## In Memory of Bob Rossana

Robert “Bob” Rossana died back in February of this year. I was inadvertently left off of the university announcement email and did not hear about it until much later. Bob was my dissertation advisor. He meant a lot to me and so I want to share some thoughts.

In graduate school, there were four macroeconomics courses. Two core classes and two field classes. Bob was instrumental in getting this four-course system started, as well as a “macro lunch,” in which any graduate student interested in macro was required to attend and present.

Many of my fellow classmates found him intimidating. I cannot say for certain why other students felt this way, but I suspect that it was Bob’s high standards. He expected a lot of us, but he was always fair. Above all, he wanted us to push the boundaries of our understanding.

Those of us who chose to write in macro got to know a different side of Bob. Visits to his office involved not only serious discussions about research and economics, more generally, but current events and funny stories. He had a great sense of humor.

He loved to talk about current events and filter them through the lens of economics. During these conversations, he frequently emphasized to me that the ability to think like an economist was a powerful tool and that one should never let economics become inconvenient. What he meant was that one should never use his or her position as an economist to advocate a particular ideological position. Any sort of policy advocacy should be based on a rigorous application of economics.

In these meetings, I also learned that he had the same high standards for what he deemed good research as he did for his students. We had many conversations about what constituted good, serious scholarship. As I have gained my own experience in the field, I have also developed strong opinions about what constitutes good research. I recognize now that a lot of the opinions I have today were shaped by these discussions.

He did not believe in a distinction between theory and applied work. Applied work should be a direct test of a theory. This is reflected in his underappreciated paper that demonstrates how to test hypotheses for a system of equations. He also frequently emphasized that one must be careful about trying to explain the data-generating process when the data are often sensitive to the nature of measurement. This was evident in his own work as well. For example, Long and Plosser had found that most macroeconomic time series followed a similar time series process. This evidence was used early on as support for real business cycle theory. Bob, along with his friend and co-author John Seater, showed that this Long and Plosser result was predicated on the use of annual data. The last paper he was working on (to my knowledge) was a unique test of money neutrality. I don’t see a recent copy online, but hopefully his co-author will see it through to publication. He was a serious scholar. A relatively recent version of his CV is here.

Bob was very business-like as a professor and an advisor. Things were either done well or not. I remember when I presented him with the first complete draft of my dissertation. I naively assumed that a completed draft meant that I was basically done. He called me to his office and listed entire sections of the dissertation that needed to be re-written – most sections, if I’m being honest. I was a bit dejected, but upon reflection, he was more than kind. When I got my first revise and resubmit during my final year of grad school, he congratulated me on my progress, but told me I should’ve sent it to a better journal. (He was probably correct about that as well.)

I had so much respect for him. Even after I had defended my dissertation, I would often address him as “Dr. Rossana.” One day, he sent me a quick email that said “Josh, you need to call me Bob. We are colleagues now.”

It was his business-like nature that made praise all the more meaningful. The last time I saw him in person was when I came back to give a seminar. I was waiting on one last signature for tenure. After my seminar, we sat in his office for a couple of hours. We talked about my job and my all-but-certain tenure case. He asked me about my wife and kids and how they liked Mississippi. When I mentioned the long road trips to visit family, he shared memories of car trips with his sons. Before I left, he said, “I want you to know that we’re proud of you. You’ve done well.” That meant a lot to me. More than he probably realized.

He died in February. Cancer. I never got to say goodbye. Or one last thank you. So I wrote this as that final goodbye and that final thank you. Hopefully it reaches him.

## The Coronavirus and Lessons for Preparedness

My new Mercatus policy brief is up. It is entitled, “The Coronavirus and Lessons for Preparedness.” You can find it here.

## Macroeconomic Policy and the Coronavirus

(The post is also going to be cross-posted at Lars Christensen’s Market Monetarist blog.)

As the coronavirus spreads across the world, there is growing concern about the economic implications of the virus and what, if any, policy action is required of governments. Some have proposed traditional stimulus measures to counteract the virus while others have proposed more targeted measures. However, before one can offer policy solutions, one needs to diagnose the problem. In this post, I will outline what the potential economic consequences of the coronavirus are and the corresponding policy implications.

