The Global Diffusion of Ideas
The Global Diffusion of Ideas∗ Francisco J. Buera
Ezra Oberfield
Federal Reserve Bank of Chicago
Princeton
June 1, 2015 Preliminary and Incomplete
Abstract We provide a tractable theory of innovation and diffusion of technologies to explore the role of international trade and foreign direct investment (FDI). We model innovation and diffusion as a process involving the combination of new ideas with insights from other industries or countries. We provide conditions under which each country’s equilibrium frontier of knowledge converges to a Frechet distribution, and derive a system of differential equations describing the evolution of the scale parameters of these distributions, i.e., countries’ stocks of knowledge. In particular, the growth of a country’s stock of knowledge depends only on the its trade and FDI shares and the stocks of knowledge of its trading partners. We use this framework to quantify the dynamic gains from trade in the short and long run. We explore the model’s potential to account for cross-sectional TFP differences, long-run changes in TFP, and post-war growth miracles. ∗
1
Economic miracles are characterized by protracted growth in productivity, per-capita income, and increases in trade and FDI flows. The experiences of Japan and South Korea in the postwar period and the recent performance of China are prominent examples. These experiences suggest an important role played by openness in the process of development.1 Yet quantitative trade models relying on standard static mechanisms imply relatively small gains from openness, and therefore cannot account for growth miracles.2 These findings call for alternative channels through which openness can affect development. In this paper we present and analyze a model of an alternative mechanism: the impact of openness on the creation and diffusion of best practices across countries. We model innovation and diffusion as a process involving the combination of new ideas with insights from other industries and countries. Insights occur randomly due to local interactions among domestic producers. In our theory openness affects the creation and diffusion of ideas by determining the distribution from which domestic producers draw their insights. Our theory is flexible enough to incorporate different channels through which ideas may diffuse across countries. We focus on two main channels: (i) insights are drawn from those that sell goods to a country, (ii) insights are drawn from technologies used domestically, whether foreign or domestically owned. In our model, openness to trade and Foreign Direct Investment (FDI) affects the quality of the insights drawn by domestic producers by selecting different sellers to a country and/or affecting the technologies used to produce domestically. We use the model to explore several questions. First, we study how barriers to trade and FDI alter the learning process. At the micro level, the insights one draws depend on local interactions. At the aggregate level, the growth of a country’s stock of knowledge depends on its trade and FDI shares and the stocks of knowledge of its trading partners, and that of foreign firms operating domestically. Starting from autarky, opening up to trade and FDI results in a higher temporary growth rate, and permanent higher level, of the stock of knowledge, as the producers are exposed to more productive ideas. Nevertheless, the case of costless trade does not necessarily result in the highest short-run growth rate of the stock of knowledge. For example, a country can change the growth rate of its stock of knowledge by altering the composition of its trading partners. If 1 See Feyrer (2009a,b) for recent estimates of the impact of trade on income, and a review of the empirical literature. See also the discussion in Lucas (2009b). 2 See Connolly and Yi (2009) for a quantification of the role of trade on Korean’s growth miracle. Atkeson and Burstein (2010) also find relatively small effects in a model with innovation.
2
learning from sellers is important, a country could increase the growth rate of its stock of knowledge by tilting imports toward its higher-wage trade partners; imports from higher-wage countries tend to be produced at higher productivity, as the high wage must be overcome with a low unit labor requirement. In this case, the growth rate of the stock of knowledge is maximized when import costs perfectly offset the wage differences among its trading partners. However, this generically conflicts with maximizing the static gains from trade. We next use our model to quantify the dynamic gains from openness, studying in particular how opening to trade and FDI shape the diffusion of ideas. In a world that is generally open, if a single closed country opens to trade, it will experience an instantaneous jump in real income, a mechanism that has been well-studied in the trade literature. Following that jump, this country’s stock of knowledge will gradually improve as the liberalization leads to an improvement in the composition of insights drawn by its domestic producers. Here, the speed of convergence depends on the nature of learning process. If insights are drawn from goods that are sold to the country, then convergence will be faster, as opening to trade allows producers to draw insight from the relatively productive foreign producers. In contrast, if insights are drawn from technologies that are used locally, the country’s stock of knowledge grows more slowly. In that case, a trade liberalization leads to better selection of the domestic producers, but those domestic producers have low productivity relative to foreign firms. Opening to FDI also tends to lead to faster convergence than opening to trade. Opening to FDI provides more immediate access to the ideas of foreign producers. Over time, the exposure to more productive foreign multinationals producing domestically leads to faster growth of the stock of knowledge. In the case of opening to FDI, the stock of knowledge grows faster when learning is from technologies used locally. We also study whether trade and FDI are substitutes or complements in the diffusion of ideas. For both the static and dynamic gains from opening, this depends crucially on the correlation of multinationals’ productivities across potential production locations. When this correlation is high, trade and FDI are substitutes in increasing the speed of learning and raising real incomes. Finally, we specify a quantitative version of the model that includes non-traded goods and intermediate inputs, and equipped labor with capital and education. Using cross-country data on trade flows and GDP, we study whether the model can account for the post-war growth of South Korea. We first vary a parameter that modulates the strength of diffusion. We find that the role 3
of diffusion is largest for intermediate values of this parameter; When the Literature Review
Our work builds on a large literature modeling innovation and diffusion of
technologies as a stochastic process, starting from the earlier work of Jovanovic and Rob (1989), Jovanovic and MacDonald (1994), Kortum (1997), and recent contributions by Alvarez et al. (2008) and Luttmer (2012).3 We are particularly related to recent applications of these frameworks to study the connection between trade and the diffusion of ideas (Lucas, 2009a; Alvarez et al., 2013; Perla et al., 2013; Sampson, 2014). Our theory captures the models in Kortum (1997) and Alvarez et al. (2008, 2013) as special cases. When the contribution of insights to the development of new technologies is zero, β = 0 in our notation, our framework simplifies to a version of Kortum (1997) with exogenous search intensity. As in that paper, ours is a model with semi-endogenous growth. When insights from domestic sellers are the only input to the development of new technologies, β = 1, our framework simplifies to the model in Alvarez et al. (2008, 2013) with stochastic arrival of ideas. Beyond analyzing the intermediate cases, β ∈ (0, 1), the behavior of the model is qualitatively different from either of the two special cases β = 0 or β = 1. With β = 0, there is no diffusion of ideas and thus no dynamic gains from trade. With β = 1, changes in trade costs alter a country’s growth rate, and the equilibrium frontier of knowledge is closer to a logistic distribution. More importantly, when β = 1 and trade barriers are finite, changes in trade barriers have no impact on the tail of the distribution of productivity, and therefore, the model has a more limited success in providing a quantitative theory of the level and transitional dynamics of productivity. With β < 1, the frontier of knowledge converges to a Frechet distribution. This allows us use the machinery of Eaton and Kortum (2002), Bernard et al. (2003), and Alvarez and Lucas (2007) which have been remarkably successful as quantitative trade models. We therefore believe that studying the intermediate case of β ∈ (0, 1) is a step toward a quantitative model of the cross-country diffusion of ideas. Finally, we are able to nest alternative sources of insights, e.g., learning from those who sell goods to a country, learning from those that produce within a country, and study the role of both trade and FDI in determining the distribution of insights. Eaton and Kortum (1999) also build a model of the diffusion of ideas across countries in which 3
Lucas and Moll (2014) and Perla and Tonetti (2014) extends these models by studying the case with endogenous search effort, a dimension that we abstract from.
