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1 Earnings Inequality in Production Networks Federico Huneeus Yale University & Central Bank of Chile Kory Kroft University of Toronto & NBER Kevin Lim University of Toronto David J. Price University of Toronto Abstract Recent research on firm-to-firm trade has documented a robust fact: there is substantial heterogeneity in buyer-seller linkages between firms. To what extent does this matter for the inequality of earnings for workers in the production network? We study this question using a novel combination of two administrative datasets from Chile: (i) employer-employee data from income tax records, merged with (ii) firm-to-firm transactions data based on valueadded tax records. We develop a new theoretical framework that has two key features that facilitate analysis of this rich data: (i) imperfect competition in labor markets for workers of heterogeneous ability, and (ii) monopolistically competitive firms that produce output by combining labor with inputs sourced from other firms in a heterogeneous production network. We show that several key parameters mediate the impact of the network structure on earnings: the labor supply elasticity, the elasticity of substitution between labor and intermediates, and the price elasticity of demand. We structurally estimate these parameters and use the model to quantify the importance of production network heterogeneity for earnings inequality by simulating counterfactual equilibria in which firms match with customers and suppliers at random. We find that firm heterogeneity of connectivity in the production network accounts for 41% percent of log earnings variance in Chile. Heterogeneity in matching with both customers and suppliers is quantitatively important, but heterogeneity in matching with suppliers is quantitatively more relevant in explaining earnings inequality. Keywords: PENDING. JEL Codes: PENDING We are grateful to participants at MIT Trade workshop, the University of Toronto international trade and empirical microeconomics workshops, Yale University Cowles Trade Day conference, and the Heterogeneity in Macroeconomics workshop at the Central Bank of Chile for invaluable feedback. We also thank the Munk School of Global Affairs and Public Policy for financial support.

2 1 Introduction Recent research on firm-to-firm trade has documented a robust fact: there is substantial heterogeneity in buyer-seller linkages between firms. At the same time, there is a growing debate about the role that firm behavior has in shaping labor market outcomes. In this paper, we contribute to both of these discussions by studying how the structure of firm-level production networks matters for worker earnings inequality. We begin with a theory of imperfect competition in labor markets. If labor markets are competitive, the law of one price for worker ability should hold, implying that the identity of a worker s employer does not matter for wage determination. This has been tested extensively in the literature through the lens of a standard variance decomposition of earnings inequality into a component attributable to firms and a component attributable to workers. Several empirical studies have shown that firms matter for earnings inequality, through the variance of the firm component as well as the correlation between the firm and worker components (see, for example, Card et al. (2013), Lamadon et al. (2019), Bonhomme et al. (2019)). This motivates a presence for frictions in the labor market that lead to deviations from the law of one price. We study a common example of such a friction that is highly salient in recent public debates: the existence of firm market power in wage setting. We then conjecture that the implications of labor market power for worker earnings depends in important ways on the structure of the production network in which a firm is embedded. For instance, a firm with more productive customers receives more output demand, which could improve the firm s capacity to hire better workers at a higher wage. Similarly, a firm with more productive suppliers faces lower input costs, which has direct implications for both the quantity and type of workers that a firm wants to hire. Given these rich potential interactions, studying the interaction between firm-to-firm connections and earnings inequality is crucial to understanding how changes in the organization of production affects workers. This is especially true in the context of an increasing role for value chains in production and a trend towards higher earnings inequality in the developed world. This paper makes three contributions. The first contribution is to develop a new model of 1

3 imperfect competition in labor markets where firms purchase intermediate goods from suppliers in their network and sell their output to final consumers and to other firms in their customer networks. We derive the equilibrium wage function in this model, which is a function of the standard markdown (depending on the labor supply elasticity facing the firm) and the marginal revenue product of labor (MRPL). The MRPL, in turn, is the product of worker productivity (which is permitted to vary across firms) and a term which is common to all workers at the firm. The production network affects wages through this common term in two ways. First, there is a scale effect which operates directly through the firm s network demand and acts to increase wages. This effect is increasing in the demand elasticity. Second, there is an effect which operates through the unit cost of materials facing the firm. On one hand, a lower cost of materials leads to an increase in materials used which leads to more worker output, holding constant labor, and hence higher wages. On the other hand, the firm may optimally reduce (increase) labor if labor and materials are substitutes (complements) and this would act to further decrease (increase) wages. Hence, the net effect of the input cost on wages can be positive or negative, depending critically on how substitutable labor and materials are relative to how substitutable products are with each other. Note that critical for production networks to matter is the firm faces an upward-sloping labor supply curve. If labor supply was perfectly elastic, differences in network productivity would not matter for wages; firms would take the market wages as given and hire workers until the MRPL was equal to the market wage. The advantage of modeling the formal connection between MRPL and the production network is that it allows one to consider how production networks matter for wage determination. Within the theoretical framework, we derive comparative static results for how firm-level productivity shocks propagate through the network and affect wages at all other firms. We show that the strength of these effects can be measured to a first-order approximation by network observables, such as the shares of sales and expenditures accounted for by firms in each others input and output markets. The results also highlight that the fundamental parameters governing these linkages are the labor supply elasticity, the demand elasticity, and the elasticity of substitution between labor and materials. 2

4 The second contribution of this paper is to develop a strategy for identification and estimation of these structural parameters. Central to our analysis are two key structural equations, one for worker wage and another for the ratio of intermediate spending to total labor costs. The wage equation depends on a time-invariant firm fixed effect, a worker-firm interaction, time-varying worker characteristics and the firm s wage bill. The pass-through of shocks in the aggregate wage bill of a firm to the worker-level wage identifies the labor supply elasticity, similar to Lamadon et al. (2019). 1 The equation for the ratio of intermediate spending to total labor costs depends on relative factor prices, with the coefficient proportional to the substitutability between labor and intermediate goods. In order to estimate this equation, one has to estimate the firm component of wages and the unit price of intermediate goods. To empirically implement these equations, we follow the strategies proposed by Bonhomme et al. (2019) (labor) and Bernard et al. (2019) (intermediate goods). We make use of administrative employee-employer matched data and firm-to-firm transaction data for the full population of Chile. Our estimates of the wage equation indicate that firm fixed effects account for roughly 16 percent of the variance of log earnings across workers. We estimate a firm-specific labor supply elasticity around 6.7. In order to estimate the elasticity of substitution between materials and labor, we estimate intermediate input costs using a two-way reduced-form fixed effects regression based on firm-to-firm sales. We recover unbiased estimates of buyer and seller fixed effects as well as relationship-specific productivity residuals and establish the mapping between these fixed effects and the input costs. Our estimates indicate that the elasticity of substitution between labor and intermediates is 0.91 implying that workers and suppliers are complements. With these estimates in hand, we make our third contribution by showing quantitatively how changes in the connections between firms in the production network affect earnings inequality in the labor market. To do so, we perform a simple counterfactual: we compare the equilibrium under the empirically-observed network with the equilibrium that obtains under a random network (in which all buyer-seller pairs are equally likely to match), holding constant the number of matches in the network. In the empirical network, more productive firms match 1 A key difference with Lamadon et al. (2019) is that they use value-added shocks to identify the labor supply elasticity. We show that with intermediate goods, value-added shocks do not identify the labor supply elasticity. 3

5 with more productive firms. Thus, in the counterfactual, more (less) productive firms will face higher (lower) input costs and lower (higher) output demand. By reducing inequality in the productivity of matches in the production network, the inequality of input costs and output demand between firms is reduced. This, in turn, is passed-through to workers by a reduction in inequality of earnings between firms. Overall, we find that the inequality of the firm component of wages is reduces by 76% and the overall inequality of earnings by 41%. The Gini coefficient of worker earnings is reduced by 14 points. This paper seeks to bridge the literature on labor market power and the literature on firms production networks. Market power in labor markets can arise as a result of concentration (Berger et al. (2019), Jarosch et al. (2019)), differentiated jobs (Sorkin (2018), Card et al. (2018), Lamadon et al. (2019), Chan et al. (2019)), or search frictions (Burdett and Mortensen (1998), Postel-Vinay and Robin (2002), Taber and Vejlin (2018)). Most models of imperfect competition share the common feature that wages at the firm are a markdown below the value of the marginal product. The markdown is a function of a the firm-specific labor supply elasticity which is constant when firms are atomistic or variable when there are a finite number of firms and hence strategic interactions between them. 2. These models all have the implication that heterogeneity across firms due to technology differences and/or differences in demand leads to heterogeneity in wages and hence earnings inequality. At the same time, a growing literature in macro and international trade has recently highlighted the role of production networks in shaping, amplifying or dampening disparities between firms. On one hand, production networks can be important in understanding the microfoundations of heterogeneity between firms as shown in Oberfield (2018) and Bernard et al. (2019). In particular, which type of buyers and suppliers the firm has, influences the size of firms through the creation of heterogeneous input costs and heterogeneous demand shifters. On the other hand, production networks can be relevant for propagating idiosyncratic shocks (Caliendo et al., 2017; Baqaee and Farhi, 2019a,b; Lim, 2019; Huneeus, 2019). This literature has shown how geography, dynamics and misallocation can generate propagation of idiosyncratic shocks through 2 Several papers have estimated the firm-specific labor supply elasticity. See Staiger et al. (2010), Azar et al. (2019), Kline et al. (2019), Lamadon et al. (2019), Dube et al. (2019), and Kroft et al. (2019). 4

6 production networks. In particular, the literature shows how the economic setup and idiosyncratic shocks interact with upstream and downstream linkages to either amplify or dampen heterogeneity of outcomes across firms. The remainder of the paper is organized as follows. Section 2 describes the theoretical framework. Section 3 describes the main analytical implications of the theory. Section 4 presents the data, implements the main empirical analysis and estimates the model. Section 5 implements the counterfactuals and Section 6 concludes. 2 Model 2.1 Labor market Workers Workers in the economy are heterogeneous in ability a with the set of worker abilities denoted by A R d +. 3 For each ability type a A, there is a continuum of workers of exogenous measure L t (a) at time t. We assume that workers have idiosyncratic preferences for working at different firms, where the set of firms is also exogenous and denoted by Ω. Specifically, the utility that a worker with ability a obtains from working at firm i at time t is given by: u it (a) = log w it (a) + log g i (a) + log τ t + β 1 ɛ it (2.1) where w it (a) denotes the wage paid by firm i to workers of ability a, g i (a) denotes the value that workers of ability a derive from amenities of firm i, τ t is a lump-sum income transfer received by each worker independent of employer, and ɛ it is an idiosyncratic preference shock for employment at firm i with β an inverse measure of the shock dispersion. There are several important features of this utility specification. First, we model amenities in order to rationalize heterogeneity in compensating differentials across firms for workers of a 3 The theoretical results that we will establish in this section do not require restrictions on the dimension d of the worker ability space. However, in the empirical estimation discussed in section 4, we will follow Lamadon et al. (2019) and assume d = 2 with worker ability comprised of a permanent component ā and a time-varying component â. 5

