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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. On the x axis, take a vector of productivity shocks for each firm, and scale it up or down. The x axis represents this overall scale. The y axis is GDP. The right hand graph is Ian's point: for large shocks, log GDP becomes linear in log productivity -- really simple. Why? Because for large enough shocks, all the networky stuff disappears. Each firm's output moves up or down depending only on one critical input. To see this, we have to dig deeper to complements vs. substitutes. Suppose the price of an input goes up 10%. The firm tries to use less of this input. If the best it can do is to cut use 5%, then the firm ends up paying 5% more overall for this input, the "expenditure share" of this input rises. That is the case of "complements." But if the firm can cut use of the input 15%, then it pays 5% less overall for the input, even though the price went up. That is the case of "substitutes." This is the key concept for the whole question: when an input's price goes up, does its share of overall expenditure go up (complements) or down (substitutes)? Suppose inputs are complements. Again, this vector of technology shocks hits the economy. As the size of the shock gets bigger, the expenditure of each firm, and thus the price it charges for its output, becomes more and more dominated by the one input whose price grows the most. In that sense, all the networkiness simplifies enormously. Each firm is only "connected" to one other firm. Turn the shock around. Each firm that was getting a productivity boost now gets a productivity reduction. Each price that was going up now goes down. Again, in the large shock limit, our firm's price becomes dominated by the price of its most expensive input. 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. Suppose instead that inputs are substitutes. Now, as prices change, the firm expands more and more its use of the cheapest input, and its costs and price become dominated by that input instead. Again, the network collapsed to one link. Ian: "negative productivity shocks propagate downstream through parts of the production process that are complementary ($\sigma_i < 1$), while positive productivity shocks propagate through parts that are substitutable ($\sigma_i > 1$). ...every sector's behavior ends up driven by a single one of its inputs....there is a tail network, which depends on $\theta$ and in which each sector has just a single upstream link."Equations: Each firm's production function is (somewhat simplifying Ian's (1)) \[Y_i = Z_i L_i^{1-\alpha} \left( \sum_j A_{ij}^{1/\sigma} X_{ij}^{(\sigma-1)/\sigma} \right)^{\alpha \sigma/(\sigma-1)}.\]Here $Y_i$ is output, $Z_i$ is productivity, $L_i$ is labor input, $X_{ij}$ is how much good j firm i uses as an input, and $A_{ij}$ captures how important each input is in production. $\sigma>1$ are substitutes, $\sigma<1$ are complements. Firms are competitive, so price equals marginal cost, and each firm's price is \[ p_i = -z_i + \frac{\alpha}{1-\sigma}\log\left(\sum_j A_{ij}e^{(1-\sigma)p_j}\right).\; \; \; (1)\]Small letters are logs of big letters. Each price depends on the prices of all the inputs, plus the firm's own productivity. Log GDP, plotted in the above figure is \[gdp = -\beta'p\] where $p$ is the vector of prices and $\beta$ is a vector of how important each good is to the consumer. In the case $\sigma=1$ (1) reduces to a linear formula. We can easily solve for prices and then gdp as a function of the technology shocks: \[p_i = - z_i + \sum_j A_{ij} p_j\] and hence \[p=-(I-\alpha A)^{-1}z,\]where the letters represent vectors and matrices across $i$ and $j$. This expression shows some of the point of networks, that the pattern of prices and output reflects the whole network of production, not just individual firm productivity. But with $\sigma \neq 1$ (1) is nonlinear without a known closed form solution. Hence approximations. You can see Ian's central point directly from (1). Take the $\sigma<1$ case, complements. Parameterize the size of the technology shocks by a fixed vector $\theta = [\theta_1, \ \theta_2, \ ...\theta_i,...]$ times a scalar $t>0$, so that $z_i=\theta_i \times t$. Then let $t$ grow keeping the pattern of shocks $\theta$ the same. Now, as the $\{p_i\}$ get larger in absolute value, the term with the greatest $p_i$ has the greatest value of $ e^{(1-\sigma)p_j} $. So, for large technology shocks $z$, only that largest term matters, the log and e cancel, and \[p_i \approx -z_i + \alpha \max_{j} p_j.\] This is linear, so we can also write prices as a pattern $\phi$ times the scale $t$, in the large-t limit $p_i = \phi_i t$, and \[\phi_i = -\theta_i + \alpha \max_{j} \phi_j.\;\;\; (2)\] With substitutes, $\sigma<1$, the firm's costs, and so its price, will be driven by the smallest (most negative) upstream price, in the same way. \[\phi_i \approx -\theta_i + \alpha \min_{j} \phi_j.