4
the distribution of productivities in each country is Frechet, and where the evolution of the scale parameter of the Frechet distribution in each country is governed by a system of differential equations. In their work insights are drawn from the distribution of potential producers in each country, according to exogenous diffusion rates which are estimated to be country-pair specific, although countries are assumed to be in autarky otherwise. Therefore, changes in trade and FDI costs do not affect the diffusion of ideas. The model shares some features with Oberfield (2013) which models the formation of supply chains and the economy’s input-output architecture. In that model, entrepreneurs discover methods of producing their goods using other entrepreneurs’ goods as inputs.4
1
Technology Diffusion with a General Source Distribution
We begin with a description of technology diffusion in a single country given a general source distribution. The source distribution describes the set of insights that domestic producers might access. In the specific examples that we explore later in the paper, the source distribution will depend on the profiles of productivity across all countries in the world, but in this section we take it to be a general function satisfying weak tail properties. Given the assumption on the source distribution, we show that the equilibrium distribution of productivity in a given economy is Frechet, and derive a differential equation describing the evolution of the scale parameter of this distribution. We consider an economy with a continuum of goods s ∈ [0, 1]. For each good, there are m producers. We will later study an environment in which the producers engage in Bertrand competition, so that (barring ties) at most one of these producers will actively produce. A producer is characterized by her productivity, q. A producer of good s with productivity q has access to a labor-only, linear technology y(s) = ql(s),
(1)
where l(s) is the labor input and y(s) is output of good s. The state of technology in the economy is described by the function Mt (q), the fraction of producers with knowledge no greater than q. We 4 Here, the evolution of the distribution of marginal costs depends on a differential equation summarizing the history of insights that were drawn. In Oberfield (2013), the distribution of marginal costs is the solution to a fixed point problem, as each producer’s marginal cost depends on her potential suppliers’ marginal costs.
5
call Mt the distribution of knowledge at t. The economy’s productivity depends on the frontier of knowledge. The frontier of knowledge is characterized by the function F˜t (q) ≡ Mt (q)m . F˜t (q) is the probability that none of the m producers of a good have productivity better than q. We now turn to a description of the dynamics of the distribution of knowledge. We model diffusion as a process involving the random interaction among producers of different goods or countries. We assume each producer draws insights from others stochastically at rate αt . However there is randomness in the adaptation of that insight. More formally, when an insight arrives to a producer with productivity q, the producer learns an idea with random productivity zq 0β and adopts the idea if zq 0β > q. The productivity of the idea has two components. There is an insight ˜ t (q 0 ). The second drawn from another producer, q 0 , which is drawn from the source distribution G component z is drawn from an exogenous distribution with CDF H(z). We refer to H(z) as the exogenous distribution of ideas.5 This process captures the fact that interactions with more productive individuals tend to lead to more useful insights, but it also allows for randomness in the adaptation of others’ techniques to alternative uses. The latter is captured by the random variable z. An alternative interpretation of the model is that z represents an innovator’s “original” random idea, which is combined with random insights obtained from other technologies.6 ˜ t (q 0 ), and the Given the distribution of knowledge at time t, Mt (q), the source distribution, G exogenous distribution of ideas, H(z), the distribution of knowledge at time t + ∆ is Z Mt+∆ (q) = Mt (q) (1 − αt ∆) + αt ∆
∞
˜ t (x) H q/xβ dG
0
The first term on the right hand side is the distribution of knowledge at time t, which gives the fraction of producers with productivity less than q. The second term is the probability that a ˜ t (q) and H(z) are exogenous. The distinction between these distriFrom the perspective of this section, both G butions will become clear once we consider specific examples of source distributions, in which the source distribution will be an endogenous function of the countries’ frontiers of knowledge. 6 If β = 0 our framework simplifies to a version of the model in Kortum (1997) with exogenous search intensity. The framework also nests the model of diffusion in Alvarez et al. (2008) with stochastic arrival of ideas if β = 1, H ˜ t = F˜t . is degenerate, and G 5
6
producer did not have an insight between time t and t + ∆ that raised her productivity above q. This can happen if no insight arrived in an interval of time ∆, an event with probability 1 − αt ∆, or if at least one insight arrived but none resulted in a technique with productivity greater than q, R∞ ˜ t (x). an event that occurs with probability 0 H q/xβ dG Rearranging and taking the limit as ∆ → 0 we obtain Mt+∆ (q) − Mt (q) d ln Mt (q) = lim = −αt ∆→0 dt ∆Mt (q)
Z
∞h
i ˜ t (x). 1 − H q/xβ dG
0
With this, we can derive an equation describing the frontier of knowledge. Since F˜t (q) = Mt (q)m , the change in the frontier of knowledge evolves as: d ln F˜t (q) = −mαt dt
Z
∞h
i ˜ t (x). 1 − H q/xβ dG
0
To gain tractability, we make the following assumption about the exogenous component of ideas.
