Nominal Rigidity and the Idiosyncratic Origin of Aggregate Fluctuations Ernesto Pasten, Raphael Schoenle, Michael Weber Februay 2016 PRELIMINARY AND INCOMPLETE PLEASE DO NOT CITE Abstract We study the aggregate propagation of idiosyncratic, sectoral shocks in a multi-sector new-keynesian model with intermediate inputs featuring sectoral heterogeneity in price stickiness, sectoral GDP, and input-output linkages. Heterogeneity of price rigidity distorts the "granular" effect of the fat-failed distribution of sectors size as well as the "network" effect of the centrality of some sectors in the production network. This distortion involves the strength of the aggregate volatility generated by sectoral shocks as well as the identity of the most important sectors. The granular and the network effects may in fact be completely irrelevant while the empirical distribution of price stickiness may generate by itself sizable aggregate volatility from sectoral shocks. We calibrate our model to 350 sectors using US data to quantify the strenght and interaction of "granular", "network" and "frictional" sources of the aggregate propagation of sectoral shocks. JEL codes: Keywords: Pasten thanks the support of the Université de Toulouse Capitole during his stays in Toulouse. Weber gratefully acknowledges financial support from the University of Chicago, the Neubauer Family Foundation, and the Fama-Miller Center. The views expressed herein are those of the authors and do not necessarily represent the position of the Central Bank of Chile. Central Bank of Chile and Toulouse School of Economics. Email: ernesto.pasten@tse-fr.eu Brandeis University. Email: schoenle@brandeis.edu. Booth School of Business, University of Chicago. Email: michael.weber@booth.edu..
1 Introduction Recent literature has stressed that idiosyncratic, sectoral shocks at disagregated level may be behind sizable aggregate fluctuations contradicting what is known as the "diversification argument." Lucas (1977) argues that idiosyncratic shocks at a very disagregated level should "wash out" in the aggregate by the application of the law of large numbers. Similar view has been taken by Dupor (1999) and virtually anyone that conceptualizes business cycles fluctuations as the result of aggregate shocks. In contrast, Gabaix (2011) argues that when the contribution of firms in total GDP is fat-tailed (in what follows, sectors), the standard law of large numbers does not readily apply. Intuitively, when some sectors remain "granular", i.e., disproportional large as disaggregation becomes finer, shocks to those sectors matter for the aggregate. In turn, Acemoglu, Carvalho, Ozdaglar and Tahbaz-Salehi (2012) argue that the diversification argument neither applies when some sectors remain central in the production network for any level of disaggregation. Both, Gabaix (2011) and Acemoglu et al (2012), respectively show that the granularity and the network centrality of sectors in US data are enough for disaggregated sectoral shocks to generate sizable aggregate fluctuations. This paper asks about role of frictions in this debate in particular, price frictions. No matter whether because a sector is large or central in the network, its ability to adjust prices after a shock is crucial for the aggregate propagation of such a shock. To fix ideas, suppose that firms in a given sector do not adjust their prices at all after receiving a shock specific to their sector. Demand for goods produced in the sector would remain unchanged. Thus, the contribution of the shock to aggregate fluctuations would be zero regardless of the size of the sector. Besides, production costs of all firms that use the goods from the shocked sector as inputs would also remain unchanged, so their prices and thus their own demand would also be unchanged. Production in the shocked sector remains unchanged, so no firm in the sector would change its demand of the goods that use as inputs. Thus, regardless the centrality of the shocked sector in the production network, the aggregate effect of the shock would be just zero. Interestingly, it is well documented in the data the quite substantial heterogeneity of price rigidity across sectors. How does this heterogeneity in price rigidity affect the violation of the diversification argument either by the granular effect proposed by Gabaix (2011) or the network effect proposed by Acemoglu et al (2012)? To answer this question, this paper studies a new-keynesian model with multiple sectors where produc- 2