7 given ability. This will be quantitatively important in matching the observed firm size-wage premium. Second, lump-sum transfers τ t will account for firm profits in the economy. These are assumed to be rebated to workers in proportion to their income, which is necessary to ensure that transfers do not affect the sorting of workers across firms. Third, we assume that worker abilities are perfectly observable by firms but that firms cannot condition wages on the idiosyncratic preference shocks ɛ it. This will imply the existence of inframarginal workers at every firm who enjoy positive rents from their employment. Fourth, we assume for analytic tractability that the distribution of idiosyncratic preference shocks is characterized as follows. Assumption 2.1. The distribution across workers of idiosyncratic worker preference shocks, ɛ t {ɛ it } i Ω, in every period is a multivariate Gumbel distribution with cumulative distribution function: F ɛ (ɛ t ) = exp i Ω e ɛ it ρ ρ (2.2) where ρ (0, 1]. Note that Assumption 2.1 imposes structure on the cross-sectional distribution of worker preference shocks but does not otherwise restrict the time-series properties of these shocks for a given worker. Also, note that the parameter ρ controls the correlation of preference shocks across firms: as ρ approaches zero, workers view all firms as perfect substitutes, whereas as ρ approaches one, preference shocks across firms become independent random variables. Assumption 2.1 implies that the probability that a worker with ability a chooses to work at firm i at time t is given by: [ ] gi (a) w it (a) γ P it (a) = (2.3) I t (a) where γ β/ρ is a composite parameter that depends on both the preference shock dispersion and correlation of these shocks across firms, and I t (a) is a wage index given by: I t (a) [g i (a) w it (a)] γ i Ω 1 γ (2.4) Note that I t (a) captures the extent of labor market competition for workers of ability a. Hence, 6

8 we refer to this as the competition index for ability a workers Firm-level labor supply Given that the set of workers of ability a is continuous, the worker-choice probabilities (2.3) imply that the total supply of workers of ability a for firm i can be written as: L it (a) = κ it (a) w it (a) γ (2.5) where κ it (a) is a firm-specific labor supply shifter: κ it (a) κ t (a) g i (a) γ (2.6) and κ t (a) is the component of this shifter that is common to all firms: κ t (a) L t (a) I t (a) γ (2.7) Now, we assume that the cardinality of the set of firms Ω is large enough such that each firm views itself as atomistic in the labor market. Specifically, in choosing wages for workers of any ability a A, each firm views the competition index I t (a) as invariant to the firm s choices. Hence, equation (2.5) implies that every firm behaves as though it faces an upward-sloping labor supply curve with a constant elasticity γ. Furthermore, this elasticity is common to all firms. In addition, note from (2.6) that conditional on wages, the supply of workers of a given ability only differs across firms due to differences in the value of firm amenities. 2.2 Final demand We assume that every worker has identical preferences over the consumption of all goods produced by firms in the economy. These preferences are described as follows. 7

9 Assumption 2.2. The utility that a worker of ability a obtains from final consumption is: v t (a) = x it (a) σ 1 σ i Ω σ σ 1 (2.8) where x it (a) is consumption of firm i s output by each worker of ability a and σ > 1 denotes the elasticity of substitution across varieties. We take the CES price index of final consumption as the numeraire, which hence rationalizes the worker utility specification described in section Furthermore, (2.8) implies that aggregate final demand for firm i s output can be written as: x F it = F t p σ F it (2.9) where p F it is the price of firm i s output for final sales and F t is aggregate consumer income (labor income plus profits): F t = ˆ w it (a) L it (a) da + π it (2.10) i Ω A i Ω with π it denoting profit earned by firm i. Note that F t is also equivalent to aggregate valueadded in the economy. 2.3 Production technology Firms produce output by combining labor with materials (intermediate inputs). Combining one worker of ability a with m it (a) units of materials at firm i produces f [φ it (a), m it (a)] units of output, where φ it : A R + maps worker ability into productivity. We assume that the properties of the worker-level production function f are as follows. Assumption 2.3. The worker-level production function f: 1. is increasing in both arguments (f φ > 0 and f m > 0); 2. is concave in materials per worker (f mm < 0); 8

10 3. is homogeneous of degree one. Note also that we allow φ it to vary by firm, which will capture potential worker-firm complementarities in productivity. Total output of firm i is then given by: X it = T it ˆ A L it (a) f [φ it (a), m it (a)] da (2.11) where T it denotes TFP of firm i. To model firm-to-firm trade, we assume that firms are potentially heterogeneous in their buyer-seller connections with other firms. We denote the set of firm i s customers and suppliers at date t by Ω C it Ω and ΩS it Ω respectively. Note that we do not restrict the network to be bipartite: firms can simultaneously be buyers and sellers. Materials for firm i are then produced by combining inputs from all of its suppliers as follows. Assumption 2.4. Production of materials by firm i is given by: M it = j Ω S it ijt (x ijt ) σ 1 σ ψ 1 σ σ σ 1 (2.12) where x ijt denotes the quantity of inputs purchased by i from j, ψ ijt is an idiosyncratic relationshipspecific productivity shock and, σ > 1 denotes the elasticity of substitution across inputs. As is standard in the literature, we assume the same elasticity of substitution across varieties in production as in final demand. 4 Note that the total allocation of materials to a firm s workers must be equal to total materials production by the firm: ˆ m it (a) L it (a) da = M it (2.13) A 2.4 Output market structure and profit maximization We assume a market structure of monopolistic competition in output markets: each firm in the economy chooses a price of its output for each of its customers taking as given the prices set by 4 This simplifies the firm s profit maximization problem as it ensures that total (final plus intermediate) demand for a firm s output has a constant price elasticity. 9

11 all other firms, but does not internalize the effect of its choices on aggregate outcomes in either the labor market or output market. In choosing output prices, firms also take as given the labor supply function (2.5), the production technologies (2.11)-(2.12), and the final demand function (2.9). To simplify exposition of the firm profit-maximization problem, we first note that since each buying firm takes its suppliers prices as given, demand by firm i for inputs from firm j takes the standard form implied by the CES production technology (2.12): x ijt = it ψ ijt p σ ijt (2.14) where p ijt is the price charged by seller j to buyer i, it is a firm-specific intermediate demand shifter that we refer to as network demand, given by: it = M it (Z it ) σ (2.15) and Z it is the unit cost of materials for firm i: Z it = j Ω S it ψ ijt Φ jt 1 1 σ (2.16) Here, we have defined Φ it as the network productivity of firm i, an inverse measure of the firm s price: Φ it p 1 σ it (2.17) Next, note that the upward-sloping labor supply curves faced by each firm imply that marginal costs of production are increasing in output. Hence, a firm s decisions to supply each of its customers are inherently interlinked: lowering prices to increase demand from one customer increases marginal cost and hence affects the choice of prices charged to another customer. However, even though firms are able to charge different prices to different customers, the following result establishes - perhaps somewhat surprisingly - that it is never optimal for them 10

12 to do so. 5 Proposition 1. The profit-maximizing prices charged by any firm i to its customers (including final consumers) do not vary across customers: p jit = p it, j Ω C it {F } Intuitively, each firm maximizes its profits by choosing prices such that marginal revenue from each customer is equal to marginal cost. From (2.9) and (2.14), demand from each customer features a constant and common price elasticity of σ. Hence, marginal revenue is proportional to price. Furthermore, even though marginal cost is increasing, it depends only on total output of the firm and hence is common across customers. As a result, each firm optimally chooses to charge a common price to each of its customers in equilibrium. With this result, total demand for firm i s output is then given from (2.9) and (2.14) as: X it = D it p σ it (2.18) where D it is a demand shifter for the firm given by the sum of final demand (common to all firms) and the network demands of the firm s customers: D it = F t + jt ψ jit (2.19) j Ω C it As with the labor market, we assume that each firm i behaves atomistically in each of its customers input markets, taking the demand shifters { jt } j Ω C it {F } as given when making choices. 6 Finally, we can now write the profit-maximization problem for firm i concisely as a choice 5 All proofs are relegated to Section A of the appendix. 6 This assumption is less restrictive when the cardinality of a firm s customer set is large. 11

13 over its production inputs: π it = { max D 1 σ it X σ 1 σ it {w it (a),m it (a)} a A ˆ s.t. X it = T it A ˆ ˆ } w it (a) L it (a) da Z it L it (a) m it (a) da A A (2.20) L it (a) f [φ it (a), m it (a)] da (2.21) L it (a) = κ it (a) w it (a) γ (2.22) The first-order conditions for this problem can be expressed as: w it (a) = ηmrp L it (a) (2.23) Z it = MRP M it (a) (2.24) where we define η γ 1+γ for brevity. MRP L it(a) denotes the marginal revenue product of workers of ability a and MRP M it (a) denotes the marginal revenue product of materials allocated to workers of ability a. 7 These are in turn given by: MRP L it (a) = 1 µ p itt it φ it (a) f φ [φ it (a), m it (a)] (2.25) MRP M it (a) = 1 µ p itt it f m [φ it (a), m it (a)] (2.26) where we define µ σ σ 1 for brevity. Equation (2.23) hence states the familiar result that wages are a constant markdown η (0, 1) over the marginal revenue products of the respective worker types, while equation (2.24) states that the unit price of materials is equal to the marginal revenue product of materials. Furthermore, since the price of materials is invariant with respect to worker ability, equation (2.24) implies that the marginal revenue product of materials must be equalized across worker ability types. Equation (2.26) then implies 8 that materials are allocated to workers in proportion to their productivity: m it (a) = ν it φ it (a) (2.27) 7 The second order conditions for the firm s profit maximization problem are satisfied under assumption Recall that x is homogeneous of degree one and hence x m is homogeneous of degree zero. 12

14 where ν it is an endogenous constant of proportionality that can be interpreted as materials per efficiency unit of labor. From equations (2.23) and (2.25), wages are then given by: w it (a) = ηφ it (a) W it (2.28) where W it is the component of wages that is common to all workers employed at firm i: W it = 1 µ D 1 σ it X 1 σ it T it f φ (1, ν it ) (2.29) We hence refer to W it as the firm-level wage. Similarly, the first-order condition for materials can be expressed as: Z it = 1 µ D 1 σ it X 1 σ it T it f m (1, ν it ) (2.30) while equilibrium output for firm i is given by (2.11) as: X it = η γ W γ it T itf (1, ν it ) φ it (2.31) Here, we have defined φ it as: ˆ φ it A κ it (a) φ it (a) 1+γ da (2.32) which is endogenous only through the labor supply shifters κ it (a). Hence, we refer to φ it as labor productivity supply for firm i. Finally, we note that sales, profits, labor costs, and material costs for firm i are respectively: R it = it Φ it (2.33) [ 1 π it = it Φ it σ + 1 η [ ]] fl (1, ν it ) µ f (1, ν it ) (2.34) E L it = η 1+γ φit W 1+γ it (2.35) E M it = η γ ν it Z it φit W γ it (2.36) 13