Assumption 1 The exogenous distribution has a Pareto right tail with exponent θ, so that
lim
z→∞
1 − H(z) = 1. z −θ
We thus assume that the right tail of the exogenous distribution of ideas is regularly varying. The restriction that the limit is equal to 1 rather than some other positive number is without loss of generality; we can always choose units so that the limit is one. We also assume that the strength of diffusion, β is strictly less than one. Assumption 2 β ∈ [0, 1). ˜ t has a sufficiently For this section we make one additional assumption: the source distribution G thin tail.7 ˜ t (q)] = 0. Assumption 3 At each t, limq→∞ q βθ [1 − G 7
In later sections when we endogenize the source distribution, this assumption will be replaced by an analogous assumption on the right tail of the initial distribution of knowledge, limq→∞ q βθ [1 − M0 (q)] = 0 and the restriction that β < 1.
7
We will study economies where the number of producers for each good is large. As such, it will be convenient to study how the frontier of knowledge evolves when normalized by the number of 1 1 ˜ t m (1−β)θ q producers for each good. Define Ft (q) ≡ F˜t m (1−β)θ q and Gt (q) ≡ G Proposition 1 Suppose that Assumption 1, Assumption 2 and Assumption 3 hold. Then in the limit as m → ∞, the frontier of knowledge evolves as: d ln Ft (q) = −αt q −θ dt
Z
∞
xβθ dGt (x)
0
Motivated by the previous proposition, we define λt ≡
Rt
−∞ ατ
R∞ 0
xβθ dGτ (x)dτ . With this,
one can show that the economy’s frontier of knowledge converges asymptotically to a Frechet distribution. 1/θ
−θ
Corollary 2 Suppose that limt→∞ λt = ∞. Then limt→∞ Ft (λt q) = e−q . 1/θ
−θ
Proof. Solving the differential equation gives Ft (q) = F0 (q)e−(λt −λ0 )q . Evaluating this at λt q 1/θ
1/θ
−1 −θ q
gives Ft (λt q) = F0 (λt q)e−(λt −λ0 )λt e−q
1/θ
. This implies that, asymptotically, limt→∞ Ft (λt q) =
−θ
Thus, the distribution of productivities in this economy is asymptotically Frechet and the dynamics of the scale parameter is governed by the differential equation λ˙ t = αt
Z
∞
xβθ dGt (x).
(2)
0
We call λt the stock of knowledge. In the rest of the paper we analyze alternative models for the source distribution Gt . A simple example that illustrates basic features of more general cases is Gt (q) = Ft (q). This corresponds to the case in which diffusion opportunities are randomly drawn from the set of domestic best practices across all goods. In a closed economy this set equals the set of domestic producers and sellers. In this case equation (2) becomes λ˙ t = αt Γ(1 − β)λβt
where Γ(u) =
R∞ 0
xu−1 e−x dx is the Gamma function. Growth in the long-run is obtained in this 8
framework if the arrival rate of insight grows over time, αt = α0 eγt . In this case, the scale of the Frechet distribution λt grows asymptotically at the rate γ/(1 − β), and per-capita GDP grows at ˆ t = λt eγ/(1−β)t the rate γ/[(1−β)θ]. In general, the evolution of the de-trended stock of knowledge λ can be summarized in terms of the de-trended arrival of ideas α ˆ t = αt eγt ˆ˙ t = α ˆβ − λ ˆ t Γ(1 − β)λ t
γ ˆ λt , 1−β
and on a balanced growth path on which α ˆ is constant, the de-trended stock of ideas is 1 1−β α ˆ (1 − β) ˆ= λ Γ(1 − β) . γ
In the model that follows, potential producers engage in Bertrand competition. In that environment, an important object is the joint distribution of the productivities of best and second best producers of a good. We denote the CDF of this joint distribution as F˜t12 (q1 , q2 ), which, for q1 ≥ q2 , equals8 F˜t12 (q1 , q2 ) = Mt (q2 )m + m [Mt (q2 ) − Mt (q1 )] Mt (q2 )m−1 . Since the frontier of knowledge at t satisfies F˜t (q) = Mt (q)m , the joint distribution can be written as F˜t12 (q1 , q2 )
= 1+m
1/m F˜t (q1 )/F˜t (q2 ) − 1 F˜t (q2 ),
q1 ≥ q2 .
1 1 Normalizing this joint distribution by the number of producers, Ft12 (q1 , q2 ) ≡ F˜t12 m (1−β)θ q1 , m (1−β)θ q2 , we have that for large m,
Ft12 (q1 , q2 ) = [1 + log Ft (q1 ) − log Ft (q2 )] Ft (q2 ),
2
q1 ≥ q2 .
International Trade
We first consider a world where n economies interact through trade, and ideas diffuse through the contact of domestic producers with those who sell goods to the country as well as with those that 8 Intuitively, there are two ways the best and second best productivities can be no greater than q1 and q2 respectively. Either none of the productivities are greater than q2 , or one of the m draws is between q1 and q2 and none of the remaining m − 1 are greater than q2 .
9
produce within the country. Given the results from the previous section, the static trade theory is given by the standard Ricardian model in Eaton and Kortum (2002), Bernard et al. (2003), and Alvarez and Lucas (2007), which we briefly introduce before deriving the equations which characterize the evolution of the profile of the distribution of productivities of countries in the world economy. In each country, consumers have identical preferences over a continuum of goods. We use ci (s) to denote the consumption of a representative household in i of good s ∈ [0, 1]. Utility is given by u(Ci ), where the the consumption aggregate is Z Ci =
1
ci (s)
ε−1 ε
ε/(ε−1) ds
0
so goods enter symmetrically and exchangeably. We assume that ε − 1 < θ, which guarantees the price level is finite. Let pi (s) be the price of good s in i, so that i’s ideal price index is i 1 hR 1−ε 1 . Letting Xi denote i’s total expenditure, i’s consumption of good s is Pi = 0 pi (s)1−ε ds ci (s) =
pi (s)−ε Xi . Pi1−ε
In each country, individual goods can be manufactured by many producers, each using a laboronly, linear technology (1). As discussed in the previous section, provided countries share the same exogenous distribution of ideas H(z), the frontier of productivity in each country is described by a −θ
Frechet distribution with curvature θ and a country-specific scale λi , Fi (q) = e−λi q . Transportation costs are given by the standard “iceberg” assumption, where κij denotes the units that must be shipped from country j to deliver a unit of the good in country i, with κii = 1 and κij ≥ 1. We now briefly present the basic equations that summarize the static trade equilibrium given the vector of scale parameters λ = (λ1 , ..., λn ). Because the expressions for price indices, trade shares, and profit are identical to Bernard et al. (2003), we relegate the derivation of these expressions to Appendix B. Given the isoelastic demand, if a producer had no direct competitors, it would set a price with a markup of
ε ε−1
over marginal cost. Producers engage in Bertrand competition. This means that
lowest cost provider of a good to a country will either use this markup or, if necessary, set a limit price to just undercut the next-lowest-cost provider of the good. Let wi denote the wage in country i. For a producer with productivity q in country j, the cost of
10
providing one unit of the good in country i is
wj κij q .
