The Grumpy Economist: Papers: Dew-Becker on Networks

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I have been studying a number of macro currently. Partially, I am simply catching up from a number of years of e book writing. Partially, I need to perceive inflation dynamics, the hunt set forth in “expectations and the neutrality of rates of interest,” and an apparent subsequent step within the fiscal idea program. Maybe weblog readers may discover fascinating some summaries of latest papers, when there’s a nice concept that may be summarized with out an enormous quantity of math. So, I begin a sequence on cool papers I am studying.

Immediately: “Tail threat in manufacturing networks” by Ian Dew-Becker, a fantastic paper. A “manufacturing community” method acknowledges that every agency buys from others, and fashions this interconnection. It is a sizzling matter for plenty of causes, beneath. I am as a result of costs cascading via manufacturing networks may induce a greater mannequin of inflation dynamics.

(This publish makes use of Mathjax equations. For those who’re seeing rubbish like [alpha = beta] then come again to the supply right here.)

To Ian’s paper: Every agency makes use of different corporations’ outputs as inputs. Now, hit the financial system with a vector of productiveness shocks. Some corporations get extra productive, some get much less productive. The extra productive ones will develop and decrease costs, however that modifications everybody’s enter costs too. The place does all of it calm down? That is the enjoyable query of community economics.

Ian’s central concept: The issue simplifies quite a bit for giant shocks. Often when issues are sophisticated we have a look at first or second order approximations, i.e. for small shocks, acquiring linear or quadratic (“easy”) approximations.

On the x axis, take a vector of productiveness shocks for every agency, and scale it up or down. The x axis represents this general scale. The y axis is GDP. The best hand graph is Ian’s level: for big shocks, log GDP turns into linear in log productiveness — actually easy.

Why? As a result of for big sufficient shocks, all of the networky stuff disappears. Every agency’s output strikes up or down relying solely on one important enter.

To see this, we’ve got to dig deeper to enhances vs. substitutes. Suppose the value of an enter goes up 10%. The agency tries to make use of much less of this enter. If the most effective it may do is to chop use 5%, then the agency finally ends up paying 5% extra general for this enter, the “expenditure share” of this enter rises. That’s the case of “enhances.” But when the agency can reduce use of the enter 15%, then it pays 5% much less general for the enter, although the value went up. That’s the case of “substitutes.” That is the important thing idea for the entire query: when an enter’s worth goes up, does its share of general expenditure go up (enhances) or down (substitutes)?

Suppose inputs are enhances. Once more, this vector of expertise shocks hits the financial system. As the dimensions of the shock will get greater, the expenditure of every agency, and thus the value it fees for its output, turns into increasingly dominated by the one enter whose worth grows probably the most. In that sense, all of the networkiness simplifies enormously. Every agency is just “related” to 1 different agency.

Flip the shock round. Every agency that was getting a productiveness enhance now will get a productiveness discount. Every worth that was going up now goes down. Once more, within the giant shock restrict, our agency’s worth turns into dominated by the value of its most costly enter. Nevertheless it’s a distinct enter. So, naturally, the financial system’s response to this expertise shock is linear, however with a distinct slope in a single course vs. the opposite.

Suppose as an alternative that inputs are substitutes. Now, as costs change, the agency expands increasingly its use of the most affordable enter, and its prices and worth grow to be dominated by that enter as an alternative. Once more, the community collapsed to 1 hyperlink.

Ian: “detrimental productiveness shocks propagate downstream via components of the manufacturing course of which can be complementary ((sigma_i < 1)), whereas constructive productiveness shocks propagate via components which can be substitutable ((sigma_i > 1)). …each sector’s habits finally ends up pushed by a single one in all its inputs….there’s a tail community, which depends upon (theta) and during which every sector has only a single upstream hyperlink.”

Equations: Every agency’s manufacturing operate is (considerably simplifying Ian’s (1)) [Y_i = Z_i L_i^{1-alpha} left( sum_j A_{ij}^{1/sigma} X_{ij}^{(sigma-1)/sigma} right)^{alpha sigma/(sigma-1)}.]Right here (Y_i) is output, (Z_i) is productiveness, (L_i) is labor enter, (X_{ij}) is how a lot good j agency i makes use of as an enter, and (A_{ij}) captures how necessary every enter is in manufacturing. (sigma>1) are substitutes, (sigma<1) are enhances.

