Economic impacts of the coronavirus - part 2/n(?)

Posted by Dietrich Vollrath on March 21, 2020 · 19 mins read

There isn’t a real question about there being severe negative impacts of the virus on employment, household finances, or the overall growth rate. What I’m after here is trying to understand how shocks (and what kind of shocks?) to some industries ripple through the economy to influence others. For example, when you shut down hospitality businesses like restaurants, bars, and hotels, how does that affect suppliers like the food processing or farming industries?

I’ve got no chance at coming up with something coherent in a small enough time frame to speak to the current round of proposed stimulus. At this point the scale of the problems are so large, and so apparent, that the kinds of things I’m thinking about here are second order. Treasury rates are historically and ridiculously low. Borrow like crazy and push a bunch of that money out to households (without a stupid phase-in) and businesses so that the former can keep paying bills and the latter can keep making payroll. Keep the circulation going. If you want to claw back some of that by making the household payments taxable later on, or make the business payments 30-year 0% loans, okay. That is less relevant than the speed of action now.

Okay, throat clearing aside, let me step back into my second-order considerations of how industry shocks propagate through the economy. If you’re looking for a reason to keep reading, by the way, then I’d consider this useful information for thinking about how to build a more resilient economic stabilization system after the coronavirus has died down.

In the last post I started outlining what an I/O table. That shows us how much of the output of one industry is used as an input by other industries - intermediate goods and services. Once you allow for industries to supply one another, then the concepts of gross output and GDP become distinct. Gross output is, to put it simply, GDP plus intermediates. What I noted in that post was that just knowing what the I/O table looks like doesn’t tell you how much economic activity there is. And the I/O table is necessary, but not sufficient, for figuring out how shocks to one industry influence others.

This post is going to verbally lay out the implications of a baseline model. If you’re digging this, then you should look closely at Carvalho and Tahbaz-Salehi and Carvalho, who go into the math. I may post up a more formal version of this verbal model in the near future.

Before we get going, be aware that this baseline model is only kinda-sorta going to be useful for thinking about shocks like the coronavirus, because it is going to assume a lot of flexibility in prices and employment across industries, neither of which are going to be good assumptions for short-run impacts of something like the virus shutdowns. But I don’t know how to think about a better model for the shutdowns without having this baseline to work from. So here we go.

The supply side in words

Imagine that some industry A experienced a 10% decline in its total factor productivity. For the same amount of capital, labor, and intermediates its real production today is only 90% of what it was yesterday. What is the effect on total GDP?

That depends on how important the industry was to begin with, of course. You might be tempted to say that the importance could be measured by the share of GDP (value added) accounted for by industry A. If industry A’s value added was 15% of GDP yesterday, then you might predict that GDP would be lower by .1 times .15 = .015, or lower by 1.5% today. And the rough intuition is right that the “bigger” the industry, the more consequential a drop (or increase) in productivity in that industry is for GDP.

The problem is that the share of GDP is not the right way to measure the impact of industry A, because industry A might be a supplier to other industries. If their productivity goes down, then there are fewer products from A flowing to industries B, C, D, … and Z. And thus those industries cannot produce as much today as they did yesterday. And therefore the output of all of those industries will fall, and the actual impact on GDP will be more severe than the 1.5% we just calculated.

The I/O table I mentioned in the last post allows us to figure out the final effect of the productivity shock to industry A by working through the ripple effects on B, C, D, … and Z. The I/O table tells us how much spending on intermediates by B, say, comes from industry A. The larger this share, the bigger the effect on B’s production, and vice versa. And it doesn’t stop there. Because if B is a big supplier to C, then when A’s productivity falls, B’s production falls, and hence C’s production falls as well. Even worse, if C is a big supplier to A, then A’s production is going to fall even further, meaning B’s falls again, and so on.

We aren’t doomed. Using the I/O table we constructed the Leontief inverse in the last post, which is essentially the solution to following all of these paths throughout the economy. It tells us the final effect of the productivity shock to industry A (or to any industry) on all other industries. We can combine that with how much final consumers like to buy the products of industry A to come up with the right way to measure the impact of industry A on the whole economy.

Those impacts are referred to as “Domar weights” after Evsey Domar, the same guy from the Harrod/Domar model of economic growth and the papers on serfdom and slavery. What’s cool about Domar weights is that even though they are built from an underlying I/O table and all that, in the end they can be calculated very simply. The Domar weight for industry A, for example, is just the gross output of industry A divided by GDP.

Our naive calculation above (15%) was the value added of industry A divided by GDP, but that understated the role of A in the economy. The Domar weight might be something like 20%, with the extra five percentage points coming from the ripple effects of A on other industries. For an industry that has very little final demand but is used as an intermediate by lots of other industries (e.g. warehousing, ocean shipping, semiconductor manufacturing, crude oil production) the actual value-added weight could be very small but the Domar weight could be huge. For an industry that is mainly final demand but isn’t a big supplier of other industries (e.g. restaurants) the Domar weight is probably very close to the share of value-added.

Regardless, if we use the Domar weight of 20% for industry A, then the actual impact of the 10% drop in industry A’s productivity will lower GDP today by .1 times 0.2 = .02, or 2%.

This has some interesting implications for overall GDP and the impact of productivity shocks. We’ll take those Domar weights seriously, but for the moment be naive about other aspects of the economy. In particular, we’re going to assume that the labor force and capital stock stay constant in response to any productivity shocks (hey, I told you it was a baseline model). Then you can do some math and show that the percentage change in GDP is equal to the Domar-weighted percentage change in all industry productivity levels. That makes some sense, I would think. The change in GDP (since we are holding labor and capital constant) has to depend on the change in productivity of all the industries that make up the economy.