The price of good s in country i is determined
as follows. Suppose that country j’s best and second best producers of good s have productivities qj1 (s) and qj2 (s). The country that can provide good s to i at the lowest cost is given by
arg min j
wj κij qj1 (s)
If the lowest-cost-provider of good s for i is a producer from country k, the price of good s in i is pi (s) = min
wj κij ε wk κik wk κik , , min ε − 1 qk1 (s) qk2 (s) j6=k qj1 (s)
That is, the price is either the monopolist’s price or else it equals the cost of the next-lowest-cost provider of the good; the latter is either the second best producer of good s in country k or the best producer in one of the other countries. In Appendix B, we show that, in equilibrium, i’s price index is
Pi = B
X
λj (wj κij )−θ
−1/θ
j
where B is a constant.9 Let Sij ⊆ [0, 1] be the set of goods for which a producer in j is the lowest-cost-provider for country i. Let πij denote the share of country i’s expenditure that is spent on goods from country R j so that πij = s∈Sij (pi (s)/Pi )1−ε ds. In Appendix B, we show that the expenditure share is λj (wj κij )−θ πij = Pn . −θ k=1 λk (wk κik ) A static equilibrium is given by a profile of wages w = (w1 , ..., wn ) such that labor market clears in all countries. The static equilibrium will depend on whether trade is balanced and where profit from producers is spent. For now, we take the each country’s expenditure as given and solve for the equilibrium as a function of these expenditures. Labor in j is used to produce goods for all destinations. To deliver one unit of good s ∈ Sij to 9
B
1−ε
=
1−
ε−1 θ
−θ −θ ε ε 1 − ε−1 + ε−1 Γ 1−
11
ε−1 θ
.
i, the producer in j uses κij /qj1 (s) units of labor. Thus the labor market clearing constraint for country j is XZ
Lj =
κij ci (s)ds. qj1 (s)
s∈Sij
i
Similarly, the total profit earned by producers in j can be written as
Πj =
wj κij ci (s)ds. pi (s) − qj1 (s) s∈Sij
XZ i
In Appendix B, we show that these can be expressed as
wj Lj =
θ X πij Xi θ+1 i
and
Πj =
1 X πij Xi θ+1 i
Under the natural assumption that trade is balanced and that all profit from domestic producers is spent domestically, then Xi = wi Li + Πi and the labor market clearing conditions can be expressed as X
wj Lj =
πij wi Li
i
As a simple benchmark, it is useful to consider the case with costless trade, κij = 1, all j, and countries of equal size Li = Lj , all j 6= i. In this case, relative wages are wiF T = wiF0 T
λi λ i0
1 1+θ
and the relative expenditure shares are FT πij FT πij 0
=
λj λj 0
1 1+θ
.
(3)
Given the static equilibria, we next solve for the evolution of the profile of scale parameters λ =
12
(λ1 , ..., λn ) by specializing (2) for alternative assumptions about source distributions. We consider source distributions that encompass two cases: (i) domestic producers learn from sellers to the country, (ii) domestic producers learn from other producers in the country.
2.1
Learning from Sellers
Following the framework introduced in Section 1, we model the evolution of technologies as the outcome of a process where technology managers combine “own ideas” with random insights from technologies in other sectors or countries. We first consider the case in which insights are drawn from sellers to the country. In particular, we assume that insights are randomly drawn from the distribution of sellers’ productivity in proportion to the expenditure on each good.10 In this case, the source distribution is given by the expenditure weighted distribution of productivity of sellers
Gi (q) = GSi (q) ≡
XZ s∈Sij |qj (s) 0 arbitrarily. Since limz→∞ 1−H(z) z −θ 1−H(z) z −θ
1−H(z) z −θ
1−H(z) z −θ
≤K
= 1, there is a z ∗ such that z > z ∗ implies
< 1 + δ. For z < z ∗ , we have that z θ [1 − H(z)] ≤ (z ∗ )θ [1 − H(z)] ≤ (z ∗ )θ . Thus for any z, n o ≤ K ≡ max 1 + δ, (z ∗ )θ
Claim 6 Suppose that Assumption 1 and Assumption 3 hold. Then in the limit as m → ∞, the frontier of knowledge evolves as: d ln Ft (q) = −αt q −θ dt
Z
∞
xβθ dGt (x)
0
1
1 − (1−β)θ
Proof. Evaluating the law of motion at m (1−β)θ q and using the change of variables w = m
