# Measuring Misallocation across Firms

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

One of the most active area of research in macro development (let’s not call it growth economics, I guess) is on misallocations. This is the concept that a major explanation for why some countries are rich, while others are poor, is that rich countries do a good job of allocating factors of production in an efficient manner across firms (or sectors, if you like). Poor countries do a bad job of this, so that unproductive firms use a lot of labor and capital up, while high productivity firms are too small.

One of the baseline papers in this area is by Diego Restuccia and Richard Rogerson, who showed that you could theoretically generate big losses to aggregate measures of productivity if you introduced firm-level distortions that acted like subsidies to unproductive firms and taxes on produtive firms. This demonstrated the possible effects of misallocation. A paper by Hsieh and Klenow (HK) took this idea seriously, and applied the basic logic to data on firms from China, India and the U.S. to see how big of an effect misallocations actually have.

We just went over this paper in my grad class this week, and so I took some time to get more deeply into the paper. The one flaw in the paper, from my perspective, was that it ran backwards. That is, HK start with a detailed model of firm-level activity and then roll this up to find the aggregate implications. Except that I think you can get the intuition of their paper much more easily by thinking about how you measure the aggregate implications, and then asking yourself how you can get the requisite firm-level data to make the aggregate calculation. So let me give you my take on the HK paper, and how to understand what they are doing. If you’re seriously interested in studying growth and development, this is a paper you’ll need to think about at some point, and perhaps this will help you out.

This is dense with math and quite long. You were warned.

What do HK want to do? They want to compare the actual measured level of TFP in sector ${s}$, ${TFP_s}$, to a hypothetical level of TFP in sector ${s}$, ${TFP^{\ast}_s}$, that we would measure if we allocated all factors efficiently between firms.

Let’s start by asking how we can measure ${TFP_s}$ given observable data on firms. This is

$\displaystyle TFP_s = \frac{Y_s}{K_s^{\alpha}L_s^{1-\alpha}}, \ \ \ \ \ (1)$

which is just measuring ${TFP_s}$ for a sector as a Solow residual. ${TFP_s}$ is not a pure measure of “technology”, it is a measure of residual productivity, capturing everything that influences how much output (${Y_s}$) we can get from a given bundle of inputs (${K_s^{\alpha}L_s^{1-\alpha}}$). It includes not just the physical productivity of individual firms in this sector, but also the efficiency of the distribution of the factors across those firms.

Now, the issue is that we cannot measure ${Y_s}$ directly. For a sector, this is some kind of measure of real output (e.g. units of goods), but there is no data on that. The data we have is on revenues of firms within the sector (e.g. dollars of goods sold). So what HK are going to do is use this revenue data, and then make some assumptions about how firms set prices to try and back out the real output measure. It’s actually easier to see in the math. First, just write ${TFP_s}$ as

$\displaystyle TFP_s =\frac{P_s Y_s}{K_s^{\alpha}L_s^{1-\alpha}}\frac{1}{P_s} = \overline{TFPR}_s \frac{1}{P_s} \ \ \ \ \ (2)$

which just multiplies and divides by the price index for sector ${s}$. The first fraction is revenue productivity, or ${\overline{TFPR}_s}$, of sector ${s}$. This is a residual measure as well, but measures how produtive sector ${s}$ is at producing dollars, rather than at producing units of goods. The good thing about ${TFPR_s}$ is that we can calculate this from the data. Take the revenues of all the firms in sector ${s}$, and that is equal to total revenues ${P_s Y_s}$. We can add up the reported capital stocks across all firms, and labor forces across all firms, and get ${K_s}$ and ${L_s}$, respectively. We can find a value for ${\alpha}$ based on the size of wage payments relative revenues (which should be close to ${1-\alpha}$). So all this is conceptually measurable.

The second fraction is one over the price index ${P_s}$. We do not have data on this price index, because we don’t know the individual prices of each firms output. So here is where the assumptions regarding firm behavior come in. HK assume a monopolistically competitive structure for firms within each sector. This means that each firm has monopoly power over producing its own brand of good, but people are willing to substitute between those different brands. As long as the brands aren’t perfectly substitutable, then each firm can charge a price a little over the marginal cost of production. We’re going to leave aside the micro-economics of that structure for the time being. For now, just trust me that if these firms are monopolistically competitive, then the price index can be written as

$\displaystyle P_s = \left(\sum_i P_i^{1-\sigma} \right)^{1/(1-\sigma)} \ \ \ \ \ (3)$

where ${P_i}$ are the individual prices from each firm, and ${\sigma}$ is the elasticity of substitution between different firms goods.

Didn’t I just say that we do not observe those individual firm prices? Yes, I did. But we don’t need to observe them. For any individual firm, we can also think of revenue productivity as opposed to their physical productivity, denoted ${A_i}$. That is, we can write

$\displaystyle TFPR_i = P_i A_i. \ \ \ \ \ (4)$

The firms productivity at producing dollars (${TFPR_i}$) is the price they can charge (${P_i}$) times their physical productivity (${A_i}$). We can re-arrange this to be

$\displaystyle P_i = \frac{TFPR_i}{A_i}. \ \ \ \ \ (5)$

Put this expression for firm-level prices into the price index ${P_s}$ we found above. You get

$\displaystyle P_s = \left(\sum_i \left[\frac{TFPR_i}{A_i}\right]^{1-\sigma} \right)^{1/(1-\sigma)} \ \ \ \ \ (6)$

which depends only on firm-level measure of ${TFPR_i}$ and physical productivity ${A_i}$. We no longer need prices.

