This is an interview I have done on behave of iMoney Magazine, and this is the English version of the published article.

This interviews also appears on Prof Paul Romer’s official website. If anyone would like to quote this interview, please attribute iMoney as the original publication.

Q: The idea of Charter Cities originated from Hong Kong and Shenzhen, am I right?

Romer: The two most interesting precedents for Charter Cities are Hong Kong and Shenzhen, so it does have some origins here. They each played important roles in fostering reform of the Chinese economy. But it is an approach that can be used in any country that wants to implement reforms, even a developed country like the United States. It turns out that this is a unique time in human history when it is possible to start many new cities because there is an enormous, unmet demand for city life.

Q: What are the essential elements of a Charter City?

Romer: In one sense, the essence of the idea is the notion of a Startup City. You have a chance to start a city anew.

Then the question is: “What can you accomplish with that? What are the things that will be required to make it successful?” I think what is unusual about a Startup City, as opposed to an existing city, is that you can propose something new without having to go through a long process of consultation and agreement amongst the people that might be affected by a change, one that would inevitably mean that a change that some people do not want is imposed on them. With a Startup City, you can propose something entirely new and let people choose whether they want to live under its rules, as embodied in its charter, the document that specifies its founding principles. People who want to try the reform can go there, and people who don’t, they don’t have to. With a startup, you can have reform without coercion.

This is part of the insight that Deng Xiaoping had in pursuing Shenzhen. As he explained later, he wanted a way to open the Chinese economy that avoided long argument and contention about what types of change to pursue and how to pursue them.

The idea is also closely related to the idea of a special zone, but it is a specific type of special zone.

First, a Charter City has to be big. Viable cities will have millions of residents, so a zone has to be big to accommodate them. Second, it should be a Reform Zone, not a Concession Zone. Most zones are created to offer concessions to firms, not to implement reforms. The goal of a Charter City is reform, not giving out concessions, so in this sense, the motivation for a Charter City is totally different from the motivation behind most special zones.

Here are my two tests for whether a policy is a reform or a concession: Would you be happy if this policy lasts forever? Would you be happy if this policy spread to the entire country? If the answer to both questions is yes, it is a reform. If not, it is almost surely a concession, a gift to some special interest. A reform zone is a zone that implements one or more fundamental reforms.

So to summarize, a Charter City is a city-scale reform zone where a startup city could emerge.

Q: What are the problems with Urban Expansion? Why do we need to have new cities?

Romer: One of the main concerns of the people working on what we call the “Urbanization Project” at NYU, is the increase by several billion in total urban population that the world needs to accommodate. There are two ways we can do this — expand existing cities or start new cities. We are working to encourage both. You will have to have both, and most of the room for additional urban residents will come from expansion of existing cities. But when there are billions of people who want to move to cities, it’s also possible to think of starting new cities.

When you need a huge increase in the total built urban environment, you have a chance to let some of this emerge inside reform zones. Then we can get the same kind of innovation in governance that we see in an industry where startup firms can enter.

One of the conclusions I came to when I was working on the theoretical foundations of growth theory is that there is enormous scope in the private economy for discovering new and better ways to provide a higher standard of living. This same insight applies equally well to the public sector. In every country, in every society, there is enormous opportunity to find better ways to provide existing government services and to change the mix of services that the government provides, often by shifting traditional government services to the private sector because the government provides a new type of service that is a complement to private service provision. For example, when a government starts to provide regulation of the prices that a utility with monopoly power can charge, it is possible for the government to get out of the business of supplying water and electricity and to turn these over to private firms.

So we need urban expansion because billions of people will be moving to cities. We can use this window of expansion to encourage rapid innovation in the provision of government services.

Q: In which parts of the world do you think a Charter City would be most valuable?

Romer: I think almost every economic system can benefit from experimentation with new forms of government and new types of government services. The practical reality is that governments that are in more trouble may be more willing to try something new, but I think startup cities could create value anywhere.

Take the United States, for example. If someone wanted to start a Charter City there, what might they do? One possibility might be to develop a new city that from the beginning requires that every vehicle be autonomous, that is computer-controlled. Then you could experiment with a radically different type of transportation system and try out things that you could never try in an existing city. For example, you could use software, minute by minute, to shift the direction of travel in each particular lane in a street. You could make the city much safer for pedestrians. And you wouldn’t have to devote any expensive human time by police officers to watching for violations of traffic laws.

