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:*

Reviews like this with pointers to other papers (and math tricks!) are an enormously valuable public good. Many thanks. Please keep them coming.

ReplyDeleteGerman end users of Russian gas may have substituted but didn't large industrial users shut down or move production out of Germany?

ReplyDeleteFrom José Luis de la Fuente.

ReplyDeleteHi, you mention at the end of this post "Chase Abram's University of Chicago Math Camp notes here are also a fantastic resource. See Appendix B starting p. 94 for production network math." Where is that Appendix B?

Thanks.

If you intend to pursue this topic further, you may want to read N. Patterson's 2012 working paper titled "Elasticities of substitution in computable equilibrium models", Working Paper 2012-02, from the department of finance, government of Canada (Ottawa). The paper is available via this link:

ReplyDeletehttps://publications.gc.ca/site/archivee-archived.html?url=https://publications.gc.ca/collections/collection_2013/fin/F21-8-2012-02-eng.pdf

Patterson describes the difficulties in estimating the elasticity of substitution, σ, for application in computable general equilibrium model production functions. He discusses the two principal methods of estimating σ -- the Allen method and the Morishima method, and the error in each. He discusses CES vs. "trans-log" vs. linear-logit forms of the production function. He notes that the Cobb-Douglas production function and the Leontief production function are special instances of the CES production function.

With respect to Dew-Becker, Richard Bellman's 'curse of dimensionality' comes into play. The model for the physical product, Yᵢ , i = 1, 2, ..., N, requires identification of 2N² + 2N + 2 parameters and variables. Patterson notes in his paper that identifying the production function for industry sectors in the relevant range of production variation is a daunting task that few modellers undertake. Instead, he states, identification efforts are focused on determining σ from marginal cost analysis.

Patterson defines the elasticity of substitution between two inputs as σₖₗ = d(ln(xₗ/xₖ))/d(ln(fₖ(x̃)/fₗ( x̃)), where fₖ(x̃) is the production function output of node k in state x̃, etc.

Overall, this is a highly developed subject that is dependent on the correct choice of models to achieve useful results. Identification is a large challenge.

The asymptotic behavior depends on the chosen model. Dimensionality will defeat the purpose in the end. Patterson's research focus is the trade-off made between energy and capital in the production activity. This is probably feasible to undertake. Extending the effort to the full range of economic activity is not likely to be viable for a single researcher to undertake alone.

The question: Where do you see yourself taking this to?

It is often useful to delve deeper into the subject matter presented in a blog post and go beyond the published paper featured in the posting. In this case, the student lacking a thorough grounding in the theory and practice of 'production functions' is afforded a high-level overview of the subject without being necessarily au courant with the higher levels of mathematical theory to appreciate the strengths, limitations, and weaknesses of the more popular economic models of the production function. One such overview is that written by S. K. Mishra of the Dept. of Economics, North-Eastern Hill University, Shillong (India). His review is comprehensive and provides useful insights into the nature of theoretical production functions used in neo-classical economics. He sets out the pros and cons, and limitations of most, if not all, production functions introduced in the 20th century in Europe and North America. Cobb-Douglas, Leontief, and CES production functions are discussed extensively and their limitations are exposed. Mishra concludes with the critique of the neo-classical economics penchant for algebraic production functions that fail to conform to reality at the aggregate economic level--the so-called "Cambridge (U.K.) versus Cambridge (U.S.A.)" debate. Mishra's concluding remarks are quoted below in their entirety.

Delete"A Brief History of Production Functions", Mishra, SK, (2007). Munich Personal RePEc Archive, MPRA Paper No. 5254, posted 10 Oct 2007 UTC.