The Diagnosis

There are three main consequences of the coronavirus that we should be concerned about. The first is part of a flight to safety. In the face of uncertainty, people often sell risky assets, so that they can buy safe assets. People prefer to hold physical cash or put their money in an insured bank account rather than holding risky assets. What this does is create a greater demand for money. At the level of the individual, an increased demand for money is most easily accomplished by a reduction in spending. However, if money demand is rising in the aggregate (assuming there is no policy response), this reduction in spending is similar to a traffic jam. Sometimes, all it takes is for a few people to slam on the brakes and suddenly traffic is at a standstill. The widespread reduction in spending is analogous to the traffic jam in the sense that the economic activity can grind to a halt because of a significant reduction in spending.

The two other consequences of the coronavirus are related to planning and containment. Right now, in places like the United States, most people are continuing to work as usual. However, if the experience of other countries is any indication, this might not continue. Those who are infected will be expected to miss work, even if the virus is not a threat to their health. In addition, there might be forced quarantines on particular areas.

In the event of self-quarantine or forced public quarantine, consumers will need to have food and medicine since they cannot leave their home. Furthermore, firms will need to be able to pay for non-operating costs during a quarantine. This includes making debt payments. Firms who are currently cash-strapped and that are relying on cash flow to make debt payments might find themselves unable to make debt payments because of quarantines. This could result in default and bankruptcies, which exacerbate the problem through the implications on financial markets.

The Prescription

So what is to be done?

The first thing to do is for the Federal Reserve to commit to do whatever it takes to maintain stability in aggregate nominal spending. As David Beckworth and I have shown, nominal spending crashes are rare, but the common cause is monetary and such events are associated with severe recessions. The Federal Reserve should commit itself to lowering interest rates and engaging in quantitative easing to the extent necessary to prevent a decline in nominal spending. I have written about why this is important in theory and in practice. In addition, the Fed should be tracking properly measured monetary aggregates, which are released more frequently than data on nominal spending. I have shown in my own research that these monetary aggregates are useful in predicting inflation and nominal spending. The Fed is already failing at this. Inflation expectations have been plummeting and their response has been inadequate.

The Trump administration has suggested a temporary payroll tax cut. Some have criticized this idea, but these critics are missing the point. A payroll tax cut is something that would work if it is done immediately. The cut in payroll taxes would increase profits to firms and increase take-home pay of workers. This would allow firms to start retaining these additional earnings now in order to make debt payments and pay other non-operating costs in the event that they have to temporarily shut down. This would also allow workers to use this additional income to make purchases to stockpile food and medicine in the event of self-quarantine or a forced geographic quarantine. However, this needs to be done now and will not be as effective once things have started shutting down.

Another policy prescription is to do lump sum transfers of around \$1000 per taxpayer. Normally, these are somewhat controversial as a consequence of Ricardian equivalence. Under Ricardian equivalence, a transfer to taxpayers that is paid for through debt creation just results in a net increase in savings which offsets the debt creation and therefore has no effect on economic activity. While there is some debate about the degree to which Ricardian equivalence holds, this debate is largely irrelevant here. These checks would allow people decide how to allocate the money for its best use in preparation for the possible consequences of the virus. For some, this might mean using it as extra savings. For others, this might be a way to purchase food and medicine in anticipation of the consequences. The objective here is not to boost real GDP per se, but rather to plug holes before they have a chance to leak.

Finally, the government should implement temporary and mandatory paid sick leave financed by the government. This will give people the peace of mind that they will not lose their income in the event that they have the virus or otherwise are unable to work because of the virus. The policy also has the added benefit of preventing the virus from spreading since people are not forced making the difficult choice between working sick or losing income.

## Podcast on Maritime Policy

I recently recorded an episode of The Economics Detective podcast with Garrett Petersen. The subject of the podcast is my paper, “U.S. Maritime Policy and Economic Efficiency.” A link to the paper is here. A link to a blog post I recently wrote about the paper is here.

A couple of links to things I have been working on…

• My paper entitled, “The Riksbank, Emergency Finance, Policy Experimentation, and Sweden’s Reversal of Fortune” is now forthcoming at the Journal of Economic Behavior and Organization. The paper includes a lot of background on Swedish governance and fiscal capacity, as well as information about the early Swedish monetary system and the Riksbank. I argue that the constraints on the Riksbank left Sweden unable to use its central bank in the same way that Britain did during times of war. I also argue that policy experimentation at the Riksbank had a negative effect on economic activity. I really enjoyed working on this paper, so I hope that others find some value in it.
• I have also written a review of Saez and Zucman’s new book, The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay.