You could also use software to prioritize emergency service vehicles instead of sirens. This sounds like a small advantage, but in New York City, the number one complaint on the city’s 311 line is noise. Noise pollution seriously reduces the quality of modern life. It may even have important effects on health. It is the kind of problem that we tend to accept as being inevitable, but which the right kind of innovation could address. But the kind of innovation that is required might not be possible for an individual firm. For something like emergency service vehicles, it might take innovation at the level of the entire city.

Q: What is the difference between a Charter City and a Private City?

Romer: There is a big difference between saying you want to allow for city-scale reform zones that will encourage reform of government and innovation in government, and saying that you want to do away with government entirely and let a corporate entity run a private city. There are people who claim that cities could be run just like a business. Sometimes they claim that once you let people choose between cities they can live in, you no longer need any of the traditional mechanisms of political accountability that we impose on governments.

I agree that choice between cities is a good thing, but I do not think that it alone provides a sufficient check on the potential for bad behavior by the people who are in charge of a city.

I don’t find the claim that choice between cities is all we need credible or well considered. People invoke loose analogies – “We have private grocery stores so why can’t we have private cities?” — without thinking seriously about the details in any such proposal. For example, in a privately run city, will there be a police force and a legal system that can put people in jail? Are we willing to let a private firm incarcerate people?

The track record of private police forces and private judicial proceedings is very bad. We have some of these in the United States run by private, but non-profit, universities. If the university has a sports program that generates lots of revenue and prestige, the university tends to protect athletes, typically men, who commit sexual violence, typically against woman. They do not offer anything like “equal protection under the law.” It is a telling illustration of how police and judicial proceedings can be bent to support the mission of the organization, even one like a university that we usually think of as being well intentioned, and fail to protect the people it is responsible for.

Unless someone is willing to specify whether there is a local police chief and how he/she is appointed and held accountable, any suggestion they make about private cities can be dismissed as frivolous.

Or to put it another way, I would never agree to go live in a private city that has a private police force. I would never want my children or grandchildren to live in one. And my rule is that I will not support any public policy initiative for a new city if it is not the kind of place that I would be willing to go live or where I would want my children and grandchildren to live.

Q: Some may say that people own the shares of the city, so it is true democracy. What do you think?

Romer: This is another example of a loose analogy that falls apart when you look at the details. If the corporation can lock you up, what good is it to have an ownership share?

Q: So Reform Zones still have a government?

Romer: Any sensible reform zone will still have government, some type of government. It will be accountable in the way that governments around the world are accountable. We give government special powers, like the ability to people in jail, but we also impose special mechanisms of accountability to make sure that those powers are not abused.

I suspect that people who say that they want to get rid of government know that you can’t. What some of these people seem to be saying is that “I want to set up a new government in which I get to be dictator.” Or my friends and I get to be the aristocracy.

Q: Is it correct to say that the goal of a Charter City is to encourage competition among cities?

Romer: Almost. I would say the goal is to improve the quality of governance, and that competition and startups are one means of achieving this goal.

This is a case where the loose analogy points you in the right direction. It is good to have competition between firms in an industry, competition both in the sense of a variety of existing firms that someone can choose from, and competition in the sense that a new firm can enter and do things differently. If you look into the details, there is a feasible way to get the benefits of these two types of competition between cities.

But the analogy alone is not enough. Once it gives you an idea, you have to work out the details required to make it work.

Q: How many Charter Cities would be needed to improve governance all around the world?

Romer: Right now, there is an unmet demand for urban opportunity by billions of people. If all of this demand were met by new cities, we could have hundreds of them, each on the scale of 10 million or more residents. And to get to a situation in which there are hundreds of thriving, newly formed cities, you would probably have to start thousands of them. Just as many startup firms will fail, we should assume that many startup cities will fail too. This is still worth doing because the cost of many failures is very small compared to the benefits offered by even one success like Shenzhen.

That said, it takes an unusual set of circumstances to make it possible for a new city to get started. So I doubt that there would be many. Even in an optimistic scenario, there might end up being no more than five or ten. But even so, it is well worth trying to make this possible.

Shenzhen has been incredibly important. It changed the course of history. People in the west seem to forget that reform in China was extremely controversial; that there was lots of opposition, even violent opposition. So it was not a sure thing that reform was going to succeed there even after Deng took power.