15 while sales from firm j to firm i can be written concisely in terms of the buyer s network demand, the seller s network productivity, and relationship-specific productivity: R ijt = it Φ jt ψ ijt (2.37) Note from (2.33) and (2.34) that the ratio of firm profits to sales is variable, which is a departure from standard models with constant returns to scale in production (where η = 1). 2.5 Equilibrium definition and solution approach We can now define an equilibrium of the model as follows. Definition 1. An equilibrium of the model at time t is a list of values and functions for: (i) aggregate income, F t ; (ii) labor market competition indices, I t ( ); (iii) firm labor supply shifters, κ it ( ); (iv) network demands, it ; (v) network productivities, Φ it ; (vi) demand shifters, D it ; (vii) material costs Z it ; (viii) material production, M it ; (ix) labor productivity supplies, φ it ; (x) output prices, p it ; (xi) output, X it ; (xii) firm-level wages W it ; (xiii) materials per efficiency unit of labor, ν it ; (iv) worker-level wages, w it ( ); (xv) materials per worker, m it ( ); and (xvi) employment, L it ( ), all of which satisfy equations (2.4), (2.5), (2.6) (2.10), (2.13), (2.15), (2.16), (2.17), (2.18), (2.19), (2.20), (2.27), (2.28), (2.29), (2.30), (2.31), and (2.32). A detailed computational algorithm that solves for an equilibrium of the model is provided in Section B of the appendix. 3 Equilibrium Analysis We now study the model s equilibrium properties in more detail to sharpen intuition regarding the key economic mechanisms in the model. 14

16 3.1 Wages Between-firm inequality: firm-level wages We begin by establishing several comparative static results for firm wages, W it, in order to better understand the determinants of between-firm earnings inequality. Upstream, downstream, and firm effects. First, we examine how firm-level wages are affected by conditions at the firm. To do so, we study equations (2.29)-(2.31), which define a system in firm wages, output, and materials, {W it, X it, ν it }, given values for the firm s demand shifter, material cost, TFP, and labor productivity supply, {D it, Z it, T it, φ } it. The following proposition summarizes the main comparative static results for firm wages in this system. [ ] 1 log(fφ (1,ν it )/f m(1,ν it )) Proposition 2. Let ɛ it denote the elasticity of substitution between d log ν it labor and materials under the optimal choice of ν it for firm i. In response to marginal changes in {D it, Z it, T it, φ } it, the firm-level wage, W it, is: (a) strictly increasing in D it ; (b) strictly increasing in T it ; (c) strictly decreasing in Z it if σ > ɛ it, strictly increasing in Z it if σ < ɛ it, and independent of Z it if σ = ɛ it ; and (d) strictly decreasing in φ it. These effects on firm-level wages can be understood in terms of two economic forces: scale and substitution effects. First, an increase in the firm s demand shifter, D it, or TFP, T it, leads to an upward shift of its isorevenue curve, allowing it to achieve higher scale holding constant its input mix of labor and materials. Since firms face upward sloping labor supply curves, the increase in employment generated by these scale effects leads to higher wages at the firm. Second, lower material costs Z it also generate positive scale effects for the firm, which tends to increase wages. However, a reduction in material costs generates an additional substitution 15

17 effect, as firms respond by moving along their isorevenue curve and changing the relative input of labor to materials in production. Intuitively, the strength of the scale effect is mediated by the elasticity of substitution across goods, σ: if products are more substitutable, the same reduction in input cost allows the firm to achieve a larger increase in sales since demand is more sensitive to price. The strength of the substitution effect, on the other hand, is naturally determined by the elasticity of substitution between labor and materials, ɛ: as ɛ decreases from 1 to 0 (labor and materials are complements), the same reduction in material cost induces the firm to hire more labor, whereas as ɛ increases from 1 to (labor and materials are substitutes), the same reduction in material cost induces the firm to hire less labor. Hence the net effect of changes in input cost Z it on firm-level wages depends on how the scale effect (σ) compares with the substitution effect (ɛ). Finally, an increase in labor productivity supply, φ it, allows the firm to reduce the physical quantity of workers that it hires while maintaining the same scale and the same ratio of labor efficiency units to material inputs. Hence, the effect of an increase in φ it on firm-level wages is unambiguously negative. Note that Proposition 2 implies two important predictions about the interaction between production network linkages and firm wages. First, increases in demand D it resulting from shocks downstream in a supply chain always have positive effects on wages of firms upstream. Second, decreases in input cost Z it arising from shocks upstream in a supply chain can have positive, negative, or neutral effects on wages of firms downstream, where the key determinant of these effects is how substitutable goods are with one another (σ) relative to how substitutable workers are with materials (ɛ). Accounting for network effects. Proposition 2 characterizes how firm-level wages respond to changes in exogenous TFP, T it, as well as changes in endogenous demand and supply variables, {D it, Z it }. We now characterize how firm-level wages respond to exogenous shocks to TFP, accounting for the effect of these shocks on {D it, Z it } through the production network. We will restrict attention to shocks that do not have general equilibrium effects in order to focus on other economic mechanisms of interest. Specifically, we will consider TFP shocks that do 16

18 not affect aggregate income and the competition indices for each worker ability type, which we denote jointly by: Θ GE t { F t, I t ( )} (3.1) In what follows, we denote by Ŷit d log Y it the marginal log change in a firm-level variable Y it and by Ŷt {Ŷit } i Ω the vector of these log changes for all firms. We also define the following observables. First, let Σ C t and Σ S t denote the output and input share matrices respectively, which are Ω Ω matrices with (i, j)-elements given by: Σ C ijt Σ S ijt R jit k Ω C it {F } R kit R ijt k Ω S R ikt it (3.2) (3.3) In other words, Σ C ijt is the share of firm i s total sales accounted for by firm j, while ΣS ijt is the share of firm i s input purchases that are accounted for by firm j. Next, we define the labor share vector Λ t, which is a Ω 1 vector with i th -element equal to the share of labor in total costs for firm i: Λ it E L it E L it + EM it (3.4) Case 1: no intermediates. We begin by characterizing a special case of the model in which intermediate inputs are not used in production. The following proposition characterizes the effects of firm-level TFP shocks on firm-level wages in this setting. Proposition 3. Suppose that intermediates are not used in production. Then firm-level wages are given by: W it = ( F t Tit σ 1 ) 1 γ+σ µ σ η γ φ it (3.5) In response to marginal changes in TFP that do not affect Θ GE t, firm wages respond as follows: Ŵ t = σ 1 γ + σ ˆT t (3.6) There are three important takeaways from Proposition 3.6. First, there is positive pass- 17

19 through of firm TFP shocks to wages, which is a corollary of Proposition 2. Second, the degree of pass-through is increasing in the elasticity of substitution across goods, σ, and decreasing in the labor supply elasticity, γ. Pass-through is increasing in σ because a larger value of σ implies that the marginal revenue product of labor is more sensitive to changes in TFP, whereas pass-through is decreasing in γ because a larger value of γ implies that firms have less market power in labor markets. 9 Third, TFP shocks at one firm cannot affect wages at another firm except through general equilibrium. Case 2: no upstream effects. Next, we consider a case of the model in which intermediates are used in production, but in which only downstream effects of TFP shocks on wages are operative. As shown in Proposition 2, upstream shocks that affect firm input costs Z it have no effect on wages if σ = ɛ. Proposition 4. Suppose that σ = ɛ globally. Then in response to marginal changes in TFP that do not affect Θ GE t, firm wages respond as follows: Ŵ t = σ 1 γ + σ (I Σc t) 1 ˆTt (3.7) where I is the Ω -dimensional identity matrix. Equation (3.7) has a simple interpretation. First, note that the matrix (I Σ c t) 1 can be written in terms of the following geometric series: (I Σ c t) 1 = I + Σ c t + (Σ c t) 2 + (3.8) Hence, the (i, j)-element of this matrix summarizes the share of firm j in firm i s sales both directly and indirectly through the production network. Equation (3.7) thus implies that the pass-through from TFP shocks at firm j to changes in wages at firm i depends on how important j is as a buyer downstream of all the supply chains that i participates in, where buyer importance 9 Equation (3.6) is identical to the pass-through of firm TFP shocks to wages in the Lamadon et al. (2019) model, which does not have intermediate inputs. 18

20 Figure 1: Example of interaction between upstream and downstream effects of TFP on wages can be measured through observable sales shares Σ c ijt. Interestingly, in this case indirect buyer importance depends only on powers of the sales share matrix and does not otherwise decay with the degree of separation in the supply chain linkage between firms. General case. Finally, we characterize the general case in which σ ɛ and both downstream and upstream effects are operative. In this case, there are complex interactions between the upstream and downstream effects of TFP shocks on firm wages. To illustrate, consider three firms that are linked in a supply chain as shown in Figure 1 and suppose that there is a positive shock to TFP for the firm in the middle of the sub-chain ( ˆT 2 > 0). This shock has several effects on wages in the supply chain. First, as shown in Proposition 2, the direct effect of the shock is an increase in wages at the firm hit by the shock (Ŵ2 > 0). Second, there is upstream propagation of the shock to firm 1: it leads to higher demand for firm 1 s output ( ˆD 1 > 0) due to an increase in scale for firm 2, which increases wages at firm 1 (Ŵ1 > 0). Third, there is downstream reflection of the shock back from firm 1 to firm 2: due to increasing marginal costs, higher demand for firm 1 raises its price of output (ˆp 1 > 0), which increases firm 2 s input cost (Ẑ2 > 0) and has non-neutral effects on wages at firm 2 if σ ɛ (Ŵ2 0). Fourth, there is downstream propagation of the shock to firm 3: it reduces the price of firm 2 s output (ˆp 2 < 0), which reduces firm 3 s input cost (Ẑ3 < 0) and has non-neutral effects on wages at firm 3 if σ ɛ (Ŵ3 0). Finally, there is 19

21 upstream reflection of the shock back from firm 3 to firm 2: changes in input cost and wages at firm 3 lead to lower demand for firm 2 ( ˆD 2 < 0) which leads to lower wages at firm 2 (Ŵ2 < 0). As is evident from this simple example, predicting the effects of TFP shocks on firm wages can be complicated in general. However, the following proposition establishes that the crosselasticities of firm wages with respect to firm TFPs are completely determined by a small set of model parameters and observables. Proposition 5. In response to marginal changes in TFP that do not affect Θ GE t, firm wages respond as follows: ( ) Ŵ t = Ψ γ, σ, ɛ, Σ C t, Σ S t, Λ t ˆTt (3.9) where the matrix of cross-elasticities Ψ depends only on the model parameters {γ, σ, ɛ} and the } observables {Σ C t, Σ S t, Λ t. Hence, Proposition 5 implies that if one observes sales shares Σ C t, input shares Σ S t, and labor shares Λ t, then predicting the first-order effects of TFP shocks on wages in the production network only requires knowledge of the parameters {γ, σ, ɛ}. 10 In particular, one does not require knowledge of the relationship-specific productivities {ψ ijt } or other features of the worker-level production function f Within-firm inequality and worker sorting What determines the sorting of workers across firms? Consider a firm i and two worker ability types {a, a}. Note from (2.28) that the relative wage paid by this firm to workers of the two ability types is: w it (a ) w it (a) = φ it (a ) φ it (a) (3.10) Since labor productivity functions are exogenous, this implies that relative wages between different worker ability types within a firm are also exogenous. Nonetheless, it is possible for the 10 For a general worker-level production function f, the elasticity of substitution between workers and materials may depend on the inputs chosen and hence may vary by firm, in which case ɛ should be interpreted as an Ω 1 vector of elasticities. In the estimation below, we impose a restriction on the functional form of f that ensures ɛ is constant across all firms. 20