## On Maritime Policy

Some time ago, I was listening to NPR and the hosts were discussing the Jones Act. For those who are unaware, the Jones Act requires that any shipping done from one U.S. port to another U.S. port must be carried by a U.S.-flagged ship with a U.S. crew that was built in the U.S. The hosts claimed that they could not find an economist who could explain why this sort of law existed. The policy seemed like a run-of-the-mill protectionist giveaway. My reaction was much different.

Here we have this law that is nearly 100 years old and no economist can explain why it exists? The only plausible explanation is that this 100-year-old law is a protectionist giveaway? This seems unlikely. One would have to believe that democracy is incredibly inefficient. This is especially true when one hears the estimates of the costs of the Jones Act. Why are voters in the U.S. just leaving billions of dollars on the sidewalk?

There is an entire literature on elections and accountability that seems to suggest that elections discipline the behavior of politicians to follow the will of the voters. However, as part of this literature, this can lead politicians to give greater weight to special interest groups. These groups tend to have more information than the average voter and provide more feedback to politicians. As a result, politicians might be too responsive to special interest groups in trying to achieve re-election. Of course, what seems like an obvious question (to me) is how special interests could convince politicians to support this legislation for so long without some other special interest group emerging to get the inefficient policy repealed. In fact, I have made the argument (along with my co-authors, Brian Albrecht and Alex Salter) that, in a world with endogenous entry of special interest groups, long-lasting policies must either have modest costs or actually efficient.

Given my prior beliefs on the endogenous entry of special interest groups, I figured that something like the Jones Act must either be substantially less costly than conventionally claimed. I decided to study this more closely and what I discovered is that there is a clear and consistent pattern of subsidizing the shipping industry throughout U.S. history. In addition, there is also an argument that such subsidies are actually efficient. So, I decided to write a paper about it. For those who don’t want to read the whole paper, what follows is a synopsis of my argument.

Consider the following basic idea. During times of peace, a navy needs less capacity than during times of war. At the same time, a navy needs to have the ability to expand rapidly when war occurs. Excess capacity during times of peace could be wasteful. A potential solution to this problem is for the navy to use private ships to assist them during times of war. Of course, there are several problems with doing so. First, diverting private ships for naval use might impair international trade. Second, foreign crews might not want to have any part in another country’s conflict. Third, if the government offers to pay ships for their services once conflict has begun or is imminent, it might be subject to a hold-up problem. Fourth, if the government simply expropriates the ships for naval use, the expectation of expropriation will cause the shipping industry to underinvest.

Given the need for a naval auxiliary, it would seem like some sort of Coaseian bargain is in order. One possibility is for the government to provide peacetime subsidies to the shipping industry in exchange for this industry’s services during times of war. This sort of thing is not without precedent.

What we have here is a possible argument for why the U.S. government offers maritime subsidies. But how would we know if this is actually the case? How do we know that I haven’t crafted a just-so argument to put a thumb in the nose of those who say there’s no justification for policies like the Jones Act? Well, I would suggest that we would observe multiple things. First, we should observe a clear and consistent pattern of maritime subsidies across U.S. history. Second, we should observe similar subsidies for other transportation services.

It turns out that both observations are true. As I detail in my paper, there is a clear and consistent pattern of subsidizing shipping and revising and updating these maritime subsidies in the run-up to military conflict or in the aftermath of conflict. Furthermore, the legislation for these subsidies, as well as speeches by FDR and Richard Nixon, explicitly outline the importance of these subsidies for the role of the merchant marine as a naval auxiliary.

There is also a history of doing this with other forms of transportation. In the decade prior to World War I, the U.S. became concerned about the number horses that would be available during times of war. Again, the U.S. could potentially purchase horses from private citizens in the event of war. However, military horses do not always have the same characteristics as the horses used on farms. Furthermore, mother nature determines the “time-to-build” for horses such that no reallocation of resources or productivity improvements allow military-ready horses to be produced any faster. As a result, the U.S. government set up remount depots and subsidized horse breeding.

Similarly, during both World War II and the Korean War, the U.S. government requisitioned private planes to provide airlift during the war. The role of aircrafts was later formalized with the creation of the Civil Reserve Air Fleet in 1952. This program allows the government to use participating member planes for airlift during times of war. In exchange, the government uses the participating airlines for airlift during times of peace and the amount of business and compensation provided is a function of the complementarity of the planes for military airlift.