For the sector level ${TFP_s}$, we now have

$\displaystyle TFP_s = \overline{TFPR}_s \frac{1}{P_s} = \frac{\overline{TFPR}_s}{\left(\sum_i \left[\frac{TFPR_i}{A_i}\right]^{1-\sigma} \right)^{1/(1-\sigma)}}. \ \ \ \ \ (7)$

At this point, there is just some slog of algebra to get to the following

$\displaystyle TFP_s = \left(\sum_i \left[A_i \frac{\overline{TFPR}_s}{TFPR_i}\right]^{\sigma-1} \right)^{1/(\sigma-1)}. \ \ \ \ \ (8)$

If you’re following along at home, just note that the exponents involving ${\sigma}$ flipped sign, and that can hang you up on the algebra if you’re not careful.

Okay, so now I have this description of how to measure ${TFP_s}$. I need information on four things. (1) Firm-level physical productivities, ${A_i}$, (2) sector-level revenue productivity, ${\overline{TFPR}_s}$, (3) firm-level revenue productivities, ${TFPR_i}$, and (4) a value for ${\sigma}$. Of these, we can appeal to the literature and assume a value of ${\sigma}$, say something like a value of 5, which implies goods are fairly substitutable. We can measure sector-level and firm-level revenue productivities directly from the firm-level data we have. The one big piece of information we don’t have is ${A_i}$, the physical productivity of each firm.

Before describing how we’re going to find ${A_i}$, just consider this measurement of ${TFP_s}$ for a moment. What this equation says is that ${TFP_s}$ is a weighted sum of the individual firm level physical productivity terms, ${A_i}$. That makes some sense. Physical productivity of a sector must depend on the productivity of the firms in that sector.

Mechanically, ${TFP_s}$ is a concave function of all the stuff in the parentheses, given that ${1/(\sigma-1)}$ is less than one. Meaning that ${TFP_s}$ goes up as the values in the summation rise, but at a decreasing rate. More importantly, for what HK are doing, this implies that the greater the variation in the individual firm-level terms of the summation, the lower is ${TFP_s}$. That is, you’d rather have two firms that have similar productivity levels than one firm with a really big productivity level and one firm with a really small one. Why? Because we have imperfect substitution between the output of the firms. Which means that we’d like to consume goods in somewhat rigid proportions (think Leontief perfect complements). For example, I really like to consume one pair of pants and one shirt at the same time. If the pants factory is really, really productive, then I can lots of pants for really cheap. If the shirt factory is really un-productivie, I can only get a few shirts for a high price. To consume pants/shirts in the desired 1:1 ratio I will end up having to shift factors away from the pants factor and towards the shirt factory. This lowers my sector level productivity.

There is nothing that HK can or will do about variation in ${A_i}$ across firms. That is taken as a given. Some firms are more productive than others. But what they are interested in is the variation driven by the ${TFPR_i}$ terms. Here, we just have the extra funkiness that the summation depends on these inversely. So a firm with a really high ${TFPR_i}$ is like having a really physically unproductive firm. Why? Think in terms of the prices that firms charge for their goods. A high ${TFPR_i}$ means that firms are charging a relatively high price compared to the rest of the sector. Similarly, a firm with a really low ${A_i}$ (like our shirt factory above) would also be charging a relatively high price compared to the rest of the sector. So having variation in ${TFPR_i}$ across firms is like having variation in ${A_i}$, and this variation lowers ${TFP_s}$.

However, as HK point out, if markets are operating efficiently then there should be no variation in ${TFPR_i}$ across firms. While a high ${TFPR_i}$ is similar to a low ${A_i}$ in its effect on ${TFP_s}$, the high ${TFPR_i}$ arises for a fundamentally different reason. The only reason a firm would have a high ${TFPR_i}$ compared to the rest of the sector is if it faced higher input costs and/or higher taxes on revenues than other firms. In other words, firms would only be charging more than expected if they had higher costs than expected or were able to keep less of their revenue.

In the absence of different input costs and/or different taxes on revenues, then we’d expect all firms in the sector to have identical ${TFPR_i}$. Because if they didn’t, then firms with high ${TFPR_i}$ could bid away factors of production from low ${TFPR_i}$ firms. But as high ${TFPR_i}$ firms get bigger and produce more, the price they can charge will get driven down (and vice versa for low ${TFPR_i}$ firms), and eventually the ${TFPR_i}$ terms should all equate.

For HK, then, the level of ${TFP_s}$ that you could get if all factors were allocated efficiently (meaning that firms didn’t face differential input costs or revenue taxes) is one where ${TFPR_i = \overline{TFPR}_s}$ for all firms. Meaning that

$\displaystyle TFP^{\ast}_s = \left(\sum_i A_i^{\sigma-1} \right)^{1/(\sigma-1)}. \ \ \ \ \ (9)$

So what HK do is calculate both ${TFP^{\ast}_s}$ and ${TFP_s}$ (as above), and compare.