22 composition of workers within a firm to vary endogenously, which then gives rise to endogenous within-firm wage dispersion. To see this, note from (2.5), (2.6), (2.7), and (2.28) that relative employment of workers of two ability types a and a is: L it (a ) L it (a) = L t (a ) L t (a) }{{} supply [ It (a ) I t (a) ] γ }{{} competition [ gi (a ] [ ) φit (a ] ) γ g i (a) φ it (a) }{{}}{{} amenities productivity (3.11) Relative labor supply, amenity values, and labor productivities are exogenous. However, relative labor market competition for the two worker ability types, captured by the relative competition indices, is potentially endogenous. Specifically, from (2.4) and (2.28), we can express the relative competition indices as: [ It (a ] ) γ i Ω = [g i (a ) φ i (a ) W i ] γ I t (a) i Ω [g i (a) φ i (a) W i ] γ (3.12) Now, observe from (3.12) that if amenity values do not vary across firms (g i (a) = g (a) a A) and there are no worker-firm complementarities (φ i (a) = φ (a) a A) then the firm-level wages in (3.12) cancel out and relative competition is exogenous. With heterogeneous amenity values and worker-firm complementarities, however, relative competition depends on the distribution of firm-level wages across firms. To illustrate this, consider a simple example in which the economy consists of two firms, 1 and 2. For brevity, let h i (a) g i (a) φ i (a) denote the composite of amenities and labor productivity. Then relative competition for two worker types a and a is: [ It (a ] ) γ = [h 1 (a ) W 1 ] γ + [h 2 (a ) W 2 ] γ I t (a) [h 1 (a) W 1 ] γ + [h 2 (a) W 2 ] γ (3.13) It is straightforward to show that the right-hand side of (3.13) is strictly increasing in the relative firm wage W 2 /W 1 if and only if the following condition is satisfied: h 2 (a ) h 2 (a) > h 1 (a ) h 1 (a) (3.14) 21

23 Intuitively, if the comparative advantage (in terms of amenities and labor productivity) of a - ability workers versus a-ability workers is higher at firm 2 than firm 1, then any shock to the economy that makes firm 2 larger than firm 1 in the labor market (in the sense that W 2 increases relative to W 1 ) will increase relative competition for a -ability workers relative to a- ability workers. What is the effect of an increase in W 2 /W 1 on the within-firm earnings distribution in this example? Suppose for concreteness that W 2 > W 1 and φ i (a ) > φ i (a) for all i Ω, so that a -ability workers are more productive than a-ability workers in all firms and hence earn higher wages. Then (3.14) implies that high-ability workers have a comparative advantage in high wage firms. Now, note from (3.11) that an increase in relative labor market competition reduces relative employment within all firms. This implies that an increase in between-firm wage dispersion reduces within-firm employment dispersion if the comparative advantage condition holds. Since relative wages are exogenous, this implies a reduction in within-firm wage dispersion as well. 3.2 Aggregate Welfare and Value-added Aggregating utility (2.1) over workers and accounting for the optimal sorting of workers to firms gives us the following measure of welfare for workers at each firm. Proposition 6. The average utility of workers of ability a that are employed at firm i is: ū it (a) = log I t (a) + log τ t + const. (3.15) where the value of the transfer τ t is equal to the aggregate ratio of value-added to labor income: τ t = F t Ē L t (3.16) with ĒL t i Ω EL it. Average utility across all workers in the economy is hence: ˆ Ū t = A [ L (a) L ] log I t (a) da + log F t log ĒL t + const. (3.17) 22

24 Next, we consider the effects of firm-level TFP shocks on average utility, Ūt, and aggregate value-added, F t. Recall that we take the final consumption price index as the numeraire and hence F t is equivalent to aggregate welfare from consumption. Here, we define a Ω 1 vector of Domar weights S R t value-added: with i th element equal to firm i s total sales as a share of aggregate S R it R it F t (3.18) In addition, we define a Ω 1 vector of labor market shares S L t with i th element equal to firm i s share of employment aggregating across all worker ability types: S L it A L it (a) da A L t (a) da (3.19) The following proposition then establishes the first-order effects of TFP shocks on aggregate welfare and value-added in a special case of the model without intermediates and without heterogeneous sorting of workers across firms. Proposition 7. Suppose that intermediates are not used in production. Suppose also that amenities and labor productivities do not vary across firms, such that g i (a) = g (a) and φ it (a) = φ t (a) for all a A. Then the first-order effect of changes in TFP on average utility is: dūt = [ γ + 1 ( ) S R σ 1 ( ) ] t + St L ˆT t (3.20) γ + σ γ + σ and the first-order effect of changes in TFP on aggregate value-added is: [ σ (γ + 1) ˆ F t = γ + σ ( ) St R γ (σ 1) ( ) ] St L ˆT t (3.21) γ + σ 23

25 In addition, the Domar weight and total employment share for firm i are given respectively by: S R it = S L it = (T it ) γ+1 γ+σ (σ 1) j Ω (T jt) γ+1 γ+σ (σ 1) (3.22) (T it ) γ γ+σ (σ 1) j Ω (T jt) γ γ+σ (σ 1) (3.23) Proposition 7 shows that the introduction of imperfect competition in labor markets alone breaks the classic Hulten (1978) result, in the sense that firm-level Domar weights are no longer sufficient statistics for the first-order effects of firm-level TFP shocks on either aggregate welfare or value-added. In particular, predicting these aggregate effects also requires knowledge of firm employment shares as well as the parameters σ and γ. Furthermore, if there is heterogeneous sorting of workers across firms (due to either heterogeneous amenities through g i or workerfirm complementarities through φ i ), then one also needs to know firm employment shares for each worker ability. Finally, note that in the limit as γ, firm shares of value-added and employment are identical, hence we recover the Hulten (1978) result under perfectly competitive labor markets. 4 Estimation We now turn towards estimating the key parameters of the model. We discuss the estimation in four steps. First, we describe the data we use. Second, we impose additional assumptions to connect the model described in Section 2 to the data. Third, we outline the estimation strategy. We end this section by presenting the estimation results and fit of the model to the data, both in terms of targeted and non-targeted moments. 4.1 Data We use three administrative datasets from tax records that firms report to the Internal Revenue Service from Chile (SII, for its acronym in Spanish). These datasets cover the entire formal private sector in Chile. First, we use a matched employer-employee dataset that has annual 24

26 and monthly earnings from each job that a worker has for the period. 11 General descriptive statistics of this dataset can be found in Aldunate et al. (2019). Second, we use a firm-to-firm dataset that is provided due to the value-added tax (VAT) for the period. In this dataset, each firm reports the full list of intermediate buyers and suppliers each year, together with the value of the flow of the transaction for each pair reported. General descriptive statistics of this dataset can be found in Huneeus (2019). Finally, we use an administrative dataset that has a set of characteristics of the balance sheet of firms for the period. This dataset includes, for example, total sales of the firm, the main industry, location of the headquarter, investment. All these datasets can be easily merged by using the unique tax ID that firms have. 12 Table 1 presents the size of the different samples of these datasets we use for different sections of the estimation and Table 2 presents basic descriptive statistics of the dataset aggregated at the firm-level. 11 Note that the measure of labor income that this dataset has includes wages and benefits. 12 Note that all the tax forms are reported at the headquarter level, so we do no have plant level information. Also, it might be that a firm has several tax IDs. We currently do not have ownership characteristics to consolidate the information at the firm level. 25

27 Table 1: Sample Sizes Panel A: Firm-to-Firm Dataset Links Sellers Buyers Sample Unique Observation-Years Unique Observation-Years Unique Observation-Years Full 9,698,676 18,948,347 78, , , ,402 Estimation 9,676,735 18,904,269 65, , , ,201 Estimation/Full (%) Panel B: Employer-Employee Dataset Workers Firms Sample Unique Observation-Years Unique Observation-Years Full 7,875,082 52,970, , ,490 Prime Age Male 4,000,906 27,318, , ,982 Prime Male Stayers Only 412,996 3,784,733 4,029 47,043 Stayers/Prime Age Male (%) Panel C: Firm Dataset Firms Sample Unique Observation-Years Full 184, ,490 Estimation 65, ,741 Estimation/Full (%) Notes: This table presents basic descriptive statistics of the size of the different samples used throughout the paper. Panel A presents statistics of the sample size of the firm-to-firm transaction dataset. These statistics are presented for the full sample and the sample used in estimating the two-way fixed effect model from Equation (4.18). For each of these samples, we document the unique number of firm-to-firm links, the total number of links-year observations, the unique number of suppliers and buyers, and the total number of buyers-year and suppliers-year observations. Panel B presents statistics of the sample size of the employer-employee dataset. These statistics are presented for the full sample, the sample of prime age (25-60) male workers which is the main sample we use for estimating the two-way fixed effect model from Equation (4.7), and the sample of balanced panel of firms and prime age male workers that are stayers in those firms throughout the period of analysis. For each of these samples, we document the unique number of workers, the total number of worker-year observations, and the unique number of firms and the total number of firm-year observations. Finally, Panel C presents statistics of the sample size of the firm dataset. These statistics are presented for the full sample and the sample used for estimation, e.g., Equation (4.8). Table 2: Basic Statistics Between Firms Moments Value Added Employment Average Wage SD of Within Firm Wages Labor Share Number Buyers Number Suppliers Average 1,404, P10 49, P50 246, P90 2,431, , SD 4,423, Notes: This table presents basic statistics of the dataset used in the paper. In particular, it presents information of firm characteristics coming from the three administrative datasets: the firm-to-firm dataset, the employer-employee and the firm dataset. The statistics are presented at the firm-level and the sample used in the one of Estimation from Panel C of Table 1. 26

28 4.2 Additional Assumptions Before describing the estimation strategy in detail, we impose three sets of additional assumptions in order to connect the model described in Section 2 to the data described in section 4.1: (i) functional form assumptions; (ii) orthogonality conditions; and (iii) a steady-state assumption Parametric assumptions Worker-level production function. First, we impose structure on the worker-level production function f. Assumption 4.1. The production function f takes the following CES form: f (φ, m) = [λ 1 ɛ φ ɛ 1 ɛ ] + (1 λ) 1 ɛ ɛ 1 ɛ 1 ɛ m ɛ (4.1) where λ [0, 1] and ɛ (0, ). If λ = 1, output is produced using labor alone and the model simplifies to a version of the model studied in Lamadon et al. (2019), which does not include intermediate inputs. With this specification, the elasticity of substitution between labor and materials, ɛ, is constant. We allow this elasticity to take on any value: if ɛ > 1, labor and intermediates are gross substitutes; if ɛ (0, 1), labor and intermediates are gross complements; and if ɛ = 1, production takes a Cobb-Douglas form. Worker ability. To estimate the model using worker-level data, we follow Lamadon et al. (2019) and assume that worker ability is characterized as follows. Assumption 4.2. The ability of worker m at time t, a mt, is comprised of a permanent (timeinvariant) component ā m and a transient (time-varying) component â mt. The transient component of worker ability is given by: log â mt = β χ mt + ξ mt (4.2) 27