In other words, what we see is a clear and consistent pattern of Coaseian bargains across time and different methods of transport as it relates to creating additional transportation capacity during times of war.

This finally brings me back to the Jones Act. As I detail in my paper, the Jones Act is part of a larger subsidization effort on the part of the U.S. government since its founding. In fact, the cabotage restrictions that are so notoriously criticized actually originated in 1817.

As I stated at the beginning of this post, much of the criticism of the Jones Act is based on its perceived astronomical costs. If one takes these costs as given, then when can understand why it is so hard to find an economist who can explain why it exists. However, what I have found is that these costs are dramatically overblown because they fail to consider the proper counterfactual. When estimating the costs of the Jones Act, most critics compare the cost of U.S.-flagged ships engaged in U.S.-based trade to the costs of foreign-flagged ships engaged in international trade. When you do so, what you find is that the cost of U.S. ships is almost 3 times as high as the foreign-flagged ships. This makes the Jones Act seem really expensive! Shouldn’t we just let the foreign-flagged ships do the work?

What this comparison ignores is that these foreign-flagged ships often fly so-called flags of convenience. These flags allow them to avoid the sort of taxes and regulations that they would face by flying the flag of a major developed country. Port-to-port U.S. shipping is considered domestic production. As a result, any ship engaged in U.S. port-to-port shipping would be subject to U.S. tax law, immigration law, labor law, and environmental laws. Perhaps most important among these compliance issues is immigration law. Under current U.S. law, foreign crews could not work on a ship conducting U.S. port-to-port shipping. This means that the labor costs of the crews on foreign-flagged ships would be substantially higher. In fact, back-of-the-envelope calculations suggest that once labor costs are taken into account, the cost of U.S. ships would only be 8% higher than foreign-flagged ships rather than nearly 3 times higher – and this is without taking into account taxes and other regulatory costs.

So, perhaps the reason that voters in the U.S. aren’t reaching down to grab billions of dollar bills on the sidewalk is that they recognize these dollars as a mirage.

None of this is to say that current maritime policy is optimal. In fact, I detail criticisms of current policy in my paper and offer suggestions for reform. Nonetheless, my paper illustrates that there is a compelling reason that countries offer maritime subsidies. This significantly changes the relevant counterfactual. So there is at least one economist who can now explain the existence of the Jones Act to the good people at NPR.

## On Drawing the Wrong Lessons from Theory: The Natural Rate of Unemployment

Economic theory is important. Theory provides discipline. Economists write down a set of assumptions and follow those assumptions to their logical conclusions. The validity of a particular theory is then tested against observed data. Modern economic theory is often mathematical, but theory comes in a variety of forms. Sometimes theory is used to develop and test specific empirical predictions. Other times, economic theory acts as a type of sophisticated thought experiment. These thought experiments generate broader empirical predictions. In fact, some of these sophisticated thought experiments contain important lessons for monetary policy.

In the late 1960s, Milton Friedman suggested that monetary policy was limited in its ability to influence the unemployment rate. Friedman argued this point by discussing the concept of a natural rate of unemployment. The idea is that there is some unemployment rate that would exist in the economy based on the fundamentals of the economy. If the unemployment rate is equal to the natural rate, the central bank cannot permanently reduce the unemployment rate. The only way in which the central bank can lower the unemployment rate is by producing higher than expected inflation. The temporary decline in real wages would lead to an increase in output and lower unemployment. Ultimately, real wages rise and employment returns to its original level.

The theory proposed by Friedman is very much in the thought experiment variety. If we accept the idea of a natural rate of unemployment pinned down by real factors, then nominal changes will not have any long-run effect on the unemployment rate. This conclusion is a version of what economists call the classical dichotomy – the idea that nominal variables only affect other nominal variables in the long-run and real factors determine resource allocation.

Subsequent economists explored this concept of the natural rate of unemployment. Finn Kydland and Ed Prescott developed a model to consider what would happen if a discretionary central bank had a lower target for the unemployment rate than the natural rate. What they found is that, in equilibrium, the unemployment rate would equal its natural rate, but the rate of inflation would be higher than if their target for the unemployment rate was equal to the natural rate.

This sort of sophisticated thought experiment contains important lessons for monetary policy. For example, what the model shows is that a preference for unemployment to be lower than its natural rate does not allow discretionary policymakers to achieve this lower rate in equilibrium. Instead, the economy will always end up at the natural rate. Discretion will only lead to higher inflation. The broad lesson is that rules-based policy is better than discretionary policy because a rule would avoid this tendency to try to manipulate the unemployment rate.

