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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.

# Cochrane on Growth and Macro

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John Cochrane recently ran a little review of his experience at NBER (h/t to Noah Smith). It’s got a really interesting observation on growth versus macro.

A last thought. Economic Fluctuations merged with Growth in the mid 1990s. At the time there was a great confluence of method as well as interest. Growth theorists were studying growth with Bellman equations, dynamic general equilibrium models of innovation and transmission of ideas, thinking about where productivity shocks came from. Macroeconomists were using Bellman equations, and studying dynamic general equilibrium models with stochastic technology, along with various frictions and other propagation mechanisms.

That confluence has now diverged. I enjoyed spending an hour or two thinking about how religion has blocked or adapted to ideas over the centuries, and Paul’s view on social norms or neuroeconomics. But I don’t really have any expertise to contribute to that debate. Questions like whether young CEOs head more innovative companies, or whether, like deans, what matters is the age of the faculty are a little closer to home, since I spend a lot of time consuming corporate finance. But the average sticky-price macro type does not. Likewise, when Daron Acemoglu, who seems to know everything about everything, has to preface his comments on macro papers with repeated disclaimers of lack of expertise, it’s clear that the two fields really have gone their separate ways. Perhaps it’s time to merge fluctuations with finance, where we seem to be talking about the same issues and using the same methods, and growth to merge with institutions and political or social economics.

This is similar in flavor to John Seater’s comment that I wrote about here. Has growth economics become different enough from mainstream macro that we should separate them from one another?

I’d argue yes. Growth is about development now – meaning that it’s motivating question is “Why are some countries rich and some poor?”. (See my earlier post on this topic here). The exploration of answers to this question are much more about big static differences in institutions, cultures, technologies, and the like, and less about transition paths and dynamics.

On what growth would look like if it did separate (literally at NBER and intellectually as a field) from macro, Cochrane gave us perhaps a pointer:

I’m not sure in the end though whether Paul[Romer] was approving or bemoaning the shift back towards literature in economic analysis. Certainly his vision for the future of growth theory, centered on values, social norms, biology, and so forth, does not lend itself easily to quantification.

Is this a feature or a bug? Perhaps the big question of “Why are some countries rich and some poor?” is not answerable in any solid empirical way. Perhaps the highest achievement here is “literature” in the sense of some overarching theory that one uses to examine history. Think of Pomeranz’s The Great Divergence or Robert Allen’s The British Industrial Revolution in Global Perspective as examples. While both books certainly appeal to economic intuition and occasionally something approaching formal theory, neither considers anything like a Bellman equation.

The counter would be that we can do better than just “literature” in growth by writing down model (perhaps static models, but no matter) that allow us to quantify the forces that people like Pomeranz and Allen propose as relevant. That is, write down an explicit model, and calibrate or simulate it to assess whether a proposed explanation has a plausibly large quantitative effect on output per worker. The issue here is, as Cochrane says, it’s essentially impossible to quantify religion or values. What is the parameter you stick in your quantitative model that captures the effect of a belief in the afterlife on your willingness to work today? If you cannot possibly hope to measure that parameter, then you cannot quantify it’s effect on output per worker.

So if we’ve entered the world where we think that values (or culture or religion) are fundamental to development, then we may be left with “literature” as the only valid form of research output. My guess is that growth economists will resist this kind of transition, mainly because we’ve invested a lot in knowing fancy dynamic models and calibration techniques, and we don’t want those skills to become worthless.

# Are We Doomed?

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The Guardian ran a piece on a forthcoming paper in Ecological Economics by Safa Motesharrei, Jorge Rivas, Eugenia Kalnay. The article is titled “Human and Nature Dynamics (HANDY): Modeling Inequality and Use of Resources in the Collapse or Sustainability of Species”. The model they construct has the feature that under certain conditions, either extreme inequality in wealth or overuse of resources will result in the collapse of society, in the sense that the number of people goes to zero.

This is a fairly standard “We’re all gonna die!” kind of model. It mechanically delivers the possibility that society could collapse. This is not some kind of blazing hot insight, it’s the equivalent of saying that you could get in a car wreck today if the conditions work out just right.

Here’s a simple way of thinking about this kind of model. Assume that you are driving on a one-way road, with a car in front of you and one following you. Those other cars are going a constant 40 mph and do not deviate from that speed ever. You drive according to two simple rules. (1) If you are getting closer to the car in front, slow down. (2) If you are getting closer to the car behind you, speed up.

Now, if your accelerator and brakes are sensitive enough, and you have particularly good reflexes, then this system is sustainable. You’ll find yourself travelling 40 mph as well, exactly between the two cars. But, if your accelerator and braking skills are a little sluggish, then eventually you are going to hit one of the cars.