To do this, I already mentioned that the one piece of data we are missing is the ${A_i}$ terms. We need to know the actual physical productivity of firms. How do we get that, since we cannot measure physical output at the firm level? HK’s assumption about market structure will allow us to figure that out. So hold on to the results of ${TFP_s}$ and ${TFP^{\ast}_s}$ for a moment, and let’s talk about firms. For those of you comfortable with monopolistic competition models using CES aggregators, this is just textbook stuff. I’m going to present it without lots of derivations, but you can check my work if you want.

For each firm, we assume the production function is

$\displaystyle Y_i = A_i K_i^{\alpha}L_i^{1-\alpha} \ \ \ \ \ (10)$

and we’d like to back out ${A_i}$ as

$\displaystyle A_i = \frac{Y_i}{K_i^{\alpha}L_i^{1-\alpha}} \ \ \ \ \ (11)$

but we don’t know the value of ${Y_i}$. So we’ll back it out from revenue data.

Given that the elasticity of substitution across firms goods is ${\sigma}$, and all firms goods are weighted the same in the utility function (or final goods production function), then the demand curve facing each firm is

$\displaystyle P_i = Y_i^{(\sigma-1)/\sigma - 1} X_s \ \ \ \ \ (12)$

where ${X_s}$ is a demand shifter that depends on the amount of the other goods consumed/produced. We going to end up carrying this term around with us, but it’s exact derivation isn’t necessary for anything. Total revenues of the firm are just

$\displaystyle (P_i Y_i) = Y_i^{(\sigma-1)/\sigma} X_s. \ \ \ \ \ (13)$

Solve this for ${Y_i}$, leaving ${(P_i Y_i)}$ together as revenues. This gives you

$\displaystyle Y_i = \left(\frac{P_i Y_i}{X_s}\right)^{\sigma/(\sigma-1)}. \ \ \ \ \ (14)$

Plug this in our equation for ${A_i}$ to get

$\displaystyle A_i = \frac{1}{X_s^{\sigma/(\sigma-1)}}\frac{\left(P_i Y_i\right)^{\sigma/(\sigma-1)}}{K_i^{\alpha}L_i^{1-\alpha}}. \ \ \ \ \ (15)$

This last expression gives us a way to back out ${A_i}$ from observable data. We know revenues, ${P_i Y_i}$, capital, ${K_i}$, and labor, ${L_i}$. The only issue is this ${X_s}$ thing. But ${X_s}$ is identical for each firm – it’s a sector-wide demand term – so we don’t need to know it. It just scales up or down all the firms in a sector. Both ${TFP_s}$ and ${TFP^{\ast}_s}$ will be proportional to ${X_s}$, so when comparing them ${X_s}$ will just cancel out. We don’t need to measure it.

What is our ${A_i}$ measure picking up? Well, under the assumption that firms in fact face a demand curve like we described, then ${A_i}$ is picking up their physical productivity. If physical ouput, ${Y_i}$, goes up then so will revenues, ${P_i Y_i}$. But not proportionally, as with more output the firm will charge a lower price. Remember, the pants factory has to get people to buy all those extra pants, even though they kind of don’t want them because there aren’t many shirts around. So the price falls. Taking revenues to the ${\sigma/(\sigma-1)}$ power captures that effect.

Where are we? We now have a firm-level measure of ${A_i}$, and we can measure it from observable data on revenues, capital stocks, and labor forces at the firm level. This allows us to measure both actual ${TFP_s}$, and the hypothetical ${TFP^{\ast}_s}$ when each firm faces identical factor costs and revenues taxes. HK compare these two measures of TFP, and find that in China ${TFP^{\ast}_s}$ is about 86-115% higher than ${TFP_s}$, or that output would nearly double if firms all faced the same factor costs and revenue taxes. In India, the gain is on the order of 100-120%, and for the U.S. the gain is something like 30-43%. So substantial increases all the way around, but much larger in the developing countries. Hence HK conclude that misallocations – meaning firms facing different costs and/or taxes and hence having different ${TFPR_i}$ – could be an important explanation for why some places are rich and some are poor. Poor countries presumably do a poor job (perhaps through explicit policies or implicit frictions) in allocating resources efficiently between firms, and low-productivity firms use too many inputs.

* A note on wedges * For those of you who know this paper, you’ll notice I haven’t said a word about “wedges”, which are the things that generate differences in factor costs or revenues for firms. That’s because from a purely computational standpoint, you don’t need to introduce them to get HK’s results. It’s sufficient just to measure the ${TFPR_i}$ levels. If you wanted to play around with removing just the factor cost wedges or just the revenue wedges, you would then need to incorporate those explicitly. That would require you to follow through on the firms profit maximization problem and solve for an explicit expression for ${TFPR_i}$. In short, that will give you this:

$\displaystyle TFPR_i = \frac{\sigma}{\sigma-1} MC_s \frac{(1+\tau_{Ki})^{\alpha}}{1-\tau_{Yi}}. \ \ \ \ \ (16)$

The first fraction, ${\sigma/(\sigma-1)}$, is the markup charged over marginal cost by the firm. As the elasticity of substitution is assumed to be constant, this markup is identical for each firm, so generates no variation in ${TFPR_i}$. The second term, ${MC_s}$, is the marginal cost of a bundle of inputs (capital and labor). The final fraction are the “wedges”. ${(1+\tau_{Ki})}$ captures the additional cost (or subsidy if ${\tau_{Ki}<0}$) of a unit of capital to the firm relative to other firms. ${(1-\tau_{Yi})}$ captures the revenue wedge (think of a sales tax or subsidy) for a firm relative to other firms. If either of those ${\tau}$ terms are not equal to zero, then ${TFPR_i}$ will deviate from the efficient level.

