**Introduction**

#### One of the oldest and most central problems of macroeconomics is the aggregation problem — how should we reason about the behavior of aggregates composed of many heterogeneous and interacting parts. For example, how doe s a shock to oil production affect real GDP, accounting for the role of oil as an input into the rest of the economy? Can firms like general motors become systemically important because of their position in their supply chains? How important is the reallocation of resources across different producers for explaining aggregate productivity growth? How do tariffs and trade barriers in different parts of global supply chains affect welfare and real GDP? Of course, these are not new questions, and in recent decades, there has been an explosion of theoretical and empirical work on macroeconomic models with heterogeneity.

#### In this essay, I do not attempt to summarize all the important work in this area (see, for example, Guvenen (2011) or Kaplan and Violante (2018) for household and Carvalho and Tahbaz-Salehi (2019) for production heterogeneity). Instead, I describe the framework Emmanuel Farhi and I, along with coauthors like Ariel Burstein and Kunal Sangani, use for thinking through these types of problems. We try to understand macroeconomic quantities by explicitly aggregating over microeconomic details. The objective is to develop a general approach to the theory and measurement of propagation and aggregation with similar and unifying equations that apply in a wide variety of contexts like international trade, business cycles, long-run growth, and cross-country comparisons. Emmanuel, who was an irreplaceable friend and collaborator in this project, tragically passed away in July 2020. This essay serves partially as my tribute to his contributions to this part of economics.

**1. The Fundamental Theorem of Aggregation**

#### To begin with, consider a competitive neoclassical but complicated economy in general equilibrium. This economy has different consumers with heterogeneous preferences and incomes, different producers with complex and nonlinear supply chains, and exists as part of a greater world economy with financial and production interlinkages that cross its borders.

#### How does a small productivity shock to one part of this economy affect aggregate total factor productivity (TFP)? At first glance, one may imagine that the answer to this question should depend on all sorts of details about the underlying production structure. How easy is it to find substitutes for the good being shocked? How complex are the supply chains leading to and the demand chains emanating from this good? What are returns to scale? How quickly and easily can factor markets work to reallocate labor and capital? How big are adjustment costs?

#### Surprisingly, the answer turns out to be deceptively simple: the elasticity of aggregate TFP to a microeconomic TFP shock is equal to the sales of the producer being shocked divided by GDP. In other words, to a first-order approximation, none of the considerations listed above matter as long as the economy is perfectly competitive. Furthermore, if labor supply is inelastic or if the definition of GDP is expanded to include the market value of leisure, then this irrelevance result also applies to real GDP (or under some additional assumptions to welfare).

#### This result, oftentimes known as Hulten’s Theorem (Hulten, 1978), is a consequence of the first welfare theorem, and therefore, is remarkably general. For a while, the surprising generality of this result led economists to de-emphasize the role of microeconomic and network production structures in macroeconomic models. After all, if sales summarize the macroeconomic impact of microeconomic shocks and we can directly observe sales, then why do we need to concern ourselves with the details of the underlying disaggregated system that gave rise to these sales? As with other irrelevance results in economics, like the Modigliani-Miller Theorem or Ricardian Equivalence, much of the economics of aggregation can be understood in terms of deviations from Hulten’s theorem.

**2. Nonlinear Aggregation and Forward and Backward Propagation**

#### At some level, the idea that sales shares are the only statistics that matter clashes with our basic intuition about the way the world works. Of course, sales matter, but surely that is not all that matters. For example, the entire electricity production industry in the US has the same sales share as Walmart (roughly 3% of nominal GDP). Surely shocks to electricity production affect the economy differently than shocks to Walmart. Further, we would expect a negative shock to electricity production to be much more costly than a positive shock is beneficial.

#### In Baqaee and Farhi (2019a), we extended Hulten’s theorem beyond a first-order linear approximation. We showed that the disaggregated details, like the fragility of supply chains, complementarities, returns to scale, and factor market reallocation, details that do not matter to a first-order, do matter for understanding the nonlinear effect of shocks.

#### The key conceptual breakthrough is to recognize that nonlinearities are captured by changes in sales shares. Intuitively, in response to a negative shock to oil or electricity, we expect the sales shares of oil or electricity to skyrocket. On the other hand, in response to a negative shock to Walmart, we expect the sales share of Walmart to decline (perhaps rapidly). The sign and magnitude of changes in sales shares tell us that output is very concave with respect to energy shocks and convex with respect to Walmart shocks. In Baqaee and Farhi (2019a), we characterize in very general and abstract terms the equations that determine changes in sales shares (and hence higher-order effects on aggregate output and productivity).