29 where χ mt is a vector of year and age effects. The stochastic process of the worker ability innovation ξ mt is iid across workers. The separation of worker ability into permanent and transient components will be important for allowing worker-firm complementarities in the estimation below. To perform model counterfactuals, we will also need to impose structure on the cross-sectional distribution of worker abilities in the economy. Assumption 4.3. The distribution across workers of log permanent and transient worker ability in period t is a multivariate normal distribution with zero mean and covariance matrix Σ a,t. Labor productivity. We next assume that worker ability maps into labor productivity as follows. Assumption 4.4. The productivity function φ it is time-invariant and takes the following form: φ it (a mt ) = φ i (a mt ) = ā 1+θ i m â mt (4.3) where θ i is an exogenous parameter. The returns to permanent worker ability are hence allowed to vary by firm through θ i, but the transient component of worker ability affects all firms in the same way. This will be important for two reasons. First, it guarantees that interaction terms between worker and firm effects in worker-level regressions that we estimate below are time-invariant. Second, it implies that innovations to the transient ability of a worker do not induce differences in the relative productivity of the worker across firms and hence do not generate sorting on this dimension of ability. Firm amenities. As with labor productivity above, we also impose a restriction on the dependence of firm amenity values on worker ability. 28

30 Assumption 4.5. The firm amenity function depends only on the permanent component of worker ability and takes the following form: log [g i (a mt )] = δ 0i + δ 1i log ā m (4.4) where δ 0i and δ 1i are exogenous parameters. This restriction is necessary for the same reasons as Assumption 4.4 above. Relationship-specific productivity. Next, we assume that relationship-specific productivity can be decomposed as follows. Assumption 4.6. Log relationship-specific productivity between buyer i and seller j is given by: log ψ ijt = log ψ it + log ψ jt + log ψ ijt (4.5) The relationship productivity residual log ψ ijt is a normal random variable with zero mean and standard deviation σ ψ,t, is iid across buyer-seller pairs within a period, and is independent across time within a buyer-seller pair. Following Bernard et al. (2019), we refer to the firm-specific component of relationship productivity, ψ it, as relationship capability. As discussed in Bernard et al. (2019), allowing for two dimensions of firm heterogeneity (TFP and relationship capability) is important for fitting key moments in production network data, especially the fact that firms with more customers have higher sales but lower sales per customer, which is a feature of the Chilean firm-to-firm transactions data as well. Distribution of firm types. Under the above assumptions, firms are heterogeneous in two dimensions: TFP T it and relationship capability ψ it. We henceforth refer to this pair of firm characteristics as firm type, denoted by χ it {T it, ψ it }. We now impose structure on the crosssectional distribution of firm types in the economy. 29

31 Assumption 4.7. The distribution across firms of log firm type, log χ it, in period t is a normal distribution with mean vector m χ,t and covariance matrix Σ χ,t. The production network. To connect the observed production network with the network Ω S it in the model, we assume that matching between firms depends only on the exogenous firm types of the buyer and seller. Assumption 4.8. The probability that buyer i matches with seller j in the production network at date t is given by m t (χ it, χ jt ) for some matching function m t Orthogonality conditions Next, we impose an assumption about the orthogonality of worker-level and firm-level shocks that will be important for identification. Assumption 4.9. Idiosyncratic worker preference shocks, ɛ mit, worker ability shocks, ξ mt, and firm types, χ it, follow independent Markov processes Steady-state Finally, we assume that the data is characterized by a steady-state of the model in which the general equilibrium terms Θ GE in (3.1) do not vary over time. Assumption Aggregate income, F t, and the labor market competition indices, I t ( ), are time invariant. Importantly, Assumption 4.10 allows us to treat labor productivity supplies φ it for each firm as time-invariant as well, so that these are absorbed by firm fixed effects in the regression specifications discussed below. 4.3 Estimation Strategy The exogenous parameters of the model that we estimate can now be summarized as: (i) the labor supply elasticity, γ; 13 (ii) the elasticity of substitution between labor and materials, ɛ; 13 Recall that γ β/ρ. We will not require separate identification of the idiosyncratic preference shock dispersion β and the correlation of shocks across firms ρ. 30

32 (iii) the standard deviation of the log relationship productivity residual, σ ψ,t ; (iv) the mean and covariance matrix of the firm type distribution, {m χ,t, Σ χ,t }; (v) the matching function m t ; (vi) the elasticity of substitution between firm products, σ; (vii) the weight on labor in the worker-level production function, λ; (viii) the worker-firm productivity parameters, θ i ; (ix) the covariance matrix of the worker ability distribution, Σ a,t ; and (x) the firm amenity parameters, δ i. The outline of our estimation strategy is as follows. First, we estimate {γ, ɛ, σ ψ,t } using reduced-form regressions that are derived from the model. Second, we back out firm types χ = {T, ψ} from the data using the structure of the model and use these to calibrate the moments of the firm type distribution, {m χ, Σ χ }, as well as the matching function, m t. Third, all remaining parameters are estimated via a simulated method of moments (SMM) algorithm since the model does not provide closed form solutions to estimate or calibrate these parameters directly Labor Supply Elasticity To estimate the labor supply elasticity γ, first note that under Assumptions 4.2 and 4.4, the log productivity of a worker m with ability a mt {ā m, â mt } at time t is given by: log φ i(m,t) (a mt ) = log ā m + θ i(m,t) log ā m + β χ mt + ξ mt (4.6) where i (m, t) denotes the firm chosen by worker m at time t. Combining this with (2.28) and (2.35), we can then express worker-level wages as: log w i(m,t) (a mt ) = log ā }{{ m + θ } i(m,t) log ā m 1 }{{} 1 + γ log φ i(m,t) worker FE }{{} worker-firm interaction firm FE + β χ mt }{{} worker characteristics γ log EL i(m,t)t }{{} firm shocks + ξ mt }{{} residual (4.7) Note that the pass-through of changes in the aggregate wage bill of a firm to worker-level 31

33 1 wages identifies the coefficient 1+γ and hence the labor supply elasticity γ. Intuitively, γ controls the extent of imperfect competition in the labor market and hence mediates the pass-through of firm-level shocks to wages. Equation (4.7) resembles the pass-through equation estimated by Lamadon et al. (2019). An important difference, however, is that in the LMS model, the wage bill E L is a constant fraction of value-added for any given firm. Hence, the pass-through coefficient 1 1+γ in (4.7) can be estimated using value-added data. In contrast, with both imperfect competition in output markets and intermediate inputs in our model, E L is no longer proportional to firm value-added. Hence, estimation of 1 1+γ will require data on the wage bill instead of value-added.14 Furthermore, note that if θ i is common to all firms, equation (4.7) fits within the Abowd et al. (1999) framework. If instead there are worker-firm complementarities and θ i differs across firms, then equation (4.7) can be estimated via the procedure in Bonhomme et al. (2019). We hence estimate the labor supply elasticity, γ, by implementing the reduced-form regression implied by (4.7). Under Assumptions 4.4 and 4.5, workers sort to firms based only on the permanent component of worker ability. Hence, innovations to transient worker ability, ξ mt, are independent of the type of the worker s firm, χ i(m,t)t, which determines firm-level equilibrium variables such as the wage bill, Ei(m,t)t L. Furthermore, under Assumption 4.9, the stochastic processes for transient worker ability innovations and firm types are independent. Hence, estimation of (4.7) using ordinary least squares (OLS) yields an unbiased estimate of 1 1+γ. The sample used for estimating γ is with a balanced panel of firms and workers. That is, we use firms operating every year of the sample and workers that stay in the firm throughout the period of analysis. After applying this filter, we exclude the first and last year of employment spells of each worker. Table 1 presents descriptive measures of the size of this sample of stayers relative to the full sample used in the remaining analysis. We present evidence of this regression in two different strategies. First, we follow Lamadon et al. (2019), and show this elasticity as a result from a difference-in-difference approach (DiD). For this, we follow a three step procedure. First, for each year, we order firms according to log 14 In ongoing work, we also estimate (4.7) using value-added to determine the extent of the bias arising from misspecification. 32

34 changes of the wage bill of the firm. Second, we identify the treatment when firms have log changes of their wage bill above the median of log changes of wage bill across firms. Finally, we plot difference in wage bill of treated and control firms both at each year (t = 0) and years before (t < 0) and after (t > 0). We perform this step for each calendar year. We perform the same exercise for other characteristics of the firm, such as moments of the earnings distribution within the firm. Note that whenever we use the wage bill, we follow a leave-out-one strategy. That is, we exclude from the wage bill the wage of the worker used in the left-hand side in order to avoid overestimating the pass-through coefficient (and thus avoiding underestimating γ) due to a mechanical positive correlation between between the left and the right-hand side of (4.7). Results are presented in Panel A of Figure 2. By construction, the treatment and control groups differ in the wage bill from period t = 1 to t = 0. On average, firms in the treatment group face an increase of 25 log points growth in their wage bill relative to firms in the control group. The effect of the treatment appears to be permanent in levels up to 5 years after the treatment. Figure 2 also shows the effect on the average earnings of firms. On average, firms in the treatment group face an increase of 3.25 log points of their average earnings relative to firms in the control group. Once again, the effect of the treatment appears to be permanent in levels up to 5 years after the treatment. Finally, firms in the treatment and control group do not experience statistically significant differences up to 5 years before the treatment, for both the wage bill and the average earnings. Through the lens of a DiD design, these results imply a pass-through rate of firms shocks of around 0.13 (= /0.25). From equation (4.7), this implies ˆγ =

35 Figure 2: Pass-Through of Firm Shocks to Workers Earnings Log Wage Bill - No Mean Effect Log Average Wage Effect Average Wage Wage P25 Wage P50 Wage P90 A. DiD Strategy B. Continuous Strategy Notes: This figure presents the results of the two strategies used for estimating γ. Panel A shows the evidence of the difference-in-difference strategy outlined in Section Panel B shows the evidence of the continuous strategy also outlined in Section Our results are in line with previous evidence of the pass-through of firms shocks to workers earnings. For example, Lamadon et al. (2019) find that the pass-through elasticity is 0.15 and Card et al. (2018) find an elasticity in the range of Note that these estimates often refer to different sources of variation. For example, Lamadon et al. (2019) use value added as the firm characteristic where the shock comes from, whereas Card et al. (2018) use value added per worker. We add another variety to these strategies: the wage bill. In a model with intermediate inputs such as ours, the relevant measure for firm shocks is the wage bill, rather than value added or value added per worker. Intuitively, value added includes variation in the profits of the firm (and in a more general model, potentially variation in other inputs such as capital), and thus does not correctly capture the margin of variation that is relevant for workers. The second strategy that we follow to estimate γ is to run the regression from equation (4.7) directly with OLS. To test for pre-trends and the dynamics of the pass-through of firm shocks, we write the specification from equation (4.7) in first differences: log w mt = β χ mt γ log EL i(m,t)t + ξ mt (4.8) 34

