I’m washing my hands, avoiding crowds (even more than normal), and thinking about the economic effects of the coronavirus. There is no question there has already been a significant effect, and there will be more effects to come. I’m thinking about what those effects are, how they are distributed across industries (e.g. airlines versus trucking), and how the affected industries influence *other* industries and consumers.

I think this is going to take a few posts, as I don’t have a solid feel for how to do this. And this is going to start theoretically before I (hopefully) get to putting some numbers on things. If you’re interested in some kind of real-time estimate of the effect on GDP and the stock market, this is very much *not* the blog for you.

Let’s get some definitions out of the way. We talk a lot about GDP ($Y$), which is the value of all *final* goods and services produced in a year (or quarter, or whatever time frame). But many, many transactions are for *intermediate* goods and services. Those intermediates are purchased by one firm from another in order to produce something else. The classic example here would be a bakery buying wheat (the intermediate good) from a farmer in order to sell bread (the final good). GDP only measures the value of the bread produced. *Gross output* ($Q$) would measure the value of the bread *and* the value of the wheat.

The distinction between gross output and GDP, by itself, doesn’t tell us anything about the effects of the coronavirus. But it does remind us that final goods depend on the production of intermediate goods, and so anything that disrupts the production of those intermediates will affect GDP even if nothing directly impacts our demand for those final goods and services. To trace the impact of the shocks from the coronavirus we need to understand the input/output (I/O) relationships linking producers of final goods and services to producers of intermediates.

Let’s think through some ways the virus would influence the economy in our toy farmer/baker example. If everyone stays home and cooks their own food, then demand for bread falls. The baker will sell less, demand less wheat, and in turn the farmer will suffer a loss of sales. We can use the I/O relationships to tell us how a shock to final demand influences intermediate producers.

On the other hand, we could have a situation where the farmer gets the virus, say, and cannot work. Then this lowers the amount of wheat produced and hence the baker cannot make as much bread. Here the I/O relationships tell us how a shock to an intermediate producer influences the production of final goods and services.

Of course both of these shocks are possible, and in reality both kinds of shocks are occurring. To get my head around how big of an impact those shocks could have, I want to build up the math behind the structure of these I/O relationships. This is going to be math-heavy relative to most posts, ultimately reaching matrix algebra. I’m sorry, I don’t know any other way to talk about this stuff. I’m adapting this from notes I’ve used in the past for a grad course. I have tried to keep in some simplified examples of what I’m doing, but it can get a little hairy.

The connection of GDP ($Y$) to gross output ($Q$) depends on the input/output relationships of all the different units of production. Input/output refers to the fact that some of the output of one unit of production ($i$) is used as an input by another unit of production ($j$). As noted, those intermediate transactions are not part of GDP. But they do determine how much gross output is necessary to produce a given amount of GDP.

Let’s set up a simple example. Consider an oil drilling company that produces barrels of oil. The barrels are used by a refining company that uses them to produce gasoline. But it is also the case that the oil drilling company needs to buy gasoline in order to run their equipment. In addition, there are consumers who have final demand for gasoline, but there is no final demand for oil (i.e. no one consumes oil directly). This is more complex than the wheat/bread example, as the two industries serve as intermediate good providers for each other. But that is something we want to allow for, as most industries are going to operate both as a user and provider of intermediate goods.

Let’s say that 100 barrels of oil can produce 1,000 gallons of gasoline. And let’s say that consumers demand 1,000 gallons of gasoline. But it is *also* the case that it takes 50 gallons of gasoline to produce 100 barrels of oil. So how much oil production is there? To get 1,000 final gallons of gasoline, we need 100 barrels of oil produced. But that requires an additional 50 gallons of gas, which requires an additional 5 barrels of oil. But that 5 additional barrels of oil requires 2.5 gallons of gas, which requires and additional 0.25 barrels of oil. And so on and so on.

