NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

Do we care if productivity growth is “broad-based”, meaning that all sectors or firms tend to be getting more productive? Or is it better to have a few sectors or firms experience massive productivity increases, even at the expense of other sectors? Think of it as an allocation problem – I’ve got a fixed amount of resources to spend on R&D, so should I spread those out across sectors or spend them all in one place?

The answer depends on how willing we are to substitute across the output of different types of goods. If we are willing to substitute, then it would be better to just load up and focus on a single sector. Make it as productive as possible, and just don’t consume anything else. On the other hand, if we are unwilling to substitute, then we would prefer to spread around the productivity growth so that all sectors produce goods more cheaply.

That’s the intuition. Here’s the math, which you can skip past if you’re not interested. Let the price people will pay for output from sector ${j}$ be ${P_j = Y_j^{-\epsilon}}$, so that ${\epsilon}$ is the price elasticity (in absolute value). As ${\epsilon}$ goes to one, demand is inelastic, and the price is very responsive to output. As ${\epsilon}$ goes to zero, demand is elastic, and in fact fixed at ${P_j = 1}$.

There are ${J}$ total sectors. Each one produces with a function of

$\displaystyle Y_j = \Omega_j Z_j^{1-\alpha} \ \ \ \ \ (1)$

where ${\Omega_j}$ is a given productivity term for the sector. ${Z_j}$ is the factor input to production in sector ${j}$. ${Z_j}$ can capture labor, human capital, and/or some physical capital. Raising it to ${1-\alpha}$ just means there are diminishing marginal returns to moving factors into sector ${j}$. There is some total stock of ${Z}$, and units of ${Z}$ are homogenous, so they can be used in any sector. So you could think of an element of ${Z}$ being a laptop, and this can be used by someone to do work in any sector. If ${Z}$ is labor, then this says that workers are equally capable of working in any sector. There are no sector-specific skills.

Now we can ask what the optimal allocation of ${Z}$ is across the different sectors. By “optimal”, I mean the allocation that maximizes the total earnings of the ${Z}$ factor. Each sector is going to pay ${w}$, the “wage”, for each unit of ${Z}$ that it uses. What maximizes total earnings, ${wZ}$?

Within each sector, set the marginal product of ${Z_j}$ equal to the wage ${w}$, which each sector takes as given. This allows you to solve for the optimal allocation of ${Z_j}$ to each sector. Intuitively, the higher is productivity ${\Omega_j}$, the more of the input a sector will employ. If we put the optimal allocations together, we can solve for the following,

$\displaystyle wZ = \left(\sum_j \Omega_j^{(1-\epsilon)/(1-(1-\alpha)(1-\epsilon))}\right)^{1-(1-\alpha)(1-\epsilon)} Z^{1-\alpha} \ \ \ \ \ (2)$

which is an unholy mess. But this mess has a few important things to tell us. Total output consists of a productivity term (the sum of the ${\Omega_j}$ stuff) multiplied through by the total stock of inputs, ${Z}$. Total earnings are increasing with any ${\Omega_j}$. That is, real earnings are higher if any of the sectors get more productive. We knew that already, though. The question is whether it would be worth having one of the ${\Omega_j}$ terms be really big relative to the others.

The summation term over the ${\Omega_j}$‘s depends on the distribution of the ${\Omega_j}$ terms. Specifically, if

$\displaystyle \frac{1-\epsilon}{1-(1-\alpha)(1-\epsilon)} > 1 \ \ \ \ \ (3)$

then ${wZ}$ will be higher with an extreme distribution of ${\Omega_j}$ terms. That is, we’re better off with one really, really productive sector, and lots of really unproductive ones.

Re-arrange that condition above into

$\displaystyle (1-\alpha) > \frac{\epsilon}{1-\epsilon}. \ \ \ \ \ (4)$

For a given ${\alpha}$, it pays to have concentrated productivity if the price elasticity of output in each sector is particularly low, or demand is elastic. What is going on? Elastic demand means that you are willing to substitute between sectors. So if one sector is really productive, you can just load up all your ${Z}$ into that sector and enjoy the output of that sector.

