Online appendix for Innovation, Firm Dynamics, and International Trade

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1 Online appendi for Innovation, Firm Dynamics, and International Trade Andrew Atkeson UCLA and Ariel Burstein UCLA, Feruary This appendi is composed of two parts. In the first part, we provide details for two analytic results from our model that are discussed in Section 4 ut not proved in the paper: the model with no productivity dynamics with Pareto distriuted productivities, and the model with assymetric countries. In the second part of this appendi, we provide additional details on the quantitative model and its numerical solution. We also provide an overview of the Matla codes that are used to compute the model, which are availale in our wesite. 1. Analytical results In this section we present proofs for results that are discussed in Section 4 of the paper Section 4.2: No productivity dynamics, Pareto distriution and β < 1 We now show that, if we allow for any β < 1 and assume that G is such that ep z is distriuted Pareto, we get that ζ = Π d Z d + Z 1 + D 1 ρ /Υ is unchanged with D. Thus, Lemma 1 applies, and hence, L r is unchanged with D. Therefore, as in the version of the model with no productivity dynamics and β 1, the ratio of the indirect effect to the direct effect of changes in trade costs on aggregate productivity is given y 4.4 in the paper. In particular, we assume that the cumulative distriution function of epz is σ ep z0 Ḡ z = 1, for ep z > ep z 0 ep z where σ > 1. Under this distriution, we have that Z d = ep z ep z Z = σ ep z 0 σ δ ep z σ ep z Using the cutoff definitions, we get that d ep z = σ ep z 0 σ δ σ 1 ep z 1 σ ep z 1 σ and σ ep z 0 σ δ ep z σ d ep z = σ ep z 0 σ δ σ 1 ep z 1 σ. Z d = σ ep z 0 σ δ σ 1 Π d σ 1 n 1 σ n 1 σ f, and D 1 ρ 1 We thank Javier Cravino for super research assistance in putting together this material.

2 Therefore, we have Π d Zd + Z 1 + D 1 ρ = Z = σ ep z 0 σ δ σ 1 Π d σ 1 n D 1 ρ 1 σ. Using the cutoff definitions, we can epress Υ as σ δ σ 1 ep z 0 Π d σ n 1 σ f + n 1 σ D 1 ρ σ. 1.1 Υ = n e + 1 δ ep z 0 Π d σ n 1 σ f Comining 1.1 and 1.2, we otain Π d Zd + Z 1 + D 1 ρ = + n 1 σ σ σ 1 Υ n e D 1 ρ σ. 1.2 Comined with A7 in the paper, this implies that oth Π d Z d + Z 1 + D 1 ρ and Υ are independent of D. Therefore, Lemma 1 applies. Therefore, the ratio of the indirect effect to the direct effect of changes in trade costs on aggregate productivity is given y 4.4 in the paper. Note that in this case, if λ = 1, M e is invariant to changes in D see epression A2 in the paper together with the fact that L r and Υ are unchanged with D. Hence, it is possile to prove this result without the use of the free-entry condition, ut instead fiing the numer of firms in each country. One does require the free-entry condition to prove our result when λ < Section 4.5: Asymmetric countries In section we etend the analytic results to the version of the model with asymmetric countries. We focus on the special cases that we can solve analytically in Section 4 all firms eport, suset of firms eport/ no productivity dynamics, elastic process innovation/eogenous selection. We continue to use the assumption that the real interest rate approaches zero, β 1, to ensure that the allocation of laor remains constant. Note that this assumption that β 1 is not necessary in all of our special cases e.g. it is not necessary when all firms eport or with no productivity dynamics and Pareto distriuted productivities, ut we use it here to unify the presentation. We first show that, if we assume trade alance etween countries, to a first-orderapproimation, the ratio of the indirect effect to the direct effect of a change in marginal trade costs on aggregate productivity is the same as with symmetric countries, and given y indirect effect direct effect = 1 λ ρ + λ 2 2

3 in all special cases of the model. With asymmetric countries, a new effect arises in our comparative statics across steady-states after a change in trade costs that is not present in the symmetric case. We term this effect the terms of trade effect. As we will see elow, to otain the result aove in the asymmetric case, we must e careful to include this terms of trade effect as part of what we call the direct effect of a change in variale trade costs on aggregate productivity. The magnitude of the terms of trade effect in response to given changes in trade costs can potentially differ across our model specifications. Hence, in general the change in aggregate productivity, output and consumption in each country will vary across model specifications. We net show, however, that with trade alance, to a first-order-approimation the growth of world output and consumption defined as an ependiture weighted average of the growth of output and consumption of individual countries is equal across model specifications. Hence, even though changes in eit, eport, and process innovation decisions can lead to different responses of output and consumption in individual countries, once the change in product innovation and terms of trade are taken into account, changes in these decisions do not affect the gloal growth in output and consumption. That is, changes in these decisions can lead to a redistriution of output and consumption across countries, ut not to changes in world output and consumption. Finally, we consider a version of our model that does not assume trade alance, ut instead assumes risk sharing etween countries. The equilirium allocations coincide with those of the planning prolem. We show that, to a first-order approimation, the growth of world consumption is also equal across our alternative model specifications. Moreover, in this case our measure of aggregate consumption growth is equal to the change in welfare of a gloal planner. Hence, to a first-order-approimation, changes in eit, eport, and process innovation decisions of firms in response to changes in trade costs do not affect gloal welfare, once changes in product innovation and terms of trade are taken into account The structure of the model in steady-state We allow our two countries to differ in the size of the laor force L, trade costs D and n, fied costs n f, entry costs n e, and the parameters of the innovation cost function c q and z. We will assume that the elasticity of sustitution parameter ρ and the share of laor in the research production function λ are the same in oth countries. We will first assume trade alance period y period and then assume risk sharing, and we only focus on 3