Friedman’s concept of a natural rate of unemployment was inspired by Knut Wicksell’s natural rate of interest. According to Wicksell, the natural rate of interest is pinned down by the marginal productivity of capital in the economy. When the market interest rate is below the natural rate, this leads to an expansion of money and credit and therefore inflation rises. When the market interest rate is above the natural rate, money and credit contract and inflation declines. Both of these concepts – the natural rate of unemployment and the natural rate of interest – continue to play a role in the way that policy is discussed and conducted.

While the sophisticated thought experiments that draw upon these concepts contain important lessons for policy, it is important to remember that they are thought experiments. In reality, there is no empirically observable natural rate of unemployment nor a natural rate of interest. These are theoretical concepts used to motivate the thought experiment.

Economists, however, seem to have drawn the wrong lesson from such thought experiments. Since policy is neutral when the market interest rate is equal to its natural rate or when the unemployment rate is equal to its natural rate, economists have sought to estimate these natural rates. This is a problem because these are theoretical constructs. Estimates of these natural rates cannot be compared to some observable counterpart to assess their goodness of fit. Estimation often requires the use of some sort of structural model. The extent to which the estimate is useful depends on the external validity of the model.

This is worrisome because references to the natural rate of unemployment or the natural rate of interest have become more common among policymakers. The Federal Reserve consistently refers to purported inflationary “pressures” that come from declining rates of unemployment (which, by the way, reverses the direction of causation described by Friedman).

Rather than judging the stance of monetary policy by the proximity of the unemployment rate or the interest rate to their respective natural rates, central banks should rely on an explicit target of a measureable macroeconomic variable that the central bank can directly influence with policy. With an explicit target, there is no need to estimate the natural rate of interest or natural rate of unemployment. For example, suppose the central bank targeted a five percent growth rate for nominal income in the economy. If nominal income growth is higher than five percent, this indicates that policy is too expansionary. If nominal income growth is below five percent, then monetary policy is too contractionary. When nominal income growth is approximately equal to its target, policy is neutral. There is no need to estimate any natural rate.

The Federal Reserve’s dual mandate of stable prices and maximum employment is partly to blame for the emphasis on the unemployment rate and attempts to estimate a natural rate. However, an explicit target would provide a sense of neutral monetary policy in a much more straightforward and easily observable way. In addition, achieving maximum employment need not require explicitly targeting the unemployment rate or some other measure of employment. The objective of the central bank should be to achieve nominal stability, such as the stability of the growth of nominal income. With nominal stability, relative prices will adjust to allocate resources to their most productive use. This is the main lesson of Friedman’s thought experiment. Attempts to estimate a natural rate of unemployment draw the wrong lesson. Such estimates are an unwelcome diversion pursued under the guise of being scientific.

## On Exhaustible Resources, Part 2

Yesterday’s post on exhaustive resources has drawn a lot of ire from critics. Some have argued that I didn’t address the problem of economic growth. In short, the argument is that there are two sources of economic growth. The first is that increased efficiency of resources allows us to produce more stuff with the same amount of resources. The second is that because resources are more productive we tend to use more of them. Others have argued that algebra is irrelevant to the problem.

I’d like to address both of these concerns because they are wrong. First, let’s address the algebra issue. The model I presented in my previous post is an example of using formal economic theory to make a point that is apparently not obvious to people. If society has exhaustible resources, will markets completely deplete those resources and leave us with nothing? What the model shows is that this will not happen. It doesn’t happen because as the resource is depleted, the price of the resource rises thereby encouraging people to use less of it. (Correspondingly, if resources are near the point of depletion shouldn’t energy prices be a lot higher?) So attacking me for using algebra will get applause from a certain type of audience and “algebra doesn’t solve environmental calamity” makes a really good bumper sticker, but it is not a valid critique. The model is an exercise in maintaining consistent logic.

Now to the more substantive critique. This is the critique that growth not only comes from changes in productivity but that these changes in productivity lead to greater resource use. So let’s tackle this problem head-on using a modified version of the Solow Model.

Before going through the model let’s recall the crux of the debate:

• George Monbiot claimed that perpetual growth is not possible in a world of finite resources.
• I replied that perpetual growth comes from finding more efficient ways to use resources (the ability to produce the same amount of stuff with fewer resources).