#### Changes in sales shares are determined by what we call forward and backward propagation equations. Forward propagation equations show how a shock to the marginal cost of a producer propagates through forward linkages, from suppliers to consumers, to change prices downstream. The backward equations show how a shock to the sales of a producer propagates through backward linkages, from consumers to their suppliers, to change sales upstream. Together, the forward and backward equations pin down changes in sales shares in response to shocks.

#### These equations not only help answer questions about the nonlinearities in output in efficient environments, but they can also be used to answer microeconomic questions including, for example, how shocks propagate from one firm to another in general equilibrium, or how the distribution of factor income shares responds to shocks (Baqaee and Farhi, 2019b). Furthermore, unlike Hulten’s theorem itself, the forward and backward propagation equations straightforwardly generalize to more complex environments where the first welfare theorem does not hold, and these generalizations will allow us to extend our analysis beyond efficient equilibria.

#### In a business-cycle calibration with sectoral shocks, we show that nonlinearities magnify negative shocks and attenuate positive shocks, resulting in an aggregate output distribution that is asymmetric (negative skewness) and fat-tailed (excess kurtosis), with a negative mean, even when shocks are symmetric around zero and thin-tailed. Average output losses due to short-run sectoral shocks are an order of magnitude larger than the welfare cost of business cycles calculated by Lucas (1987). Nonlinearities can also cause shocks to critical sectors to have disproportionate macroeconomic effects, almost tripling the estimated impact of the 1970s oil shocks on aggregate output. Furthermore, the extent of nonlinearity in output with respect to a shock is a macroeconomic property in the sense that it depends critically on assumptions outside of the particular market being shocked. For example, assumptions about the factor market, and how quickly factors can be withdrawn from one sector and sent to another critically changes the extent of nonlinearity in how output responds to shocks, holding fixed production functions.

**3. Inefficiencies, Reallocation, and Misallocation**

#### This discussion so far has taken the first welfare theorem and marginal-cost pricing for granted. Therefore, a natural question is: what can we say about the aggregate consequences of a shock if the economy is inefficient and the first welfare theorem does not hold? Hulten’s theorem derives its deceptive simplicity from two facts: (i) marginal-cost pricing ensures that the expenditures by firms on every input measures the elasticity of output with respect to that input (Shephard’s lemma); (ii) marginal-cost pricing ensures production is efficient, meaning that reallocating resources from one user to another does not change real GDP to a first order. Since reallocation effects can safely be ignored to a first-order, (ii) implies that the elasticity of aggregate output to shocks can be computed by assuming that the allocation of resources stays constant and resources simply scale up or down proportionally according to initial shares. From (i) we know that this will change each firm’s output by that firm’s expenditure share on the input being scaled. This “mechanical” effect of scaling resources by initial shares when summed over all input users yields sales, which is the Hulten formula.

#### Inefficient economies break Hulten’s theorem in two ways. First, sales shares no longer capture the “mechanical” effect of scaling up input usage because of wedges between output elasticities and expenditures shares. Second, reallocation effects, which are first-order irrelevant in efficient equilibria, now matter to a first-order and must be solved for.

#### In Baqaee and Farhi (2019d), we extend Hulten’s theorem to distorted economies and provide a structurally interpretable decomposition of changes in aggregate TFP into the mechanical effect of changes in technology, holding fixed the allocation of resources, and the (endogenous) changes in allocative efficiency due to reallocations, holding fixed technologies. In other words, when a producer becomes more productive, the impact on aggregate TFP can be broken down into two components.

#### First, given the initial distribution of resources, the producer increases its output, and this, in turn, increases the output of its direct and indirect customers; this is the mechanical effect that would be equal to sales shares in the absence of wedges. Second, there are reallocation effects that can raise or lower aggregate output holding fixed the level of technology. We show that this reallocation effect can be measured by a specific weighted average of changes in wedges and changes in factor income shares (in an economy with a single factor, say labor, this is simply the labor income share). Intuitively, if a shock reallocates resources in such a way that boosts aggregate output, then this shock will “save” on factor usage. This reallocation makes factors less scarce and causes factor prices and, ceteris paribus, factor income shares to decline on average. The fact that factor income shares decline on average therefore captures changes in aggregate TFP due to reallocation effects.