4 steady-state equiliria. As we did in our paper, we find it useful to reak the equations characterizing the steadystate into four groups: 1 those characterizing the Bellman Equation of the firm and free entry used to pin down profitaility" in each country relative to the price of the research good Π d /W r and Π d /W r. It is in differentiating these equations that we invoke Envelope Theorems to argue that the derivatives of these equations do not depend on the specification of the firms decision prolems as long as we are differentiating at a point at which these decisions are chosen optimally. 2 those static equations relating profitaility relative to the price of the research good Π d /W r and Π r/wr to domestic real wages and output. 3 those equations that determine the allocation of laor in steady-state. Here, we use our result that the allocation of laor is fied in steady-state if β 1. 4 those equations that relate real wages and output to the components of productivity and the allocation of laor. This is the set of equations in which all of the new action arises in the case of asymmetric countries. Bellman Equation and Free Entry The only impact that asymmetric countries has on our Bellman Equations and Free Entry conditions is that a term that we call the terms of trade" effect comes into the definition of variale profits within the period. This term, denoted y u, reflects the impact of differences in market size and price levels across countries. Thus, in the asymmetric case, we can write steady-state variale profits in units of the local research good in the two countries as follows Πs W r = Π d Πd ep z + ma ud 1 ρ ep z n, 0 W r W r Π ep z + ma d Wr 1 u D1 ρ ep z n, 0. Π s = Π d Wr Wr Note that we will relate u to market size and price levels in the second set of equations elow. With symmetry, we have u = 1. In the asymmetric case, we must solve for it endogenously. Also note that in the paper, we chose the home research good as the numeraire, so W r = 1 y definition and W r must solve for W r W r = 1. = 1 y symmetry. Here we can set W r = 1 as a numeraire, ut we in equilirium. To keep the notation easy to read, we will not impose 4

5 Using the same logic of the paper under our special cases in Section 4, differentiating the free-entry condition in the home country and using the envelope conditions of process innovation, eit and eporting decisions, yields so Π d W r Zd ud 1 ρ Z + Π d W r Z 1 + ud 1 ρ = 0 Z log Π d W r = 1 + ud 1 ρ Z d ud 1 ρ Z 1.3 = s log u + ρ 1 s log D, where s and s denote the shares of eports in the output of intermediate good firms, given y s = uz D 1 ρ Z d + Z + uz D 1 ρ and s = ZD 1 ρ u Zd Z + ZD 1 ρ It is straightforward that the envelope condition can e applied here even with asymmetric countries. In the case with endogenous process innovation, we are using the fact that the real interest rate is zero so that the hyrid share of eports s equals the actual share of eports s. Note that epression 1.3 coincides with the one in the paper, with the addition of the change in relative country sizes, s log u, which affects profits from sales in the foreign country. Analogously, in the foreign country, differentiating the free-entry condition and using the envelope conditions for process innovation, eit and eporting decisions yields log Π d W r = s log u + s ρ 1 log D. 1.5 Static relations: profitaility, output, and wages The following equations otained from static profit maimization and the definitions of W r and W r, do not change with the assumption that countries are asymmetric: Π d = λλ 1 λ 1 λ W r ρ ρ ρ 1 1 ρ W/P 1 ρ λ Y, 1.6 Π d Wr = λλ 1 λ 1 λ ρ ρ ρ 1 1 ρ W /P 1 ρ λ Y. Clearly, these equations are already log-linear and hence any derivatives associated with them do not depend on other aspects of the specification of the model. 5

6 The variale u, which summarizes the size and price level differences across countries that are relevant for firms profits, is defined y u = Y P ρ Y P ρ. 1.7 We will refer to changes in relative sizes and prices across countries, as summarized y u, as the terms of trade effect. This equation is already log-linear as well. Aggregate allocation of laor Here we follow eactly the same steps as in the symmetric model. From the CES aggregator, payments to production employment is a fied ratio of variale profits: W L L r = ρ 1 Π d Me Z d + Z + D 1 ρ M e Z u. Co-Douglas production of the research good implies a constant cost share on research laor W L r = λw r ΥM e, where Υ denotes the use of research good per entering firm as defined in equation 3.7 of the paper. Dividing these two equations yields: L L r = ρ 1 L r λ Π d M e Z d + Z + D 1 ρ M e Z u W r ΥM e With β 1, the free entry condition implies Π Z d d +Z +D 1 ρ M ez u W r β 1, so Υ The same relation holds in the foreign country. L L r L r = ρ 1 λ. 1.8 Hence, the aggregate allocation of laor is unchanged across steady states when the real interest rate approaches zero. Note that in the special versions of our model in which all firms eport or with no productivity dynamics and Pareto distriuted productivities, one does need to assume β = 1 for the aggregate allocation of laor to e unchanged across steady-state. Aggregate output, wages and productivity under trade alance The final set of equations relate aggregate output and wages to productivities. Here, with asymmetric countries, the algera is messier and new terms related to the change in u enter. The equations differ slightly etween the cases of trade alance and risk sharing. 6