36 The results are presented in Panel B of Figure 2. We conclude three facts from this figure. First, the effect of wage bill changes on impact are around 0.13, which confirms the result from the DiD strategy. 15 Second, the figure shows both no clear pre-trends and also that the effect is transitory in changes, which means it is permanent in levels. Finally, one concern about using the wage bill instead of value added, is that there might be a mechanical relationship between the wage bill and average earnings driven by outliers. To control for this, we implement Equation (4.8) also with other moments of the earnings distribution within the firm, such as the percentile 25, 50 and 90. The results of these alternative elasticities are presented in Panel B of Figure 2. We find that there are no significant differences between the behavior of the average earnings within the firm and the percentiles 25 and 50. There is a significant difference with the 90th percentile. Workers in those groups have a pass-through elasticity of This implies that high-wage-within-the-firm workers face a higher γ and therefore belong to markets that are closer to being competitive than workers that are lower in the within-firm wage distribution Worker and Firm Wage Effects To estimate the worker and firm effects in equation (4.7), we follow a two-step procedure proposed by Bonhomme et al. (2019). First, we cluster firms into K groups by using moments of the earnings distribution of workers within the firm. Second, we use those K groups as the relevant firm identifier. We implement this strategy with K = 10. In order to account for the role of firm shocks, we include time-varying firm effects which is more flexible than the structural version of firm shocks in Equation (4.7) and run OLS on Equation (4.7) with time-varying firm effects. This yields estimates of the firm fixed effect φ i and the complementarity parameter θ i for every firm i Ω, as well as the permanent component of ability ā m for every worker. 15 As we said above, we follow a leave-out one strategy in order to avoid a mechanical correlation between the left and the right-hand side. In practice, the effect of doing this is not economically significant: it increases the pass-through elasticity to instead of

37 Table 3: Wage Variance Decomposition: BLM vs AKM AKM BLM BLM BLM BLM Share explained by: (1) (2) (3) (4) (5) 1. Worker Effects (%) Firm Effects (%) Sorting (%) Interactions (%) Time-Varying FE (%) Residual (%) Sorting Correlation Notes: This table presents the results from estimating the two-way fixed effect model of wages outlined in Section The results of the estimation are presented in terms of a variance decomposition of log earnings outlined in Lamadon et al. (2019). Column 1 presents the results from the estimation strategy proposed in Abowd et al. (1999). Column 2 presents the results of the basic version of the estimation strategy of Bonhomme et al. (2019) that clusters firms to solve the limited mobility bias problem. Column 3 adds worker-firm interactions to the model in Column 2. Column 4 adds time-varying firm effect to the model in Column 2. Column 5 adds both worker-firm interactions and time-varying firm effects to the model in Column 2. We present results from this estimation by reporting the standard decomposition of the variance of log earnings in Table 3. In order to benchmark our results relative to the literature, we build relative to the standard Abowd et al. (1999) estimation strategy and then add sequentially the different pieces that are extensions in Bonhomme et al. (2019) and Lamadon et al. (2019). We find that when going from the standard AKM (column 1) procedure to the basic BLM procedure (column 2), which is basically fixing the problem of limited mobility bias, the role of the worker fixed effects increases significantly and the role of both the firm fixed effects and sorting decreases. Adding worker-firm interactions (column 3) increases the role of firm fixed effects and reduces the role of sorting. Adding time varying firm effects increases the role of sorting. Finally, adding worker-firm interactions and time-varying firm effects (column 5), our preferred and model-consistent specification, reduces the role of sorting. Comparing our preferred specification to AKM (column 1) shows that the role of worker fixed effect increases at the expenses of a reduction in the role of sorting. The role of worker fixed effects in column 5 is similar to the one documented in the US (69% in Chile versus 72% in the US), the firm effect is bigger (14% in Chile versus 4% in the US) and the role of sorting is smaller (6% in Chile versus 36

38 13% in the US). The role of the interaction component is also small both in Chile and the US (around 2%). Finally, the overall fit of the model increases but only 2 percentage points and is similar to the one in the US (91% in Chile versus 90% in the US) Amenities Under Assumption 4.5, the value of firm amenities varies across workers only in relation to permanent ability. Hence, it will be helpful here to be explicit about the distinction between permanent and transient worker ability. We begin with the the labor supply equation (2.5): L it (ā, â) = L t (ā, â) [g i (ā) w it (ā, â)] γ j Ω [g j (ā) w jt (ā, â)] γ (4.9) Next, consider the equilibrium wage equation (2.28). We can write this as: w it (ā, â) = ηā θiâw it (4.10) where we have used the functional form for productivity φ i in Assumption 4.4. The average wage paid by firm i to workers with permanent ability ā is hence: w it (ā) = ηā θ i W it E [â ā] (4.11) where E [â ā] denotes the average value of transient ability conditional on permanent ability being equal to ā. Notice that since workers do not sort to firms based on transient ability â, this average does not vary by firm. Combining (4.10) and (4.11), we then have: w it (ā, â) = w it (ā) â E [â ā] (4.12) 37

39 Substituting this into (4.13): L it (ā, â) = L t (ā, â) [g i (ā) w it (ā)] γ j Ω [g j (ā) w jt (ā)] γ (4.13) Now notice that the employment share of workers of ability {ā, â} varies across firms only in relation to permanent ability ā. This is a direct implication of Assumptions 4.4 and 4.5, which ensure that workers do not sort to firms based on transient ability â. Hence, the share of workers of permanent ability ā employed at firm i is: s L it (ā) = [g i (ā) w it (ā)] γ Ī t (ā) γ (4.14) where Īt (ā) [ ] 1/γ j Ω [g j (ā) w jt (ā)] is a wage index for workers of permanent ability ā. Writing (4.14) in terms of amenities, we then obtain: g i (ā) = sl it (ā)1/γ w it (ā) Īt (ā) (4.15) From the estimation procedure described in Section 4.3.2, we obtain estimates of permanent ability ā for every worker. Hence, we are able to observe in the data both employment shares and wages for every firm and permanent worker ability type. 16 We then choose a normalization of firm amenities such that Īt (ā) = 1 for every permanent worker ability type and use equation (4.15) to back out g i (ā). Projecting this on permanent worker ability then allows us to estimate the amenity parameters {δ 0i, δ 1i } as defined in Assumption Elasticity of Substitution between Labor and Materials To estimate the elasticity of substitution between labor and materials, ɛ, we first note from equations (2.29), (2.30), (2.35), and (2.36) that the ratio of intermediate spending to total labor 16 In practice, we cluster both firms and workers. Firms are clustered as described in Section 4.3.2, while workers are clustered based on permanent ability into K A = 10 groups. 38

40 costs depends on relative factor prices: log ( E M it E L it ) [ 1 = log η ( 1 λ λ )] + (ɛ 1) log W it Z it + ϕ it (4.16) We treat the residual ϕ it as measurement error and hence estimate (4.16) via OLS to recover ɛ. However, note that in order to implement this regression, we first require measures of firmlevel wages, W it, and intermediate input costs, Z it. Since we do not observe these measures directly in the data, we estimate them as follows. First, from the estimation procedure described in Section 4.3.2, we obtain estimates of the firm fixed effects φ i. We then use these to recover W it by using the following relationship from our model: log W it = log η γ log φ i γ log EL it (4.17) To recover Z it, we follow a three-step procedure. First, we write log sales from firm j to firm i using equations (2.37) and (4.5) as: where log R ijt = γ log η + log }{{ it + log Φ } jt + log ψ ijt (4.18) }{{} buyer effect seller effect it it ψ it (4.19) Φ it Φ it ψ it (4.20) Under Assumption 4.6, the assignment of buyers to sellers is exogenous with respect to ψ ijt. Hence, as discussed in Bernard et al. (2019), estimation of equation (4.18) using a two-way fixed { effects estimator delivers unbiased estimates of the buyer and seller fixed effects, it, Φ } it, as well as the relationship-specific productivity residuals, log ψ ijt, and their standard deviation, σ ψ,t Note that equation (4.18) can be estimated using cross-sectional data. This is in contrast with two-way fixed effect models of earnings such as (4.7), which are identified based on workers moving between firms. The difference in intermediate input markets is that matching occurs many-to-many: each seller can have several buyers at once 39

41 Note that to estimate (4.18), firms must have multiple connections. To identify seller fixed effects, each seller needs to have at least two buyers. Similarly, to identify buyer fixed effects, each buyer needs to have at least two sellers. In the data, some firms have either one supplier or one seller. Hence, we implement the aforementioned restriction using an iterative approach known as avalanching. Specifically, we first drop firms with one supplier or seller. Doing this may result in additional firms that have one supplier or seller, hence in the next step, we drop these firms as well. We continue this process until firms are no longer dropped from the sample. Table 1 presents descriptive statistics of the effect of this sample restriction. The algorithm takes three iterations to converge in practice and reduces the sample size of firm-to-firm linkages from a total of 31.6 million transactions to 31.4 million transactions, that is, a reduction of 1% of transactions. Hence, the avalanching algorithm has little impact on our sample size. 18 Second, we recover firm relationship capabilities, ψ it. Note from (2.9) and (2.14) that the share of a firm s total sales that come from the network (i.e. excluding final sales) is: Solving for ψ it, we obtain: s F it = = ψ it = F t ( j Ω C jtψ jit it F t + j Ω C (4.21) jtψ jit it ψ it j Ω C jt ψjit it (4.22) F t + ψ it j Ω C jt ψjit it s F it 1 s F it ) j Ω C it 1 (4.23) jt ψjit Since s F it is observable and we can construct j Ω C it jt ψjit from estimates of (4.18), we can identify ψ it up to a normalizing constant. 19 Given ψ it, we can then parse the buyer and seller { fixed effect estimates it, Φ } it to obtain estimates of the network characteristics { it, Φ it }. We also construct relationship-specific productivity from (4.5). and each buyer can have several sellers. 18 Bernard et al. (2019) report that avalanching also eliminates around 1% of firm-to-firm links in the production network for Belgium. 19 The intuition here is that a higher value of ψ it increases sales only within the network but not to final consumers. Hence, after controlling for total final expenditure F t and characteristics of a firm s customers within the network through j Ω jtψ jit, variation in s F C it is informative about ψ it. it 40