tion needs labor and intermediate inputs. Critically, our model allows for sectoral heterogeneity of price rigidity, size of sectors measured by sectoral GDP, and fully flexible input-output linkages across sectors. We first study a simple version of our model that allows for close-form solution to transparently show the interaction of price frictions and the granular and network mechanisms. The solution of this simple model nests the solution of the models of Gabaix (2011) and Acemoglu et al (2012) as special cases. Our main result is that price rigidity is central the aggregate propagation of idiosyncratic shocks. First, price rigidity distorts the strenght of the granular and network effects as well as the identity of the most important sectoral shocks for aggregate fluctuations. Under certain conditions, the granular and network mechanisms may be completely irrelevant for the violation of the diversification argument. We then explore the extrapolation of these results in a more elaborate version of our model and calibrate it to US data to quantify the aggregate volatility generated by idiosyncratic, sectoral shocks. For this we merge data for 350 sectors from the Input-Output Tables constructed by the Bureau of Economic Analysis and the micro data undelying the Producer Price Index constructed by the Bureau of Labor Statistics. Empirically, the sectoral distribution of price rigidity has the ability to generate by itself a mechanism to violate the diversification argument. However, although the negative correlation between price rigidity and sectoral size, and price rigidity and network centrality are weak, it is enough to generate quite large aggregate volatility from idiosyncratic, sectoral shocks. [ADD LITERATURE REVIEW] [ADD LAY OUT] 2 Model This section displays our multi-sector model with households and firms demanding goods from all sectors respectively for consumption and as intermediate inputs. There are three key ingredients of our model: (1) Households aggregate sectoral consumption using heterogeneous weights; (2) firms across sectors use different combination of goods for production; and (3) sectoral heterogeneity of price stickiness. 3
2.1 Households The utility of a representative household is subject to ( max E 0 β t Ct 1 σ 1 1 σ t=0 ) 1+ϕ Lkjt g k I k 1 + ϕ dj k=1 W kt L kjt dj + Π kt + I t 1 B t 1 B t = Pt I c C t k k=1 k=1 W kt L kjt dj 1 I k k=1 where C t and P c t respectively are aggregate consumption and aggregate prices to be specified below, L kjt and W kt are labor employed and wages paid by firm j in sector k = 1,..., K, Π kt is transfers from firms in sector k, I t 1 is the gross interest rate paid by bonds holding at the beginning of period t, B t 1. Aggregate consumption is C t [ K k=1 ] η 1 1 η 1 η ωck C1 η kt where C kt is the aggregation of sectoral consumption given by (1) C kt [ n 1/θ k C 1 1 θ kjt I k ] θ dj θ 1. (2) where C kjt is the demand by households of the good produced by firm j in sector k. There is a continuum of goods indexed by j [0, 1] with total measure one. These goods, each produced by a given firm also indexed by j, belong to one of the K sectors in the economy. Mathematically, the set of goods is partitioned into subsets {I k } K k=1 respectively with measure n k such that K k=1 n k = 1. Although the sectoral subindex is redundant, we consider it clarifies exposition. We allow the elasticity of substitution across sectors η to differ from the elasticity of substitution within sectors θ. The first key feature of ingredient in our model is the vector of weights Ω c [ω c1,..., ω ck ] in the 4