That’s it. That’s the model. The Motesharrei et al model does this, except renaming the various components of the model. But in the end all they are asking is: given the existence of these other 40 mph cars, is it possible you will crash? The answer is, of course, yes. In fact, it’s almost certain you will unless your reactions are calibrated exactly right.

So why don’t we see widespread mayhem on highways? Massive fifty-car pileups multiple times a day in every city? Because the assumption that all the other cars will always go 40 mph is ridiculous. The rest of the system, the other cars, will all react to the situation as well. If you put on your brakes because you get to close to the car ahead, the car behind you will slow down as well, preventing you from being rear-ended.

Models like Motesharrei et al, in order to focus on some simple dynamics, ignore the possibility that the actors in their model will change behavior. They assume all the other cars just go 40 mph all the time. But just as other cars respond to your actions, technology can change (for better or worse in terms of using resources), people will alter their consumption behavior, the composition of the elite and commoner groups will change, and the distribution of wealth will be shifted. The system responds.

This is why economists always scream “you ignored prices!” when they see models like these. Because prices are like brake lights and turn signals, they provide information to those around you. They inform the system about what is scarce and what is abundant. They induce changes in behavior in the rest of the actors in the system. Behavior changes mean it is not inevitable that the system will collapse. Just like it is not inevitable that every time you get in a car you are going to get into a wreck.

Could we create some ecological disaster that dooms the planet? Sure. The ecology of Earth is so complex that I’m sure if we did something wrong we could unravel the whole thing. But this is not inevitable, whatever the equations in Motesharrei et al tell you.

# The Solow Model

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This is another idea for modifying how to teach the Solow model. One thing I’d like to do is go immediately to including productivity – it follows cleanly from the simplest growth model. Second, I think it might be nice to work with the K/Y ratio immediately. In this way, I think you can actually skip using the whole “k-tilde” thing. And, *gasp*, do away with the traditional Solow diagram.

The simplest growth model doesn’t allow for transitional growth, and this due to the fact that it does not allow for capital, a factor of production that can only be slowly accumulated over time. The Solow Model is a standard model of economic growth that includes capital, and will be better able to account for the transitional growth that we see in several countries.

Production in the Solow Model takes place according to the following function

$\displaystyle Y = K^{\alpha}(AL)^{1-\alpha}. \ \ \ \ \ (1)$

${K}$ is the stock of physical capital used in production, and ${A}$ and ${L}$ are defined just as they were in our simple growth model. So the production function here is just a modification of the simple model to include capital. The coefficient ${\alpha}$ is a weight telling us how important capital or ${AL}$ are in determining output.

To analyze this model, we’re going to rewrite the production function. Divide both sides of the function by ${Y^{\alpha}}$, giving us

$\displaystyle Y^{1-\alpha} = \left(\frac{K}{Y}\right)^{\alpha} (AL)^{1-\alpha} \ \ \ \ \ (2)$

and then take both sides to the ${1/(1-\alpha)}$ power, which gives us the following expression

$\displaystyle Y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} AL. \ \ \ \ \ (3)$

In per capita terms, this is

$\displaystyle y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} A. \ \ \ \ \ (4)$

Output per worker thus depends not just on ${A}$, but also on the capital-output ratio, ${K/Y}$.

So to understand the role of capital in economic growth, we need to understand the capital-output ratio and how it changes over time. We’ll start by looking at the balanced growth path, and then turn to situations where the economy is not on the balanced growth path (BGP).

One fact about the BGP is that the return to capital, ${r}$, is constant. The return to capital is ${r = \alpha Y/K}$, which depends (negatively) on the capital-output ratio (the return to capital is just the marginal product of capital). If ${r}$ is constant on the BGP, then it must be that ${K/Y}$ is constant on the BGP as well. What does this mean? It means that ${K/Y}$ can have a level effect on output per worker, but has no growth effect. To see this more clearly, take logs of output per worker,

$\displaystyle \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(t) \ \ \ \ \ (5)$

and then plug in what we know about how ${A(t)}$ moves over time,

$\displaystyle \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(0) + gt. \ \ \ \ \ (6)$

The capital-output ratio affects the intercept of this line — a level effect — alongside ${A(0)}$. The slope of this line — the growth rate — is still ${g}$.

