David Baqaee in SED Newsletter, November 2021.

Hulten’s theorem:

… the elasticity of aggregate TFP to a microeconomic TFP shock is equal to the sales of the producer being shocked divided by GDP. … 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. … 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.

Nonlinearities:

… disaggregated 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. … we characterize in very general and abstract terms the equations that determine changes in sales shares

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 …

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 … 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.

… 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)

Frictions:

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 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.

… 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.

… 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).

… 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.

Non-convexities:

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.

Household heterogeneity:

… 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.