42 Third, given estimates of Φ it and ψ ijt, we then construct firm-level input costs using equation (2.16) as Z 1 σ it = j Ω S i Φ jtψ ijt. Figure 3 plots the distribution of the firm fixed effect of wages, the buyer and seller fixed effect, log W it and log Z it. Three features of these distributions are worth mentioning. First, all these variables are well-behaved and follow a normal distribution. Second, the adjusted R 2 from Equation (4.18) is 45% which highlights the role of buyer-supplier match quality for firmto-firm transactions. 20 Finally, we find a negative correlation between the seller and the buyer fixed effect. This happens because firms that have more and better inputs, can afford a lower production costs, which implies a lower marginal cost and therefore becoming more attractive to buyers and a greater capacity of reaching more buyers, which implies more sales. 20 The R 2 in our data is similar to the one found in Bernard et al. (2019), which is 38%. 41

43 Figure 3: Distribution of Firm Estimates Fraction Fraction A. Seller Effects, log Ψ it B. Buyer Effects, log it Fraction Fraction C. log W it D. log Z it Notes: This figure presents the distributions of different characteristics of firms that we estimate. Panel A and B plots seller and buyer fixed effects estimates of log Ψ it and log it, respectively. The estimation strategy of these estimates is outlined in Section Panel C and D plots estimates of log W it and log Z it, respectively, which are described in Section With estimates of W it and Z it in hand, we can now estimate equation (4.16) via OLS. Note that a typical challenge encountered in estimating substitution elasticities by regressing relative expenditures on relative prices as in (4.16) is that the residuals of such regressions often contain factor-augmenting productivities. 21 To illustrate this issue, suppose that instead of estimating the firm wage W it, we use the average earnings of the firm as a regressor in (4.16). From equation (2.28), the average wage at 21 For example, see León-Ledesma et al. (2010). 42

44 firm i is: w it = η ˆφ it W it (4.24) where ˆφ it is a measure of aggregate labor productivity at firm i: ˆφ it A κ it (a) φ it (a) 1+γ da A κ it (a) φ it (a) γ da (4.25) Hence, we can rewrite the factor choice equation (4.16) as: ( ) log Eit M /Eit L = log [ 1 η ɛ ( 1 λ λ )] + (ɛ 1) log w it Z it (ɛ 1) log ˆφ it (4.26) Notice that if we were to estimate Equation (4.26) using average wages w it instead of the firm wage component W it, the labor productivity term ˆφ it would lead to a downardly-biased estimate of ɛ as it is positively correlated with the average wage and the coefficient in front of the residual is negative. To highlight the strength of our estimation strategy, we also estimate (4.26) using average firm wages according to Equation (4.26). Table 4 presents the results. We find that ˆɛ 1 = and thus ˆɛ = 0.915, which means that workers and suppliers are complements in our model. For this result we use the full version of the wage equation given by our model and estimate it using the procedure from Bonhomme et al. (2019). Column 2 presents the results when using the estimating procedure from Abowd et al. (1999). In that case, we find ˆɛ = Finally, if instead, we ran the version of Equation (4.26), we find that ˆɛ B 1 = 0.131, which is around 1.5 times the unbiased estimate. This biased estimates leads to ˆɛ = 0.869, which is significantly lower than our preferred estimate. Note that this result confirms the bias predicted by the theory. 43

45 Table 4: Estimation of Elasticity of Substitution between Materials and Labor (ɛ) log E M /E L (1) (2) (3) log W/Z (0.009) (0.011) log w/z (0.007) Epsilon Model for Wage Component BLM: K = 10 AKM Average Wage Adjusted R Number of Observations 107, , ,888 Notes: This table presents the results of the estimation of ɛ outlined in Section Column 1 and 2 presents the estimates according to Equation (4.16) and Equation (4.26), respectively Firm TFP Recall that estimates of relationship capabilities for each firm are obtained from (4.23) in the process of estimating ɛ. To estimate the remaining dimension of firm heterogeneity - TFP, T it - we proceed as follows. Using equations (2.17), (2.18), (2.29), and (2.31), we can express TFP as: T ɛ 1 it = µɛ 1 λ ( it D it ) ( ) ɛ σ σ 1 Φ it Zit 1 σ (4.27) Now using (2.16) and (2.19), we write this as: ( Tit ɛ 1 = µɛ it 1 λ snetwork it j Ω C jtψ jit it ) ( Φ it j Ω S it ψ ijtφ jt dj ) ɛ σ σ 1 (4.28) Hence, given estimates of ɛ from (4.16) and { it, Φ it, ψ ijt } from (4.18) and (4.23), we can estimate TFP at the firm-level up to a normalizing constant. Intuitively, we infer variation in a firm s TFP from variation in its network demands and productivities, { it, Φ it }, after controlling for the network demands and productivities of its customers and suppliers respectively through j Ω C it jtψ jit and j Ω S it ψ ijtφ jt dj. Figure 4 shows histograms of our estimates of log TFPs and relationship capabilities. As is 44

46 evident, the distributions are well-approximated by normal distributions. From this, we obtain estimates of the mean and covariance matrix of log firm types, {m χ, Σ χ }. For the covariance matrix, we use only the relative variance of log TFP to log relationship capability, σt 2 /σ2 ψ and the correlation between the two variables, ρ T ψ. We estimate the level of log TFP variance in the SMM procedure below in order to match the firm size distribution observed in the data. Figure 4: Distribution of Productivity: Firms and Links Fraction Fraction A. log T it B. log ψ it Notes: This figure presents the estimates of firm-level TFP, log T it, and firm-level relationship capability, log ψ it. The estimation of these objects is outlined in Section (4.3.5) Matching function Given the parameterization of the model described thus far, firms are heterogeneous along five dimensions: (i) TFP, T it ; (ii) relationship capability, ψ it ; (iii) worker-firm production complementarities, θ i ; (iv) the base value of firm amenities, δ 0i ; and (v) the rate at which the value of amenities increases with worker permanent ability, δ 1i. To estimate the matching function m t described in Assumption 4.8, we first implement a dimensional reduction procedure in anticipation of the numerical counterfactuals. Specifically, we parameterize {θ i, δ 0i, δ 1i } as functions of firm TFP. Since these parameters are time-invariant whereas TFP is time-varying, we project these on firm TFP averaged across years in our sample, T i : 45

47 log θ i = b θ log T i (4.29) log δ 0i = b δ0 log T i (4.30) log δ 1i = b δ1 log T i (4.31) This allows us to treat firms are heterogeneous in only two dimensions: TFP and relationship capability. To estimate the matching function, we then bin firms by percentiles of firm type {T, ψ} and measure empirically the fraction of firm pairs within each bin that match. We treat this as an estimate of the matching function. In practice, we discretize the TFP and relationship capability space into 10 grid points each. This implies 10, 000 bins of the matching function. This illustrates the need for a large dataset such as ours: with between 200m to 400m firm-to-firm transactions per year, we have an average of between 200 to 300 matches per bin even with such a large number of bins Remaining Parameters: SMM The remaining parameters of the model to be estimated are: (i) the elasticity of substitution between firm products, σ; (ii) the weight on labor in the worker-level production function, λ; and (iii) the variance of log TFP, σt 2. We estimate these via a simulated method of moments approach. Although the model does not admit closed-form solutions that allow direct estimation or calibration of these parameters, they are intuitively connected to key moments in the data. First, the elasticity of substitution between products, σ, controls the extent of imperfect competition in output markets. Consequently, σ has a strong influence on firm profits. With this parameter in mind, we include as a targeted SMM moment the share of labor in aggregate value-added. Second, the weight on labor in the worker-level production function, λ, naturally has a strong influence on the relative input of labor versus materials at each firm. With this parameter in mind, we include as a targeted SMM moment the ratio of value-added to gross output. Note 46

48 that as λ approaches 1, for example, intermediates are not used in production and hence the ratio of value-added to gross output approaches 1. Third, the variance of log TFP, σt 2, controls the degree of firm heterogeneity in the model.22 With this parameter in mind, we include as a targeted SMM moment the standard deviation of log firm sales. In addition, although we are able to estimate the worker-firm production complementarity and amenity parameters {b θ, b δ0, b δ1 } directly, as described in sections 4.3.2, 4.3.3, and 4.3.6, we adopt a different approach for the numerical counterfactuals and include these parameters in the SMM estimation as well. This allows us to better fit two key moments in the data. First, the worker-firm complementarity parameter b θ influences the importance of betweenfirm variation in wages relative to within-firm variation. When b θ is large, higher ability workers are more productive at firms with higher TFP, which would increase the importance of betweenfirm variation. With this parameter in mind, we include as a targeted moment the share of log worker earnings variance accounted for by between-firm variation, computed as in Song et al. (2019). Second, the amenity parameter b δ0 controls the correlation between amenity values and firm TFP. Since TFP is directly related to firm size, b δ0 has a strong influence on the firm size premium for firm-level average earnings. With this parameter in mind, we include as a targeted moment the correlation between firm sales and average wages at the firm. In the SMM procedure, we assume that amenity values do not vary by worker ability and set b δ1 = Estimation Results and Model Fit Table 5 shows our estimated parameter values. There are several points of note. First, we estimate a labor supply elasticity of γ = 6.7, which is close to (but slightly higher than) typical estimates in the literature. 24 Second, we estimate an elasticity of substitution between labor and materials of ɛ = 0.91, implying that labor and materials are complements in production. Third, 22 Recall from Section that we fix the relative variance of log TFP and log relationship capability as well as the correlation of the two varaibles given our estimtates of {T, a} for every firm. 23 This is work in progress. 24 For example, see the references cited in footnote 2. 47

49 we estimate a product substitution elasticity of σ = 4.5, implying output market markups over average costs of around 28%. Note also that our estimates imply σ > ɛ, which from Proposition 2 implies that reductions in material input costs have positive effects on wages. Fourth, we estimate a strong negative correlation between firm TFP and relationship capability of This is in line with the findings of Bernard et al. (2019), who also estimate TFP and relationship capability to be negatively correlated in Belgian production network data. Fifth, we estimate a positive value of b θ = 7.27, indicating that higher ability workers are more productive in firms with higher TFP. Finally, we estimate a positive value of b δ0 = 9.05, indicating that firms with higher TFP also have amenities that are more valuable to workers. This acts to reduce the firm-size wage premium, since large firms with high TFP do not have to offer as high wages in order to attract workers. Table 5: Parameter Estimates Description Parameter Value labor supply elasticity γ 6.7 labor-materials substitution elasticity ɛ 0.91 product substitution elasticity σ 4.5 weight on labor in production λ 0.29 sd(log TFP) s T 0.15 sd(log relationship capability) s ψ 2.28 corr(log TFP, log relationship capability) ρ T ψ sd(log worker permamnent ability) sā 0.50 sd(log worker transient ability) sâ 0.20 worker-firm production complementarity b θ 7.27 firm amenities, base value b δ firm amenities, worker ability premium b δ Notes: Estimates are based on 2005 data. Table 6 shows the fit of the estimated model to several key moments in the data. There are 48