The capital/output ratio is constant along the BGP, and has no effect on the growth rate on the BGP. But what if the economy is not on the BGP? Then it will be the case that ${K/Y}$ affects the growth rate of output per worker, because the ${K/Y}$ ratio will not be constant. More precisely, the growth rate of capital/output is

$\displaystyle \frac{\dot{K/Y}}{K/Y} = \frac{\dot{K}}{K} - \frac{\dot{Y}}{Y}. \ \ \ \ \ (7)$

So the ${K/Y}$ ratio will change if capital grows more quickly or more slowly than output. First, capital accumulates as follows

$\displaystyle \dot{K} = s Y - \delta K \ \ \ \ \ (8)$

where ${\dot{K}}$ is the change in the capital stock. ${s}$ is the savings rate, the fraction of output that the economy sets aside to invest in new capital goods, so that ${sY}$ is the total amount of new investment. ${\delta}$ is the depreciation rate, the fraction of the existing capital stock that breaks or becomes obsolete at any given moment.

To find the growth rate of capital, divide through the above equation by ${K}$ to get

$\displaystyle \frac{\dot{K}}{K} = s\frac{Y}{K} - \delta. \ \ \ \ \ (9)$

You can see that the growth rate of capital depends on the capital/output ratio itself.

The growth rate of output is

$\displaystyle \frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1-\alpha)\frac{\dot{A}}{A} + (1-\alpha)\frac{\dot{L}}{L}. \ \ \ \ \ (10)$

Now, with (7), and using what we know about growth in capital and output, we have

$\displaystyle \frac{\dot{K/Y}}{K/Y} = (1-\alpha)\left(s\frac{Y}{K} - \delta - g - n \right) \ \ \ \ \ (11)$

where we’ve plugged in that ${\dot{A}/A = g}$, and ${\dot{L}/L = n}$.

Re-arranging a bit, the capital output ratio is growing if

$\displaystyle \frac{K}{Y} < \frac{s}{\delta + n + g}, \ \ \ \ \ (12)$

and growing if the capital/output ratio is larger than the value on the right-hand side. In other words, if the capital stock is relatively small, then it will have a tendency to grow faster than output, raising the ${K/Y}$ ratio. Eventually ${K/Y = s/(\delta+n+g)}$, the steady state value, and the ${K/Y}$ ratio stops changing.

What is happening to growth in output per worker? If ${K/Y < s/(\delta+n+g)}$ then the ${K/Y}$ ratio is growing, and so output per worker is growing faster than ${g}$. So the temporarily fast growth in output per worker in Germany or Japan would be because they found themselves with a ${K/Y}$ ratio below their steady state value. How would this occur? It’s easier to see how this works if we re-write the ${K/Y}$ ratio slightly

$\displaystyle \frac{K}{Y} = \frac{K}{K^{\alpha}(AL)^{1-\alpha}} = \left(\frac{K}{AL}\right)^{1-\alpha}. \ \ \ \ \ (13)$

From this we can see that the ${K/Y}$ ratio would be particularly low if the capital stock, ${K}$, were to be reduced. This is what happened in Germany, to a large extent, after World War II. The capital stock was destroyed, so ${K/AL}$ fell sharply. This made ${K/Y}$ fall below the steady state value, which meant that there was growth in the ${K/Y}$ ratio, and so growth in output per worker greater than ${g}$.

A slightly different situation describes South Korea. There, we can think of there being a level effect on ${A}$, an advance in productivity. This also makes ${K/AL}$ fall sharply, and again causes growth in ${K/Y}$ and growth in output per worker faster than ${g}$. But in both this case and in Germany’s, as the ${K/Y}$ ratio grows it approaches the steady state value and growth in output per worker slows down to ${g}$ again.

# The Simplest Growth Model

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This is an idea for a new way of introducing growth theory. Given that productivity growth is the source of long-run growth, it seems to make sense to start with that, rather than with the Solow model.

Let’s write down a very simple model of economic growth. Let total output ${Y}$ be determined by

$\displaystyle Y = A L \ \ \ \ \ (1)$

where ${A}$ is a measure of labor productivity, and ${L}$ is the number of workers. If we divide through by ${L}$, then we get a measure of output per worker. To keep notation clean, let ${y = Y/L}$ be output per worker, so that now we have

$\displaystyle y = A \ \ \ \ \ (2)$

as our model of economic growth. Basically, output per worker is simply equal to labor productivity ${A}$.

From this we know that the time path of output per worker is simply the same as the time path of labor productivity, ${A}$. So what determines the time path of labor productivity? We’ll assume that it is growing at a constant rate, meaning that it goes up by the same percent every period of time,

$\displaystyle A(t) = A(0) e^{g t}. \ \ \ \ \ (3)$

Here, we’ve written ${A(t)}$ to be clear that we mean labor productivity at any given time ${t}$. ${A(0)}$ is labor productivity in the initial moment of time. The exponential term says that labor productivity grows at the rate ${g}$ over time.