Let’s imagine that there is an aggregate production function that is given as

$Y = (AR)^{\alpha}K^{1 - \alpha}$

where $Y$ is output, $R$ is the quantity of exhaustible resources, $K$ is capital, $\alpha \in (0,1)$ is a parameter, $A$ is the productivity of energy use. So $AR$ has the interpretation of “effective units of resources.” Now let’s assume that

$dR = -cRdt$

where $c$ is the rate of resource extraction. Note here that I am assume no uncertainty. The amount of resources are known and declining with use.

Also, I will assume that

$dA = gAdt$

where $g$ is the growth rate of the productivity of energy use.

Finally, the law of motion of the capital stock is given as

$dK = (sY - \delta K)dt$

where $s \in (0,1)$ is the savings rate and $\delta \in (0,1)$ is the depreciation rate on capital.

Define $e = AR$ as effective units of resources and $k = K/e$ as capital per effective unit of resources. The corresponding law of motion for capital per effective unit of resources is given as

$dk = [sk^{1 - \alpha} - (\delta + g - c)k]dt$

From this equation, there is a stable and unique steady state equilibrium for $k$ if $\delta + g - c > 0$. A sufficient condition for this to hold is $g - c > 0$.

Now, let $y = Y/e = k^{1 - \alpha}$. Note that this implies that in the steady state, $dy = dk = 0$. Thus, output per effective unit of resources should be constant in the steady state. This implies that the growth rate of output itself satisfies

$\frac{dY}{Y} = (g - c)dt$

It follows that in the steady state equilibrium, we can experience perpetual economic growth so long as the productivity of energy use rises by more than enough to offset the rate of resource extraction. Put differently, we can experience long-run economic growth even in a world of finite resources as long as we continue to use those resources more efficiently. Recall that Monbiot argued that it is impossible. I, on the other hand, argued that this is incorrect because growth is the result of being able to produce the same amount of stuff with fewer resources. This is precisely what I meant.

Of course, we might wonder if this is actually going on in reality. So let’s go to the data. We can measure the productivity of resource use by plotting GDP relative to energy consumption. The following figure is from the World Bank.

As one can see from the graph, there has been a considerable productivity increase in the use of energy over the last few decades. This is not the whole story since this graph only measure $g$. One would need to compare this to $c$ to determine whether we are currently on a sustainable path. Nonetheless, the claim made by Monbiot was that perpetual growth is not possible in a world of finite resources. What I have shown is that this is wrong as a logical statement. Furthermore, my basic model in this post actually understates our ability for perpetual growth since I assumed that it is not possible to substitute from the exhaustible resource to either another exhaustible resource or to a renewable resource.

## On Exhaustible Resources

Yesterday, George Monbiot wrote in the Guardian that the survival of capitalism relies on persistent economic growth and persistent economic growth is impossible in the long-run because there are finite resources in the world. In response, I made the following popular, but sarcastic tweet.

The tweet was meant to be funny. The format itself is a meme. Nonetheless, it does drive home the point that the source of economic growth is finding more efficient uses of resources. With this being the internet, however, I started receiving replies telling me that I was an idiot who doesn’t understand exhaustible resources and even had one person recommend that I read up on resource economics. As it turns out, I know a little bit about resource economics — and wouldn’t you know it, resource economics actually supports my position. So I thought it was worth a blog post.

Let’s imagine that we have an exhaustible resource. Suppose that the quantity of the exhaustible resource at time $t$ is given by $R(t)$, where $R(0) = R_0 > 0$. Now let’s suppose that $R(t)$ follows a geometric Brownian motion:

$dR = -cR dt + \sigma R dz$

where $c$ is the rate of resource extraction, $\sigma$ is the standard deviation, and $dz$ is an increment of a Wiener process. The intuition of this assumption is as follows. First, zero is an absorbing barrier here. What I mean is that once $R(t) = 0$, it is permanently there. This is the exhaustible resource part. Second, on average the amount of the resource that is available is declining by the consumption of the resource. Third, there is some uncertainty about the quantity of the resource that is actually available. For example, one might observe positive or negative shocks to the supply of the resource. In other words, there are times when new supplies of the resource are discovered. There are other times in which there is less supply than had been estimated. In addition, one could also include “technology shocks” as a source of positive movement in the supply of resources in the sense that better production processes tend to economize on the use of resources, which is basically the same thing as a discovery new amounts of the resource. In short, what we have here is a reasonable representation of how the supply of an exhaustible resource is changing over time.