* A note on multiple sectors * HK do this for all manufacturing sectors. That’s not a big change. Do what I said for each separate sector. Assume that each sector has a constant share of total expenditure (as in a Cobb-Douglas utility function). Then

$\displaystyle \frac{TFP^{\ast}_{all}}{TFP_{all}} = \left(\frac{TFP^{\ast}_1}{TFP_1}\right)^{\theta_1} \times \left(\frac{TFP^{\ast}_2}{TFP_2}\right)^{\theta_2} \times ... \ \ \ \ \ (17)$

where ${\theta_s}$ is the expenditure share of sector ${s}$.

# Productivity Pessimism from Productivity Optimists

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The projected future path of labor productivity in the U.S. is perhaps the most important input to the projected future path of GDP in the U.S. There are lots of estimates floating around, many of them pessimistic in the sense that they project labor productivity growth to be relatively slow (say 1.5-1.8% per year) over the next few decades compared to the relatively fast rates (roughly 3% per year) seen from 1995-2005. Robert Gordon has laid out the case for low labor productivity growth in the future. John Fernald has documented that this slowdown probably pre-dates the Great Recession, and reflects a loss of steam in the IT revolution starting in about 2007. This has made Brad DeLong sad, which seems like the appropriate response to slowing productivity growth.

An apparent alternative to that pessimism was published recently by Byrne, Oliner, and Sichel. Their paper is titled “Is the IT Revolution Over?”, and their answer is “No”. They suggest that continued innovation in semi-conductors could make possible another IT boom, and boost labor productivity growth in the near future above the pessimistic Gordon/Fernald rate of 1.5-1.8%.

I don’t think their results, though, are as optimistic as they want them to be. A different way of saying this is: you have to work really hard to make yourself optimistic about labor productivity growth going forward. In their baseline estimate, they end up with labor productivity growth of 1.8%, which is slightly higher than the observed rate of 1.56% per year from 2004-2012. To get themselves to their optimistic prediction of 2.47% growth in labor productivity, they have to make the following assumptions:

1. Total factor productivity (TFP) growth in non-IT producing non-farm businesses is 0.62% per year, which is roughly twice their baseline estimate of 0.34% per year, and ten times the observed rate from 2004-2012 of 0.06%.
2. TFP growth in IT-producing industries is 0.46% per year, slightly higher than their baseline estimate of 0.38% per year, and not quite double the observed rate from 2004-2012 of 0.28%
3. Capital deepening (which is just fancy econo-talk for “build more capital”) adds 1.34% per year to labor productivity growth, which is one-third higher than their baseline rate of 1.03% and, and double the observed rate from 2004-2012 of 0.74%

The only reason their optimistic scenario doesn’t get them back to a full 3% growth in labor productivity is because they don’t make any optimistic assumptions about labor force quality/participation growth.

Why these optimistic assumptions in particular? For the IT-producing industries, the authors get their optimistic growth rate of 0.46% per year by assuming that prices for IT goods (e.g. semi-conductors and software) fall at the fastest pace observed in the past. The implication of very dramatic price declines is that productivity in these sectors must be rising very quickly. So essentially, assume that IT industries have productivity growth as fast as in the 1995-2005 period. For the non-IT industries, they assume that faster IT productivity growth will raise non-IT productivity growth to it’s upper bound in the data, 0.62%. Why? No explanation is given. Last, the more rapid pace of productivity growth in IT and non-IT will induce faster capital accumulation, meaning that its rate rises to 1.34% per year. This last point is one that comes out of a simple Solow-type model of growth. A shock to productivity will temporarily increase capital accumulation.

In the end here is what we’ve got: they estimate labor productivity will grow very fast if they assume labor productivity will grow very fast. Section IV of their paper gives more detail on the semi-conductor industry and the compilation of the price indices for that industry. Their narrative explains that we could well be under-estimating how fast semi-conductor prices are falling, and thus under-estimating how fast productivity in that industry is rising. Perhaps, but this doesn’t necessarily imply that the rest of the IT industry is going to experience rapid productivity growth, and it certainly doesn’t necessarily imply that non-IT industries are going to benefit.

Further, even rapid growth in productivity in the semi-conductor industry is unlikely to create any serious boost to US productivity growth, because the semi-conductor industry has a shrinking share of output in the U.S. over time. The above figure is from their paper. The software we run is a booming industry in the U.S., but the chips running that software are not, and this is probably in large part due to the fact that those chips are made primarily in other countries. If you want to make an optimistic case for IT-led productivity growth in the U.S., you need to make a software argument, not a hardware argument.