When reform was underway and Deng went on his Southern Tour in 1992, the visible success of Shenzhen protected reform from the reactionaries. Without Shenzhen, reform in China might have stalled before it had a chance to take hold.

So even if Shenzhen is the last startup city we ever see, history will be different because it was possible to start a new city in China.

Q: If we compare Hong Kong and Shenzhen, which is the better model for a Charter City?

Romer: I think it depends on the circumstances. If you are thinking about setting up a Charter City in the United States, Shenzhen is a better model. If you are thinking about ISIS controlled Syria, Hong Kong would probably be better.

There are also mixtures of these two that we should consider. Think about an alternative history of Hong Kong. Imagine that it had been the result of a negotiated agreement between Britain and China in which China decided to import some local government services. It could have started a Shenzhen-like city where the national government in China retained ultimate control, but still imported some government services from different countries.

Q: But don’t you think Shenzhen is a bit more like a Concession Zone?

Romer: I think there was some risk that it would be used as Concession Zone and who knows what kinds of compromises it took to make sure that it took off. But it seems clear to me that the fundamental motivation in establishing the four initial special zones, of which Shenzhen was the most successful, was to achieve local reform quickly in hopes that this would induce reform throughout China. To an overwhelming degree, the measures implemented in Shenzhen pass my two tests for reform: they have been adopted as permanent policies and they have spread to the rest of China.

Q: What about the Shanghai Free Trade Area?

Romer: I don’t know enough about the details [of the Shanghai Free Trade Area], but I think you could treat this as a continuation of a strategy of gradual experimentation to implement reforms. Much of the focus in this zone seems to be on piloting reforms to the financial market that could ultimately spread to the entire country. In this sense, I think it is a new example of a Reform Zone.

Q: What kinds of reform zones do the world need the most?

Romer: Depending on the context, the types of reform may be very different. In the United States, we could obviously benefit from some financial reforms as well, so we don’t cause another worldwide financial crisis. And as I mentioned, we might be at a time when we could profoundly change the logistics of moving goods and people around in a city.

In China today, I think it would be interesting to start a new city that competes for residents by offering the cleanest air in the country. It might do this by banning, from the start, all diesel fuel, all coal and all gasoline; in fact, by banning the combustion of all liquid and solid fuel. The fuel sources that would be allowed could include nuclear, wind, solar, and natural gas. This is already technologically feasible. We know how to produce cost-competitive electricity from a mix of these sources. We also know how to make small modifications to internal combustion engines so that they can run on compressed natural gas. It could be a place where battery-powered electric vehicles could try to compete for market share, or soon where new electric vehicles powered by fuel cells could try to compete. If a city tried this, I suspect it could have cost-competitive electricity and transport and hence, at no additional cost, would have the cleanest air in the developing world. And it could do so without having to have in place a very intrusive and complicated system of regulation and inspection of individual cars like the one we use in the United States.

This new city could then be a model for all developing countries.

I think that China will be able to reduce pollution in many cities in the next 10 years because it has a relatively competent government. When people really want something, the government is able to respond. It could do so by trying to copy the system in the United States, but it would be much better for the rest of the developing world, where government capacity is scarce, if China demonstrated a strategy for getting clean urban air that did not make such stringent demands for regulatory and enforcement capacity from the local government.

Q: I would like to discuss one very important question a little bit: What is the end game of the Charter City? In the case of Hong Kong, it seems that after the handover in 1997, Hong Kong turned into some kind of Concession Zone. If we generalize this observation, when the sovereign becomes a developed country with the help of the Charter City, there could be tendency to pull the Charter City back to its own system of governance. What do you think?

Romer: If you think of Shenzhen, there hasn’t been much problem integrating it into the rest of the Chinese system. Frankly, this is because the whole country has adapted to the model of Shenzhen rather than the other way.

The problem in Hong Kong may be that the British tried to force on the Chinese government some political reforms that the Chinese government didn’t want. The British did that because they thought that was the right thing to do for people in Hong Kong. But we have to be realistic that when you try to force a government to do things it does not have a domestic incentive to do, you are likely to fail. What might have been more productive would have been to think about how to create incentives that encourage them to move in the right direction.

An example of this was when the US embassy in Beijing started putting out hourly reports of particulate levels in Beijing, particularly the most dangerous small particulates, what they call PM 2.5. This created an awareness about pollution that is now inducing government action on pollution.