On the other hand, if your demand is inelastic (${\epsilon}$ is close to one), then you are unwilling to substitute between sectors. Think of Leontief preferences, where you demand goods in very specific bundles. Now having one really productive sector does you no good, because even though you can produce lots of agricultural goods (for example) cheaply, no one wants them. You’d be better off with all sectors having similar productivity levels, so that each was about equally cheap.

So where are we? Well, I’d probably argue that across major sectors, people are pretty unwilling to substitute. Herrendorf, Rogerson, and Valentinyi (2013) estimate that preferences over value-added from U.S. sectors is essentially Leontief. Eating six bushels of corn is not something I’m going to do in lieu of binge-watching House of Cards, no matter how productive U.S. agriculture gets. With inelastic demand, it is better to have productivity in all sectors be similar. I’d even trade off some productivity from high-productivity sectors (ag?) if it meant I could jack up productivity in low-productivity sectors (services?). I don’t know how one does that, but that’s the implication of inelastic demand.

But while demand might be inelastic, that doesn’t mean prices are necessarily inelastic. If we can trade the output of different sectors, then the prices are fixed by world markets, and it is as if we have really elastic demand. We can buy and sell as much output of each sector as we like. In this case, it’s like ${\epsilon=0}$, and now we really want to have concentrated productivity. I’m better off with one sector that is hyper-productive, while letting the rest dwindle. If I could, I would invest everything in raising productivity in one single sector. So a truly open economy that traded everything would want to load all of its R&D activity into one sector, make that as productive as possible, and just export that good to import everything else it wants.

Now, we do have lots of open trade in the world, but for an economy like the U.S. the vast majority of GDP is still produced domestically. So we’re in the situation where we’d like to spread productivity gains out across all sectors and/or firms.

Part of productivity is the level of human capital in the economy. If aggregate productivity is highest when productivity improvements are spread across lots of sectors, then we want to invest in broad-based human capital that is employable anywhere. That is, we don’t want to put all our money into training a few nuclear engineers with MD’s and an MBA, we want to upgrade the human capital of the whole range of workers. I think this is an argument for more basic education, as opposed to focusing so heavily on getting a few people through college, but I’m not sure if that is just an outcome of some implicit assumption I’ve made.

# Scale, Profits, and Inequality

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After my post last week on inequality, I got a number of (surprisingly reasonable) responses. I pulled one line out of a recent comment, not to call out that particular commenter, but because it encapsulates an argument for *not* caring about inequality.

Gates and the Waltons really did probably add more value to humanity than the janitor at my school.

The general argument here is about incentives. Without the possibility of massive profits, people like Bill Gates or Sam Walton will not bother to innovate and create Microsoft and Walmart. So we should not raise taxes because those people deserve, in some sense, the fruits of their genius. More important, without them innovating, the economy wouldn’t grow.

But if we take seriously the incentives behind innovation, then it isn’t simply the genius of the individual that matters for growth. The scale of the economy is equally relevant. In any typical model of innovation and growth, the profits of a firm are going to be something like Profits = Q(Y)(P-MC), where (P-MC) is price minus marginal cost. Q(Y) is the quantity sold, and this depends on the aggregate size of the economy, Y.

The markup of price over marginal cost (P-MC), is going to depend on how much market power you have, and on the nature of demand for your product. This markup depends on your individual genius, in the sense that it depends on how indispensable people find your product. Apple is probably the better example here. They sell iPhones for way over marginal cost because they’ve convinced everyone through marketing and design that substitutes for iPhones are inferior.

The scale term, Q(Y) does not depend on genius. It depends on the size of the market you have to sell to. If we stuck Steve Jobs, Jon Ive, and some engineers on a remote island, they wouldn’t earn any profits no matter how many i-Devices they invented, because there would be no one to sell them to.

People like Gates and the Waltons earn profits on the scale effect of the U.S. economy, which they did not invent, innovate on, or produce. So the “rest of us”, like the janitor mentioned above, have some legitimate reason to ask whether those profits are best used in remunerating Bill Gates and the Walton family, or could be put to better use.