I appreciate that Byrne, Oliner, and Sichel want to provide an optimistic case for higher productivity growth. But that case is just a guess, and despite the fact that they can lay some numbers out to come up with a firm answer doesn’t make it less of a guess. Put it this way, I could write a nearly exact duplicate of their paper which makes the case that expected labor productivity growth is only something like 0.4% per year simply by using all of the lower bound estimates they have.

Ultimately, there is nothing about recent trends in labor productivity growth that can make you seriously optimistic about future labor productivity growth. But that doesn’t mean optimism is completely wrong. That’s simply the cost of trying to do forecasting using existing data. You can always identify structural breaks in a time series after the fact (e.g. look at labor productivity growth in 1995), but you cannot ever predict a structural break in a time series out of sample. Maybe tomorrow someone will invent cheap-o solar power, and we’ll look back ten years from now in wonder at the incredible decade of labor productivity growth we had. But I can’t possibly look at the existing time series on labor productivity growth and get any information on whether that will happen or not. Like it or not, extrapolating current trends gives us a pessimistic growth rate of labor productivity. Being optimistic means believing in a structural break in those trends, but there’s no data that can make you believe.

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

So I spotted this article in the Guardian, by one Damian Carrington, who gives us an example of how not to write about new research. The article is about the release of this paper in Science, by Gerland et al.

Let’s take a little walk through the article to see how Mr. Carrington mangles nearly everything important about this research.

1. The title of the article is “World population to hit 11bn in 2100”. The Gerland et al article is about using Bayesian techniques to arrive at confidence intervals for the size of global population, meaning that their article is about how much uncertainty there is in a population projection. The entire point of their work is that statements like “World population to hit 11bn in 2100” are stupid because they do not tell you about the uncertainty in that estimate.
2. “A ground-breaking analysis….” is how the Gerland et al article is introduced. I’m sure Gerland and his co-authors are very capable scholars. But this is not ground-breaking analysis. How do I know that? Because they base all their work on the existing 2012 United Nations Population Projections, so they do not fundamentally change our estimates of future population. What they do add is the Bayesian analysis to give more accurate confidence intervals to those United Nations projections. This technique was developed by some of the co-authors a while ago, see this paper. *That* technique could arguably called ground-breaking, but the current Science paper is not.
3. “The work overturns 20 years of consensus that global population, and the stresses it brings, will peak by 2050 at about 9bn people.” What consensus? The UN’s 2012 population projection that this Science article is based on predicts that population will be 11 billion by 2100, and that it will still be growing at that point. The UN population projection in 2010 also predicted population would be 11 billion in 2100. Population projects by the UN from around 2000 suggest that population would hit 9 billion in 2050, but never said it would max out there. The UN just didn’t project out populations past 2050 back then.
4. “Lack of healthcare, poverty, pollution and rising unrest and crime are all problems linked to booming populations, he [Prof. Adrian Raftery, U. of Washington] said.” Mr. Carrington does not feel compelled to support these statements by citing any evidence that (a) the links exist and (b) are causal. I’d like to think that Prof. Raftery at least tried to provide this kind of evidence, but we don’t know.
5. “The research, conducted by an international team including UN experts, is published in the journal Science and for the first time uses advanced statistics to place convincing upper and lower limits on future population growth.” Statistics? No – advanced statistics. See the difference? One is more advanced. Convincing? Convincing of what? That upper and lower limits exist? Of what relevance is it that the team was international? Do the advanced statistics require people with different passports to run the code? This is just such a ridiculous sentence. The stupid, it burns us.
6. “But the new research narrows the future range to between 9.6bn and 12.3bn by 2100. This greatly increased certainty – 80% – allowed the researchers to be confident that global population would not peak any time during in the 21st century.” They didn’t increase certainty at all. The prior UN projections were mechanical, and had no uncertainty associated with them at all. Gerland et al have created confidence intervals where none existed. This didn’t increase certainty, it quantified it.

By the way, it took me all of about 10 minutes on the Google machine to find the references I just cited here, and to look up the old UN projections. And I didn’t use any special PhD kung-fu to do this. So I don’t want to hear “well, science is hard for the layman to understand”. This is click-bait crap reporting, period.

# Slow Growth in Potential GDP for the U.S.?

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Robert Gordon released a paper recently where he presents his estimates of potential GDP for the U.S. going forward. I had planned on writing a longer post discussing about why you should take his projections seriously, and maybe some speculation about what would have to happen to reverse his conclusions. Seriously, I had a few paragraphs written, a couple figures cut out and ready to go, and then Jim Hamilton put up precisely the post I wanted to write. So the first thing you should do is go read Jim’s post.

…and if you are still here, let me provide an uber-quick summary of my own before I talk about how you could convince yourself that Gordon is overly-pessimistic (he’s probably not, but if you want to rock yourself to sleep at night, this might help).

So, what does Gordon find? Basically, GDP growth is equal to growth in GDP per hour worked (productivity) times growth in hours worked. Hours worked are unlikely to grow much, given that unemployment has already fallen back to 6%, that average hours per worker have recovered much of their decline, and that the labor force participation rate is unlikely to recover much of its recent decline. So the only way for GDP to grow fast enough to hit our prior level of potential GDP (which is essentially what the CBO projects it will do) is for productivity (GDP per hour) to grow much faster than it has at any time in the last decade.