aggregator of sectoral consumption in (1). These weights show up in households demand: ( ) η Pkt C kt = ω ck Pt c C t. In steady state (solved in the Appendix) all prices are identical, so these weights govern the sectoral in total consumption C. These weights satisfy ι Ω c = 1 where ι denotes a column-vector of ones. In our model, aggregate consumption is the best proxy to value-added production. Thus we interpret Ω c as the vector of sectoral shares in steady state GDP which we simply refer to as "GDP shares". Outside steady state, households sectoral demand is distorted by the respective sectoral price relative to the consumption aggregate prices P c t. P c t = [ K k=1 ω ck P 1 η kt ] 1 1 η (3) Given the interpretation of C as GDP, we call P c t as GDP deflactor. Within sectors, households demand for goods is given by C kjt = 1 ( ) θ Pkjt C kt for k {1,..., K}, n k P kt such that, in steady state, all firms in sector k share equally households demand for goods categorized in sector k. Outside the steady state, the demand of goods is distorted by its price and the sectoral price [ ] 1 1 P kt = P 1 θ n kjt dj 1 θ for k {1,..., K}. k I k From the first order conditions for households we also get labor supply and the Euler equation: W kt P c t = g k L ϕ kjt Cσ t for all k, j, (4) E t [β ( Ct+1 C t ) σ I t P c t P c t+1 ] = 1 (5) Implicitly in our specification for disutility of labor is the sectoral segmentation of the labor market, so labor supply in (4) holds for a sector specific wage {W kt } K k=1. In the Appendix we set the parameters 5
{g k } K k=1 to ensure a symmetric steady state across sectors. In turn, the Euler equation is standard. 2.2 Firms Goods are monopolistically produced. We refer to a firm producing good j categorized in sector k as "firm kj" which production function is given by Y kjt = e a kt L 1 δ kjt Zδ kjt, where a kt is a productivity shock specific to sector k following an AR(1) process with unconditional expectations E [a kt ] = 0, unconditional variance V [a kt ] = v 2 and persistence ρ a for all k, L kjt is labor hired and Z kjt is an aggregator of intermediate inputs demanded by firm k, j. The aggregator of intermediate inputs used by firm kj is Z kjt [ K r=1 ] η 1 η ωkr Z kjt (r) 1 1 η 1 η where, in turn, Z kjt (r) is the amount of goods from sector r demanded by firm k, j at period t. The second key ingredient of our model are the aggregator weights {ω kr } k,r denoted in matrix form by Ω. These weights satisfy ι Ω = 1 where ι denotes a column-vector of ones. Importantly, these weights are allowed to be different across sectors and with those specified for households, Ω c. Z kjt (r) aggregates demand of firm kj for goods in sector r according to Z kjt (r) [ n 1/θ ( r Z kjt r, j ) ] θ 1 1 θ dj θ 1 I r where Z kjt (r, j ) is the amount of goods produced by firm j in sector r that is demanded as input by firm kj. The demand of firm kj across and within sectors is ( ) η Prt Z kjt (r) = ω kr Pt k Z kjt, Z kjt ( r, j ) = 1 n r ( Prj t P rt ) θ Z kjt (r). 6
In steady state, when all prices are identical, the share of goods from sector r in the total costs of a firm in sector k is given by Ω k [ω k1,..., ω kk ]. We thus refer to Ω as the input-output structure of the economy and Ω k as the input-output linkages of sector k (as demander). Within each sector, demand of firm kj for goods in sector r in steady state is shared equally across firms in sector r. Outside the steady state, sectoral demand is distorted by sectoral prices relative to the aggregate price P k t relevant for firms in sector k which is given by P k t = [ K r=1 ω kr P 1 η rt ] 1 1 η for k {1,..., K} that uses the input-output linkages of sector k (as demander). Since input-output linkages are heterogeneous across sectors, there are K + 1 aggregate price indexes (one for households and one for each sector). In turn, relative demand within a given sector depends on firms prices relative to their respective sectoral price given by [ ] 1 1 P rt = P 1 θ 1 θ n rj t dj for r {1,..., K} r I r which are the same regardless of the demanding firm and are also the same to the one relevant for households demand. The third key ingredient of our model is sectoral heterogeneity in price rigidities. Specifically, we model price rigidities á la Calvo with parameter {α k } K k=1. Thus, the objective of firm j, k is max P kjt E t Q t,t+s α s k [P kjty kjt+s MC kt+s Y kjt+s ] s=0 ( ) δ where marginal costs are given by MC kt = 1 δ 1 δ 1 δ A 1 kt W 1 δ ( ) kt P k δ t in reduced form after imposing the optimal mix of labor and intermediate inputs: δw kt L kjt = (1 δ) P k t Z kjt. (6) In principle firms may like to discriminate among the different customers they have, either households or other firms. We assume that the elasticity of substitution across and within sectors, although allowed to be different among them, are the same for all households and all firms. This assumption shuts down the incentives of firms to discriminate among different customers, so the optimal pricing problem takes the 7