There isn’t necessarily any kind of efficiency loss from raising taxes on Gates, Walton, and others with large incomes. They may, on the margin, be slightly less willing to innovate. But if the taxes are put to use expanding the scale of the U.S. economy, then we might easily increase innovation by through the scale effect on profits. Investing in health, education, and infrastructure all will raise the aggregate size of the U.S. economy, and make innovation more lucrative. Even straight income transfers can raise the effective scale of the U.S. economy be transferring purchasing power to people who will spend it.

Can we argue about exactly how much of the profits are due to “genius” (the markup) and how much to scale? Sure, there is no precise answer here. But you cannot dismiss the idea of taxing high-income “makers” because their income represents the fruits of their individual genius. It doesn’t. Their incomes derive from a combination of ability and scale. And scale doesn’t belong to individuals.

The value-added of “the Waltons” is particularly relevant here. Sam Walton innovated, but the profits of Walmart are almost entirely derived from the scale of the U.S. (and world) economy. It’s the presence of thousands and thousands of those janitors in the U.S. that generates a huge portion of Walmart’s profits, not the Walton family’s unique genius.

Alice Walton is worth around \$33 billion. She never worked for Walmart. She is a billionaire many times over because her dad was smart enough to take advantage of the massive scale of the U.S. economy. I’m not willing to concede that Alice has added more value to humanity than anyone in particular. So, yes, I’ll argue that Alice should pay a lot more in taxes than she does today. And no, I’m not afraid that this will prevent innovation in the future, because those taxes will help expand the scale of the economy and incent a new generation of innovators to get to work.

# Why I Care about Inequality

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

“Inequality” is a term that has been tossed about quite a bit. The Occupy movement, to Piketty’s book, to debates over the minimum wage, to Greg Mankiw‘s defense of the 1%. Just today Mark Thoma published an op-ed on inequality. A few days ago John Cochrane had a post about why we care about inequality.

One of Cochrane’s main points is that the term “inequality” has been used in so many contexts, and to refer to so many different things, that it is ceasing to lose meaning. I’ll agree with him on this completely. If you want to talk about “inequality”, you have to be very clear about what precisely you mean.

There are three things that people generally mean by “inequality”:

1. The 1% versus the 99%. That is, the difference in average annual income of the top 1% of all households versus the average annual income of the bottom 99%.
2. The stagnation of median real wages and those below the median.
3. The college premium, or the gap in earnings between those who finished college and those who did not (or did not attend).

When I say I care about inequality, I mean mainly the second – the stagnation of median wages – but this is going to take me into territory covered by the first – the growth in top 1% income. There are things to say about the college premium, but I’m not going to say them here.

Why do I care about the stagnation of median wages?

• Because I’m going to be better off if everyone shares in prosperity. I want services like education, health care, and home repairs to be readily available and cheap. The way to achieve that is to invest in developing a large pool of skilled workers – teachers, nurses, electricians, carpenters. Those at the bottom of the distribution don’t have sufficient income to make those investments privately, so that requires public provision of those investments (i.e. schools) or transfers to support private investments. You want to have an argument about whether public provision or transfers are more efficient? Okay. But the fact that there is an argument on implementation doesn’t change the fact that stagnant wages are a barrier to these investments right now.
• Because people at the bottom of the income distribution aren’t going to disappear. We can invest in these people, or we can blow our money trying to shield ourselves from them with prisons, police officers, and just enough income support to keep them from literally starving. I vote for investment.

One response to this is that I don’t care about inequality per se, I care about certain structural issues in labor markets, education, and law enforcement. So why don’t we address those fundamental structural issues, rather than waving our hands around about inequality, which is meaningless? Because these strutural issues are a problem of under-investment. The current allocation of income/wealth across the population is not organically producing enough of this investment, so that allocation is a problem. In short, if you care about these structural issues, you cannot escape talking about the distribution of income/wealth. In particular, you have to talk about another kind of inequality, the 1%/99% kind.