You can see his implications in the above figure. The red line is Gordon’s projection for potential GDP based on his assumed lower productivity growth rate, which is closer to recent averages. The CBO potential GDP path is driven by what Gordon says are aggressive assumptions about how fast productivity will grow.

In short, we aren’t going to recover back to the pre-Great Recession trend line for potential GDP any time soon. One might quibble a little about Gordon’s assumptions, and perhaps we won’t diverge from the prior trend line (the yellow one) as much as he suggests. But it’s really hard to come up with reasonable evidence that the CBO is making the right assumption regarding productivity growth.

Now, if I want to go to bed at night believing that we might be able to get back on that prior trend line, what should I tell myself? I’m not going to tell myself that we’ll be magically saved by some kind of technology boom. It could happen, I guess, but that’s not something I could rely on, or having any way of reliably predicting.

What I might tell myself is that productivity – because of the way it is backed out of the data – is not simply a measure of technical productivity, it’s a measure of revenue productivity. I talked about this in a prior post, but the difference is that revenue productivity measures firms ability to generate dollars, not their ability to generate widgets. Revenue productivity can thus experience a temporary burst of growth if firms are able to exert some market power, in the same way that revenue productivity can experience a temporary sag if firms lose pricing power during a recession. So if some of the distinct drop in measured productivity growth over the 2008-2014 period was because firms lost pricing power (and not because of a slowdown in innovation/technology growth), then this could be recovered if firms are able to reassert that pricing power.

A few points on this. Why would firms gain (or why did they lose) pricing power? My guess is that it depends on the willingness of consumers to “shop around”, which in turn is based on economic conditions. When things get bad in 2008, people become more sensitive to price changes, and so firms lose pricing power, and hence revenue productivity falls. If consumers were to recover in the sense of becoming less sensitive to price changes, then firms could gain pricing power and that would raise measured productivity. Will that happen? My guess is yes, it will, I just don’t know when. Will that boost to revenue productivity be sufficient to put potential GDP back on the pre-Great Recession path? I don’t know.

The last thing to point out is that revenue productivity is exactly what we want to measure in Gordon’s case, where he ultimately is worried about Debt/GDP ratios. Because the debt is denominated in dollars, what I care about is the economy’s ability to generate dollars, not widgets. There’s an entirely different post to be written about why the Debt/GDP ratio is a stupid way to measure the debt burden, but I’ll leave that alone for now.

In short, if you want to be optimistic about bouncing back to the pre-Great Recession trend for potential GDP, then part of that optimism is that firms regain lost pricing power, and thus experience a boost to their revenue productivity. This can occur in the absence of any change in the underlying pace of real technological change, and isn’t tied to our expectations about the usefulness or arrival of new technologies.

# Taxes and Growth

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

William Gale and Andy Samwick have a new Brookings paper out on the relationship of tax rates and economic growth in the U.S. [Apologies to whichever blog/site led me to the paper, I can’t remember.] Short answer, there is no relationship. They do not identify any change in the trend growth rate of real GDP per capita with changes in marginal income tax rates, capital gains tax rates, or any changes in federal tax rules.

One of the first pieces of evidence they show is from a paper by Stokey and Rebelo (1995). This plots taxes as a percent of GDP in the top panel, and the growth rate of GDP per capita in the lower one. You can see that the introduction of very high tax rates during WWII, which effectively became permanent features of the economy after that, did not change the trend growth rate of GDP per capita in the slightest. The only difference after 1940 in the lower panel is that the fluctuations in the economy are less severe that in the prior period. Taxes as a percent of GDP don’t appear to have any relevant relationship to growth rates.

The next piece of evidence is from a paper by Hungerford (2012), who basically looks only at the post-war period, and looks at whether the fluctuations in top marginal tax rates (on either income or capital gains) are related to growth rates. You can see in the figure that they are not. If anything, higher capital gains rates are associated with faster growth.

The upshot is that there is no evidence that you can change the growth rate of the economy – up or down – by changing tax rates – up or down. Their conclusion is more coherent than anything I could gin up, so here goes:

The argument that income tax cuts raise growth is repeated so often that it is sometimes taken as gospel. However, theory, evidence, and simulation studies tell a different and more complicated story. Tax cuts offer the potential to raise economic growth by improving incentives to work, save, and invest. But they also create income effects that reduce the need to engage in productive economic activity, and they may subsidize old capital, which provides windfall gains to asset holders that undermine incentives for new activity.

The effects of tax cuts on growth are completely uncertain.

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

I saw this article on the Atlantic by Jeremy Caradonna, a professor of history at the U. of Alberta. It’s about whether “progress” is good for humanity. The article takes particular aim at “progress” as a concept associated with sustained economic growth since the Industrial Revolution.

The first point to make is that Caradonna mischaracterizes the conclusions that economic historians and growth economists make about the moral character of growth after the Industrial Revolution. None of them, at least the ones I’ve read, and I’ve read a lot of them, have ever suggested that humanity is morally superior for having achieved sustained growth. Here’s the quote he pulls from Joel Mokyr’s The Enlightened Economy

Material life in Britain and in the industrialized world that followed it is far better today than could have been imagined by the most wild-eyed optimistic 18th-century philosophe—and whereas this outcome may have been an unforeseen consequence, most economists, at least, would regard it as an undivided blessing.