We can construct this interaction in a little system of equations.

\[\begin{aligned} y_B &= 0.1\times y_G + 0 \\ y_G &= 0.5\times y_B + 1,000 \end{aligned}\]where $y_B$ are barrels produced, which are equal to one-tenth of the total gallons of gas produced, as that is the demand from the refinery. The zero in the first line represents the fact that there is zero final demand for oil (no one uses oil directly, it is only an intermediate good here). The second line shows that the total gallons of gas produced are equal to one-half the number of barrels produced (that is the demand from oil producers) plus the 1,000 gallons demanded by final consumers. This is just a two-equation, two-unknown situation. It can be solved to find $y_B = 105.26$ is the gross output of the oil drilling company, and $y_G = 1,052.63$ is the gross output of the refinery. That 1,000 gallons of final demand for gas leads to an extra 52-ish gallons produced, along with 105-ish barrels of oil as intermediate good.

This same logic can be extended to the case of $J$ arbitrary units of production. Ultimately we’d have a $J$ equation system, with $J$ unknown gross outputs of $p_j y_j$, taking the final demands $p_j c_j$ of each unit as given. This is tedious, but it is a straightforward linear algebra problem, and can be set up in matrix form. The only difference with the little example of oil and gas is that here we’re going to use values (i.e. with relative prices included) rather than raw quantities.

\[\begin{aligned} p_1 y_1 &= p_1 y_{11} + p_1 y_{21} + ... + p_1 y_{J1} + p_1 c_1 \\ p_2 y_2 &= p_2 y_{12} + p_2 y_{22} + ... + p_2 y_{J2} + p_2 c_2 \\ &...& \\ p_J y_J &= p_J y_{1J} + p_J y_{2J} + ... + p_J y_{JJ} + p_J c_J \\ \end{aligned}\]where recall that $p_i y_{ji}$ is the intermediate demand by unit $j$ for output from unit $i$. Depending on the level of analysis you are working with - establishment, firm, sector - you might be able to assert that $y_{ii} = 0$. But we don’t have to assume this. It is okay if we think of units of production purchasing their own output to use as an intermediate good. Think of a refinery buying its own gas to run its own trucks and equipment. Now we can simplify this by building it into matrix form.

Define the matrices as follows

\[\mathbf{q} = \begin{bmatrix} p_1 y_1 \\ p_2 y_2 \\ ... \\ p_J y_J \end{bmatrix} \mathbf{c} = \begin{bmatrix} p_1 c_1 \\ p_2 c_2 \\ ... \\ p_J c_J \end{bmatrix} \mathbf{A} = \begin{bmatrix} \frac{p_1 y_{11}}{p_1 y_1} & ... & \frac{p_1 y_{J1}}{p_J y_J} \\ \frac{p_2 y_{12}}{p_1 y_1} & ... & \frac{p_2 y_{J2}}{p_J y_J} \\ ... & \ddots & ... \\ \frac{p_J y_{1J}}{p_1 y_1} & ... & \frac{p_J y_{JJ}}{p_J y_J} \\ \end{bmatrix}\]The vector $q$ captures the gross output of each unit, the vector $c$ the final demand for each units output. The matrix $A$ is called a “technical coefficients” matrix. The raw data for these matrices come from something called a “use table”, which are part of input/output accounts produced by statistical agencies. I’ll probably go through one of those in a future post.

$A$ tells us how much of each intermediate good is necessary to produce one (real) dollar of additional output from a given unit of production. It takes $p_2 y_{12}/p_1 y_1$ in purchases of intermediate good 2 to produce one real dollar of output of good 1, for example. $A$ thus tells us the “recipe” for producing gross output.

With these matrices, we can simplify our system of equations down to

\[\mathbf{q} = \mathbf{A}\mathbf{q} + \mathbf{c},\]and solve this using normal matrix operations,

\[(\mathbf{I} - \mathbf{A})\mathbf{q} = \mathbf{c}\]where $\mathbf{I}$ is an identity matrix, and so

\[\mathbf{q} = (\mathbf{I} - \mathbf{A})^{-1}\mathbf{c}.\]This is the matrix equivalent of our earlier example with the barrels and gas. The $(\mathbf{I} - \mathbf{A})^{-1}$ matrix is a Leontief inverse”. It isn’t obvious here, but this Leontief inverse summarizes all of the direct and indirect effects of final demand for $c$ on gross output $q$.