Because of the decisions that the British took just before handover, it makes sense for the Chinese government to refrain from using Hong Kong as the test bed for new market reforms or new political reforms. They perceive the situation there as being too dangerous. So they are doing the free trade zone in Shanghai rather than doing something here [in Hong Kong]. They will experiment with other forms of political reform in other places.

It is a harsh thing to say, but the British may have ruined the chance for Hong Kong to continue as a test bed for reform.

There is one other point worth making. Reform is a process that never ends, just as innovation is a process that never ends. It is disappointing that Shenzhen has not reformed every aspect of Chinese society. I suppose you could even say that it is a disappointment because it hasn’t removed the threat of climate change or brought peace to the Middle East. But this would of course be foolish. Shenzhen is a step along a long path of progress. Hong Kong was another important step along this path. It is important to be ambitious and to strive for big achievements, but then to accept small successes as steps along a longer path.

Q: Then which model of Charter City is the best? Other than the Hong Kong and Shenzhen models, actually Singapore is quite a model for a Charter City. Singapore did a great job bringing up the whole Malaysian region while staying independent from Malaysia. Do you agree that this is maybe the best model for new cities?

Romer: I think that you should be open to both kinds of models. In general, I think that competition between cities with different systems can be a good thing. So I don’t think you have to have an overarching government that forces all the cities to be the same. I think it’s actually okay to have cities that are different.

On the other hand, you do have to allow some flexibility in the government structure so it can adapt over time. You want this flexibility because it is very hard to know what the world is going to look like in a hundred years. You shouldn’t try to lock in, either like a Singaporean version, which is always independent, or a Shenzhen version, which is always connected. I think you want to try and start with some independence and then have some mechanism that can adjust over time.

For example, I stopped working on a project in Honduras because a group of people there is trying to create a system that establishes a type of aristocracy that will never be subject to local electoral control. They are doing this by establishing a government board that will re-appoint its own members. It will not be subject to political control by the people in the zone, nor by the citizens of Honduras, nor even voters elsewhere as was the case in Hong Kong. They are trying to create a true aristocracy in a small group of twenty or so people, who will appoint their own replacements, and who will always be in charge.

There will be no flexibility, no ability to respond to future developments. And no accountability in the event that this small circle of self-appointed aristocrats misuses their powers. A core of appointees, all from the current governing party in Honduras, controls this board. As a result, the proposal there no longer passes my test: “Would I want to live there or want my children or grandchildren to live there?” So I have refused to have any further involvement in this project.

Q: So is the Honduras project still on going?

Romer: They say that they are still moving forward. They have passed a new law that crosses my bright red line. It removes all possibility of electoral accountability for the people who run the zone. This was not in the law that I supported. They have appointed the members of the board.

What they are trying to do is so wrong that I can’t tell if what we are watching is a farce or a tragedy. It could end up as farce because I can’t imagine a thoughtful investor would go in and invest under these circumstances, in a place controlled by a crony aristocracy. I predict that this effort will collapse on its own. But legally the structure now exists.

Q: Are there any other Charter City projects in which you are current involved?

Romer: No, none at an implementation stage that I am involved in right now. There are some preliminary discussions and some thinking about these ideas, but nothing that is ready for implementation.

Q: Do you think any government can succeed without the help of experts like you?

Romer: Of course! The idea is not that complicated. Deng did just fine without me! And, for that matter, without the advice from experts at places like the World Bank, who have always been very hostile to special zones.

All someone like me can do is plant a seed, encourage some new ideas and hope that it gets combined with some other ideas that eventually get put to work some time in the future.

Q: What can ordinary people like our readers do for the Charter City project?

Romer: That’s a good question. Maybe one thing your readers might do is to make some suggestions about how they can help. It is an idea that has excited a number of people; that’s kind of an encouraging sign. But I think everybody is unsure about what to do next. We should be open to many suggestions.

It might be interesting to have more detailed kinds of discussions. For example, it might be useful to have a forum where people can propose types of reforms that might be worth trying in different places. I suggested a focus on air pollution in China. Others might have much better suggestions.