Companies are aggressive, so worth equals marginal value, and every agency’s worth is [ p_i = -z_i + frac{alpha}{1-sigma}logleft(sum_j A_{ij}e^{(1-sigma)p_j}right).; ; ; (1)]Small letters are logs of huge letters. Every worth depends upon the costs of all of the inputs, plus the agency’s personal productiveness. Log GDP, plotted within the above determine is [gdp = -beta’p] the place (p) is the vector of costs and (beta) is a vector of how necessary every good is to the patron.

Within the case (sigma=1) (1) reduces to a linear method. We are able to simply resolve for costs after which gdp as a operate of the expertise shocks: [p_i = – z_i + sum_j A_{ij} p_j] and therefore [p=-(I-alpha A)^{-1}z,]the place the letters symbolize vectors and matrices throughout (i) and (j). This expression reveals among the level of networks, that the sample of costs and output displays the entire community of manufacturing, not simply particular person agency productiveness. However with (sigma neq 1) (1) is nonlinear and not using a recognized closed kind answer. Therefore approximations.

You’ll be able to see Ian’s central level immediately from (1). Take the (sigma<1) case, enhances. Parameterize the dimensions of the expertise shocks by a set vector (theta = [theta_1, theta_2, …theta_i,…]) instances a scalar (t>0), in order that (z_i=theta_i instances t). Then let (t) develop holding the sample of shocks (theta) the identical. Now, because the ({p_i}) get bigger in absolute worth, the time period with the best (p_i) has the best worth of ( e^{(1-sigma)p_j} ). So, for big expertise shocks (z), solely that largest time period issues, the log and e cancel, and [p_i approx -z_i + alpha max_{j} p_j.] That is linear, so we are able to additionally write costs as a sample (phi) instances the dimensions (t), within the large-t restrict (p_i = phi_i t), and [phi_i = -theta_i + alpha max_{j} phi_j.;;; (2)] With substitutes, (sigma<1), the agency’s prices, and so its worth, might be pushed by the smallest (most detrimental) upstream worth, in the identical method. [phi_i approx -theta_i + alpha min_{j} phi_j.]

To specific gdp scaling with (t), write (gdp=lambda t), or if you need to emphasize the dependence on the vector of expertise shocks, (lambda(theta)). Then we discover gdp by (lambda =-beta’phi).

On this huge worth restrict, the (A_{ij}) contribute a continuing time period, which additionally washes out. Thus the precise “community” coefficients cease mattering in any respect as long as they aren’t zero — the max and min are taken over all non-zero inputs. Ian:

…the boundaries for costs, don’t rely on the precise values of any (sigma_i) or (A_{i,j}.) All that issues is whether or not the elasticities are above or beneath 1 and whether or not the manufacturing weights are larger than zero. Within the instance in Determine 2, altering the precise values of the manufacturing parameters (away from (sigma_i = 1) or (A_{i,j} = 0)) modifications…the degrees of the asymptotes, and it may change the curvature of GDP with respect to productiveness, however the slopes of the asymptotes are unaffected.

…when fascinated with the supply-chain dangers related to giant shocks, what’s necessary is just not how giant a given provider is on common, however fairly what number of sectors it provides…

For a full answer, have a look at the (extra fascinating) case of enhances, and suppose each agency makes use of slightly bit of each different agency’s output, so all of the (A_{ij}>0). The most important enter worth in (2) is identical for every agency (i), and you’ll shortly see then that the largest worth would be the smallest expertise shock. Now we are able to resolve the mannequin for costs and GDP as a operate of expertise shocks: [phi_i approx -theta_i – frac{alpha}{1-alpha} theta_{min},] [lambda approx beta’theta + frac{alpha}{1-alpha}theta_{min}.] We’ve solved the large-shock approximation for costs and GDP as a operate of expertise shocks. (That is Ian’s instance 1.)

The graph is concave when inputs are enhances, and convex when they’re substitutes. Let’s do enhances. We do the graph to the left of the kink by altering the signal of (theta). If the id of (theta_{min}) didn’t change, (lambda(-theta)=-lambda(theta)) and the graph can be linear; it might go down on the left of the kink by the identical quantity it goes up on the proper of the kink. However now a completely different (j) has the biggest worth and the worst expertise shock. Since this should be a worse expertise shock than the one driving the earlier case, GDP is decrease and the graph is concave. [-lambda(-theta) = beta’theta + frac{alpha}{1-alpha}theta_{max} gebeta’theta + frac{alpha}{1-alpha}theta_{min} = lambda(theta).] Due to this fact (lambda(-theta)le-lambda(theta),) the left facet falls by greater than the proper facet rises.