x
we get 1 ∂ = −mαt ln F˜t m (1−β)θ q ∂t
Z
∞
0
Z
∞
= −mαt 0
1
"
m (1−β)θ q 1−H xβ
!#
1 (1−β)θ
˜ t (x) dG
1 q ˜ (1−β)θ m d G m w 1 − H t β 1 m (1−β)θ w
1 1 ˜ t m (1−β)θ q which have the From above, we have that Ft (q) ≡ F˜t m (1−β)θ q , and Gt ≡ G following derivatives 1 1 ˜ 0t m (1−β)θ q G0t (q) = m (1−β)θ G 1 1 Ft0 (q) = m (1−β)θ F˜t0 m (1−β)θ q 1 ˜t m (1−β)θ q ∂ F ∂Ft (q) = ∂t ∂t The equation becomes ∂ ln Ft (q) = −mαt ∂t
Z 0
∞
1 (1−β)θ
q m 1 − H β dGt (w) 1 m (1−β)θ w 46
This can be rearranged as ∂ ln Ft (q) = −αt q −θ ∂t
Z 0
∞
"
1 − H m1/θ qw−β −θ m1/θ qw−β
# wβθ dGt (w)
We want to take a limit as m → ∞. To do this, we must show that we can take the limit inside the integral. By Lemma 5, there is a K < ∞ such that for any z, 1−H(z) ≤ K. Further, given the z −θ R∞ ˜ t (w) is finite. Thus we can take the limit assumptions on the tail of Gt , the integral 0 Kwβθ dG inside the integral using the dominated convergence theorem to get ∂ ln Ft (q) = −αt q −θ ∂t
B
Z
∞
wβθ dGt (w)
0
Trade
B.1
Equilibrium
This section derives expressions for price indices, trade shares, and market clearing conditions that determine equilibrium wages. The total expenditure in i is Xi . Throughout this section, we h i −θ maintain that Fi12 (q1 , q2 ) = 1 + λi q1−θ − λi q2−θ e−λi q . For a variety s ∈ Sij (produced in j and exported to i) that is produced with producw κ
tivity q, the equilibrium price in i is pi (s) = jq ij , the expenditure on consumption in i is 1−ε 1−ε pi (s) pi (s) 1 X , consumption is Xi , and the labor used in j to produce variety s i Pi Pi pi (s) 1−ε κ /qj1 (s) pi (s) Xi for i is ijpi (s) Pi Define πij ≡
λj (wj κij )−θ P −θ . k λk (wk κik )
We will eventually show this is the share of i’s total expenditure
that is spent on goods from j. We begin with a lemma which will be useful in deriving a number of results. Lemma 7 Suppose τ1 and τ2 satisfy τ1 < 1 and τ1 + τ2 < 1. Then # τ2
"
Z qj1 (s)
τ1 θ
−τ2 θ
pi (s)
˜ 1 , τ2 ) ds = B(τ
s∈Sij
X k
47
λk (wk κik )
−θ
πij
λj πij
τ1
˜ 1 , τ2 ) ≡ where B(τ
1−
τ2 1−τ1
+
τ2 1−τ1
ε ε−1
−θ(1−τ1 )
Γ (1 − τ1 − τ2 )
Proof. We begin by defining the measure Fij to satisfy q2
Z Fij (q1 , q2 ) = 0
Y
Fk12
k6=j
wk κik x wk κik x , wj κij wj κij
Fj12 (dx, x)+
q1
Z
Y
Fk12
q2 k6=j
wk κik q2 wk κik q2 , wj κij wj κij
Fj12 (dx, q2 ) (16)
Fij (q1 , q2 ) is the fraction of varieties that i purchases from j with productivity no greater than q1 and second best provider of the good to i has marginal cost no smaller than
wj κij q2 .
There are
two terms in the sum. The first term integrates over goods where j’s lowest-cost producer has productivity no greater than q2 , and the second over goods where j’s lowest cost producer has productivity between q1 and q2 . The corresponding density
∂2 ∂q1 ∂q2 Fij (q1 , q2 )
will be useful because
it is the measure of firms in j with productivity q that are the lowest cost providers to i and for which the next-lowest-cost provider has marginal cost wj κij /q2 . We first show that i − 1 λ q−θ h j Fij (q1 , q2 ) = πij + λj q2−θ − q1−θ e πij 2
The first term of equation (16) can be written as Z
q2
0
Y
Fk12
k6=j
wk κik x wk κik x , wj κij wj κij
Fj12 (dx, x) =
q2 − P k6=j λk
Z
e
wk κik wj κij
−θ
x−θ
−θ
θλj x−θ−1 e−λj x dx
0
−θ
=
λj (wj κij ) e P −θ λ (w κ ) k k ik k
−
P
k λk
wk κik wj κij
−θ
q2−θ
λ
− π j q2−θ
= πij e
ij
The second term is Z
q1
Y
q2 k6=i
Fk12
wk κik q2 wk κik q2 , wj κij wj κij
Fj12 (dx, q2 )
−
P
= e
k6=j
λk
wk κik wj κij
−θ
q2−θ
Z
q1
q2 −
P
= e
k
λ
λk
− π j q2−θ
= e
48
ij
wk κik wj κij
−θ
q2−θ
h i λj q2−θ − q1−θ i
h λj q2−θ − q1−θ
−θ
θλj x−θ−1 e−λj q2 dx
Together, these give the expression for Fij , so the joint density is − 1 λ q−θ 1 ∂2 j Fij (q1 , q2 ) = θλj q1−θ−1 θλj q2−θ−1 e πij 2 ∂q1 ∂q2 πij We next turn to the integral a markup of
ε ε−1
R s∈Sij
qj1 (s)θτ1 pi (s)−θτ2 ds. Since the price of good s is set at either
over marginal cost or at the cost of the next lowest cost provider, this integral
equals ∞Z ∞
Z 0
q2 ∞Z ∞
Z = 0
q2
q1θτ1
wj κij ε wj κij , q2 ε − 1 q1
−θτ2
wj κij ε wj κij , q2 ε − 1 q1
−θτ2
min
q1θτ1 min