It might be useful to have a journal or a newsletter, or something like that where people could propose and exchange ideas. There is some chance that out of this might come a decision by some policy makers to evaluate some of these ideas. To use an analogy, when John Kennedy was the president in the United States, someone persuaded him to pursue the idea of putting a man on the moon. Maybe this discussion could lead to a decision by a government to try a new “moon shot.”

To be honest, I had no idea how this idea would be received or where it would go. I just wanted to start a discussion that was not stuck in the usual rut. I was especially focusing on the poorest countries around the world. I hadn’t thought about what might work in China now, or the United States now, but I’m persuaded that we should be thinking more broadly and consider these possibilities.

So maybe there is a way to encourage more participation. It’s kind of fun to think about; thinking about new possibilities, even moon shots, tends to make one a little bit optimistic. At a personal level, and at the broader political level, it is good to have something that offsets the many disappointments that anyone who tries to make the world a better place inevitably faces.


Q: Do you agree that growth theory is not a hot topic in economics research right now? And why?

Romer: For reasons that I don’t think we understand, economics goes through phases. A topic can be intensively studied for a while, and then it will suddenly go out of fashion. This is not necessarily a bad thing because, in a way, you let it go quiet for a while, and when you restart it, it is almost like a start-up. We take a fresh look. We explore new directions.

We had a burst of growth theory in the late 1950s and 1960s. Then growth went out of fashion. Endogenous growth theory came back “in” starting in the early 1980s and pursued a new direction. Since 2000, growth has been pretty quiet. There is ongoing work on the empirical application of growth theory, but frankly, I think that much of the work that is trying to examine the foundations of growth is pursuing a dead-end. So it’s not necessarily a bad thing for work in this area to go quiet for a while. We have seen this in other areas too.

As you can tell, what interests me right now is urbanization. There was a lot of work in urban economics and on the process of urbanization in the 1960s and 1970s, and then it too went quiet. Now the pendulum is swinging in the other direction. People are realizing that there are lots of important theoretical and practical questions that we can pursue about urbanization.

Q: Do you agree that Growth Theory and Development Economics are basically the same?

Romer: No. I think it is helpful to distinguish Growth, which is concerned with the progress at the technological frontier, and Development, which is about catch-up growth. One of the problems with the empirical work that coincided with the theoretical work on endogenous growth is that it did not distinguish between these two.

Endogenous growth gives you tools to think about speeding up technological progress in places like the United States and European Union. It is actually the kind of theory that we should be using to think about how to deal with Global Warming because if carbon emissions turn out to be as damaging as many fear, the only feasible response will be to encourage technological progress that lowers the cost of zero carbon energy sources.

There is a separate question about what I think we should call catch-up growth. How can a country that starts far from the existing technological frontier catch-up? This is the question that faced leaders in China in the 1980s. It is the question facing most countries in the world, and based on China’s experience, there is a growing sense that if a government does the right things, it can enjoy rapid catch-up growth. I’m persuaded that one of those things it has to do right, perhaps the single most important thing that it has to do right, is to facilitate successful urbanization.

Q: In recent years, development economists focused on three research areas, namely the effects of international aid as favored by Jeffrey Sachs, the political economy effects on growth as promoted by Daron Acemoglu and James Robinson, or the experimental economics on growth championed by Esther Dulfo. How do you view the three theories of catch-up growth that are now the hottest in research?

Romer: You know, I actually don’t agree with any of these three. I think urbanization is the thing we’ve got to study. Consider the case of India. If it wants to get better access to the technology that already exists around the world, and make this technology available not just to an elite, but to all workers, even workers who are getting their first formal sector job that pays them a wage, they will have to have gateway cities that invite foreign workers and firms to come operate in India. And gateway cities that invite people from rural areas throughout India to come join them. Unfortunately, India is not building gateway cities that work. If you look at any of the Indian cities right now, they are as polluted as Chinese cities right now. The traffic congestion is horrible. They exclude new residents from rural areas.

In this context, think about what Jeff Sachs has proposed about providing more aid. There is nowhere near enough aid available to bribe Indians into building cities that work if they don’t want them. Change the political economy of India? Great if you can. But how? Where is the evidence that you can change things in a way that raises standards of living? As my friend Edwin Lim has observed, democracy in India has totally failed to provide such elementary basics as child nutrition or education for girls, failed even compared to neighbors like Bangladesh. And experimental economics? Cameras for teachers? Or as Lant Pritchett has put it in a recent blog post, do we really want to devote so much academic effort to experimental analysis of policies like women’s self help groups? [] Where is the evidence that you can generate Chinese rates of growth from measures like these?