Let me be very clear about this too, because I don’t want anyone to think I’m trying to be clever and hide something. I would take some of the income and/or wealth from people with lots of it, and then (a) give some of that to currently poor people so they can afford to make private investments and (b) use the rest to invest in public good provision like education, infrastructure, and health care.

Would I use a pitchfork and torches to do this? No. Would I institute “confiscatory taxation” on rich people? No, that’s a meaningless term that Cochrane and others use to suggest that somehow rich people are going to be persecuted for being rich. I am talking about raising marginal income tax rates and estate tax rates back to the archaic levels seen in the 1990s.

• Because rich people spend their money on useless stuff. Not far from where I live, there is a new house going up. It will be over 10,000 square feet when it is complete. 2,500 of those square feet will be a closet that has two separate floors, one for regular clothes and one for formal wear. If that is what you are spending your money on, then yes, I believe raising your taxes to fund education, infrastructure, and health spending is a net gain for society.

Don’t poor people spend money on stupid stuff? Of course they do. Isn’t the government an inefficient provider of some of these goods, like education? Maybe. But even if both those things are true, public investment and/or transfers to poor people will result in some net investment that I’m not currently getting from the mega-closet family. I’m happy to talk about alternative institutional settings that would ensure a greater proportion of the funds get spent on actual investments.

• Because I’m not afraid that some embattled, industrious core of “makers” will decide to “go Galt” and drop out of society, leaving the rest of us poor schleps to fend for ourselves. Oh, however will we figure out how to feed ourselves without hedge fund managers around to guide us?

This is actually a potential feature of higher marginal tax rates, by the way, not a bug. You’re telling me that a top tax rate at 45% will convince a number of wealthy self-righteous blowhards (*cough* Tom Perkins *cough*) to flee the country? Great. Tell me where they live, I’ll help them pack. And even if these self-proclaimed “makers” do stop working, the economy is going to be just fine. How do I know? Imagine that the entire top 1% of the earnings distribution left the country, took all of their money with them, and isolated themselves on some Pacific island. Who’s going to starve first, them or the remaining 300-odd million of us left here? The income and wealth of the top 1% have value only because there is an economy of another 300-odd million people out there willing to provide services and make goods in exchange for some of that income and wealth.

So, yes, I care about 1%/99% inequality itself, because I cannot count on the 1% to privately make good investment decisions regarding the human capital of the bottom 99%. And the lack of investment in the human capital of the bottom part of the income distribution is a colossal waste of resources.

# Subsistence and Self-perpetuating Inequality

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It’s common in development or growth to think about subsistence constraints. From a macro perspective, we think of them as an explanation for the low income-elasticity of food expenditures, and therefore a cause of structural change away from agriculture. The idea has there in development for years. I don’t know the full intellectual history here, but my understanding of it goes back to The Moral Economy of the Peasant by James C. Scott. He says you can understand a lot about the peasant mind-set by realizing they face subsistence constraints, and that this makes them incredibly risk averse. Kevin Donovan has a recent paper that looks at the macro consequences of this risk-aversion for agricultural development.

I think the concept of subsistence constraints and risk aversion is useful for thinking about inequality in general, even outside of developing countries. In particular, it offers an explanation for why inequality will be self-perpetuating and how mitigating inequality will be growth-enhancing.

If there is some subsistence constraint in your utility function, then your risk aversion is declining with income. You can take my word for it, or if you’d like to see it more clearly, let utility be

$\displaystyle U = \ln{y-\overline{y}} \ \ \ \ \ (1)$

and the coefficient of relative risk aversion is then

$\displaystyle \frac{-yU''}{U'} = \frac{y}{y-\overline{y}}. \ \ \ \ \ (2)$

Risk aversion approaches infinity as income approaches the subsistence constraint, meaning people will refuse to take any gamble that might put them below ${\overline{y}}$. Risk aversion approaches 1 as income rises. The value 1 itself isn’t important, what’s important is that as people get richer, they are willing to accept bigger and bigger gambles with their income, because they are less and less danger of falling below ${\overline{y}}$. Richer people are more risk-tolerant.