To see what is going on, go back to the simple example involving barrels and gasoline. Write the matrix form of this as

\[\begin{bmatrix} y_B \\ y_G \end{bmatrix} = \begin{bmatrix} 0 & 0.1 \\ 0.5 & 0 \\ \end{bmatrix} \begin{bmatrix} y_B \\ y_G \end{bmatrix} + \begin{bmatrix} c_B \\ c_G \end{bmatrix}\]where I’ve allowed both oil, $c_B$, and gas, $c_G$, to have some final demand. Our general setting says that we should form the Leontief inverse first, which in this case is

\[(\mathbf{I} - \mathbf{A})^{-1} = \begin{bmatrix} 1.052 & 0.105 \\ 0.526 & 1.052 \\ \end{bmatrix}.\]The entries in this table tell us the total effect of final demand, whatever that may be. For example, the upper right entry (0.105) tells us that every additional gallon of gas demanded induces 0.105 barrels of oil to be produced. We know that technically, only 0.1 barrels are necessary, but an extra 0.005 are produced because oil production requires some gasoline itself. The bottom left entry says that every additional barrel of oil demanded (if anyone wanted oil as a final good) would lead to 0.526 gallons of gas being produced, 0.5 because it takes that much gas to produce a barrel, and an additional 0.026 because that gas requires some oil itself.

Given that Leontief inverse, we know that

\[\begin{bmatrix} y_B \\ y_G \end{bmatrix} = \begin{bmatrix} 1.052 & 0.105 \\ 0.526 & 1.052 \\ \end{bmatrix} \begin{bmatrix} c_B \\ c_G \end{bmatrix}.\]Note that just knowing the matrix $\mathbf{A}$, and thus the Leontief inverse, doesn’t tell us how big GDP or gross output will be. It only allows us to solve for the relationship of gross output and GDP. Determining actual GDP or gross output would still depend on things like the supply of factors of production (e.g. labor, capital), the productivity of individual units of production, and the preferences of people involved. Nevertheless, this structure is still useful because it gives us the basis for understanding how to account for the interactions of different units of production.

Ignoring all those other relevant factors (preferences, productivity, factor supply), let’s see a little of what we can do with the Leontief matrix.

Let’s say that prior to the virus that final use of oil was $c_B = 100$ and final use of gasoline was $c_G = 1000$, and treating these as measured in real values, that means GDP was 1,100. Gross output (again in real values) of oil was $y_B = 210.2$ and of gasoline was $y_G = 1104.6$, for total gross output of 1314.8.

If the virus meant a shock to demand for gasoline because we all stayed home to watch Netflix, we can work out the impacts. Let’s say that now $c_G = 500$, but that demand for oil stays the same at $c_B = 100$. This drops GDP to 600. But it also drops gross output to $y_B = 157.7$ and $y_G = 578.6$, for a total drop to gross output of 736.3. Because of the drop in gasoline demand, oil production overall falls from 210.2 to 157.7. A full analysis would account for whether there was any impact on final demand for oil due to the virus, whether the drop in gross output in the oil industry or gasoline industry led people to get fired and hence to even lower final demand, and so on and so on.

We can make this harder. What if the virus means a drop in the production of oil, say because the virus gets loose on a drilling rig in the Gulf? Let’s say that total production of oil is forced to drop from 210.2 to $y_B = 100$. What happens? We have to solve this for the level of final demand such that the gross output of oil is only 100. We’re going to have to scale down everything in order to accommodate that drop in oil production. But how? There are a lot of solutions to this system such that $y_B = 100$, and nothing in the Leontief inverse itself pins down exactly which solution we’d end up at. One *possible* answer is that we have $c_B = 0$ and $c_G = 950$, so that the 100 barrels of oil are used solely to provide final consumption of gasoline, but there is no oil left over for final use. Or, we could have $c_B = 20$ and $c_G = 750$ if people really, really want to consume some oil directly.

The point is that we’re going to have to account for people’s spending preferences as well in order to parse out the shocks. Which is why this is probably going to take me a long time and a few posts to figure out how to think about this right. For now, let’s pretend that I’ll be all over that project, and you’ll see more information over the next few days.

Until then, wash your hands and stay the f&#@ home.