7 Laor market equilirium in the home country requires ρ ρ 1 W ρ P ρ Y M e Z d + Z + P ρ Y M e Z D 1 ρ = L L r, ρ which using the definition of u yields ρ ρ ρ 1 W Y M e Zd + Z + uz D 1 ρ = L L r. 1.9 ρ P The price level in the home country is or P = ρ Me Z d + Z W 1 ρ + Me Z W 1 ρ D 1 ρ 1/1 ρ, ρ 1 W P = ρ 1 M e Z d + Z + Me Z ρ Trade alance requires that eports equal imports, or Sustituting 1.11 into 1.10 yields Comining 1.9 and 1.12 yields W 1 ρ 1/ρ 1 D 1 ρ W W 1 ρ D 1 ρ M e Z P ρ Y = W 1 ρ D 1 ρ M e Z P ρ Y W P = ρ 1 Me Z d + Z + D 1 ρ M e Z u 1/ρ ρ Y = M e Z d + Z + D 1 ρ um e Z 1 ρ 1 L L r 1.13 Note that 1.12 and 1.13 coincide with that in our symmetric model, ecept that now aggregate productivity of eporting firms is adjusted with the term u. We define aggregate productivity in the home country as: are Z = M e Z d + Z + D 1 ρ um e Z 1 ρ The analogous epressions for the real wage and aggregate output in the foreign country W P = ρ 1 ρ Y = 1/ρ 1 Me Zd + Z + D 1 ρ Me Z u 1 ρ 1 L L r M e Z d + Z + D 1 ρ 1 u M e Z Equilirium consumption in each country is C = Y 1 λ W λ P, and C = Y 1 λ λ 7 W P. 1.17

8 Effects of changes in trade costs under trade alance We first calculate, under trade alance, the ratio of the indirect effect to the direct effect of a change in trade costs on aggregate productivity, to a first-order approimation. Net, we calculate the impact of such a change in trade costs on growth of world output and consumption. Aggregate productivity Here we study the effects of a small change in marginal trade costs on aggregate productivity across steady-states. Following the logic in our paper, to a first-order approimation, the change in aggregate productivity of the home country in response to a change in marginal trade costs is log Z = s log D + 1 ρ 1 s log u } {{ } Direct Effect ud s 1 ρ 1 + ud log Z ρ 1 ud 1 ρ ρ s log Z ud 1 ρ d + log M e. } {{ } Indirect Effect 1.18 The direct effect corresponds to the change in aggregate productivity from the change in trade costs and the susequent change in relative country sizes and price levels as summarized y u i.e. we can refer to this as the terms of trade effect, with firms eit, eport, process, and product innovation decisions held fied. With symmetric countries, u remains constant. The indirect effect arises from changes in firm decisions, which are themselves responding to the trade cost and the change in u. We first evaluate the size of the indirect effect relative to the direct effect. Log-differentiating 1.6 and using the fact that with zero interest rates the aggregate allocation of laor remains unchanged across steady-states, we otain: log Π d = 2 ρ λ log M e Z d + Z + D 1 ρ um e Z W r ρ 1 = 2 ρ λ Direct + Indirect 1.19 Comining 1.3 and 1.19 as in our paper, we otain: Indirect Effect Direct Effect = 1 λ ρ + λ 2 8

9 The same epression holds in the foreign country. So, the ratio of the indirect to the direct effect is the same as with symmetric countries, with the caveat that direct effect now includes the change in relative country sizes and price levels as summarized y u i.e. the terms of trade effect. Note that when λ = 1, the indirect effect is equal to zero. Therefore, changes in product innovation log M e eactly offset changes in eit, eport, and process innovation decisions as summarized y log Z d and log Z. The magnitude of the change in aggregate productivity and output in each country now depends on the terms of trade effect, as summarized y log u, which is determined in general equilirium. The magnitude of the change in the terms of trade can potentially vary across model specifications. To see this, note that the trade alance condition, 1.11, depends on changes in eporters aggregate productivity log Z and log Z and not on changes in domestic producers aggregate productivity log Z d and log Zd, and these can vary across model specifications. One can pick parameters across some of the model specifications to ensure that the change in log Z and log Z are the same i.e.: for eample elastic process innovation - inelastic eport participation, or inelastic process innovation - elastic eport participation. We now show that even though the change in output and consumption of each individual country can vary across model specifications due to the terms of trade effect, the growth in world output and consumption and the welfare of a gloal planner under complete markets does not. World output and consumption We define a measure of the growth of world consumption as an ependiture-weighted average of the growth of consumption in each country, log C W = k log C + k 1 + k log C 1.20 where k is the ratio of foreign consumption ependitures to home consumption ependitures: k = P C P C. Using 1.8, 1.12, 1.13, 1.15, and 1.16, and 1.17, we have that the ratio of consumption ependitures is equal to the ratio of final output ependitures this is straightforward with λ = 1, ut otherwise it relies on the fact that real wages are proportional to final 9

10 output: k = P C P C = P Y P Y These epressions also imply that log Y = log W/P in each country, so using 1.17, we have log C = log Y, log C = log Y, and log Y W = log C W We otain log Y and log Y using 1.3, 1.5, 1.19 and the analogous of epression 1.19 in the foreign country: s log u + ρ 1 s log D = log Π d W r = 2 ρ λ log Y 1.21 s log u + ρ 1 s log D = log Π d = 2 ρ λ log Y 1.22 Wr Using the definitions s =Eports/P Y and s =Eports /P Y, and the trade alance condition Eports=Eports we have k = s /s. Hence, we can calculate log C W multiplying 1.22 y s /s cancel-out: log C W = and adding it to 1.21, noting that the terms with log u s log Y + log Y s 1 + s s Note that with symmetric countries, log C W paper. = ρ 1 s s log D + log D ρ + λ 2 s + s y = ρ 1 / ρ + λ 2 log D, as in our Therefore, conditional on choosing the model parameters to match eport shares s and s in each country, the steady-state world growth in output and consumption in response to changes in trade costs are, to a first-order approimation, equal across our alternative model specifications. Changes in eit, eport, and process innovation decisions that have differential impact on the terms of trade affect the distriution of world output across countries, ut not the growth in world output and consumption Effects of changes in trade costs under risk sharing The previous discussion assumed trade alance etween countries i.e. financial autarky We now sustitute this assumption with risk sharing, which emerges under complete financial markets etween countries. We show that, to a first-order approimation, the steady-state world growth in consumption is still the same across our model specifications as long as we choose parameters to target the same trade shares and relative ependitures across countries. 10