simple, standard form: Q t,t+s α s k Y kjt+s s=0 [ P kt where Y kjt+s is the total production of firm k, j at period t + s. θ ] θ 1 MC kt+s = 0 Since idiosyncratic shocks {a kt } K k=1 are defined at the sectoral level, the optimal adjusting price, P kt, is the same for all firms in a given sector. Thus, aggregating among all prices within sectors yields P kt = [ ] 1 (1 α k ) P 1 θ kt + α k P 1 θ 1 θ kt 1 for k = 1,..., K. (7) 2.3 Monetary policy Monetary policy controls I t which is set according to the Taylor rule: I t = 1 β ( P c t P c t 1 ) φπ ( ) φy Ct (8) C Note that monetary policy reacts to inflation using the GDP deflactor P c t and deviations from steady state of value-added production C t. There is no monetary shock. 2.4 Equilibrium conditions and definitions B t = 0, L kt = L kjt dj, I k Y kjt = C kjt + W t L t k =1 n k W kt, k=1 L kt. k=1 I k Z k j t (k, j) dj, The first of these equations is the equilibrium in the assets market. The second equation simply aggregates labor within sectors by summing up hours worked in all firms in a given sector. The last equation is 8
Walras law. The llast two equations are definitions of the aggregate wage and aggregate labor. 3 The core idea To provide intuition for our main results, in this section we solve in closed-form a simplified version of our model. Since our goal is to study the effect of sectoral idiosyncratic shocks on aggregate fluctuations, we focus on solving for the log-linear deviation from the steady state of value-added production c t. As standard, we denote log-deviations from steady state in small cases. Assumptions for this section. We make five assumptions. The first is that households have log utility of consumption. The second is that their disutility of labor is linear, ϕ = 0. This assumption implies that there is no sectorally segmented labor markets and labor supply is inelastic. The third assumption is that monetary policy does not follow the Taylor rule specified in (8) but P c t C t = M which targets of a given level of nominal GDP. Our forth assumption is that sectoral technology shocks a kt are i.i.d for all k. Our fifth, and last, assumption is a modified version of the pricing friction. All prices are assumed fully flexible although there is probability λ k that a firm in sector k must set its price before observing the realization of shocks, otherwise it does after. Thus the optimal price of a firm kj is P kjt = and sectoral prices are given by θ θ 1 E t 1 [MC kt ] with probability λ k, θ θ 1 MC kt with probability 1 λ k, P kt = θ [ (1 λ k ) MC 1 θ θ 1 kt + λ k E t 1 [MC kt ] 1 θ] 1 1 θ for k = 1,..., K (9) instead of the expression in (7). We allow for sectoral heterogeneity in the "degree of price flexibility" {1 λ k } K k=1. 9
Solution. The next proposition solves for the log-linear deviation of value-added output c t. Proposition 1 Under the assumptions imposed in this section, c t = χ a t where χ (I Λ) [I δ (I Λ) Ω ] 1 Ω c is the "influence vector", Λ is a diagonal matrix with the vector [λ 1,..., λ K ] as diagonal, and a t [a 1,..., a K ] is the vector of realizations of sectoral technology shocks. Proof. See Appendix. The solution for the log-deviation of value-added output is simply a linear combination of the realizations of sectoral shocks. Thus, given our assumption that V [a kt ] = v 2 for all k, we get v c = v K χ 2 k = χ 2 v (10) k=1 with χ 2 denoting the Euclidean norm of the influence vector χ. As Gabaix (2011) and Acemoglu et al (2012), we are interested in the rate of decay to zero of v c as the number