https://mpra.ub.uni-muenchen.de/5254/1/MPRA_paper_5254.pdf

"IV. Concluding Remarks: It is said that once Stanislaw Ulam very earnestly sought for an example of a theory in social sciences that is both true and nontrivial. Paul Samuelson, after several years supplied one: the Ricardian theory of comparative advantages. If this is true then it speaks enough about the position of ‘production function’ in economics. Anwar Shaikh almost conclusively proved that the properties of the Cobb-Douglas production function are devoid of any economic content; they stem from the algebraic properties of the function. Measurement of capital was put in jeopardy by the capital controversy, not only for aggregate production function, but also for any production function even at the firm level. If one has interacted with engineers and the managers who are decision makers at certain level, one might know their view of the utility of production functions. Possibly, Mrs. Joan Robinson was right to comment that the production function has been a powerful instrument of miseducation. The student of economic theory is taught to write Q= f (L, K ) … and that is its sole utility. The era of classical economics had ended (sometime in 1870’s) with the criticism of it by Marx and the birth of neoclassicism. The era of neoclassical economics possibly ended with the capital controversy sometime in 1970’s."

In modelling the national economy of any large or medium size open economy, the assumption of a production function of a manufactory (producer) that produces multiple product lines (yₗ, y₂, ..., yₘ) from multiple productive factor inputs (x₁, x₂, ..., xₙ) the definition of the appropriate theoretical production function is not clear. When a productive sector of the economy includes multiple manufactories each producing multiple product lines consuming multiple productive factor inputs in doing so, aggregation of the production functions of the individual manufactories into an aggregate production function for the sector is said to be problematic insofar as each individual manufacturer may be operating sub-optimally according to its own preferences and under its own idiosyncratic constraints, and none producing an identical product or using an identical process. Assumption of a particular aggregate production function for ease of computation does not guarantee that the result will conform to practice (which some might refer to as 'reality').

DeleteTheoretical work in generating models of actual economies are only useful to the extent that the model answers specific questions of some importance faithfully, i.e., the answers are reasonable analogues of what the economy modelled would likely do if the model's inputs and constraints are based on conditions that the economy might face under similar circumstances.

To say that such-and-such a production network model when a scalar quantity, say a coefficient of proportionality, " t ", is increased indefinitely towards infinity, will behave like so-and-so, in the limit, is only a curiosity if it is physically impossible for such a circumstance to arise in the first place. One generally looks to applied mathematics (as opposed to theoretical mathematics) for guidance in such instances.

Resources are fixed in the short-term; reconfiguration of networks feeding finished products and intermediate products into the U.S. economy take time to achieve, and may not all be achievable; CES production functions cannot faithfully model the elasticity of substitution between more than two productive factor inputs. Mishra (2007), states: "Uzawa (1962) and McFadden (1962, 1963) proved that it is impossible to obtain a functional form for a production function that has an arbitrary set of constant elasticities of substitution if the number of inputs (factors of production) is greater than two. Mathematical enunciations of these assertions are now known as the impossibility theorems of Uzawa and McFadden."

The student, reading the literature, is cautioned that neo-classical economics with its emphasis on mathematical models of economic systems is fraught with inconsistencies. Practitioners continue to improve the models, with more mathematics. But, mathematics alone will not make macroeconomics more accurate or improve the fidelity of the models used to predict future economic behavior.

We live in interesting times, in more ways than one.

In an article for the BBVA lecture of the European Economic Association and the BBVA Foundation, David Baqaee and Emmanuel Farhi set down aggregate production functions for networks. The mathematical treatment is dense, but not abstruse to the point of being incomprehensible. P. Samuelson's 1966 concession to J. Robinson et al., re: aggregation of capital productive input factors to conclude the Cambridge-Cambridge debate is discussed in the context of networked production functions. The article is the complement to the Dew-Becker paper.

ReplyDeleteD. Baqaee and E. Farhi, "The Microeconomic Foundations of Aggregate Production Functions", Journal of the European Economic Association, July 22, 2022.

Abstract

Aggregate production functions are reduced-form relationships that emerge endogenously from input-output interactions between heterogeneous producers and factors in general equilibrium. We provide a general methodology for analyzing such aggregate production functions by deriving their first- and second-order properties. Our aggregation formulas provide non-parameteric characterizations of the macro elasticities of substitution between factors and of the macro bias of technical change in terms of micro sufficient statistics. They allow us to generalize existing aggregation theorems and to derive new ones. We relate our results to the famous Cambridge-Cambridge controversy

https://scholar.harvard.edu/farhi/publications/microeconomic-foundations-aggregate-production-functions