Here’s the weird/cool/scary part of this. Because of the interactions between industries in the I/O table, the Domar weights need not add up to one. And this means that the effect on GDP of productivity shocks can be bigger than any of the individual industry shocks. The interrelated nature of the industries is going to amplify productivity shocks beyond their initial size.

Let’s take an example. If we had 3 industries A, B, and C that are all interrelated, we could easily have Domar weights for the three of them that are 20%, 60%, and 50%. And if the coronavirus or anything else were to lower productivity in each industry by exactly 10% then the interrelated nature of the economy tells us that the drop in GDP is equal to .2 times .1 + .6 times .1 + .5 times .1 = .13, or 13%. No industry experiences more than a 10% drop in productivity, but GDP falls by 13%. Why? Industry B gets hit by its own 10% drop in productivity but it also loses inputs from A and C, meaning its production goes down by more than 10%. And that in turn means A and C have fewer inputs, which lowers their production by more than 10%.

If you think of the virus as some sort of common productivity shock across industries (or firms, or whatever) then it is their relationships with one another as suppliers and customers that create a massive economic shock in the aggregate. The more related industries are, the larger the sum of the Domar weights, and the larger the aggregate effect of the initial productivity shock.

One thing you might have picked up on here is that I’ve described the effects of the productivity shocks in any industry in terms of their effect on its “downstream” customers. That is, I described how if A has a negative productivity shock, this influences B because B needs inputs from A. I didn’t speak directly about the “upstream” effects of the productivity shock in A, meaning I didn’t describe the fact that A may buy fewer inputs itself in response to its productivity shock, hurting its suppliers.

The concept that changes in GDP are a Domar-weighted combination of changes in productivity by industries is usually known as “Hulten’s Theorem”, and holds in many more generic cases than what I’ve described. My focus on the “downstream” effects only was partly for exposition, and partly based on assumptions in the baseline model I’m working off of. There has been a bunch of work on this recently, so see things like Gabaix, Baqaee and Farhi, or the papers I linked above to dig into the generality or non-generality of that Theorem.

The demand side in words

Thinking through the supply side was just half the problem. And to be honest, I’m not sure that for the specifics of the corona-virus we should be thinking in terms of productivity shocks. Restaurants did not suddenly forgot how to cook when people got sick. This was more about significant changes in demand, whether those were chosen by individuals or ordered by governments.

With this baseline model, think about a change in the share of their spending that people choose to do on the products of different industries. So everyone decides to decrease the fraction of their spending they want to do on industry B (e.g. restaurant meals) and increase the spending they want to do on industry A (insert toilet paper joke here). What affect does this change in the composition of demand have on an economy where A and B and C are linked through the I/O table?

In short, what is going to matter is how this change in demand changes the Domar weights. Those weights, recall, are based not only on the I/O relationships which capture intermediate use but also depend on final demand for products. Our change in what final consumers want to spend their money on is going to change the Domar weights.

This could be good or it could be bad for aggregate output, in theory. If industry A is not a big user of intermediate goods for whatever reason, but industry B is a big user of intermediate goods, then this shift in spending could be very bad for aggregate output. Why? If we decide to spend all of our money on products from industry A this creates very little “upstream” demand for intermediates from other industries, which limits their economic activity. When we spend more on industry A we’re going to push up it’s Domar weight, but not by very much.

The drop in spending on industry B reduces its Domar weight directly. But because this industry uses a lot of intermediates that reduces even further the upstream demand by industry B from other industries, reducing B’s Domar weight further. And don’t forget industry C, who is standing there trying to act like nothing is happening. It’s Domar weight will be pulled down to some extent because of the change in demand for its products by both A and B (and that will feed back to the Domar weights of A and B).

If industry A does not have a lot of upstream demand then the increase in its Domar weight is going to be more than offset by the drop in the Domar weights of B and C. Remember from above how the Domar weights could sum to a number more than one? Here, the shift of spending from B to A is going to lower that sum. And maybe that will lower aggregate economic activity.

Why maybe? With the demand shift we have to consider the level of productivity across industries and how that interacts with the change in Domar weights. Yes, shifting spending to industry A from industry B may lower the sum of the Domar weights. But if the level of productivity in industry A is very high and productivity in B and C is very low then deciding to spend more money on A would be good for aggregate output.

The ultimate effect of a demand shock from the virus will depend on (1) which industries our spending shifts into and (2) whether those industries have relatively high or low productivity.

Notice that the effect of demand shocks in this baseline model are all about the “upstream” connections of an industry to its suppliers. The productivity shocks were all about the “downstream” connections of an industry to its customers. It’s not true that the distinction is that clean. Demand shocks can have upstream and downstream effects, as can productivity shocks. But in this baseline version things separate out cleanly. That said, Acemoglu, Akcigit, and Kerr find evidence that shocks do tend to propagate this way. So the baseline model is not a terrible first pass.

We’re not done yet

The big missing link here is that the baseline model is built on the assumption that the labor and capital supplies are held constant. And that is obviously not what we’re seeing in the short-run response to the coronavirus, where layoffs and firings are proceeding at a staggering pace. What comes next in figuring this out is shutting down the ability of labor to shift freely between industries, so that the productivity and/or demand shocks that hit the economy lead to labor being unemployed. Then the I/O network is going to tell us how the demand shock that is killing restaurants, for example, not only leads to unemployment in that industry, but propagates to unemployment in other industries that supply it. But I still need to figure that out, so stay tuned.