#### Changes in factor income shares are relatively straightforward to measure and in combination with measures of wedges can be used to non-parametrically measure aggregate productivity growth and to decompose it into its technical and allocative efficiency components. We implement this decomposition in the U.S. over the period 1997-2014. Focusing on firm-level markups as a source of distortions, we find that the improvements in the allocation of resources across firms account for about 50% of the cumulated growth in aggregate TFP. A rough intuition for this surprising result is that average markups have been increasing primarily due to a between-firm composition effect, whereby firms with high markups have been getting larger, and not to a within-firm increase in markups. From a social perspective, these high-markup firms were too small to begin with, and so the reallocation of resources towards them increases aggregate TFP over time.

#### The way aggregate output responds to shocks in an inefficient environment is very dissimilar to the way aggregate output responds in an efficient environment. Luckily, unlike changes in aggregate output, the forward and backward equations in efficient and inefficient economies are very similar. Hence, the same results that can be used to understand nonlinearities in aggregate output in efficient economies can be used to understand first-order reallocation effects in inefficient economies. This is because reallocation effects in inefficient economies depend on how factor shares change in equilibrium and changes in factor income shares generically depend on the details of the production structure. Therefore, for inefficient economies, details like the shape of the production network, the elasticities of substitution, and returns to scale matter to a first- rather than second-order.

#### Using these equations, we characterize, to a first-order, the response of aggregate output to microeconomic shocks in inefficient economies as a function of microeconomic primitives, which can be used to answer counterfactual questions. To do this, we solve for the changes in factor income shares, which summarize reallocation effects, via the forward and backward propagation equations. We can also use these results to provide analytical formulas for the social cost of distortions up to a second-order approximation, generalizing misallocation formulas like those of Hsieh and Klenow (2009) to economies with arbitrary input-output network linkages, arbitrary numbers of factors and returns to scale, microeconomic elasticities of substitution, and distributions of distorting wedges.

#### We show that these generalizations matter quantitatively in empirical applications. For example, we find that in the U.S. in 2015, eliminating markups would raise aggregate TFP by about 20% (depending on the markup series). This increases the estimated cost of monopoly distortions by two orders of magnitude compared to the famous estimate of 0.1% of Harberger (1954). There are several reasons for this dramatic difference. First, we use firm-level data, whereas Harberger only had access to sectoral data, and dispersion of markups is higher across firms within a sector than across sectors. Second, the relevant elasticity of substitution is higher in our exercise than in Harberger’s since it applies across firms within a sector rather than across sectors. Finally, the use of firm-level data and higher elasticities of substitution is not enough: accounting for the existence of input-output linkages, instead of assuming value-added production functions, almost triples the losses.

#### Understanding reallocation and misallocation effects is important for understanding long-run growth. However, our non-parametric decomposition of TFP growth in the US shows that aggregate TFP and the contributions of reallocation to aggregate TFP are strongly procyclical. Hence, reallocation and changes in misallocation potentially play an important role in understanding cyclical behavior as well. Building on Baqaee and Farhi (2019d), in Baqaee et al. (2021), we show that changes in aggregate demand, for example, monetary policy shocks, can naturally affect an economy’s TFP due to reallocation effects. In particular, we propose a supply-side channel for the transmission of aggregate demand shocks by showing that in an economy with heterogeneous firms and endogenous markups, demand shocks can have first-order effects on aggregate productivity.

#### Intuitively, if high-markup firms have lower pass-throughs than low-markup firms, as is consistent with the empirical evidence, then an aggregate demand shock, like a monetary easing, generates an endogenous positive “supply shock” that amplifies the positive “demand shock” on output. The result is akin to a flattening of the Phillips curve. We derive a tractable four-equation dynamic model, disciplined by four sufficient statistics from the distribution of firms, and use it to show that a monetary easing generates a procyclical hump-shaped response in aggregate TFP and countercyclical dispersion in firm-level TFPR. In the data, in line with the model’s predictions, we observe both procyclical aggregate productivity and reallocations to high-markup firms during expansions. A calibration using firm-level pass-throughs suggests that the supply-side channel is quantitatively as important as real rigidities, and amplifies both the impact and persistence of monetary shocks. Moreover, this channel becomes stronger, and the Phillips curve becomes flatter, with increases in industrial concentration.