11 Moreover, the growth in world consumption in this case is proportional to the change in the welfare of a gloal planner. We assume preferences over consumption in each country of the form U C = 1 1 σ C1 σ, with σ 0. The risk sharing condition that emerges under complete markets is C σ P = χc σ P 1.23 where χ is a constant that is determined y the initial level of wealth in each country. Equation 1.23 replaces the trade alance condition The ratio of foreign consumption ependitures to domestic consumption ependitures is given y k = χ C1 σ C 1 σ Define the welfare function of a world social planner y U W = 1 1 σ C1 σ + χ 1 1 σ C1 σ. Note that the weight on foreign utility depends on the initial level of wealth in each country. The log-percentage change in welfare is proportional to the ependiture-weighted average of consumption growth in each country: { } log U W C 1 σ χc 1 σ = 1 σ log C + log C 1 σ + χc1 σ C 1 σ C + χc1 σ { 1 = 1 σ 1 + k log C + k } log C = 1 σ log C W 1 + k We now show that, conditional on targeting eport shares in each country s and s and relative ependitures k, the steady-state world growth of consumption, log C W, is independent of the response of eit, eport, and process innovation decisions, and hence is equal across our alternative model specifications. In deriving our results, we assume that firms receive a per-unit production susidy τ 1, so that revenues of a firm selling y units at a price p are equal to τpy. With this perunit susidy, the domestic price set y a home firm with productivity inde ep z is p = 1 τ ρ W. ρ 1 epz 1/ρ 1 If λ = 1, the level of τ does not affect the steady-state growth in world consumption and hence, we can assume τ = 1 as in our aseline model. If λ < 1, we need to set τ = τ = ρ/ ρ 1 for our result to hold. This is related to the fact that, with λ < 1, markups distort the equilirium level of entry. With τ = τ, markups are eliminated and the effi cient level of entry is restored. Our central equivalence result holds for any value of τ if λ = 1, and τ = τ if λ < 1. 11

12 Loglinearized system of equations With risk sharing, we can make use of many of the previous steps under alanced trade. In particular, the change in the constant on variale profits is still given y 1.3 and 1.5, the constant in variale profits is related to output and the real wage y 1.6, the laor market clearing condition is given y 1.9, the price level is determined y 1.10 with the addition of a multiplicative constant that reflects the production susidy τ, and the level of consumption is determined y When real interest rates are zero, the aggregate allocation is constant and given y 1.8. Note, however, that we cannot make use of equations 1.12 and 1.15 ecause in deriving these we made use of the trade alance equation 1.11, which is now replaced y With this in mind, the log-linear equations that determine the first-order aggregate effects of changes in trade costs are: 1.3, 1.5, the loglinear versions of 1.6 log Π d W r = 1 ρ λ log W/P + log Y 1.24 log Π d = 1 ρ λ log W /P + log Y, 1.25 Wr the loglinear version of 1.9 and the analogous epression in the foreign country ρ log W/P = log Y + s log u + s 1 ρ log D + log M e ud ρ 1 + ud 1 ρ s log Z ud 1 ρ d + s log Z ud 1 ρ, ρ log W /P = log Y s log u + s 1 ρ log D + log Me s 1 + u 1 D 1 ρ log Z u 1 D 1 ρ d + s 1 + u 1 D 1 ρ log Z u 1 D 1 ρ the loglinear version of 1.10 and the analogous epression in the foreign country log W/P = log M e + Z log W /P = 1 ρ 1 1 s m the loglinear versions of 1.17 Z d Z d + Z log Z d + log Z 1.28 Z d + Z d + 1 ρ 1 s m log M e + log Z s m log W /W s m log D 1 ρ 1 1 s m log Me + Z d Zd + log Z Z d + Z Zd + log Z1.29 Z + 1 ρ 1 s m log M e + log Z + s m log W /W s m log D, log Y = C Y log C + 1 C Y log log W/P

13 log Y = C Y log C + and the loglinear versions of 1.7 and C Y log W /P, 1.31 log W /W = σ log C log C + log W /P log W/P 1.32 log u = log Y log Y σρ log C log C 1.33 Here, we define s m and s m to e the share of imports in final good ependitures: s m = Imports P Y = M e Z W W M e Z d + Z + M e Z 1 ρ D 1 ρ W 1 ρ W D 1 ρ s m = Imports P Y = Under trade alance, s m = s and s m = s. M e Z W W 1 ρ D 1 ρ M e Z d + Z + M e Z W W 1 ρ D 1 ρ 1+uD Moreover, s n = s 1 ρ in 1.26 denotes the employment share of eporters in the ud 1 ρ home country, and s 1+u n = s 1 D 1 ρ in 1.27 denotes the employment share of eporters u 1 D 1 ρ in the foreign country. The ratio Z d /Z in 1.28 is equal to 1 sn s n s, and the ratio Zd /Z in 1.29 is equal to 1 s n. s n s The 12 loglinear equations 1.3, 1.5, 1.24, 1.25, 1.26, 1.27, 1.28, 1.29, 1.30, 1.31, 1.32, and 1.33 can e used to solve for the twelve variales log Π d W r, log Π d, Wr log C, log C, log Y, log Y, log W/P, log W /P, log M e, log Me, log u, log S, given changes in trade cost { log D, log D }, changes in aggregate productivity indices { log Z d, log Z, log Z d, log Z }, the model parameters {ρ, λ, σ}, eport shares s, s, import shares s m, s m, ratio of consumption ependitures k, and employment share of eporters s n, s n. Given log C and log C, we can otain log C W using At the end of this document Additional appendi 1 we show that import shares s m, s m, and the ratio of consumption to output, C/Y, C /Y are related to eport shares s, s and the ratio of consumption ependitures across countries k as follows: s k k+1 s s k 1 k 1 +1s s m = 1 s + s k k+1 s 1 k+1 s C + Y = 1 + s 1 k+1 s, s m = 1 s + s k 1 k 1 +1 k k+1 s s k k+1 s, s 13