Using the change of variables x1 =
(wj κij )−θτ2 πij
R ∞ R x2
˜ 1 , τ2 ) ≡ Define B(τ
0
0
λj πij
τ1 +τ2 Z
qj1 (s)
∞ Z x2
0
1 x−τ 1
Z
λj −θ πij q1
min x2 ,
θτ1
−θτ2
pi (s)
0
ε ε−1
∂ 2 Fij (q1 , q2 ) dq1 dq2 ∂q1 ∂q2 − 1 λ q−θ 1 j θλj q1−θ−1 θλj q2−θ−1 e πij 2 dq1 dq2 πij λj −θ πij q2
and x2 =
( x1−τ1 min x2 , −τ2
θ
x1
, this becomes
ε ε−1
λj (wj κij )−θ P −θ , k λk (wk κik )
ds = B(τ1 , τ2 ) (wj κij )
we have (wj κij )−θτ2
49
λj πij
e−x2 dx1 dx2
x1
e−x2 dx1 dx2 , so that the integral is
−θτ2
s∈Sij
Using πij =
)−τ2
θ
τ 2
=
hP
πij
λj πij
τ1 +τ2
−θ k λk (wk κik )
iτ2
. Finally we com-
˜ 1 , τ2 ): plete the proof by providing an expression for B(τ
˜ (τ1 , τ2 ) = B
Z
∞ Z x2
0
Z
0
1 min x2 , x−τ 1
∞ Z x2
= (
0
Z
(
−θ ε x2 ε−1
)
1−τ1 ∞ x2
−
ε ε−1
−θ
1−
ε ε−1
)−τ2
θ
−θ(1−τ1 ) Z
e−x2 dx1 dx2
x1 Z
−θ
ε ( ε−1 )
∞Z
+ 0
x2
0
1 x−τ 1
x2 2 −x2 dx2 x−τ 2 e
∞
+ ε ε−1
ε ε−1
−θτ2 Z
ε ε−1
∞
Z 0
∞
)−τ2
θ
e−x2 dx1 dx2
x1 ε ε−1
0
−θ(1−τ1 )
1 −τ2 −x2 e dx2 + x1−τ 2 1 − τ1 1 − τ1 − τ2 0 −θ(1−τ1 ) −θ(1−τ1 ) ε ε 1 − ε−1 ε−1 = + Γ (2 − τ1 − τ2 ) 1 − τ1 1 − τ1 − τ2 ( −θ(1−τ1 ) ) τ2 τ2 ε 1− + Γ (1 − τ1 − τ2 ) = 1 − τ1 1 − τ1 ε − 1
=
(
1−τ1
1 − τ1
0
ε ε−1
1 −τ2 −x2 dx1 dx2 x−τ 1 x2 e
=
−θ
1−τ1 −τ2 x2
1 − τ1 − τ2
x21−τ1 −τ2 e−x2 dx2
where the final equality uses the fact that for any x, Γ(x + 1) = xΓ(x). We first use this lemma to provide expressions for the price index in i and the share of i’s expenditure on goods from j. Claim 8 The price index for i satisfies #− 1 1 "X θ 1−ε−1 ε − 1 ˜ 0, Pi = B λk (wk κik )−θ θ k
where B ≡
1−
ε−1 θ
−θ −θ ε ε 1 − ε−1 + ε−1 Γ 1−
ε−1 θ
. πij =
λ (w κ )−θ P j j ij −θ k λk (wk κik )
is the share
of i’s expenditure on goods from j. Proof. The price aggregate of goods provided to i by j is
R s∈Sij
pi (s)1−ε ds. Using Lemma 7, this
equals # ε−1 "X θ ε − 1 −θ 1−ε ˜ pi (s) ds = B 0, λk (wk κik ) πij θ s∈Sij
Z
k
50
e−x2 dx2
The price index for i therefore satisfies
Pi1−ε =
# ε−1 "X θ ε − 1 ˜ 0, λk (wk κik )−θ pi (s)1−ε ds = B θ s∈Sij
XZ j
k
and i’s expenditure share on goods from j is R s∈Sij
pi (s)1−ε ds Pi1−ε
= πij
We next turn to the market clearing conditions. Claim 9 Country j’s expenditure on labor is
θ θ+1
P
i πij Xi .
i Proof. i’s consumption of good s is pi (s)−ε PX1−ε . If j is the lowest-cost provider to i, then j’s i
κ
expenditure on labor per unit delivered is wj qj1ij(s) . The total expenditure on labor in j to produce R w κij i goods for i is then s∈Sij qj1j (s) pi (s)−ε PX1−ε ds. Using Lemma 7, the total expenditure on labor in j i
is thus XZ i
s∈Sij
Z X wj κij Xi Xi pi (s)−ε 1−ε ds = qj1 (s)−1 pi (s)−ε ds wj κij 1−ε qj1 (s) Pi P s∈Sij i i " #ε X − 1 θ X θ λj 1 ε X i −θ ˜ − , = B wj κij 1−ε λk (wk κik ) πij θ θ πij Pi i k
˜ −1, ε = The result follows from B θ θ
C
θ ˜ θ+1 B
0, ε−1 and θ
wj κij Pi1−ε
hP
−θ k λk (wk κik )
iε θ
λj πij
− 1 θ
˜ 0, ε−1 =B θ
Source Distributions
This appendix derives expressions for the source distributions under various specifications. We begin by describing learning from sellers.
51
−1
.
C.1
Learning from Sellers
Here we characterize the learning process when insights are drawn from sellers in proportion to the expenditure on each seller’s good. Consider a variety that can be produced in j at productivity q. Since the share of i’s expenditure on good s is (pi (s)/Pi )1−ε , the source distribution is
Gi (q) =
XZ j
(pi (s)/Pi )1−ε ds
{s∈Sij |qj1 (s)≤q}
The change in i’s stock of knowledge depends on Z
∞ βθ
q dGi (q) =
XZ
0
qj1 (s)βθ (pi (s)/Pi )1−ε ds
s∈Sij
j
Using Lemma 7, this is Z
∞
q βθ dGi (q) =
0
j
=
C.1.1
X ˜ B ˜ B
# ε−1 "X β θ λj ε − 1 −θ ˜ λk (wk κik ) πij 1−ε B β, θ πij Pi k β X β, ε−1 λj θ πij ε−1 πij 0, θ 1
(17)
j
Alternative Weights of Sellers
Here we explore two alternative processes by which individuals can learn from sellers. In the first case, individuals are equally likely to learn from all active sellers, independently of how much of the seller’s variety they consume. In the second case, insights are drawn from sellers in proportion to consumption of each sellers’ goods. In each case, the speed of learning is the same as our baseline (equation (17)) up to a constant. Learning from All Active Sellers Equally If producers are equally likely to learn from all active sellers, the source distribution is P R Gi (q) =