Combine this conception of subsistence and risk aversion with the commonly understood relationship of risk and reward. In finance and entrepreneurship, we generally believe that those willing to absorb higher risks are able to reap higher rewards. Entrepreneurs earned that money by taking a risk in starting a company. Aside from entrepreneurship, big fixed investments in education — quitting your job to go back to school — are very risky moves that in turn have high expected rewards.

Together, subsistence and the risk/reward correlation imply that inequality will be self-perpetuating. Poor people will not take on big risks (starting a business) because they cannot handle even a small probability of failure that takes them below subsistence. So they stay in low-wage jobs and don’t undertake investments that might make them better off.

Well-off people are able to tolerate bigger risks, and so also earn higher rewards. They start businesses, get more education, or move across country for a new job. If it doesn’t work out, they won’t starve, so it’s worth the risk. And because they undertake big risks, they tend to earn high rewards. The rich can expand their incomes and/or wealth at a faster rate than the poor, because of their higher risk tolerance. This naturally acts to expand inequality over time.

The crucial point is that there is no pathology to the poor refusing to take big risks. With subsistence constraints, the poor don’t remain poor because they are lazy or stupid, but because they are rationally avoiding big risks that might push them below subsistence.

The conceit of people who espouse a “just deserts” theory of inequality (Greg Mankiw, Sean Hannity, et al) is that they would have been well-off regardless of where they start in life. They believe that they possess qualities — smarts, skills, work ethic — that make them valuable to the market, and that they are rewarded for that. But start them off in a truly poor household, one where the next meal is uncertain, and with 99.99% certainty they would not end up a professor at Harvard or, well, whatever Hannity is. There were big risks taken in their lives that had big payoffs. Risks they or their family would not have been able to conceive of taking if they were truly poor.

Subsistence constraints also imply that acting to mitigate inequality — whether by raising incomes of the poor or making their incomes less uncertain — would have a distinct positive effect on economic growth. Ensuring that people won’t fall to subsistence (or below) means that more people are willing to start a business, and we get more economic activity, more competition, and more innovation. If more people are willing to go for advanced training (college or vocational school) then we get a more skilled workforce. Acting to insure or support poor incomes has positive spillovers.

Won’t providing income support just incent all the poor people to stop working? Remember that most of the people screaming loudest about this – Casey Mulligan – are tenured professors who cannot be fired. Despite having no incentives whatsoever to continue working on research or provide more than perfunctory teaching, Mulligan continues to work. Why? Why does he not rationally show up to collect his check and then go home to eat Cheetos and watch Dr. Phil? If Casey Mulligan continues to work despite a guaranteed income, I’m fairly confident that the vast, vast majority of people will continue to work even if they are no longer at risk of falling into destitution.

# Wealth and Capital are Different Things

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

Piketty’s book is like a giant attention-sucking vortex. I can’t seem to escape it. This time I’m thinking about the criticism of Piketty’s analysis that has to do with rates of return on capital. Piketty says that if ${r > g}$, where ${r}$ is the return to capital, and ${g}$ is the growth rate of aggregate GDP, then wealth will become more and more concentrated.

Critiques of Piketty have questioned the assumptions underlying this conclusion. The most recent one I’ve seen is in Larry Summers’ review piece. Let’s let him sum up the issues:

This rather fatalistic and certainly dismal view of capitalism can be challenged on two levels. It presumes, first, that the return to capital diminishes slowly, if at all, as wealth is accumulated and, second, that the returns to wealth are all reinvested. Whatever may have been the case historically, neither of these premises is likely correct as a guide to thinking about the American economy today.

With respect to the first assumption regarding the rate of return, here is what Summers says:

Economists universally believe in the law of diminishing returns. As capital accumulates, the incremental return on an additional unit of capital declines.

But Summers has fallen into what I think is a really common trap for economists. He presumes that his second statement (“As capital accumulates, the incremental return on an additional unit of capital declines”) contradicts Piketty’s assumption (“that the return to capital diminishes slowly, if at all, as wealth is accumulated”). These two statements are not mutually exlusive.