14 C s + k 1 k 1 +1s Y = 1 k 1 +1 s s s k 1 +1 k 1 k 1 +1s s s where = 1 1 λ τ. Hence, conditional on matching eport shares and a ratio of consumption ρ ependitures across countries, import shares and the ratio of consumption to output are also uniquely pinned down. World consumption Assume that the production susidy is given y τ = ρ/ ρ 1 if λ < 1, and takes any value τ > 0 if λ = 1. To a first-order approimation, the growth in world consumption, log C W, in response to changes in trade costs is given y log C W = ρ + λ 1 ρ k 2 ρ λ 1 λ s m + s m + λ + ρ {ρ 1 s m log D + ks m log D + 1 λ s m 1 s m k s m log D + ks m 1 s m s m log D + ρ + λ 2 s m ks ρ 1 m ρ + λ 1 s log D s log D } + ρ k s log D + ks log D Note that with symmetric countries, log C W paper. = ρ 1 / ρ + λ 2 log D, as in our Hence, log C W is only a function of the model parameters ρ, λ, eport shares s, s, and the ratio of consumption ependitures across countries k recall that s m and s m are pinned down y s, s, and k. Hence, changes in eit, eport and process innovation decisions, which impact { log Z d, log Z, log Z d, log Z }, have no effects on log C W and welfare. The derivation of 1.34 is presented at the end of this document Additional appendi 2 2. Quantitative model This section is divided in two parts. In the first part, we first present results that we use when numerically solving our model. In the second part, we descrie the algorithms used to compute the steady-state and the transition dynamics of our model Preliminaries Discrete time approimation Recall that z denotes the step size in logs and q denotes the proaility of taking a step up. We choose these two parameters as functions of the 14

15 length of a time period as follows. Let t denote the time interval for one step. Let α denote the epected change in z per unit time and σ 2 the variance of z per unit time. We choose z and q so that z t+ t z t is distriuted normal with mean α t and variance σ 2 t. To do that, we set z = σ t and q = α t σ This is ecause Ez t+ t z t = q z 1 q z = 2q 1 z = α t and E z t+ t z t 2 = q 2 z + 1 q 2 z = 2 z so V ar z t+ t z t = 2 z 1 2q 1 2 = 4q1 q 2 z = σ 2 t α 2 t 2 Note that this formula works eactly if α = 0 no drift and it is approimate if the time interval is small so t 2 is really small. Thus, the real parameters that we choose are α, the mean growth of size, and σ 2, the variance of growth of size. In our Matla codes, we define the parameter per = 1/ t as the numer of periods in a year. The higher is per, the shorter is each time period. Choosing elasticity parameter Recall that in our formulation, the cost of process innovation is given y ep q. We want to calirate so that the relevant elasticity defined elow is unchanged as we vary the length of the time period,. In order to do so, suppose that the cost of process innovation is epressed in terms of the epected growth rate of z per unit of time, α: ep α. In this case, the optimal choice of α for large firms solves the prolem: ma α h ep + β 1 δ V q ep z + 1 q ep z α,q s.t. q = α t 2 z where we used the fact that s = gs t, and that for these large firms, V z = V ep z ecause fied costs for these firms represent a trivial portion of revenues and hence can e 15

16 ignored in the value functions. The FOC w.r.t. q is: 2 z t h ep α = β 1 δ V ep z ep z where we used α = 2 z. Hence, q t α = log V 1 In our current formulation, the FOC with respect to q is Sustituting the definition of q : h ep 1 + α t 2 z h ep q = β 1 δ V ep z ep z = β 1 δ V ep z ep z Therefore, So, we will choose so that α log V = 1 2 z t = 2 z t Inelastic process innovation In the paper, we refer to the case with inelastic process innovation as that with = 1200 high curvature of the process innovation cost function. We also considered an alternative specification in which q is assumed to e eogenously given and equal for all firms. That is, we impose q z = q calirated for all z, and we set the costs of process innovation equal to zero for all firms. In the codes we call this case y setting = Both alternative assumptions give almost identical results the key is that with = 1200, q z is almost invariant with z. Timing of product innovation In our quantitative model, as we reduce the period length i.e. smaller, we keep the entry period of new firms at one year and we set the annual entry cost at ne t so that n e denotes the entry cost per unit of time. Initial guess of value functions In "Fied.m" we use V z = V ep z as the initial guess when solving for the fied point of V z in the initial steady state. 2 Here we are using 2 For the new steady state, we use as our initial guess V SS0, the value function in the initial steady state. 16

17 the fact that, for large firms, fied costs ecome insignificant as a share of total value, so value functions are proportional to ep z. We solve for V as follows. We have V = Π d 1 + D 1 ρ h ep q + V β 1 δ q ep z + 1 q ep z and the foc for q is h ep q = V β 1 δ ep z ep z Replacing the FOC into V, we have: V = Π d 1 + D 1 ρ V β 1 δ ep z ep z +V β 1 δ q ep z + 1 q ep z or V = Π d 1 + D 1 ρ 1 + β 1 δ ep z ep z 1 q β 1 δ ep z So, given Π d, we set V z = V ep z as the initial guess. Calirating h We calirate h to target a slope of the employment-ased distriution of 0.2 for firms ranging etween 1, 000 and 5, 000 employees in the version of the model with inelastic process innovation high. The model then implies a value of process innovation q for large firms. As we lower, we adjust the model parameters to keep the value of q for large firms constant and thus keep the dynamics of large firms unchanged. When we implement this procedure, we guess for a slope coeffi cient for large firms, then otain the value of q consistent with this slope coeffi cient, and calirate h so that large firms choose this value of q. We follow this indirect procedure ecause it depends less on our choice of the length of a time period. To see this, here we derive an epression of the slope coeffi cient for suffi ciently large firms. We define the slope coeffi cient as the slope of the line log 1 dm z ep z /z, where we used the fact that employment is proportional to z ep z, so the log of employment is proportional to z. Denote y m n de mass of firms with productivity inde z = z 0 + n z, normalized y the 17