Does all of this matter? Nicely, absolutely extra for questions when there could be a giant shock, resembling the large shocks we noticed in a pandemic, or huge shocks we would see in a warfare. One of many huge questions that community idea asks is, how a lot does GDP change if there’s a expertise shock in a selected business? The (sigma=1) case during which expenditure shares are fixed offers a regular and pretty reassuring consequence: the impact on GDP of a shock in business i is given by the ratio of i’s output to whole GDP. (“Hulten’s theorem.”) Industries which can be small relative to GDP do not have an effect on GDP that a lot in the event that they get into bother.

You’ll be able to intuit that fixed expenditure shares are necessary for this consequence. If an business has a detrimental expertise shock, raises its costs, and others cannot cut back use of its inputs, then its share of expenditure will rise, and it’ll impulsively be necessary to GDP. Persevering with our instance, if one agency has a detrimental expertise shock, then it’s the minimal expertise, and [(d gdp/dz_i = beta_i + frac{alpha}{1-alpha}.] For small corporations (industries) the latter time period is more likely to be a very powerful. All of the A and (sigma) have disappeared, and mainly the entire financial system is pushed by this one unfortunate business and labor.

Ian:

…what determines tail threat is just not whether or not there may be granularity on common, however whether or not there can ever be granularity – whether or not a single sector can grow to be pivotal if shocks are giant sufficient.

For instance, take electrical energy and eating places. In regular instances, these sectors are of comparable measurement, which in a linear approximation would indicate that they’ve comparable results on GDP. However one lesson of Covid was that shutting down eating places is just 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 may anticipate {that a} important discount in obtainable electrical energy would have strongly detrimental results – and that these results can be convex within the measurement of the decline in obtainable energy. Electrical energy is systemically necessary not as a result of it will be significant in good instances, however as a result of it might be necessary in dangerous instances.

Ben Moll turned out to be proper and Germany was capable of substitute away from Russian Fuel much more than folks had thought, however even that proves the rule: if it’s arduous to substitute away from even a small enter, then giant shocks to that enter indicate bigger expenditure shares and bigger impacts on the financial system than its small output in regular instances would counsel.

There is a gigantic quantity extra within the paper and voluminous appendices, however that is sufficient for a weblog evaluate.

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Now, a number of limitations, or actually ideas on the place we go subsequent. (No extra on this paper, please, Ian!) Ian does a pleasant illustrative computation of the sensitivity to giant shocks:

Ian assumes (sigma>1), so the primary elements are what number of downstream corporations use your merchandise and a bit their labor shares. No shock, vans, and power have huge tail impacts. However so do attorneys and insurance coverage. Can we actually not do with out attorneys? Right here I hope the subsequent step appears arduous at substitutes vs. enhances.

That raises a bunch of points. Substitutes vs. enhances absolutely depends upon time horizon and measurement of shocks. It could be straightforward to make use of rather less water or electrical energy initially, however then actually arduous to cut back greater than, say, 80%. It is often simpler to substitute in the long term than the quick run.

The evaluation on this literature is “static,” which means it describes the financial system when all the pieces has settled down. The responses — you cost extra, I exploit much less, I cost extra, you utilize much less of my output, and so on. — all occur immediately, or equivalently the mannequin research a future the place this has all settled down. However then we speak about responses to shocks, as within the pandemic. Absolutely there’s a dynamic response right here, not simply together with capital accumulation (which Ian research). Certainly, my hope was to see costs spreading out via a manufacturing community over time, however this construction would have all worth changes immediately. Mixing manufacturing networks with sticky costs is an apparent concept, which among the papers beneath are engaged on.

Within the idea and knowledge dealing with, you see a giant discontinuity. If a agency makes use of any inputs in any respect from one other agency, if (A_{ij}>0), that enter can take over and drive all the pieces. If it makes use of no inputs in any respect, then there isn’t a community hyperlink and the upstream agency cannot have any impact. There’s a huge discontinuity at (A_{ij}=0.) We would like a idea that doesn’t leap from zero to all the pieces when the agency buys one stick of chewing gum. Ian needed to drop small however nonzero parts of the input-output matrix to produces smart outcomes. Maybe we should always regard very small inputs as at all times substitutes?