The issue is that Summers is confounding wealth and capital. This is not helped by Piketty, who uses “capital” in his title and in the book the way that normal people use it, as a synonym for “wealth”. But from the perspective of an economist, these two concepts are not the same thing. The capital that Summers refers to in his critique (often denoted ${K}$) is a subset of the measure of national wealth (${W}$, as I’ll call it) that Piketty documents.

Without going too deep into this, Piketty’s measure of wealth consists of three parts: real estate, corporate capital, and financial assets. Only real estate and corporate capital are what economist have in mind when they say capital (${K}$). Wealth, however, consists of all three parts, so that Piketty’s wealth is ${W = K + F}$, where ${F}$ is the value of financial assets. Asserting that the return to capital falls as the capital stock increases – as Summers does – does not imply that the return to wealth falls as the stock of wealth increases. Even if we assume that financial markets work so efficiently that the return to capital and the return to financial assets are identical, this does not mean that the return to wealth necessarily falls as wealth accumulates.

To see this, consider a really slimmed down version of the “bubble asset” model from Blanchard and Fischer (1989, p. 228). We have that the return on capital is ${r = f'(K)}$, where ${f'(K)}$ is the marginal product of capital. The ${f'(K)}$ is the derivative of the production function, and represents the marginal increase in output we’d get from adding one more unit of capital. Under our typical assumptions about diminishing returns, as ${K}$ goes up ${r}$ goes down. This is what Summers is using as his critique.

An efficient financial market would ensure that financial assets (F) would also have a return of ${r}$. If they did not, then people would buy/sell financial assets until the return was equal. (Yes, I’m ignoring risk entirely, but that doesn’t change the main point here). So the return on all wealth is equal to ${r}$, and note that this is pinned down by the value of ${K}$ alone.

Now, we have assumed that ${r}$ falls as ${K}$ increases. Does this imply that ${r}$ falls as wealth (${W}$) increases? No. The relationship between ${r}$ and ${W}$ depends entirely on the composition of the change in ${W}$. If ${W}$ rises because ${K}$ rises (say ${F}$ stays constant), then the rate of return on wealth falls because the marginal product of capital has declined. This is what Summers and others have in mind.

However, it’s perfectly plausible that ${W}$ rises even though ${K}$ falls, because the value of financial assets (${F}$) are increasing even more quickly. In this case, the marginal product of capital has increased, and the rate of return on wealth has increased. In this case, the rate of return rises with wealth.

Is it reasonable for an economy to experience falling capital but a rising value of financial assets? Sure. The point of Blanchard and Fisher’s model of bubbles is that even though all individuals are acting rationally at all times, the economy can take off onto a weird path where the stock of capital (${K}$) gets run down while the value of financial assets (${F}$) rises. Eventually this is unsustainable, as we’d run out of capital, but there is no reason that a situation like this cannot persist for a while.

Will the return to wealth necessarily rise as wealth accumulates? No. There are other equally reasonable paths that the economy could take where wealth accumulation is driven mainly by capital accumulation and the rate of return falls as wealth accumulates, consistent with the Summers critique. The point I want to make is that there is no particular reason to believe in a fixed relationship between wealth and the return on capital. They can move completely independently of each other.

So Piketty can easily be right that we are currently in a world where both the wealth/income ratio is increasing and the rate of return on wealth is rising (or remaining roughly constant), and that this could persist for some indefinite period. On the other hand, it was not inevitable that this was going to happen, and it could just as easily end tomorrow as in 100 years.

I think the story that is milling around beneath the surface of Piketty’s book is that recent wealth accumulation has been primarily of financial assets, not capital. Hence the return has stayed high and the concentration of wealth has continued. If the returns on that wealth are continually reinvested in financial assets as opposed to capital, then Piketty’s death spiral of wealth concentration would likely be the outcome. To avoid that death spiral, you’d want to get the returns on wealth reinvested into real capital so that the return on capital (and hence wealth) gets pushed down.