And here is Caradonna’s reaction to that quote:

The idea that the Industrial Revolution has made us not only more technologically advanced and materially furnished but also better for it is a powerful narrative and one that’s hard to shake.

The only sense in which Mokyr means “we’re better for it” is precisely that it made us more materially furnished. We are superior in real consumption. Full stop. Nowhere does Mokyr make a claim that this superiority in real consumption implies any kind of superiority in virtue, morality, or ethics.

We are shockingly, amazingly, well off on a material basis compared to our ancestors not only 200 years ago even thirty years ago. This despite the fact that the population of the earth is now roughly 7-8 times higher than it was when the Industrial Revolution started.

So Caradonna has set up a straw man to take down. Fine, he’s hardly the first person to do that. What’s his real argument, then? Let me take a stab at summarizing it. After the Industrial Revolution, bad things happened in addition to good things. Caradonna thinks those bad things are particularly bad, and thinks we should give up some of the good things (gas-powered cars) in order to alleviate the bad things (global warming).

Okay. Great. I’m with you Prof. Caradonna. Seriously, I’m in for a carbon tax and expanded spending on alternative energy R-and-D. I want to drive around either an electric car, or one powered by hydrogen, or using gas produced by algae that actually pulls CO2 from the atmosphere.

But the idea that economic growth – progress – is somehow the enemy of that goal is misguided. To paraphrase Homer Simpson: “To economic growth, the cause – and solution – to all of life’s problems”. Economic growth created the conditions that allowed us to alleviate evils like starvation and infant mortality while at the same time giving us more clothes, better housing, faster ways to get around, means of communication, Diet Coke, and gigantic-ass TV’s. It also bequeathed us technologies that heat up the atmosphere. And that sucks. But it sucks less than starving.

Economic growth means we’ve got a new kind of constrained optimization problem to solve in the 21st century: how to maximize real consumption while minimizing environmental damage. Caradonna has a particular type of solution to that optimization in mind, one tilted more towards minimizing damage than maximizing consumption. But the world seems to be making a different kind of choice, and so he’s trying to persuade others to adopt his solution. More power to him. There is no one who can tell him (particularly me) that his choice of how to solve that optimization problem is wrong. It’s just about preferences.

But anything that alleviates the constraints in this problem is welcome, regardless of preferences. Innovations that mitigate global warming (or other environmental concerns) would help us regardless of our exact preferred solution. If we can invent hyper-efficient spray-on solar panels, that would be an incredible boon to humanity. Cheap, clean power. Everyone wins. You know what I would call something like that? Progress.

The underlying issue is not a concept like progress or economic growth, but the fact that constraints exist.

# Robots as Factor-Eliminating Technical Change

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

A really common thread running through the comments I’ve gotten on the blog involve the replacement of labor. This is tied into the question of the impact of robots/IT on labor market outcomes, and the stagnation of wages for lots of laborers. An intuition that a lot of people have is that robots are going to “replace” people, and this will mean that wages fall and more and more of output gets paid to the owners of the robots. Just today, I saw this figure (h/t to Brad DeLong) from the Center on Budget and Policy Priorities which shows wages for the 10th and 20th percentile workers in the U.S. being stagnant over the last 40 years.

The possible counter-arguments to this are that even with robots, we’ll just find new uses for human labor, and/or that robots will relieve us of the burden of working. We’ll enjoy high living standards without having to work at it, so why worry?

I’ll admit that my usual reaction is the “but we will just find new kinds of jobs for people” type. Even though capital goods like tractors and combines replaced a lot of human labor in agriculture, we now employ people in other industries, for example. But this assumes that labor is somehow still relevant somewhere in the economy, and maybe that isn’t true. So what does “factor-eliminating” technological change look like? As luck would have it, there’s a paper by Pietro Peretto and John Seater called …. “Factor-eliminating Technical Change“. Peretto and Seater focus on the dynamic implications of the model for endogenous growth, and whether factor-eliminating change can produce sustained growth in output per worker. They find that it can under certain circumstances. But the model they set is also a really useful tool for thinking about what the arrival of robots (or further IT innovations in general) may imply for wages and income distribution.

I’m going to ignore the dynamics that Peretto and Seater work through, and focus only on the firm-level decision they describe.

****If you want to skip technical stuff – jump down to the bottom of the post for the punchline****

Firms have a whole menu of available production functions to choose from. The firm-level functions all have the same structure, ${Y = A X^{\alpha}Z^{1-\alpha}}$, and vary only in their value of ${\alpha \in (0,\overline{\alpha})}$. ${X}$ and ${Z}$ are different factors of production (I’ll be more specific about how to interpret these later on). ${A}$ is a measure of total factor productivity.

The idea of having different production functions to choose from isn’t necessarily new, but the novelty comes when Peretto/Seater allow the firm to use more than one of those production functions at once. A firm that has some amount of ${X}$ and ${Z}$ available will choose to do what? It depends on the amount of ${X}$ versus the amount of ${Z}$ they have. If ${X}$ is really big compared to ${Z}$, then it makes sense to only use the maximum ${\overline{\alpha}}$ technology, so ${Y = A X^{\overline{\alpha}}Z^{1-\overline{\alpha}}}$. This makes some sense. If you have lots of some factor ${X}$, then it only makes sense to use a technology that uses this factor really intensely – ${\overline{\alpha}}$.