**4. Incorporating Non-Convexities and Fixed Costs**

#### The discussion so far has abstracted away from non-neoclassical production — that is, fixed costs and increasing returns to scale. Fixed costs, and entry and exit due to the presence of fixed costs, is pervasive at the microeconomic level. Firms and products must maintain a minimum scale to be economically viable. In these situations, firm failures in one part of the economy can travel up and down supply chains causing further exits, which can amplify and reinforce the effects of the shock. In Baqaee and Farhi (2020), building on Baqaee (2018), we extend the framework above to account for fixed costs, entry/exit in disaggregated economies with non-constant returns to scale, input-output linkages, arbitrary substitution elasticities, and distortions.

#### First, we provide conditions under which the equilibrium is efficient, which happens when the net markup on each product is equal to the consumer surplus the product generates with offsetting production subsidies market-by-market. When the equilibrium is efficient, we show that Hulten’s theorem still holds. Hence, even with non-convexities, fixed costs, and product creation and destruction, the logic of Hulten (1978) applies as long as the equilibrium is efficient. That is, reallocation effects can be ignored and the elasticity of output with respect to a technology shock is equal to the sales of the shocked quantity relative to GDP.

#### Of course, in general, there is no reason to expect the decentralized equilibrium to be efficient when there are fixed costs and increasing returns to scale. In inefficient economies, we once again have to contend with reallocation effects to understand aggregate output responses. These reallocation effects depend on which markets expand and shrink, and on whether these adjustments in market sizes occur through changes in the size of existing producers or changes in the number of producers.

#### In Baqaee and Farhi (2020), we show that the resulting changes in allocative efficiency can be summarized by changes in rents and quasi-rents. Here, rent is variable profit due to either decreasing returns or markups, and quasi-rent is the part of this rent that is dissipated by the fixed cost of entry. We show that changes in rents and quasi-rents capture reallocation effects in equilibrium. This generalizes the intuition, in Baqaee and Farhi (2019d), that changes in factor shares capture reallocation effects. The equilibrium changes in rents and quasi-rents are, in turn, given by a generalized version of the forward and backward equations which accounts for fixed costs and entry and exit behavior.

#### We use our comparative static results to study the social cost of distortions and the gains from industrial policy. We generalize Harberger (1964) to economies with non-convexities and entry and exit. In particular, we show that the social cost of inefficiencies is, up to a second-order approximation, equal to the sales-weighted sum of a series of Harberger triangles. Some of these triangles are associated with production and some are associated with entry. We characterize these Harberger triangles in terms of microeconomic primitives —- elasticities of substitution, expenditure shares, and returns-to-scale parameters. In doing so, we overturn a common intuition, valid in models without entry, that the social cost of misallocation is monotone in the elasticities of substitution. Whereas a high elasticity of substitution increases the size of Harberger triangles associated with variable production, a low elasticity of substitution increases the size of Harberger triangles associated with entry. This results in the non-monotonicity of losses with respect to elasticities of substitution.

#### We provide an application by quantifying the social costs of markups using a calibrated firm-level model for the U.S. We decompose the losses into losses arising from misallocation of resources in variable production and misallocation of resources in the amount of entry versus variable production. As mentioned earlier, without entry, we find that markups estimated by a production-function approach à la De Loecker et al. (2019) reduce aggregate productivity by around 20% . Accounting for entry can double these losses. Furthermore, the exact number depends on whether entry costs are paid in units of labor or goods and whether the value of entry arises from consumer surplus or producer surplus.

#### We also study how a marginal entry or production subsidy affects output starting at the distorted equilibrium. Unlike first-best policies, which are independent of network structure and simply ensure efficiency market-by-market, the effects of second-best policies are network-dependent. In particular, for economies with increasing returns to scale, we rationalize and revise Hirschman’s influential argument that policy should encourage expansion in sectors with the most forward and backward linkages, and we give precise formal definitions for these concepts. We show that the optimal marginal intervention aims to boost the sales of sectors that have strong scale economies, but are also upstream of other sectors with strong scale economies.