Now suppose that the consumption of the resource gives us some utility, $u(cR)$ where utility has the usual properties. The objective is to maximize utility over an infinite horizon (with finite resources). Given the process followed by the resources, I can write the Bellman equation for a benevolent social planner as:

$rv(R) = \max\limits_{c} u(cR) - cR v'(R) + \frac{1}{2} \sigma^2 R^2 v''(R)$

where $r$ is the rate of time preference (or the risk-free interest rate). The first-order condition is given as

$u'(cR) = v'(R)$

Intuitively, what this says is that the marginal utility of the consumption of the resource is equal to the marginal value of the resource. Or that marginal benefit equals marginal cost. In fact, this implies that $v'(R)$ is the shadow price of the resource, or the spot price (more on this below).

Now, for simplicity, let’s suppose that consumers have the following utility function:

$u(cR) = \frac{(cR)^{1-\gamma}}{1 - \gamma}$

It is straightforward to show (after A LOT of algebra) that

$c = \frac{r}{\gamma} + \frac{1}{2}\sigma^2 (1 - \gamma)$

So the rate of resource extraction is constant and a function of the parameters of the model. Or, if we assume that there is log-utility, we can simplify this to $c = r.$ Let’s make this further simplification to economize on notation.

So we can re-write our geometric Brownian motion under log utility as

$dR = -rR dt + \sigma R dz$

So now we have the evolution of resources in terms of exogenous parameters. We might be interested in the quantity of resources in existence at any particular point in time, say time $t$. Fortunately, our stochastic differential equation has a solution of the form:

$R(t) = R_0 e^{-[r + (\sigma^2/2)]t + \sigma z(t)}$

Since exponential functions are always positive and $R_0 > 0$, it must be the case that $R(t) > 0, \forall t$.

So what does this mean in English?

What it means is that given the choice about how much to consume of a finite resource over an infinite horizon, the rate of resource exhaustion is chosen to maximize utility. Given the choice of consumption over time, the total supply of the resource will decline on average over time with the rate of resource exhaustion. However, the quantity of the resource will always be positive.

How is this possible?

$u'(cR) = v'(R)$

Recall that I defined $v'(R)$ as the marginal value of the resource, or the shadow price of the resource. Note that as time goes by, $R$ is declining on average. Since $c$ is constant, when $R$ declines, the marginal utility of consumption rises because total consumption $cR$ is declining. It must therefore be the case that shadow price of the resource increases as well. But the problem I described is a planner’s problem (i.e., how a benevolent social planner would allocate the resource given the preferences for society). Nonetheless, a perfectly competitive market for the resource would replicate the planner’s problem. What this means is that as the resource becomes more scarce, the spot price of the resource will rise so that people economize on the use of the resource. Consumption of the resource declines over time such that the resource is never completely exhausted.

Thus, and somewhat ironically given Monbiot’s point, it would be a deviation from competitive markets for the resource or poorly-defined property rights that might lead us to depart from this outcome. So it’s the markets that save us, not the people who want to save us from the markets.

## A Simple Lesson About Money and Models

Imagine you are in your high school algebra class and you are presented with the following two equations:

$x + y = 20$
$2x + 10y = 100$

Two linear equations with 2 unknowns. This is a simple problem to solve.

Now suppose that your teacher gives you the following three equations:

$x + y = 20$
$2x + 10y = 100$
$x + z = 5$

Note that this is still a simple problem to solve. The first two equations are identical to the previous example. You can use those first two equations to solve for x and y. Then, knowing x, you can solve for z. The central point is that the third equation is not important for determining the value of x. The first two equations are sufficient to solve for x and y.

So why am I bringing this up?

This is precisely how the benchmark New Keynesian model deals with money. The baseline New Keynesian model does not include money. The model is complete and a solution exists. Subsequently, to examine whether money would be important in the model, a money demand function is added to this system of equations. There is a solution to the model that exists. Money is then shown to be irrelevant in the determination of the other variables. But, then again, so was z.

UPDATE: I have updated the post to read “benchmark New Keynesian model” to reflect the fact that some have attempted to integrate money into the NK model in other ways, specifically through non-separable utility. This is, in fact, where I am going to take this argument in the future. Nonetheless, for now, see the excellent comment by Jonathan Benchimol below with some links to his related research.