On the other hand, if you have a lot of ${Z}$ compared to ${X}$, then what do you do? You do the opposite – kind of. With a lot of ${Z}$, you want to use a technology that uses this factor intensely, meaning the technology with ${\alpha=0}$. But, if you use only that technology, then your ${X}$ sits idle, useless. So you’ll run a ${X}$-intense plant as well, and that requires a little of the ${Z}$ factor to operate. So you’ll use two kinds of plants at once – a ${Z}$ intense one and a ${X}$ intense one. You can see their paper for derivations, but in the end the production function when you have lots of ${Z}$ is

$\displaystyle Y = A \left(Z + \beta X\right) \ \ \ \ \ (1)$

where ${\beta}$ is a slurry of terms involving ${\overline{\alpha}}$. What Peretto and Seater show is that over time, if firms can invest in higher levels of ${\overline{\alpha}}$, then by necessity it will be the case that we have “lots” of ${Z}$ compared to little ${X}$, and we use this production function.

What’s so special about this production function? It’s linear in ${Z}$ and ${X}$, so their marginal products do not decline as you use more of them. More importantly, their marginal products do not rise as you acquire more of the other input. That is, the marginal product of ${Z}$ is exactly ${A}$, no matter how much ${X}$ we have.

What does this possibly have to do with robots, stagnant wages, and the labor market? Let ${Z}$ represent labor inputs, and ${X}$ represent capital inputs. This linear production function means that as we acquire more capital (${X}$), this has no effect on the marginal product of labor (${Z}$). If we have something resembling a competitive market for labor, then this implies that wages will be constant even as we acquire more capital.

That’s a big departure from the typical concept we have of production functions and wages. The typical model is more like Peretto and Seater’s case where ${X}$ is really big, and ${Y = A X^{\overline{\alpha}}Z^{1-\overline{\alpha}}}$, a typical Cobb-Douglas. What’s true here is that as we get more ${X}$, the marginal product of ${Z}$ goes up. In other words, if we acquire more capital, then wages should rise as workers get more productive.

The Peretto/Seater setting says that, at some point, technology will progress to the point that wages stop rising with the capital stock. Wages can still go up with general total factor productivity, ${A}$, sure, but just acquiring new capital will no longer raise wages.

While wages are stagnant, this doesn’t mean that output per worker is stagnant. Labor productivity (${Y/Z}$) in this setting is

$\displaystyle \frac{Y}{Z} = A \left(1 + \beta \frac{X}{Z}\right). \ \ \ \ \ (2)$

If capital per worker (${X/Z}$) is rising, then so is output per worker. But wages will remain constant. This implies that labor’s share of output is falling, as

$\displaystyle \frac{wZ}{Y} = \frac{AZ}{A \left(Z + \beta X\right)} = \frac{Z}{\left(Z + \beta X\right)} = \frac{1}{1 + \beta X/Z}. \ \ \ \ \ (3)$

With the ability to use multiple types of technologies, as capital is acquired labor’s share of output falls.

Okay, this Peretto/Seater model gives us an explanation for stagnant wages and a declining labor share in output. Why did I present this using ${X}$ for capital and ${Z}$ for labor, not their traditional ${K}$ and ${L}$? This is mainly because the definition of what counts as “labor”, and what counts as “capital”, are not fixed. “Capital” might include human as well as physical capital, and so “labor” might mean just unskilled labor. And we definitely see that unskilled labor’s wage is stagnant, while college-educated wages have tended to rise.

***** Jump back in here if you skipped the technical stuff *****

The real point here is that whether technological change is good for labor or not depends on whether labor and capital (i.e. robots) are complements or substitutes. If they are complements (as in traditional conceptions of production functions), then adding robots will raise wages, and won’t necessarily lower labor’s share of output. If they are substistutes then adding robots will not raise wages, and will almost certainly lower labor’s share of output. The factor-eliminating model from Peretto and Seater says that firms will always invest in more capital-intense production functions and that this will inevitably make labor and capital substitutes. We happen to live in the period of time in which this shift to being substitutes is taking place. Or one could argue that it already has taken place, as we see those stagnant wages for unskilled workers, at least, from 1980 onwards.

What we should do about this is a different question. There is no equivalent mechanism or incentive here that would drive firms to make labor and capital complements again. From the firms perspective, having labor and capital as complements limits their flexibility, because they then depend on the other. They’d rather have the marginal product of robots and people independent of one other. So once we reach the robot stage of production, we’re going to stay there, absent a policy that actively prohibits certain types of production. The only way to raise labor’s share of output once we get the robots is through straight redistribution from robot owners to workers.

Note that this doesn’t mean that labor’s real wage is falling. They still have jobs, and their wages can still rise if there is total factor productivity change. But that won’t change the share of output that labor earns. I guess a big question is whether the increases in real wages from total factor productivity growth are sufficient to keep workers from grumbling about the smaller share of output that they earn.

I for one welcome….you know the rest.