\] To express gdp scaling with $t$, write $gdp=\lambda t$, or when you want to emphasize the dependence on the vector of technology shocks, $\lambda(\theta)$. Then we find gdp by $\lambda =-\beta'\phi$. In this big price limit, the $A_{ij}$ contribute a constant term, which also washes out. Thus the actual "network" coefficients stop mattering at all so long as they are not zero -- the max and min are taken over all non-zero inputs. Ian: ...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...For a full solution, look at the (more interesting) case of complements, and suppose every firm uses a little bit of every other firm's output, so all the $A_{ij}>0$. The largest input price in (2) is the same for each firm $i$, and you can quickly see then that the biggest price will be the smallest technology shock. Now we can solve the model for prices and GDP as a function of technology shocks: \[\phi_i \approx -\theta_i - \frac{\alpha}{1-\alpha} \theta_{\min},\] \[\lambda \approx \beta'\theta + \frac{\alpha}{1-\alpha}\theta_{\min}.\] We have solved the large-shock approximation for prices and GDP as a function of technology shocks. (This is Ian's example 1.) The graph is concave when inputs are complements, and convex when they are substitutes. Let's do complements. We do the graph to the left of the kink by changing the sign of $\theta$. If the identity of $\theta_{\min}$ did not change, $\lambda(-\theta)=-\lambda(\theta)$ and the graph would be linear; it would go down on the left of the kink by the same amount it goes up on the right of the kink. But now a 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. Does all of this matter? Well, surely more for questions when there might be a big shock, such as the big shocks we saw in a pandemic, or big shocks we might see in a war. One of the big questions that network theory asks is, how much does GDP change if there is a technology shock in a particular industry? The $\sigma=1$ case in which expenditure shares are constant gives a standard and fairly reassuring result: the effect on GDP of a shock in industry i is given by the ratio of i's output to total GDP. ("Hulten's theorem.") Industries that are small relative to GDP don't affect GDP that much if they get into trouble. You can intuit that constant expenditure shares are important for this result. If an industry has a negative technology shock, raises its prices, and others can't reduce use of its inputs, then its share of expenditure will rise, and it will all of a sudden be important to GDP. Continuing our example, if 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. Ian: ...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. Ben Moll turned out to be right and Germany was able to substitute away from Russian Gas a lot more than people had thought, but even that proves the rule: 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.There is an enormous amount more in the paper and voluminous appendices, but this is enough for a blog review. ****Now, a few limitations, or really thoughts on where we go next. (No more in this paper, please, Ian!) Ian does a nice illustrative computation of the sensitivity to large shocks:Ian assumes $\sigma>1$, so the main ingredients are how many downstream firms use your products and a bit their labor shares. No surprise, trucks, and energy have big tail impacts. But so do lawyers and insurance. Can we really not do without lawyers? Here I hope the next step looks hard at substitutes vs. complements.That raises a bunch of issues. Substitutes vs. complements surely depends on time horizon and size of shocks. It might be easy to use a little less water or electricity initially, but then really hard to reduce more than, say, 80%. It's usually easier to substitute in the long run than the short run. The analysis in this literature is "static," meaning it describes the economy when everything has settled down. The responses -- you charge more, I use less, I charge more, you use less of my output, etc. -- all happen instantly, or equivalently the model studies a long run where this has all settled down. But then we talk about responses to shocks, as in the pandemic. Surely there is a dynamic response here, not just including capital accumulation (which Ian studies). Indeed, my hope was to see prices spreading out through a production network over time, but this structure would have all price adjustments instantly. Mixing production networks with sticky prices is an obvious idea, which some of the papers below are working on. In the theory and data handling, you see a big discontinuity. If a firm uses any inputs at all from another firm, if $A_{ij}>0$, that input can take over and drive everything. If it uses no inputs at all, then there is no network link and the upstream firm can't have any effect. There is a big discontinuity at $A_{ij}=0.