of sectors K. Unlike these papers, we focus on the interaction of the heterogeneity of price rigidity across sectors with the features of the distribution of sectoral GDP shares in Ω c and/or the input-output structure of the economy in Ω that generate a slow rate of decay of v c. We are also interested in the effect of the distribution of price rigidity on the scale of v c and the identity of the sectors which idiosyncratic shocks have stronger effect in the aggregate. From this point on we find convenient to use the following definition. Definition 1 A given random variable X is said to follow a power-law distribution with power parameter β when Pr (X > x) = γx β for some constant γ. 3.1 The granular effect and price stickiness We now connect our work with Gabaix (2011) which demonstrates the importance of the distribution of sectoral size on the rate of decay of aggregate volatility. To do so, we shut down the effect of the input- 10
output structure of the economy by setting δ = 0 to get χ = ( I Λ ) Ω c or, simply, χ k = (1 λ k ) ω ck for all k. It is apparent from this expression that the convoluted distribution of price stickiness and sectoral size (here measured as steady-state GDP) governs the scale and rate of decay of the volatility v c of value-added output. To study this convoluted distribution, we proceed in steps. To start, we assume homogeneous price stickiness across sectors. Although Gabaix (2011) does not include price stickiness in his analysis, we can get the rate of decay of v c in this case by directly applying his Proposition 2. Proposition 2 (Granular effect) When production does not need intermediate inputs, δ = 0, all sectors have the same degree of price flexibility, so λ k = λ for all k, and steady state sectoral GDP {C k } follows a power-law distribution with power parameter β c > 1, v c v c u 0 K 1 1/β c v for β c (1, 2) u 0 K 1/2 v for β c > 2. where u 0 is a random variable independent of K and v. Proof. See Appendix. This proposition states the central idea of the "granular effect": When the size distribution of sectors is fat-tailed, which is a feature of the power law distribution when the power parameter lies in the (1, 2) interval, the rate of decay of aggregate volatility is slower than the standard rate K 1/2. If this parameter is close enough to 1, then sectoral idiosyncratic shocks have a sizable effect on aggregate volatility even if sectors are defined at a very disaggregated level. In intuitive terms, when the size distribution of sectors is fat-tailed, there are few sectors that remain disproportionally large as the number of sectors go to infinity. Lemma 1 When δ = 0 and price stickiness is homogeneous across sectors, λ k = λ for all k, v c = (1 λ) v C k K 1/2 V (C k ) + C 2 k 11
for a given number of sectors K, where C k and V ( ) respectively are the sample mean and sample variance of its argument. Since price stickiness is here homogeneous across sectors, its effect is confined only to the scale of aggregate volatility. As price stickiness increases, aggregate volatility becomes smaller in an economy with a given number of sectors. The reason is that the extent in which production of a sector changes following a technology shock crucially depends on the ability of firms in that sector to adjust their prices. We now move one step forward by allowing for heterogeneity in price stickiness that is uncorrelated with steady-state sectoral GDP. Proposition 3 When δ = 0, the degree of sectoral price flexibility {1 λ k } and steady-state sectoral GDP {C k } are mutually uncorreated, and both follow power-law distributions with respective power parameter β λ, β c > 1, v c v c u 1 K 1 1/ min{β λ,β c } v for min {β λ, β c } (1, 2) u 1 K 1/2 v for min {β λ, β c } > 2. where u 1 is a random variable independent of K and v. Proof. See Appendix. This proposition presents the first key result in this paper: the "granular effect" highlighted by Gabaix (2011) is completely irrelevant for the rate of decay of aggregate volatility when the sectoral distribution of price flexibility exhibits a fatter upper tail than the distribution of steady-state sectoral GDP, provided that they are