j {s∈Sij |qj1 (s)≤q} ds
P R
j s∈Sij
The change in i’s stock of knowledge depends on
R∞ 0
52
ds P R
q βθ dG
i (q)
=
j
s∈S
P Rij j
qj1 (s)βθ ds
s∈Sij
ds
. Using Lemma 7,
this is Z
β β X ˜ λj λj B(β, 0) X = Γ(1 − β) πij πij ˜ 0) πij πij B(0,
∞ βθ
q dGi (q) = 0
j
j
Learning from Sellers in Proportion to Consumption i . If producers learn in proportion to consumpi’s consumption of goods s is ci (s) = pi (s)−ε PX1−ε i
tion, then the source distribution is −ε Xi ds j {s∈Sij |qj1 (s)≤q} pi (s) P 1−ε
P R
i
Gi (q) =
−ε Xi ds j {s∈Sij } pi (s) Pi1−ε
P R
The change in i’s stock of knowledge depends on Z
P R
∞ βθ
q dGi (q) = 0
j s∈Sij
P R
qj1 (s)βθ pi (s)−ε ds
j s∈Sij
pi (s)−ε ds
Using Lemma 7, this is
Z
iε β hP P ˜ λj −θ θ ε λ (w κ ) π B β, ij πij k ik k k j θ iε hP P ˜ −θ θ ε πij k λk (wk κik ) j B 0, θ ˜ β, ε X B λj β θ π ij ˜ 0, ε πij B θ
∞
q βθ dGi (q) =
0
=
j
C.2
Learning from Producers
Here we characterize the learning process when insights are drawn from domestic producers in proportion to labor used in production. For each s ∈ Sij , the fraction of j’s labor used to produce the good is
1 κij Lj qj1 (s) ci (s)
i with ci (s) = pi (s)−ε PX1−ε . Summing over all destinations, the source i
distribution would then be
Gj (q) =
XZ i
s∈Sij |qj1 (s)≤q
53
1 κij Xi pi (s)−ε 1−ε ds Lj qj1 (s) Pi
The change in j’s stock of knowledge depends on ∞
Z
βθ
q dGj (q) = 0
XZ i
q βθ
s∈Sij
1 κij Xi pi (s)−ε 1−ε ds Lj qj1 (s) Pi
Using Lemma 7, this is Z
∞
0
#ε "X β− 1 θ X κij Xi θ λj 1 ε −θ βθ ˜ B β − , λ (w κ ) π q dGj (q) = ij k k ik Lj Pi1−ε θ θ πij j k
Using the expressions for Pi and πij from above, this becomes ∞
Z 0
C.2.1
β ˜ β − 1, ε B λj 1 X θ θ q dGj (q) = πij Xi ε−1 ˜ 0, wj Lj πij B θ βθ
j
Alternative Weights of Producers
Here we briefly describe the alternative learning process in which insights are equally likely to be dawn from all active domestic producers. We consider only the case in which trade costs satisfy the triangle inequality κjk < κji κik , ∀i, j, k such that i 6= j 6= k 6= i. We will show that, in this case, all producers that export also sell domestically. This greatly simplifies characterizing the learning process. Towards a contradiction, suppose there is a variety s such that i exports to j and k exports to wi κji qi (s)
i. This means that
≤
wk κjk qk (s)
and
wk κik qk (s)
≤
wi κii qi (s) .
Since κii = 1, these imply that κji κik ≤ κjk ,
a violation of the triangle inequality and thus a contradiction. R
In this case, the source distribution is Gi (q) =
s∈Sii |qi1 ≤q R s∈Sii
ds
ds
. The change in i’s stock of knowledge
depends on Z
R
∞ βθ
q dGi (q) = 0
s∈Sii
q βθ ds
R s∈Sii
ds
Using Lemma 7, this is
Z
∞ βθ
q dGi (q) = 0
˜ B(β, 0)πij
λj πij
˜ 0)πij B(0,
54
β = Γ(1 − β)πij
λj πij
β
D
Simple Examples
D.1
Symmetric Countries
If countries are symmetric, there are two possible values of πij :
πii = πij
=
1 1 + (n − 1)κ−θ κ−θ , 1 + (n − 1)κ−θ
i 6= j
Normalizing the wage to unity, the price level is − 1 1 θ P = Bλ− θ 1 + (n − 1)κ−θ
The de-trended scale parameter on a balance growth path is # 1 ε−1 −θ(1−β) 1−β Γ 1 − β − α 1 + (n − 1)κ θ ˆ λ(κ) = (1 − β) 1−β γ Γ 1 − ε−1 (1 + (n − 1)κ−θ ) θ "
Using this expression, per-capita income, yi = wi /Pi , is 1 1 θ ε − 1 − 1−ε 1 −θ y(κ) = Γ 1 − λ θ 1 + (n − 1)κ θ 1 # θ(1−β) 1 " 1 α Γ 1 − β − ε−1 ε − 1 − 1−ε −θ(1−β) θ(1−β) θ (1 − β) 1 + (n − 1)κ = Γ 1− θ γ Γ 1 − ε−1 θ The de-trended stock of knowledge and per-capita income relative to costless trade are ˆ λ(κ) ˆ λ(1) y(κ) y(1)
" =
1 + (n − 1)κ−θ(1−β)
#
1−β
1 1−β
n
β − 1−β
(1 + (n − 1)κ−θ ) 1 1 1 + (n − 1)κ−θ θ λ(κ) θ = n λ(1)
In particular, per-capita income in autarky relative to the case with costless trade β 1 y(∞) − θ(1−β) = |{z} n− θ n | {z } y(1)
static
55
dynamic
D.2
A Small Open Economy
Consider a small open economy. The economy is small in the sense that actions in the economy have no impact on other countries’ expenditures, price levels, wages, or stocks of knowledge.