How necessary is the community stuff anyway? We have a tendency to make use of business categorizations, as a result of we’ve got an business input-output desk. However how a lot of the US business input-output is just vertical: Loggers promote timber to mills who promote wooden to lumberyards who promote lumber to Residence Depot who sells it to contractors who put up your home? Power and instruments feed every stage, however do not use a complete lot of wooden to make these. I have never checked out an input-output matrix just lately, however simply how “vertical” is it?

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The literature on networks in macro is huge. One method is to choose a latest paper like Ian’s and work again via the references. I began to summarize, however gave up within the deluge. Have enjoyable.

A technique to consider a department of economics is not only “what instruments does it use?” however “what questions is it asking? Lengthy and Plosser “Actual Enterprise Cycles,” a basic, went after concept that the central defining characteristic of enterprise cycles (since Burns and Mitchell) is comovement. States and industries all go up and down collectively to a outstanding diploma. That pointed to “mixture demand” as a key driving pressure. One would suppose that “expertise shocks” no matter they’re can be native or business particular. Lengthy and Plosser confirmed that an enter output construction led idiosyncratic shocks to provide enterprise cycle widespread motion in output. Sensible.

Macro went in one other method, emphasizing time sequence — the concept that recessions are outlined, say, by two quarters of mixture GDP decline, or by the larger decline of funding and sturdy items than consumption — and within the mixture fashions of Kydland and Prescott, and the stochastic development mannequin as pioneered by King, Plosser and Rebelo, pushed by a single economy-wide expertise shock. A part of this shift is just technical: Lengthy and Plosser used analytical instruments, and have been thereby caught in a mannequin with out capital, plus they didn’t inaugurate matching to knowledge. Kydland and Prescott introduced numerical mannequin answer and calibration to macro, which is what macro has finished ever since. Perhaps it is time to add capital, resolve numerically, and calibrate Lengthy and Plosser (with updated frictions and client heterogeneity too, perhaps).

Xavier Gabaix (2011) had a distinct Massive Query in thoughts: Why are enterprise cycles so giant? Particular person corporations and industries have giant shocks, however (sigma/sqrt{N}) should dampen these on the mixture degree. Once more, this was a basic argument for mixture “demand” versus “provide.” Gabaix notices that the US has a fat-tailed agency distribution with a number of giant corporations, and people corporations have giant shocks. He amplifies his argument by way of the Hulten mechanism, a little bit of networkyiness, for the reason that impression of a agency on the financial system is gross sales / GDP, not worth added / GDP.

The big literature since then has gone after quite a lot of questions. Dew-Becker’s paper is in regards to the impact of huge shocks, and clearly not that helpful for small shocks. Bear in mind which query you are after.

The “what is the query” query is doubly necessary for this department of macro that explicitly fashions heterogeneous brokers and heterogenous corporations. Why are we doing this? One can at all times symbolize the aggregates with a social welfare operate and an mixture manufacturing operate. You could be fascinated with how aggregates have an effect on people, however that does not change your mannequin of aggregates. Or, you could be fascinated with seeing what the mixture manufacturing or utility operate appears like — is it per what we find out about particular person corporations and folks? Does the dimensions of the mixture manufacturing operate shock make sense? However nonetheless, you find yourself with only a higher (hopefully) mixture manufacturing and utility operate. Or, you may want fashions that break the aggregation theorems in a major method; fashions for which distributions matter for mixture dynamics, theoretically and (tougher) empirically. However do not forget you want a purpose to construct disaggregated fashions.

Expression (1) is just not straightforward to get to. I began studying Ian’s paper in my common method: to study a literature begin with the newest paper and work backward. Alas, this literature has advanced to the purpose that authors plop outcomes down that “all people is aware of” and can take you a day or so of head-scratching to breed. I complained to Ian, and he stated he had the identical downside when he was getting in to the literature! Sure, journals now demand such overstuffed papers that it is arduous to do, however it might be awfully good for everybody to begin together with floor up algebra for main ends in one of many countless web appendices. I finally discovered Jonathan Dingel’s notes on Dixit Stiglitz methods, which have been useful.

Replace:

Chase Abram’s College of Chicago Math Camp notes right here are additionally a unbelievable useful resource. See Appendix B beginning p. 94 for manufacturing community math. The remainder of the notes are additionally actually good. The primary half goes slightly deeper into extra summary materials than is actually essential for the second half and utilized work, however it’s a great and concise evaluate of that materials as nicely.

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