uncorrelated and the two distributions are power-law. This result is direct application of Proposition 2 after using the property that the multiplication of two independent power-law random variables is also distributed power-law wtih the smallest of the two power parameters. The next proposition completes the characterization of the interaction between the heterogeneity of price stickiness and the granular effect. Proposition 4 When δ = 0, the distribution of {(1 λ k ) C k } follows a power-law distribution with power parameter β λc > 1 and marginal distributions for {1 λ k } and {C k } also following power-law distribu- 12
tions with respective power parameter β λ, β c, v c v c u 2 K 1 1/β λc v for β λc (1, 2) u 2 K 1/2 v for β λc > 2. where u 2 is a random variable independent of K and v, and β λc < (>) min {β λ, β c } when {1 λ k } and {C k } are positively (negatively) correlated. Proof. See Appendix. If the bigger sectors are also the most flexible sectors, then shocks to these sector would have even stronger aggregate than when price stickiness and sectoral size are uncorrelated. Thus, the rate of decay of aggregate volatility would be even slower provided that the convoluted distribution of price flexibility and sectoral size follows a power law with power parameter β λc (1, 2). The opposite is true when price flexibility and sectoral GDP are negatively correlated. Proposition 4 implies that conditions on the marginal distributions of price flexibility and sectoral GDP are not sufficient to characterize the rate of decay of aggregate volatility as the number of sectors increases. As a matter of fact, it may be the case that the rate of decay of aggregate volatility v c may be slow even when price flexibility and sectoral size are not fat-tailed at all (if their correlation is positive and high enough); v c could even diverge. However, such a strong implication is not relevant empirically in our numerical exercises that follow below, so we ignore here this possibility. Conversely, it is also possible that the decay of v c may be fast (i.e., K 1/2 ) even when price stickiness and sectoral size are both heavy-tailed if they are negatively correlated enough. Finally, we state the following result regarding the effect of heterogeneity in price stickiness on the level of aggregate volatility for a given number of sectors K. Lemma 2 When δ = 0 and price stickiness is heterogeneous across sectors, v c = v C k K 1/2 V ((1 λ k ) C k ) + [ COV (1 λ k, C k ) + ( 1 λ ) C k ] 2 for a given number of sectors K, where λ and COV ( ) respectively are the sample mean of λ k and the sample covariance of its argument. 13
Comparing this expression with the one in Lemma 1, the sectoral variation of price rigidity as well as its covariance with sectoral GDP contribute to the scale of aggregate volatility for a given number K of sectors. This lemma may be seen as a natural implication of Proposition 4. 3.2 The network effect and price stickiness We now connect our work with Acemoglu et al (2012) which demostrates the importance of the input-output network structure of the economy on the rate of decay of value-added output volatility. For this we assume δ (0, 1) and shut down the granular effect, so Ω c = 1 K ι where ι is a column-vector of ones. In this case, the influence vector in Proposition 1 solves χ = 1 K (I Λ) [ I δ (I Λ) Ω ] 1 ι. This expression nests the solution for the influence vector in Acemoglu et al (2012) when prices are fully flexible, i.e. λ k = 0 for all k = 1,..., K, with the only caveat that here ι χ = 1/ (1 δ). 1 To study the interaction between price stickiness and network effects, we proceed in steps. But first we borrow some useful definitions from Acemoglu et al (2012). Definition 2 (i) The vector of outdegrees d [d 1,..., d K ] is such that d k ω k k for all k = 1,..., K. k =1 (ii) The vector of second-order outdegrees q [q 1,..., q K ] is such that q k d k ω k k for all k = 1,..., K. k =1 These definitions are statistics capturing the "centrality" of sectors in the production network. The outdegrees measure the strength of the first-order input-output linkages across sectors, this is, the importance of sectors as suppliers of other sectors. The second-order ourdegrees measure the second-order input-output linkages, i.e., the importance of sectors as suppliers weighted by the importance of "costumer" sectors as 1 Acemoglu et al (2012) normalize the scale of shocks to get the sum of the influence vector equal to one. 14
