it weakened the evaluation of assets of potential enrollees. In addition, the American Reinvestment and Recovery Act also loosed eligibility requirements by once again increasing the maximum benefit that one could receive, gave states the ability to loosen the work requirement, and further loosened income requirements.

Galston, however, downplays most of these changes and argues that macroeconomic trends explain the vast majority of the rise of SNAP spending. However, the use of this type of explanation is problematic because it is taking the actual increase in recipients and then explaining the increase in spending ex post. To understand why this is misleading, consider the following example. Suppose that there is an individual who lost his job in 2009. Prior to 2007, he would not have been eligible for SNAP whereas after the changes he is now eligible. Thus, after 2007, this increases the number of recipients of SNAP. Galston might claim that this change is the result of macroeconomic trends because this person would not have enrolled in SNAP had he not lost his job. Others might say that this change is due to eligibility requirements becaus if the worker had lost his job two years prior, he would not have been eligible. While I certainly understand Galston’s perspective on this, the relevant comparison is to the counterfactual. In other words we can’t explain the rise in SNAP recipients ex post, we need to consider what actually happened to what would have happened in the absence of a policy change.

So what do the counterfactuals say?

Again, Casey Mulligan has constructed these counterfactuals. What he finds is that between 2007 and 2010, the increase in per capita SNAP spending was 100%, adjusted for inflation. He then constructs two counterfactuals. The first counterfactual takes macroeconomic trends as given and computes the increase in per capita SNAP spending under 2007 eligibility rules. The second counterfactual does the same thing assuming that in addition to maintaining 2007 eligibility rules, the government had maintained constant real benefit rules (i.e. would not have increased the after-inflation maximum benefit).

The first counterfactual suggests that from 2007 to 2010 per capita SNAP spending would have only increased by 60%, adjusted for inflation. The second counterfactual suggests that per capita SNAP spending would have increased only 24%, adjusted for inflation. Had no policy changes been enacted in 2008 and 2009, per capita spending on SNAP would have been 62% of what it actually was in 2010. Put differently, 48% of the per capita 2010 spending is attributable to changes in eligibility. Thus, contrary to the claims of Galston, a very large fraction of the increase in SNAP spending is explained by changes in eligibility.

An entirely separate question is whether or not this increased spending is worth it. Answering that question is certainly beyond the scope of this post. However, it is important to be mindful that such analysis must consider both the costs and the benefits of the expansion. The benefits are obvious. Households receive assistance in purchasing food and feeding their families. The costs, however, are more complex. A significant fraction of the increase in spending can be explained by changes in eligibility. Thus, we need to consider the counterfactual. One big issue is to consider how much of the increased benefits are going to those who would not have qualified under the asset tests. Another issue is to consider is the effect of changes in eligibility on the labor supply of those at or near the new eligibility requirements, especially given the work requirement waiver. And there is the obvious monetary cost to the taxpayer. Too often those on each side of the debate focus on only the benefits or only the costs.

Whether the policy changes are worth it depends on a careful analysis of these questions. I will remain agnostic with regards to that type of analysis. However, to argue that those concerned about the expansion spending due to changes in eligibility are misguided and driven by “anti-government ideology”, as Galston argues, is an unfair criticism to those who have carefully looked at the data.

## 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.