**5. Enriching the Household Side**

#### All of the results mentioned so far assume that final consumer demand is well-behaved, in the sense that preferences are stable, homothetic, and aggregable across households. In Baqaee and Farhi (2019c), we show how to extend our results to environments with heterogeneous households with homothetic preferences. This is particularly relevant in international contexts where consumers have different preferences, and hence, changes in household income can be a source of changes in sales shares. Since changes in the distribution of income can change sales shares, and sales shares discipline how aggregate output responds to shocks, this means that changes in income distributions can affect how output responds to shocks (via reallocation in inefficient economies and nonlinearities in efficient economies).

#### Furthermore, in open economies, aggregate output for a country is divorced from welfare. Nevertheless, the forward and backward propagation equations, which are relatively straightforward to modify to an international context, can also be used to analyze welfare effects. Using these tools, one can quantify the effect of trade shocks and tariffs on both welfare and real GDP, decompose the effect into mechanical and reallocative forces, and understand the way input-output linkages and complementarities in global supply chains amplify the effect of trade shocks.

#### Baqaee and Burstein (2021) enriches the household-side further by considering how changes in household tastes and or non-homothetic preferences affect real GDP and welfare. These issues are important in both the long-run and the short-run. In the long-run, income elasticities differ widely by sector, so some sectors grow and others shrink as households become richer over time (this is the effect of non-homotheticity). In the short-run, consumer tastes change at the product level over time and these non-price-driven changes in expenditure shares contaminate the construction of price and quantity indices. A particularly noteworthy recent example is provided by the Covid-19 crisis where consumer tastes changed dramatically at the sector level -– that is, holding fixed prices and incomes, household spending patterns changed. In the presence of non-homotheticities or taste shocks, if we maintain perfect competition, then Hulten’s theorem still characterizes changes in aggregate productivity as it is measured in the data. However, this notion of aggregate productivity is no longer well-grounded in economic fundamentals. In particular, aggregate productivity in general equilibrium fails to capture an economically meaningful measure of productivity in the same way that consumer surplus (the area under the Marshallian demand curve) fails to measure consumer welfare in partial equilibrium.

#### Therefore, we provide a modified version of Hulten’s theorem that does answer welfare questions in general equilibrium economies with non-homothetic, non-aggregable, and unstable preferences. We show that calculating changes in welfare in response to a shock only requires knowledge of expenditure shares and elasticities of substitution and (given these elasticities) does not require income elasticities and taste shocks. We also characterize the gap between changes in welfare and changes in real consumption.

#### We apply these results to various short- and long-run applications. In each case, we show that if changes in expenditure shares over time are driven by changes in preferences or changes in income, as opposed to changes in relative prices, then the implications for welfare are different. For example, if structural transformation is caused by income effects (households getting richer) or changes in tastes (households getting older), then welfare-relevant growth is much lower than what we measure. As another example, if taste shocks are positively correlated with price changes at the product-level, something we provide evidence for using retail scanner data, then welfare-relevant measures of inflation are significantly higher than what we measure.

**6. Avenues for Additional Research**

#### There are many avenues for further research. One direction is to use the general equilibrium with wedges type of framework described above to study specific sources of inefficiencies and their interactions in complex environments. For example, financial frictions, as in Bigio and La’O (2016), nominal rigidities, as in Rubbo (2020) or La’O and Tahbaz-Salehi (2020), or taxes, tariffs, and contracting failures, as in Atkin and Donaldson (2021), or capital regulations, as in Bau and Matray (2020). Another interesting direction is to explicitly grapple with dynamic problems, which the type of analysis I outlined deals with using the Arrow-Debreu formalism (recent examples include Lehn and Winberry, 2020 or Huo et al., 2019). Finally, the type of Hulten-derived logic I outlined above is fundamentally smooth. At the most disaggregated levels, economic data is rarely smooth. Changes in the firm-to-firm network of interlinkages are lumpy, and an interesting and important question is to know the extent to which this kind of non-smooth lumpiness at the micro-level can matter for understanding aggregate phenomena — these are questions taken up by Oberfield (2018), Tintelnot et al. (2018), Taschereau-Dumouchel (2020), Ghassibe (2020), Acemoglu and Azar (2020), Acemoglu and Tahbaz-Salehi (2020), and Elliott et al. (2020) among others. All in all, there is much work left to be done and I look forward to all the progress the field will surely make in the coming years.

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