suppliers. We now start our analysis by assuming no across-sectors variation of price rigidity. Proposition 5 (Network effect) When δ (0, 1), all sectors are homogeneous in their degree of price flexibility, so λ k = λ for all k, and steady-state sectoral GDP, so Ω c = 1 K ι, the distribution of outdegrees {d k } and second-order outdegrees {q k } are independent and follow power-law distributions with respective power parameter β d, β q > 1 v c v c u 3 K 1 1/ min{β d,β q} v for min { } β d, β q (1, 2) u 3 K 1/2 v for min { } β d, β q > 2. where u 3 is a random variable independent of K and v. Proof. See Appendix. This proposition summarizes the essence of the "network effect": The rate of decay of aggregate volatility is bounded below by the fattest-tailed distribution between the outdegrees and the second-order outdegrees, provided that they are independent and each follow power-law distributions. Thus, if there are some sectors disproportionally central in the production network, sectoral idiosyncratic shocks have sizable effect on aggregate volatility even if sectors are defined at a very disaggregated level. Although Acemoglu et al (2012) do not study price rigidities, we can get this proposition by directly combining their main results. 2 The intuition comes from noticing that, when price rigidities and sectoral GDP in steady state are homogeneous across sectors, the solution for the influence vector is χ = 1 λ K ι [I (1 λ) δω] 1. (11) = 1 λ (1 λ) τ δ τ ι Ω τ K τ=0 given that all elements of Ω lie outside the unit circle. Further, truncating the expression for χ to τ = 2 and 2 In particular, theorems 2 and 3 and corollaries 1 and 2 in their paper. 15
using d = ι Ω and q = ι Ω 2 yields χ 1 λ K [ ] 1 + (1 λ) δd + (1 λ) 2 δ 2 q (12) ehere the inequality follows from Ω > 0. Thus, from (10), the rate of decay of v c is at least the fattest-tailed distribution between outdegrees and sectoral outdegrees. We now use equation (11) to look at the scale effect on v c of the (homogeneous) price rigidity λ. Lemma 3 For δ (0, 1), Ω c = 1 K ι and λ k = λ for all k, and given a number of sectors K: (i) Aggregate volatility v c is decreasing in λ (ii) λ has stronger effect on the contribution to v c of the second-order outdegrees q than the contribution of outdegrees d. For part (i) of this lemma, it is clear from (11) that χ is decreasing in λ. For instance, if the production network is symmetric, i.e., Ω = 1 K ιι, then v c = 1 λ 1 (1 λ) δ K 1/2 v. For part (ii) of this lemma, note from (10) that the cross-sectional variation of d and q contribute to v c and from (12) that d is multiplied by 1 λ while q is multiplied by (1 λ) 2. Therefore, even when the rate of decay of aggregate volatility may be slow because of the second-order outdegrees, the scale of their effect is smaller as price rigidity increases. This result is important to highlight because in the empirical analysis of Acemoglu et al (2012) it is the second-order outdegrees what have the strongest network effect. In the data, producer prices are in average quite rigid, so this result suggest that second-order outdegrees are irrelevant quantitatvely. [CONFIRM THIS IN NUMERICAL EXERCISES WITH THE FULL MODEL]. As another intermediate step, we now assume that price rigidity is heterogeneous across sectors but the production network is symmetric. Proposition 6 For δ (0, 1), Ω c = 1 K ι, a symmetric production network, so Ω = 1 K ιι, and the degree of 16