50 several important takeaways. First, the model provides a good fit of the distribution of log earnings across workers: the predicted standard deviation, percentile ratios, and Gini coefficient are very close to what we observe empirically. This is despite the fact that we do not target any moment of the worker earnings distribution directly in the SMM procedure and instead estimate worker abilities directly from the data. The successful fit of the model likely partially reflects the fact that the empirical earnings distribution distribution is well-approximated by a log-normal parametric form, which is the same functional form that we assume for worker abilities. Second, the model also fits fairly well with several key firm-level variables. The predicted dispersion of log average earnings across firms is close to what we observe empirically, although slightly smaller. The model also fits the correlation between firm sales and supplier counts almost exactly, although it overpredicts the correlation between sales and customer counts. Finally, at the firm-to-firm level, the model predicts negative assortative matching between firms when matching is characterized in terms of sales, employment, or degree (customer and supplier counts). This is qualitatively in line with what we observe empirically, although in several cases the extent of negative matching in the model is greater than in the data. This is particularly true for matching patterns with a firm s customers. When firm-to-firm matching is characterized in terms of average earnings, we see instead positive assortative matching in the data: firms with higher average worker earnings are more likely to match with customers and suppliers that have higher average earnings themselves. For matching with customers, the model replicates this pattern almost exactly. However, it predicts negative assortative matching with suppliers on average earnings, which is at odds with the data. 49

51 Table 6: Model Fit Moment Model Data Targeted aggregate variables value-added to gross outut ratio labor share of value-added between-firm share of var. log earnings worker variables s.d. log earnings /10 earnings Ratio /25 earnings Ratio /50 earnings Ratio /10 earnings Ratio earnings Gini coefficient firm variables sd(log sales) sd(log value-added) sd(log wage bill) sd(log avg. earnings) corr(log sales, log #cus) corr(log sales, #sup) corr(log sales, log avg. earnings) corr(log sales, s.d. earnings) firm-to-firm variables sd(log firm-to-firm sales) corr(log sales,log avg. customer sales) corr(log sales,log avg. supplier sales) corr(log employment, log avg. customer employment) corr(log employment, log avg. supplier employment) corr(log avg. earnings, log avg. customer earnings) corr(log avg. earnings, log avg. supplier earnings) corr(log #cus, log avg. customer #cus) corr(log #cus, log avg. supplier #cus) corr(log #sup, log avg. customer #sup) corr(log #sup, log avg. supplier #sup) Notes: Empirical moments are from Model moments are based on parameters estimated from 2005 data. Targeted column indicates if moment was targeted in simulated method of moments algorithm. For firm-to-firm variables, customer and supplier averages are weighted based on sales and expenditure shares respectively. 50

52 Figure 5: The empirical network versus a random network 5 Counterfactuals Several policies in developing economies attempt to improve the matching between agents: for example, improving the matching in business-to-business relationships through business fairs. We use the structure of our model to study how changes in the matching between firms in the production network affect workers labor earnings. Specifically, we study the importance of heterogeneity in matching between firms in the production network for earnings inequality. To quantify this, we perform a simple counterfactual exercise: we solve for the model s equilibrium under the empirically-observed network and compare this to the equilibrium that would obtain under a random network, while keeping fix the aggregate mass of linkages. This change in the network structure is illustrated in Figure 5. The empirical network in Chile exhibits positive assortative matching (PAM) in TFP. This means that firms with higher TFP are more likely to buy from and sell to firms with higher TFP. In contrast, under the counterfactual random network, matching between firms is independent of TFP. Hence, randomizing the network results in less inequality in matching between firms in the production network which in turn implies less inequality in the productivity of suppliers and buyers. We use our framework to study how these changes in the productivities of trading partners are passed through to workers labor earnings and hence how they affect earnings inequality. 51

53 5.1 Aggregate Effects Figure 6 shows the distribution of log worker wages w mt and log firm average wages w it under the empirical and random networks. As expected, randomizing the production network reduces the dispersion of wages both across workers and across firms. Figure 7 presents inequality statistics for both the empirical and counterfactual network. The variance of log worker wages is 41% lower under the random network as compared with the empirical network, while the variance of log firm average wages w it is 76% lower. Since the decrease in the variance of the firm component of wages is bigger than the variance of earnings at the worker level, the between firm share of earnings variance decreases from 40% to 16%. The reduction in inequality of wages comes mostly at the top of the earnings distribution since 90/10 wage ratios decreases more than the 50/10 wage ratios. The Lorenz curve for the empirical network lies everywhere below the curve for the random network, indicating a more unequal wage distribution with network heterogeneity. The Gini coefficient of worker earnings falls from 0.49 under the empirical network to 0.35 under the random network. Figure 6: Worker wage and firm wage distribution under empirical and random networks A. log w mt B. log w it Notes: This figure presents the distribution of different variables under the empirical network and the counterfactual random network. Panel A presents those distributions for workers wages, w it. Panel B presents those distributions for firm average wages, w it. 52

54 Figure 7: Inequality measures under empirical and random networks A. Inequality Statistics B. Lorenz Curve Notes: This figure presents inequality statistics under the empirical network and the counterfactual random network. Panel A presents several inequality statistics. Panel B presents Lorenz curves. 5.2 Mechanisms The key aggregate result is that a reduction in the inequality of matching between firms is transmitted into a reduction in the inequality of worker earnings. There are two mechanisms that generate this result: (i) changes in earnings inequality between firms; and (ii) changes in earnings inequality within firms. The between-firm effect arises from the fact that under the counterfactual network, firms with higher TFP now connect with suppliers and buyers with relatively lower TFP. This implies that they now face higher input costs and lower output demand. The opposite is true for firms with lower TFP. Thus, the counterfactual network reduces inequality in input prices and output demand between firms, as shown in Figure 8. These reductions in inequality between firms in input costs and output demand then generate a reduction in inequality in the firm component of earnings according to the predictions in Proposition 2. As discussed in Section 2, the pass-through depends critically on the elasticity of substitution in demand and the elasticity of substitution between labor and materials. For example, in our case we have that σ > ɛ and thus positive shocks to the input costs of firms imply larger increases in the firm component of wages, as discussed in Section 2. 53

55 Figure 8: Input cost and output demand under empirical and random networks A. log Z it B. D it Notes: This figure presents the distribution of input costs (panel A) and output demand (panel B) under the empirical and random network. The within-firm effect, on the other hand, is more subtle and arises through general equilibrium. First, recall that we estimate positive worker-firm complementarities in production (b θ > 0), which implies that workers of higher ability are more productive at firms with higher TFP. Second, given that the empirical production network exhibits positive assortative matching on TFP, randomizing firm-to-firm matches in the production network reduces the heterogeneity between high-tfp and low-tfp firms. These two results together imply that randomizing the production network leads to a reduction in the relative competition for high-ability versus low-ability workers. This is illustrated in Figure 9, which shows the relative change in wage indices I (a) for workers of different permanent abilities under the counterfactual network. As is evident, the percentage decline in the wage index is greater for high ability workers verus low ability workers. 54

56 Figure 9: Sorting Indices Notes: This figure shows the ratio of the wage index under the random network, I ran (a), to the wage index under the empirical network, I emp (a), for workers of different permanent abilities a. As discussed in section 3.1.2, this change in relative competition leads to an increase in the relative employment of high-ability versus low-ability workers in all firms. Since relative wages within a firm are fixed (as shown in equation (3.10)), the within-firm effect of randomizing the production network thus occurs purely through compositional changes in the workforce of a firm. These changes are illustrated in Figure 10, which shows how the share of a firm s employment that is accounted for by workers of a given permanent ability varies with the TFP of the firm, both in the baseline equilibrium and under the counterfactual random network. As can be seen, under the random network, all firms increase the share of their employment that is attributable to higher ability workers. 55

57 Figure 10: Relative Employment Notes: The Z-axis of each figure shows the share of a firm s work force that is accounted for by workers of a particular permanent ability. The implication of this for within-firm earnings inequality is twofold. First, for a high TFP firm that already hires more high-ability versus low-ability workers, the change in workforce composition leads to a decrease in within-firm earnings inequality, as the firm s workers now become more similar in ability. This adds to the reduction in between-firm earnings inequality. Second, for a low TFP firm that initially hires more low-ability versus high-ability workers, the change in workforce composition leads to an increase in within-firm earnings inequality, as the firm s workers now become more different in ability. This partially offsets the reduction in earnings inequality generated by the other mechanisms described above. These effects are illustrated in Figure 11, which shows how the standard deviation of log earnings across workers within a firm varies with firm TFP, both in the baseline equilibrium and under the counterfactual random network. As can be seen, within-firm earnings dispersion falls for high TFP firms but increases for low TFP firms. 56

58 Figure 11: Change in Within-firm Earnings Dispersion Notes: This figure shows the standard deviation of log wages across workers within a firm for firms with different TFP values. In sum, we find that eliminating heterogeneity in production network linkages reduces earnings inequality by a non-trivial amount and that this occurs through a reduction in between-firm earnings inequality and within-firm earnings inequality for high TFP firms. At the same time, within-firm inequality increases for low TFP firms. 5.3 Customer versus Supplier Heterogeneity Our results above establish that heterogeneity in production network linkages is quantitatively important for earnings inequality. How much of this is due to heterogeneity in connections with a firm s customers versus heterogeneity in connections with a firm s suppliers? To examine this, we repeat the same counterfactual exercise as above, but instead of randomizing the network completely, we study two separate counterfactuals in which: (i) firms meet with customers at random; and (ii) firms meet with supplies at random. First, consider the counterfactual scenario in which the identity of a firm s customers does not depend on the characteristics (TFP and relationship capability) of the customer. This production network structure is illustrated in Figure 12. Under this network, the suppliers that 57

59 Figure 12: The empirical network versus a random customer network a firm matches with does not depend on the firm s characteristics. Hence, heterogeneity in input costs Z it across firms is greatly reduced, whereas heterogeneity in demand shifters D it remains substantial. These changes are illustrated in Figure 13. Figure 13: Input cost and output demand under empirical and random networks A. log Z it B. D it Notes: This figure presents the distribution of input costs (panel A) and output demand (panel B) under the empirical and random customer network. Second, consider the counterfactual scenario in which the identity of a firm s suppliers does not depend on the characteristics (TFP and relationship capability) of the supplier. This production network structure is illustrated in Figure 14. Under this network, the customers that a firm matches with does not depend on the firm s characteristics. Hence, heterogeneity in demand shifters D it across firms is greatly reduced, whereas heterogeneity in input costs Z it remains substantial. These changes are illustrated in Figure

60 Figure 14: The empirical network versus a random supplier network Figure 15: Input cost and output demand under empirical and random networks A. log Z it B. D it Notes: This figure presents the distribution of input costs (panel A) and output demand (panel B) under the empirical and random supplier network. Figure 16 shows the change in earnings inequality statistics under the random customer and supplier networks described above, relative to the baseline equilibrium. These results imply that heterogeneity in matching with both customers and suppliers matters for earnings inequality. For example, eliminating customer heterogeneity reduces log earnings variance by 17.8%, while eliminating supplier heterogeneity reduces log earnings variance by 28.7%. This is in contrast with the overall reduction of 41.1% in log earnings variance that is predicted under a completely random production network. In sum, both customer and supplier heterogeneity are quantitatively important for worker earnings inequality, although supplier heterogeneity matters more. 59

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