$ We would prefer a theory that does not jump from zero to everything when the firm buys one stick of chewing gum. Ian had to drop small but nonzero elements of the input-output matrix to produces sensible results. Perhaps we should regard very small inputs as always substitutes? How important is the network stuff anyway? We tend to use industry categorizations, because we have an industry input-output table. But how much of the US industry input-output is simply vertical: Loggers sell trees to mills who sell wood to lumberyards who sell lumber to Home Depot who sells it to contractors who put up your house? Energy and tools feed each stage, but don't use a whole lot of wood to make those. I haven't looked at an input-output matrix recently, but just how "vertical" is it? ****The literature on networks in macro is vast. One approach is to pick a recent paper like Ian's and work back through the references. I started to summarize, but gave up in the deluge. Have fun. One way to think of a branch of economics is not just "what tools does it use?" but "what questions is it asking? Long and Plosser "Real Business Cycles," a classic, went after idea that the central defining feature of business cycles (since Burns and Mitchell) is 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. Macro went in another way, emphasizing time series -- the idea that recessions are defined, say, by two quarters of aggregate GDP decline, or by the greater decline of investment and durable goods than consumption -- and in the aggregate models of Kydland and Prescott, and the stochastic growth model as pioneered by King, Plosser and Rebelo, driven by a single economy-wide technology shock. Part of this shift is simply technical: Long and Plosser used analytical tools, and were thereby stuck in a model without capital, plus they did not inaugurate matching to data. Kydland and Prescott brought numerical model solution and calibration to macro, which is what macro has 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). Xavier Gabaix (2011) had a different Big Question in mind: Why are business cycles so large? Individual firms and industries have large shocks, but $\sigma/\sqrt{N}$ ought to dampen those at the aggregate level. Again, this was a classic argument for aggregate "demand" as opposed to "supply." Gabaix notices that the US has a fat-tailed firm distribution with a few large firms, and those firms have large shocks. He amplifies his argument via the Hulten mechanism, a bit of networkyiness, since the impact of a firm on the economy is sales / GDP, not value added / GDP. The enormous literature since then has gone after a variety of questions. Dew-Becker's paper is about the effect of big shocks, and obviously not that useful for small shocks. Remember which question you're after.My quest for a new Phillips curve in production networks is better represented by Elisa Rubbo's "Networks, Phillips curves and Monetary Policy," and Jennifer La'o and Alireza Tahbaz-Salehi's "Optimal Monetary Policy in Production Networks," If I can boil those down for the blog, you'll hear about it eventually. The "what's the question" question is doubly important for this branch of macro that explicitly models heterogeneous agents and heterogenous firms. Why are we doing this? One can always represent the aggregates with a social welfare function and an aggregate production function. You might be interested in how aggregates affect individuals, but that doesn't change your model of aggregates. Or, you might be interested in seeing what the aggregate production or utility function looks like -- is it consistent with what we know about individual firms and people? Does the size of the aggregate production function shock make sense? But still, you end up with just a better (hopefully) aggregate production and utility function. Or, you might want models that break the aggregation theorems in a significant way; models for which distributions matter for aggregate dynamics, theoretically and (harder) empirically. But don't forget you need a reason to build disaggregated models. Expression (1) is not easy to get to. I started reading Ian's paper in my usual way: to learn a literature start with the latest paper and work backward. Alas, this literature has evolved to the point that authors plop results down that "everybody knows" and will take you a day or so of head-scratching to reproduce. I complained to Ian, and he said he had the same problem when he was getting in to the literature! Yes, journals now demand such overstuffed papers that it's hard to do, but it would be awfully nice for everyone to start including ground up algebra for major results in one of the endless internet appendices. I eventually found Jonathan Dingel's notes on Dixit Stiglitz tricks, which were helpful. Update: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. The rest of the notes are also really good. The first part goes a little deeper into more abstract material than is really necessary for the second part and applied work, but it is a wonderful and concise review of that material as well.