sectoral price flexibility {1 λ k } follows a power-law distribution with power parameter β λ > 1, then v c v c u 4 K 1 1/β λ v for β λ (1, 2) u 4 K 1/2 v for β λ > 2. where u 4 is a random variable independent of K and v. Proof. See Appendix. This proposition shares its flavor with Proposition 3: When the only souce of heterogeneity across sectors is their degree of price flexibility, the rate of decay of aggregate volatility is slow when this distribution is fat-tailed. Regarding the level of aggregate fluctutations given a number of sectors, it is useful to establish the following result. Lemma 4 When δ (0, 1), Ω c = 1 K ι, and λ k is heterogeneous across sectors, v c = K 1/2 v 1 ( 1 λ ) δ V (1 λ) + ( 1 λ ) 2. Comparing this result with Lemma 3 for the case when the production network is symmetric, this lemma states that cross-sectoral variation of price rigidities contributes positively to aggregate volatility. The final step in this section is to allow for sectoral heterogeneity in price rigidity and input-output linkages such that the production network is assymetric. Before stating our main results, it is convenient to make the following definitions. Definition 3 (i) The vector of modified outdegrees d [ d1,..., d ] K is such that d k (1 λ k ) ω k k for all k = 1,..., K. k =1 (ii) The vector of modified second-order outdegrees q [ q 1,..., q K ] is such that q k (1 λ k ) d k ω k k for all k = 1,..., K. k =1 17
Keeping a similar logic with Definition 2, we here define some statistics that capture the interaction between the heterogeneity in price rigidities across sectors with centrality in the production network. The modified outdegrees measure the first-order input-output linkages weighting them by the degree of price flexibility of the "costumer" sector. Similarly, the modified second-order outdegrees measure the secondorder input-output linkages weighted by the degree of price flexibility and the modified outdegrees of the costumer sectors. For our main theoretical result, we drop the assumption of homoegenous steady-state sectoral GDP. Proposition 7 For δ (0, 1), Ω c = 1 K ι, heterogeneous price rigidity and asymmetric production network: (i) The influence vector χ satisfies χ k ω ck (1 λ k ) [1 + δ d k + δ 2 q ] k { } (ii) Assume that {1 λ k }, {ω ck } dk and { q k } follow all power-law distributions, the rate of decay of v c depends on their power parameters and their mutual correlations. Proof. See Appendix. This is the most important result in the paper regarding the network effects: The importance of centrality in the network for the aggregate propagation of sectoral shocks is distorted by its interaction with price rigidities. This interaction is captured by the modified outdegrees d and the second-order modified outdegrees q. Besides, revisiting Proposition 4, the importance of the distribution of sectoral size on the aggregate propagation of sectoral shocks. Finally, these two distorted granular and network effects interact depending on their own mutual correlation. The next corollary highlights another effect of price rigidities. Corollary 1 Heterogeneity of price stickiness may change the identity of the sectors with the largest contribution to aggregate volatility. This corollary simply states a direct implication of Proposition 7, so it needs no proof. The most important sectors for aggregate volatility do not need to be the largest sectors or the most central sectors in the production network, but a complicated mix between them and the sectoral distribution of price rigidities. 18
4 Concluding remarks TO BE ADDED. 5 References TO BE ADDED. 19
Appendix: Proofs [TO BE COMPLETED] Note: Log-linear deviations from steady state are denoted by small cases. Proposition 1. Production efficiency in (6) implies mc kt = (1 δ) w kt + δp k t a kt where w kt is the sectoral wage and p k t is the the aggregate price relevant for firms in sector k: p k t = ω kk p k t. k =1 Using labor supply in (4) with ϕ = 0 and σ = 1 yields w kt = c t + p c t where p c t is the aggregate price relevant for households (also interpreted as GDP deflactor): p c t = ω ck p k t. k =1 Given that there are no monetary policy shocks, the policy rule implies c t = p c t, so mc kt = δp k t a kt. From the assumption that {a kt } are i.i.d. follows that E t 1 [mc kt ] = 0. Combining these results with the log-linearized equation (9) yields p kt = δ (1 λ k ) p k t (1 λ k ) a kt and using again c t = p c t solves c t = ( I Λ ) [ I δ ( I Λ ) Ω ] 1 Ω c a t. Proposition 2. This proof heavily relies on Gabaix (2011), proof of Proposition 2. 20