The exponential term implies, perhaps not surprisingly, exponential growth. You get exponential growth when something goes up by the same percent every period of time. If ${g = 0.02}$, then we have 2% growth. At time zero, labor productivity is just ${A(0)}$. When ${t=2}$, then ${A(2) = A(0)e^{.02(2)} = 1.041 A(0)}$, or labor productivity is a little more than 4% higher than at time zero. When ${t=10}$, ${A(10)=A(0)e{.02(10)}=1.221}$, or labor productivity is more than 22% higher than at time zero.

It may not seem obvious, but output per worker in the U.S. and most other developed nations displays exponential growth. Our model matches that, as

$\displaystyle y(t) = A(0) e^{g t}. \ \ \ \ \ (4)$

These countries also tend to have a similar growth rate of about 1.8%, or ${g=0.018}$. Seeing this in a figure, though, is difficult. Graphing ${y}$ over time for the U.S. gives you a curve that quickly accelerates upwards and is almost off the page. Graphs like this will also make it difficult to compare countries to one another.

For that reason, among others, we like to work with the natural log of output per worker, ${\ln{y(t)}}$. Taking natural logs of ${y(t)}$ gives us

$\displaystyle \ln{y(t)} = \ln{A(0)} + g t. \ \ \ \ \ (5)$

This is an equation that says the natural log of output per worker is a linear function of time, ${t}$. If we graph ${\ln{y(t)}}$ against ${t}$, we get a straight line, similar to the trend line we drew in the figure for U.S. output per worker.

We can calculate the growth rate of output per worker by taking the derivative of (5) with respect to time. This results in the following

$\displaystyle \frac{\dot{y}}{y} = g. \ \ \ \ \ (6)$

The value of ${A(0)}$ is fixed, so the derivative of it with respect to time is just zero. The notation ${\dot{y}/y}$ is a shorthand way of writing the growth rate. ${\dot{y}}$ is the absolute change in output per worker at any given moment, and by dividing by ${y}$ we get that change relative to the level of output per worker. This means that ${\dot{y}/y}$ is essentially the percent change in output per worker at any given moment.

That’s it for the simple growth model. Output per worker depends on labor productivity ${A(t)}$, and labor productivity grows at a constant rate ${g}$, which means output per worker grows at that same rate. Despite the mechanical simplicity, this model helps us be clear when we are talking about the growth experiences of different countries. It allows us to distinguish between two forces determining output per worker.

• Level effects: These refer to ${A(0)}$, the intercept of the line in (5)
• Growth effects: These refer to ${g}$, the slope of the line in (5)

Looking at the data over the long run, the general impression we get that the growth rate ${g}$ is similar across countries, and they differ mainly because of level effects. That is, ${A(0)_{Japan}}$ appears to be lower than ${A(0)_{US}}$, but the growth rate ${g}$ is very similar. Theories of economic growth should be consistent with these facts. Things like investment rates, schooling, and social infrastructure are important determinants of level effects, ${A(0)}$, but they have no effect on the growth rate, ${g}$. Under plausible assumptions, theories of endogenous innovation will suggest that the growth rate, ${g}$, is identical across countries.

There are some facts, though, that this simple growth model cannot account for. Namely, there are notable cases where output per worker grows more quickly or more slowly than ${g}$. China, for example, over the last 30 years has grown much faster than the U.S. or Japan. South Korea had a similar growth miracle, starting in about 1960 and lasting until the 2000’s. Germany, from World War II until about 1980, grew at a very accelerated pace compared to the U.S. in the same period. How do we reconcile these facts with the assertion above that ${g}$ is the same for all countries?

The key is noting that these growth accelerations were temporary. Germany grew very quickly, but after 1980 its growth rate fell back to a value nearly identical to the U.S. South Korea’s growth rate has diminshed as well in the 2000’s. What appears to be happening is that once output per worker approaches a frontier level, generally defined by the U.S., growth slows down. While China continues to grow quickly, it has not approached the U.S. level of output per worker.

Looking at these countries, what appears to be happening is that there is a level effect, or their ${A(0)}$ has shifted up. However, it seems to take them a long time to move from their old level to the new, higher level. We call the temporary growth spurt that occurs when a country moves between levels transitional growth. Output per worker grows faster than ${g}$ temporarily – although this could last a few decades – but then growth returns to the rate ${g}$.

Our simple model doesn’t offer a way of understanding this transitional growth. The first major extension we’ll make to this simple model is to add physical capital, which has to be slowly accumulated over time. Because of this slow accumulation, the economy will take an extended time to fully respond to a level effect.