I've been reading a lot of macro lately. In part, I'm just catching up from a few years of book writing. In part, I want to understand inflation dynamics, the quest set forth in "expectations and the neutrality of interest rates," and an obvious next step in the fiscal theory program. Perhaps blog readers might find interesting some summaries of recent papers, when there is a great idea that can be summarized without a huge amount of math. So, I start a series on cool papers I'm reading.

Today: "Tail risk in production networks" by Ian Dew-Becker, a beautiful paper. A "production network" approach recognizes that each firm buys from others, and models this interconnection. It's a hot topic for lots of reasons, below. I'm interested because prices cascading through production networks might induce a better model of inflation dynamics.

(This post uses Mathjax equations. If you're seeing garbage like [\alpha = \beta] then come back to the source here.)

To Ian's paper: Each firm uses other firms' outputs as inputs. Now, hit the economy with a vector of productivity shocks. Some firms get more productive, some get less productive. The more productive ones will expand and lower prices, but that changes everyone's input prices too. Where does it all settle down? This is the fun question of network economics.

Ian's central idea: The problem simplifies a lot for *large* shocks. Usually when problems are complicated we look at first or second order approximations, i.e. for small shocks, obtaining linear or quadratic ("simple") approximations.

*when an input's price goes up, does its share of overall expenditure go up (complements) or down (substitutes)?*

*But it's a different input.*So, naturally, the economy's response to this technology shock is linear, but with a different slope in one direction vs. the other.

*smallest*(most negative) upstream price, in the same way. \[\phi_i \approx -\theta_i + \alpha \min_{j} \phi_j.\]

...the limits for prices, do not depend on the exact values of any \(\sigma_i\) or \(A_{i,j}.\) All that matters is whether the elasticities are above or below 1 and whether the production weights are greater than zero. In the example in Figure 2, changing the exact values of the production parameters (away from \(\sigma_i = 1\) or \(A_{i,j} = 0\)) changes...the levels of the asymptotes, and it can change the curvature of GDP with respect to productivity, but the slopes of the asymptotes are unaffected.

...when thinking about the supply-chain risks associated with large shocks, what is important is not how large a given supplier is on average, but rather how many sectors it supplies...

*different*\(j\) has the largest price and the worst technology shock. Since this must be a worse technology shock than the one driving the previous case, GDP is lower and the graph is concave. \[-\lambda(-\theta) = \beta'\theta + \frac{\alpha}{1-\alpha}\theta_{\max} \ge\beta'\theta + \frac{\alpha}{1-\alpha}\theta_{\min} = \lambda(\theta).\] Therefore \(\lambda(-\theta)\le-\lambda(\theta),\) the left side falls by more than the right side rises.

*one*firm has a negative technology shock, then it is the minimum technology, and [(d gdp/dz_i = \beta_i + \frac{\alpha}{1-\alpha}.\] For small firms (industries) the latter term is likely to be the most important. All the A and \(\sigma\) have disappeared, and basically the whole economy is driven by this one unlucky industry and labor.

...what determines tail risk is not whether there is granularity on average, but whether there can ever be granularity – whether a single sector can become pivotal if shocks are large enough.

For example, take electricity and restaurants. In normal times, those sectors are of similar size, which in a linear approximation would imply that they have similar effects on GDP. But one lesson of Covid was that shutting down restaurants is not catastrophic for GDP, [Consumer spending on food services and accommodations fell by 40 percent, or $403 billion between 2019Q4 and 2020Q2. Spending at movie theaters fell by 99 percent.] whereas one might expect that a significant reduction in available electricity would have strongly negative effects – and that those effects would be convex in the size of the decline in available power. Electricity is systemically important not because it is important in good times, but because it would be important in bad times.

*if*it is hard to substitute away from even a small input, then large shocks to that input imply larger expenditure shares and larger impacts on the economy than its small output in normal times would suggest.

*comovement*. States and industries all go up and down together to a remarkable degree. That pointed to "aggregate demand" as a key driving force. One would think that "technology shocks" whatever they are would be local or industry specific. Long and Plosser showed that an input output structure led idiosyncratic shocks to produce business cycle common movement in output. Brilliant.

*done*ever since. Maybe it's time to add capital, solve numerically, and calibrate Long and Plosser (with up to date frictions and consumer heterogeneity too, maybe).

*Update:*

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