18 mass of entering firms i.e. M z. The slope coeffi cient is lim k = lim log n=k+1 ep n z m n log n=k ep n z m n log n=k+1 k log = lim k n=k+1 z epn z m n epk z m k log z epn z m n epk z m k log z n=k 1 + n=k+1 epn z m n epk z m k epn z m n epk z m k We need to compute lim epn z m n k n=k+1 epk z m k. We know that suffi ciently large firms have a constant q. Then, for large enough n, and where m n = 1 δ qm n δ 1 q m n+1 m n = 1 δ q + 1 δ 1 q m n+1 m n 1 m n 1 mn m n 1 is independent of n for large firms. Define y = mn m n 1, so The slope coeffi cient is so y = 1 δ q + 1 δ 1 q y 2 log n=1 ep z y n log 1 + n=1 ep z y n z = log ep zy log 1 1 ep zy 1 ep zy z = log ep z y z log y = 1 + z Suppose we want to hit a slope coeffi cient of < 0. Then, we need 1 + log y z = y = ep 1 z What value of q do we need to hit a given y? From y = 1 δ q + 1 δ 1 q y 2 we otain q = y 1 δ y2 1 y 2 18

19 2.2. Algorithms to compute model The codes "Master.m", "Steady.m" and "Fied.m" solve the steady-state and transition dynamics of our model. Master.m is the main code to choose parameter values, and to call the other codes. The output is stored in the file "Storeresults" in the active directory. We include an ecel file, "results.ls" that displays the output for most of the parameterizations in the paper. We included comments in the code to eplain various steps. However, some steps might still e not 100% clean in the codes, so please do not hesitate to let us know if you have any questions. Steady-state As descried in section 3.4 of the paper, we solve for the steady-state in two steps. First, we find the level of Π d that is consistent with free entry, which in turn gives eit, eport and process innovation decisions. We then use these policy functions to solve for the aggregate variales. These steps are taken in the functions "Fied.m" and "Steady.m". "Fied.m" finds the steady-state value function V associated with any given Π d using value function iteration, and calculates the difference etween the entry cost and the discounted value of entry denoted y F. The free entry condition requires F = 0. In the initial steady state, "Fied.m" also calirates the parameter h to match a value of q for large firms, as descried aove. "Steady.m" solves for the steady-state. It first uses "fsolve" to find the value of Π d such that F = 0 in "Fied.m". This is the only steady state value of Π d that is consistent with the free-entry condition. Associated with Π d are the firms policy functions, which are used to find the stationary distriution of firms. Given these policy functions and the stationary distriution of firms, Steady.m solves for the remaining aggregate variales. Transition dynamics After finding the initial and final steady states, we use the following algorithm to compute the transition across steady-states: Guess the numer of periods it takes to reach the new steady state denoted y pertran. Guess sequences for {Yt tranit, Zt tranit, Ct tranit, Wt tranit t=1, where {Y t, Z t, C t, W t } denote output, aggregate productivity, consumption and the wage, respectively. } pertran Using the value function from the final period new steady state and the guess {Y tranit t, Zt tranit, Ct tranit, Wt tranit t=1, iterate ackwards to find the policy functions } pertran 19

20 every period. Use the free-entry condition to find the value of the wage {Wt tran } pertran t=1 consistent with free-entry in each period. Using the initial guess for {Yt tranit, Zt tranit, Ct tranit, Wt tranit t=1, the initial distriution } pertran of productivities and the policy functions we found in the previous step, iterate forward and solve the distriution of firms over productivities and the aggregate variales. Find updated values for {Y tran t, Z tran, C tran, W trainit t t t } pertran t=1. Make a new guess using a weighted average of our initial guess and the updated values for the aggregate variales. Make sure to give a lot of weight to the old guess low speedconv. Iterate until convergence. Remarks Choice of the numeraire: When computing the transition, we used the price of the final good P as the numeraire instead of using the price of the research good W r. This choice is made so that we can update the wage W directly from the free-entry condition as we iterate ackwards in step 2. In particular, when the value of a firm V is written in terms of the final good, the free-entry condition is: W r,t per n e per = 1 V 1 + r annual t z 0, t per where per is the numer of time periods in a year. As we iterate ackwards, we can use this 1 equation to find the value of P r,t per consistent with free entry, and use W r,t = W λ λ λ 1 λ 1 λ t to find Wt tran. Finding L p : To solve for L p, we use a weighted average of the value implied y our guess on ZZ tranit and our updated value of ZZ tran. This makes the algorithm less sensitive to deviations in the updated value of ZZ, making the convergence more stale. 3. Additional Appendi on Model with Assymetric countries Appendi 1: Deriving import shares and consumption-output ratios under risk sharing The value of production wages is W L p = ρ 1 τout 3.1 ρ 20