I think the new prime minister in India understands how important urbanization is to the future of his country. It is the make-or-break issue for India right now. But I think it’s going to be hard for him to do anything about this. They have a proposal for 100 Smart Cities that tries to address the problems they face, but it remains to be seen whether this will make any difference. I wish they were thinking seriously about letting some new jurisdictions enter as startups and compete for residents with all the failing ones that they already have.

Q: So are you not going back to work on growth theory?

Romer: Actually I am writing something about growth theory right now, but it is mostly a commentary on what happened to growth theory. To be honest, I think that a substantial fraction of the work that people are now doing on growth has to be judged a failure from a scientific perspective.

In particular – and I apologize if this relies too much on the jargon of our field — monopolistic competition turns out to be just the tool for understanding the economic ideas. (It also turns out to be the tool for understanding international trade, economic geography, and macroeconomics.) But there has been a series of models that are associated with the University of Chicago – from what some people call the freshwater camp in macroeconomics – that are continuing a fight that George Stigler started in the 1930s to keep monopolistic competition from being used in economics. It is hard to explain to an outsider why a whole group of economists have ended up on the wrong side of scientific progress, resisting the direction that all of modern economic theory is taking, but they are.

In the economics of ideas, we have to be willing to at least consider the possibility that someone could have some control over an idea, hence some monopoly power associated with ideas. This could come from patent or a copyright. It could also come from secrecy.

Then we can ask if it is a good idea or a bad idea to have more intellectual property rights or more protection of ownership of ideas. We know that the answer here is mixed. Sometimes some amount of it can be good, but it can also be harmful if the property rights are too strong or are given to the wrong types of ideas. But if you don’t even allow for the possibility of ex post monopoly rents from the discovery of ideas, you can’t even ask the question.

So it is scientifically unacceptable to have people who say, “We will never, as a matter of principle, consider a model in which there are ever any monopolies. We will dogmatically stick only to models of price-taking competition.” I think this an untenable scientific stance.

I don’t think that this critique is going to reignite interest in growth theory. But like I said, when it’s time for interest to come back, somebody have a new take on growth theory, and work in this area will start again. But in the meantime, we have to stop tolerating work that is scientifically unjustifiable.

Q: I thought the endogenous growth model paved a new direction for growth theory to further develop, yet the academic interest in this theory just stopped. And even textbooks just briefly mention the endogenous growth model. What is the problem?

Romer: Well, I think the thing we learn from endogenous growth is something very simple. It is the notion of an idea as a nonrival good. The statement that an idea is a nonrival good is very powerful because what that tells you is the value of an idea is proportional to, or at least scales with, the total number of people who can use it. So it means that scale effects are at the heart of economic activity. This is why globalization is so important, because it is now possible for any idea to be used by everyone.

The Solow model already allows for a non-rival good, but the model also made it nonexcludable – which means that no one could control or own an idea. This turns a nonrival good into a public good. What endogenous growth theory said is that, there are some nonrival goods that can be at least partially excludable. This means that incentives start to matter, both for discovering ideas and for spreading ideas. The people who want to stick with price-taking never want to allow the possibility of that a nonrival good could be even partially excludable. Because of their untenable insistence on price-taking models, they have tried to stop the spread of the key insight from endogenous growth theory. And they have been at least partially successful in doing so.

Once you admit that there are some nonrival goods, globalization becomes much more important than standard theory suggests. And if you allow that some of them are partially excludable, then incentives matter a lot more than standard theory suggests, for better and for worse.

So for example, there is a non-rival idea that a firm can control. They can keep it secret. They can take it to work in a factory in India. If they want to locate in India but there are no gateway cities there, they may go elsewhere in the world. So the policies that make a place like Mumbai so dysfunctional influence growth for the entire country.

What a government can do is influence the incentives for people to bring ideas into a country. One way to think about why Shenzhen was so powerful is that it created incentives for firms to bring ideas into China and combine those ideas with Chinese workers.

You don’t even need a formal model for that. Once you see the underlying idea, sometimes the words and the clarity of thinking are what really matters, not the math. You can use the math to get there, but once you got there, you don’t need that anymore. It is unfortunate that these ideas are not being communicated to students in our textbooks, because these are the exciting ideas.

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