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Blog Entry#15August 6, 2023

Barbies Amongst Themselves: Or, What Happens When You Make a Film about a Commodity

Blog: Unemployed Negativity

unemployed negativity

Watching Barbie reminded me of two essays that I had not read in a long time, Luce Irigaray's "Women on the Market " and "Commodities Among Themselves". In those essays Irigaray considers to what extent Marx's theory of the commodity form can be used to make sense of the status of women in society. Irigaray's texts takes as its start the idea of a society founded on an exchange of women, an idea integral to structural and psychoanalytic theories of kinship. From this it is possible to posit that relations among women would have the fantastic character of Marx's brief foray into describing the world of commodities amongst themselves. It is precisely such a world, Barbie Land, that Barbie: The Movie opens. The only difference is that women, Barbies, in this world do not so much exist as things to be exchanged, as daughters to be given away as wives, but are defined by their use value, or, more to the point, their concrete labor. It is a world of Barbie doctors, presidents, supreme court justices, and so on--a Barbie for every career and full employment for all Barbies. Greta Gerwig's film taps into an aspect of Barbie that often falls beneath the image of the Barbie stereotype, or, in the world of the film, Stereotypical Barbie, and that is the myriad number of Barbies that have been manufactured with different careers, from veterinarian to astronaut. The Barbie stereotype of blond hair, impossible proportions, and pink, well everything, dominates our image of Barbie, it is what adults think of when we think of Barbie, so much so that we forget that for a lot of girls (and boys) who play with her she that is less a supermodel than the model for every kind of activity and career. Whatever you want to be they have a Barbie for that. I remember once watching a relative's kid play Barbie and it was less a foray into a world of beauty and fashion than it was an hour of being a large animal veterinarian, giving check ups to horses. A far cry from the image of fashion and beauty that comes to mind when you say Barbie to an adult. The two sides of Barbie, the blonde and pink stereotype that adults think of and the various different Barbies of every career and hobby that kids play with, are the central contradiction of the film.The Barbie pet care centerBarbie Land is that imaginary place where Barbies amongst themselves can be anything or anyone. There are Kens in this world too, but since this world is the world of children playing, no one really knows what Ken is for. Ken is more sidekick than boyfriend. (Pietro Bianchi has offered a great Freudian reading of this world of innocence). The Barbies in Barbie Land are aware of the real world, that it exists, and as far as they are concerned they have fundamentally altered it. An imaginary world where Barbie can be anything must in some sense produce a reality where kids can be anyone. It is the logic of meritocratic role models taken to its logical conclusion. All the world needs is the right role models for the world to change. Trouble begins when stereotypical Barbie (played by Margot Robbie) begins to have some very un-Barbie thoughts, like of death, aging, and cellulite. These intrusive thoughts must be the product of the kid that is playing with her so she has to go out into the "real world" to find this kid and fix things. This brings us back to the commodity form. The commodity, as Marx tells us, is both an exchange value and a use value, it is both something with its own properties, or in the case of labor, capacities, and with a value, a capacity to stand in for other commodities, to be exchanged. In the world of the film we get two sides of Barbie, there is the Barbie Land Barbie in which there is a Barbie that can do anything, and there is the real world Barbie, where Barbie is defined not by her capacities, what she can do, but by her appearance, what she looks like. It is on arriving in the real world that Barbie finds herself not as an object of little girl's dreams, but the object of male fantasies. (As A.S. Hamrah points out in this great roundtable discussion of the film, the patriarchy that Barbie is subject to is incredibly mild and gentle, more befitting a cartoon world than the real world). If I wanted to add another grad school reference, namely Jean Baudrillard, I would say that Barbie's conflict is less between use value and exchange value as it is between use value and sign value, between what Barbie can do and what she signifies, what blonde hair, impossibly long legs, and gravity defying curves signify. To put it back in Irigaray's terms, her capacities might define what she is capable of, but her appearance for men defines her place in society. As Irigaray writes, "just as, in commodities, natural utility is overridden by the exchange function, so the properties of a woman's body have to be suppressed and subordinated to the exigencies of its transformation into an object of circulation among men." Use Value/Exchange Value, the two sides of the commodity are dominated by exchange value just as women in society are dominated by the demand to be seen, and exchanged, by men. Upon arrival in the real world, Barbie and Ken learn that making Barbie role models for every career has not ended patriarchy. Barbie and Ken react differently to the persistence of patriarchy. Barbie is horrified and confused. Ken is happy and excited. Ken finds himself being respected just because he is a man. He immediately hatches a plan to bring the patriarchy to Barbie Land with the help of some books checked out from the library. (I thought for a long time about what this particular plot point reminded me of, a story where two characters have opposed reactions to the new world they are transported to, and eventually I thought of Time after Time, The film in which H.G. Welles and Jack the Ripper end up time traveling to the seventies. Welles is horrified of the lack of social progress while Jack the Ripper revels in the violence of the twentieth century. For sake of this digression, and because I really love that film, I include the following clip.)
Back to the film in question, and skipping several plot points, by the time Barbie discovers the source of her angst, an adult playing with Barbies and returns to Barbie Land it has been transformed. The Barbie dream houses have all been remade into Mojo Dojo man caves for Ken and the Barbies have abandoned their various careers as veterinarians and the President to dote after their Kens, bringing them snacks and beer. The spell of patriarchy is broken, however, when Gloria (America Ferrara) , the adult from the real world who has brought anxiety to Barbie, spells out the contradictions of being a woman. This speech is the thematic and emotional core of the film. Since I found the whole thing online, I post it in its entirety below. "It is literally impossible to be a woman. You are so beautiful, and so smart, and it kills me that you don't think you're good enough. Like, we have to always be extraordinary, but somehow we're always doing it wrong.