21 where Out denotes the value of output y intermediate good producers eclusive of the 1 susidy τ. To see this, note that for a firm with productivity 1, p = w, y = ρ 1 τ p ρ Y P ρ, 1 ρ ρ 1 ρ py = W τ ρ 1 Y P ρ, W l = W 1 ρ 1 ρ τ ρ 1 Y P ρ, so W l = py ρ 1 τ. Summing over home ρ and foreign sales and aggregating across producers we otain 3.1. Aggregate variale profits inclusive of the susidy are ρ τπ d Me Z d + Z + D 1 ρ M e Z u = τout W L p so W L p = ρ 1 τπ d Me Z d + Z + D 1 ρ M e Z u The value of research wages is W L r = λw r ΥM e, Dividing the last two two equations yields L L r = ρ 1 L r λ τπ d M e Z d + Z + D 1 ρ M e Z u W r ΥM e With free-entry and β = 1, τπ dm ez d +Z +D 1 ρ M ez u W rυm e = 1, and we otain 1.8. The value of final goods used in research activities, P X, is P X = 1 λ λ W L r = 1 λ λ λ ρ 1 W L p = = 1 λ λ ρ 1 λ ρ 1 ρ τout = 1 λ ρ τout The total value of final goods in the home country is The value of consumptions is where = 1 1 λ ρ τ. In the foreign country P Y = Out + Imports-Eports P C = P Y P X = Out + Imports-Eports- 1 λ ρ τout = Out + Imports-Eports P C = Out + Imports -Eports = Out + Eports-Imports where we used the fact that Eports =Imports, and Imports =Eports. 21

22 Recall that so P C = kp C Out + Eports-Imports = k Out + Imports-Eports or Out = kout + k + 1 Imports-Eports Recall that s =Eports/Out and s =Imports/Out. So, thus s = Imports Out = Imports kout + k+1 Imports-Eports = Imports/Out k + k+1 Imports/Out s Imports/Out = s 1 k+1 s k k + 1 s s m = Imports P Y = s 1 k+1 s 1 + s 1 k+1 s Imports = Out + Imports Eports = Imports/Out 1 + Imports/Out s k k+1 s s k k+1 s k k+1 s s = 1 s + s k k+1 By symmetry Finally s m = s k 1 k 1 +1 s 1 s + s k 1 k 1 +1 C Y = = Out + Imports-Eports Out + Imports-Eports = + Imports/Out s 1 + Imports/Out s + s 1 k k+1 s k+1 s s 1 + s k k+1 s s 1 k+1 s and similarly C Y = + Imports-Eports Out 1 + Imports -Eports Out = + s 1 k 1 +1 s k 1 k 1 +1 s s 1 + s 1 k 1 +1 s k 1 k 1 +1 s s 22

23 Appendi 2: Deriving epression 1.34 under risk sharing To save on notation, define W = log W/P, W = logw /P, Y = log Y, Y = log Y, C = log C, C = log C, M = log M e, M = log M e, U = log U, S = log W /W, d = log D, d = log D, z d = log Z d, z = log Z, z d = log Z d, z = log Z, c = C/Y, c = C /Y. Using this notation, we can re-write the system of log-linearized equations that is used to solve for the endogenous variales as 1 ρ λ W + Y = s U + ρ 1 s d ρ λ W + Y = s U + ρ 1 s d 3.3 Y + M + s U + s 1 ρd + 1 s n z d + s n z = ρw 3.4 Y + M s U + s 1 ρd + 1 s n zd + s nz = ρw 3.5 W = 1 ρ 1 1 s m M + s m M sn 1 s m z d + s n s z + s m z s m S s m d ρ 1 1 s 1 s W = 1 ρ 1 1 s m M + s mm+ 1 1 s 1 s n ρ 1 m z 1 s d + s n s z 1 s + s mz +s ms s md 3.7 Case λ = 1 Comining 3.2 and 3.4 we otain 3.6 Y = cc + 1 c W 3.8 Y = c C + 1 c W 3.9 U = Y Y σρ C C 3.10 S = σ C C + W W 3.11 and comining 3.3 and 3.5 we otain M = 1 s n z d s n z So, we can write 3.6 as W = M = 1 s n z d s nz 1 ρ 1 1 s m 1 s n z d + s n z s m 1 s n zd + s nz sn 1 s m z d + s n s z + s m z s m S s m d ρ 1 1 s 1 s 23

24 and comining terms W = 1 s m ρ 1 By symmetry W = 1 s m ρ 1 s 1 s 1 s n z d z + s m ρ 1 1 s n z z d s m S s m d 3.12 s 1 s 1 s n z d z + s m ρ 1 1 s n z z d + s ms s md 3.13 To simplify the notation further, we summarize the changes in the aggregate productivity indices y two variales, a and a : a = 1 ρ 1 1 s n z d z, a = 1 ρ 1 1 s n z d z 3.14 Then, 3.12 and 3.13 can e re-epressed as W = 1 s m s 1 s a s m a s m S s m d s W = 1 s m a s 1 s ma + s ms s md Sustituting the epression for W into 3.2 we otain Y = ρw s U + ρ 1 s d s = ρ 1 s m a s m a 1 s s U ρs m S + ρ 1 s d ρs m d and similarly sustituting the epression for W into 3.4 we otain Y = ρ 1 s s m a s 1 s ma + s U + ρs ms + ρ 1 s d ρs md With λ = 1, log C w = log Y w, so 1 + k log C w = 1 + k log Y w = Y + ky s = ρ 1 s m a s m a + k 1 s s 1 s m a ks 1 s ma + ks s U + ρ ks m s m S + ρ 1 s ρks m d + ρ 1 ks ρs m d Using the definitions of s m and s m, we can show k = 1 s m s s m 1 s and 1 k = 1 s m s s m 1 s 24

25 Then, = s 1 s m a s m a + k 1 s 1 s m a ks 1 s ma 1 sm s s m 1 s k s ma + k 1 s m s s m 1 s 1 s m a = 0 so a and a cancel-out from log C w. Therefore, Y + ky = ks s U + ρ ks m s m S + ρ 1 s ρks m d + ρ 1 ks ρs m d We now focus on the term ks s U + ρ ks m s m S. Using 3.2, 3.3, we otain: comined with 3.10 Y Y ρ W W + s + s U = ρ 1 s d s d ρ W W = Y Y 1 + s + s ρσ 1 + ρ 1 s d s d 3.15 Therefore, using 3.10, 3.11 and 3.15, ks s U + ρ ks m s m S = ks s Y Y 1 σρ + ρσ ks m s m Y Y + ρ ks m s m W W = ks s σρ 1 + ρσ ks m s m ks m s m ks m s m s + s ρσ 1 Y Y + + ks m s m ρ 1 s d s d = ks m s m ρ 1 s d s d In the final step, we used ks s σρ 1 + ρσ 1 ks m s m ks m s m s + s ρσ 1 = σρ 1 ks s + ks m s m 1 s s = σρ 1 ks s + k s k 1 k s s k k + 1 s 1 s s 1 s s = σρ 1 ks s + s s s k + 1 s k + s k + 1 s = σρ 1 ks s + s s k = 0 s 1 s s Therefore s m = s k k + 1 s 1 s s, s m = s k 1 k s 1 s s Y + ky = ρ 1 s ρks m d + ρ 1 ks ρs m d + ks m s m ρ 1 s d s d = ρ 1 s 1 + ks m s m ρks m d + ρ 1 s k + s m ks m ρs m d 25

26 which coincides with epression 1.34 when λ = 1. Case λ < 1 From 3.2 and 3.4 we otain M = 1 λ W 1 s L z d s L z, M = 1 λ W 1 s L z d s Lz Replacing these into 3.6 W = 1 λ ρ 1 1 s m W + s m W s + 1 s m a s m a s m S s m d 1 s where a and a are defined in Symmetrically in the foreign country W = W = W = 1 λ ρ 1 1 s m W + s mw + 1 s m Solving for W and W, we otain ρ λ 1 1 λ s ρ 1 2 ρ λ m + s m ρ λ 1 1 λ s ρ 1 2 ρ λ m + s m 1 1 λ s a s 1 s ma + s ms s md 1 ρ 1 s m 1 s m s 1 s a s m a s m S s m d + 1 λ s ρ 1 m 1 s m s a s 1 s ma + s ms s md 1 1 λ 1 s ρ 1 m 1 s m s a s 1 s ma + s ms s md 1 s m s 1 s a s m a s m S s m d + 1 λ ρ 1 s m In calculating W + kw, one can show that the sum of the terms involving a and a are equal to zero: 1 λ 1 ρ 1 1 s s m 1 s m a s m a + 1 s 1 λ k 1 ρ 1 1 s m 1 s s m a s 1 s ma + = 0 1 λ ρ 1 s m k 1 λ ρ 1 s m s 1 s m 1 s 1 s m a s ma s 1 s a s m a Therefore W + kw = = ρ λ 1 λ s ρ 1 2 ρ λ m + s m λ ρ 1 1 s m s m S + d + 1 λ s ρ 1 ms m S + d + k 1 1 λ 1 s ρ 1 m s m S + d + k1 λ ρ 1 s ms m S + d ρ 1 2 ρ λ 2 1 λ s 2 ρ λ m + s m 1 ρ 1 1 λ 1 s m s m S + d + 1 λ s m s m S + d + k ρ 1 1 λ 1 s m s m S + d + 1 λ s ms m +S + d 26

27 We now calculate 1 + k log C w : C + kc = Y + ky + 1 c C W + k 1 c C W = ρ + λ 1 W + kw s U + ρ 1 s d + ks U + = +k ρ 1 s d + 1 c C W + k 1 c C W ρ + λ 1 ρ 1 2 ρ λ 2 1 λ s 2 ρ λ m + s m 1 ρ 1 1 λ 1 s m s m S + d + 1 λ s m s m S + d + k ρ 1 1 λ 1 s m s m S + d + 1 λ s ms m +S + d s U + ρ 1 s d + ks U + k ρ 1 s d + 1 c C W + k 1 c C W so In the epression for C + kc, the terms with d and d are ρ + λ 1 ρ 1 2 ρ λ 1 λ s m + s m + λ + ρ ρ 1 1 λ 1 s m s m d + + ρ 1 1 λ 1 s m ks md + 1 λ s ms m d + kd We now focus on the remaining terms of C + kc, which when collected are given y: ρ + λ 1 ρ 1 2 ρ λ 1 λ s m + s m + λ + ρ {ρ 1 s m ks m + 1 λ k 1 s m s m 1 s m s m + 1 λ s ms m k 1} S s ks U + 1 c C W + k 1 c C W Using 3.2, 3.3 we otain Using : Y Y = ρ + λ 1 W W U s + s + ρ 1 s d s d W W = Y Y ρ + λ 1 + s + s 1 ρ U + ρ + λ 1 ρ + λ 1 s d s d U = Y Y σρ C C 3.18 = Y Y 1 σρ + σρ 1 c C W 1 c C W 27

28 From epression 3.11 S = σ C C + W W = σ Y Y + σ 1 c C W 1 c C W + + Y Y ρ + λ 1 s + s ρ 1 U + ρ + λ 1 ρ + λ 1 s d s d 1 = ρ + λ 1 σ Y Y s + s ρ 1 U + ρ + λ 1 ρ + λ 1 s d s d + +σ 1 c C W 1 c C W Sustituting 3.18 we otain 1 s + s S = 1 σρ ρ + λ 1 + σ s + s σρ ρ + λ 1 σ Y Y + ρ 1 ρ + λ 1 s d s d c C W 1 c C W Using 3.18, 3.19, one can show that 3.17 is equal to: ρ + λ 1 ρ 1 ρ 1 2 ρ λ 1 λ s m + s m + λ + ρ 2 ρ + λ {ρ 1 s m ks m + 1 λ k 1 s m s m 1 s m s m + 1 λ s ms m k 1} s d s d Summing up 3.16 and 3.20 and dividing y 1 + k results in log C w given in epression