You have to be thin, but not too thin. And you can never say you want to be thin. You have to say you want to be healthy, but also you have to be thin. You have to have money, but you can't ask for money because that's crass. You have to be a boss, but you can't be mean. You have to lead, but you can't squash other people's ideas. You're supposed to love being a mother, but don't talk about your kids all the damn time. You have to be a career woman but also always be looking out for other people.""You have to answer for men's bad behavior, which is insane, but if you point that out, you're accused of complaining. You're supposed to stay pretty for men, but not so pretty that you tempt them too much or that you threaten other women because you're supposed to be a part of the sisterhood.

But always stand out and always be grateful. But never forget that the system is rigged. So find a way to acknowledge that but also always be grateful.

You have to never get old, never be rude, never show off, never be selfish, never fall down, never fail, never show fear, never get out of line. It's too hard! It's too contradictory and nobody gives you a medal or says thank you! And it turns out in fact that not only are you doing everything wrong, but also everything is your fault.

I'm just so tired of watching myself and every single other woman tie herself into knots so that people will like us. And if all of that is also true for a doll just representing women, then I don't even know."The speech is a long list of the "too contradictory" situation of women in the real world. In the film this bit of wisdom from the real world restores Barbie Land, frees Barbie from the rule of Ken. However, the film does not connect the contradictions of the real world to the contradictory unity of Barbie as a commodity, a commodity with use value, all of Barbie's various careers from doctor to president, and an exchange value, her appearance. In the film there are two worlds, Barbie Land defined by Barbie's capacities to do anything, and our world, where Barbie is defined by her appearance, but it never really reflects on the contradictory unity of those two worlds, on the fact that while Barbie dolls can do anything they still have to look like Barbie. Making a movie about Barbie is strange endeavor because the logic of Barbie is the logic of Hollywood. It is a world where women can be scientists and superheroes, at least some of the time, but in doing so they still have to look like at least one of the varieties of Barbie. Ability is subordinated to appearance, use value to exchange value.The film presents Barbie Land and the real world as two different realities, one dominated by the different abilities of Barbie and the other by the circulation of her appearance, but the reality of the commodity, of capital, is that use value and exchange value exists side by side even as they contradict each other. As Isabelle Garo puts it, "The originality of Marx's approach attaches to the dialectical nature of his analysis of contradictions, which is no mere juxtaposition of opposed tendencies: the capitalist labour process is not alienating in one respect and emancipatory in another, but it interweaves these two tendencies at the very heart of the labourer's individuality and of social relations."Or, to put it back in the terms of the film, it is not that one gets to choose between a land where Barbies are recognized for their abilities and one that they are reduced to their appearances but they are always both. This is the too contradictory situation referenced in Gloria's speech. The fact that the film does not connect these dots connects brings us back to the question the film asks but does not answer, why has a Barbie Land where dolls can be anything not transformed our world where women are all too often reduced to being dolls? That the film has no reflection on the failure of its own world of role models is its real limit. All Barbie the movie can do is diversify Barbie Land, adding a few different body types and a little more diverse product line, but it cannot address the question as to why all the positive role models in the world have not changed patriarchy. Perhaps that question is for the inevitable sequel.