D.2.1
Steady State Gains from Trade
D.2.2
Speed of Convergence
We use the notation x ˇ to denote log-deviation from of x from its steady state (or BGP) value. To derive the speed of convergence, we want expressions for how the trade shares and wages change over time. The trade shares and market clearing condition for i are
πij rji wi Li
λj (wj κij )−θ Pn −θ k=1 λk (wk κik ) Xj πji = wj Lj X = Xj πji
=
j
For a small open economy, we have ˇ i − θw π ˇii = (1 − πii ) λ ˇi ˇ i − θw j= 6 i : π ˇij = −πii λ ˇi ˇ i − θw i 6= j : π ˇji = λ ˇi ˇ i − θw j 6= i : rˇji = π ˇji − w ˇi = λ ˇi − w ˇi
The change in the wage can be found from linearizing the labor market clearing condition: X ˇ i − θw ˇ i − θw ˇi w ˇi = rii w ˇi + (1 − πii ) λ ˇi + rji λ j6=i
w ˇi (1 − πii ) w ˇi w ˇi
ˇ i − θw ˇ i − θw = πii w ˇi − πii λ ˇi + λ ˇi ˇ i − θw = (1 − πii )2 λ ˇi ˇ i − θw = (1 + πii ) λ ˇi
56
This last equation can be expressed in two ways: ˇ i − θw λ ˇi =
w ˇi =
ˇi λ 1 + θ(1 + πii )
(1 + πii ) ˇ λi 1 + θ(1 + πii )
Plugging these back into the shares, we have (1 − πii ) ˇ λi 1 + θ (1 + πii ) πii ˇi j 6= i : π ˇij = − λ 1 + θ(1 + πii ) 1 ˇi i 6= j : π ˇji = λ 1 + θ (1 + πii ) 1 (1 + πii ) ˇ −πii ˇi j= 6 i : rˇji = π ˇji − w ˇi = − λi = λ 1 + θ (1 + πii ) 1 + θ (1 + πii ) 1 + θ (1 + πii ) π ˇii =
ˇi. We now proceed to characterizing transition dynamics for the stock of knowledge, λ 1−β ˆ β πij λj P 1−β ˆ β . π k ik λk
Learning from Sellers Let ΩSij ≡
The change in the the deviation of i’s stock of
knowledge from the BGP is ˇi ∂λ ∂t
=
ˆi 1 ∂λ BS α ˆ i X 1−β ˆ β γ = πij λj − ˆ i ∂t ˆi 1−β λ λ j
Log-linearizing around the steady state gives ˇi ∂λ ∂t
≈
= =
BS α ˆ i X 1−β ˆ β ˇj − λ ˇi πij λj (1 − β)ˇ πij + β λ ˆi λ j P 1−β ˆ β ˇj − λ ˇi πij + β λ γ j πij λj (1 − β)ˇ P 1−β ˆ β 1−β j πij λj γ X S ˇj − λ ˇi Ωij (1 − β) π ˇij + β λ 1−β j
ˇi 1 − β ∂λ γ ∂t
=
X
ˇj − λ ˇi ΩSij (1 − β) π ˇij + β λ
j
57
ˇ j = 0, π For a small open economy, we have λ ˇii =
ˇi (1−πii )λ 1+θ(1+πii ) ,
and π ˇij =
ˇi −πii λ 1+θ(1+πii )
for j 6= i. The
law of motion can be written as ˇi 1 − β ∂λ γ ∂t
X ˇ i + (1 − β) ˇi = ΩSii (1 − β) π ˇii + β λ ΩSij π ˇij − λ j6=i
= = = = ˇi ∂λ ∂t
=
X ˇi ˇi (1 − πii ) λ −πii λ ˇ i + (1 − β) ˇi (1 − β) ΩSij + βλ −λ 1 + θ (1 + πii ) 1 + θ (1 + πii ) j6=i −πii (1 − π ) ii S S ˇi + β + (1 − β) (1 − Ωii ) −1 λ Ωii (1 − β) 1 + θ (1 + πii ) 1 + θ (1 + πii ) ΩSii − πii S ˇ − βΩii λi − 1 − (1 − β) 1 + θ (1 + πii ) ΩSii − πii S ˇi + β 1 − Ωii λ − (1 − β) − (1 − β) 1 + θ (1 + πii ) β ΩSii − πii S ˇi −γ 1 − + 1 − Ωii λ 1 + θ (1 + πii ) 1 − β
ΩSii
Finally, we can use this to get at the speed of convergence for real income: 1 1 ˇ (1 − πii ) ˇ i = Aλ ˇi 1− λ λi − π ˇii = θ θ 1 + θ (1 + πii ) ˇi dλ ΩSii − πii β d ˇi w ˇi − Pˇi = A = −γ 1 − + 1 − ΩSii Aλ dt dt 1 + θ (1 + πii ) 1 − β ΩSii − πii β S = −γ 1 − + 1 − Ωii w ˇi − Pˇi 1 + θ (1 + πii ) 1 − β w ˇi − Pˇi =
Learning from Producers Let ΩPij ≡
ˆ i /πji )β rji (λ P β. ˆ k rki (λi /πki )
The change in the the deviation of i’s
stock of knowledge from the BGP is ˇi ∂λ ∂t
=
β ˆi 1 ∂λ BP α ˆi X ˆ γ = rji λi /πji − ˆ ˆ 1−β λi ∂t λi j
58
Log-linearizing around the steady state gives ˇi ∂λ ∂t
≈ =
β BP α ˆi X ˆ ˇi rˇji − β π ˇij − (1 − β)λ rji λi /πji ˆi λ j γ X P ˇi Ωij rˇji − β π ˇij − (1 − β) λ 1−β j
ˇi 1 − β ∂λ γ ∂t
=
X
ˇi ΩPij [ˇ rji − β π ˇij ] − (1 − β) λ
j
Using the expressions for π ˇii , π ˇji and rˇji , along with πii = rii , the law of motion can be written as ˇi 1 − β ∂λ γ ∂t
= ΩPii [ˇ rii − β π ˇii ] +
= =
E
ˇi ΩPij [ˇ rji − β π ˇij ] − (1 − β) λ
j6=i
=
ˇi ∂λ ∂t
X
=
X −πii 1 (1 − πii ) ˇ P ˇ ˇ ˇi λi + Ωij λi − β λi − (1 − β) λ (1 − β) 1 + θ (1 + πii ) 1 + θ (1 + πii ) 1 + θ (1 + πii ) j6=i ( ) P Ωii (1 − β) (1 − πii ) + 1 − ΩPii (−πii − β) ˇi − (1 − β) λ 1 + θ (1 + πii ) ) ( ΩPii (1 − β) (1 − πii ) − πii 1 − ΩPii (1 − β) − β 1 − ΩPii (1 + πii ) ˇi − (1 − β) λ 1 + θ (1 + πii ) ( ) 1 − ΩPii (1 + πii ) ˇ ΩPii − πii β −γ 1 − + λi 1 + θ (1 + πii ) 1 − β 1 + θ (1 + πii )
ΩPii
Research
This section proves the following claim: Claim 10 If Πiτ is total flow of profit earned by entrepreneurs in i at time τ , then the flow of profit earned in i at time τ from ideas generated between t and t0 (with t < t0 < τ ) is: λit0 − λit Πiτ λiτ 0
(t,t ] Proof. For v1 ≤ v2 , let V˜jiτ (v1 , v2 ) be the probability that at time τ , the lowest cost technique
to provide a variety to j was discovered by a manager in i between times t and t0 , that the marginal cost of that lowest-cost technique is no lower than v1 , and that marginal cost of the (t,t0 ]
next-lowest-cost supplier is not lower than v2 . Just as in Section ??, we will define Vjiτ (v1 , v2 ) = − 1 − 1 (t,t0 ] limm→∞ V˜jiτ m (1−β)θ v1 , m (1−β)θ v2 . 59
(t,t0 ]
be profit from all techniques drawn in i between t and t0 . Thus total profit in i at n o−ε (−∞,τ ] ε τ is Πiτ . Defining p (v1 , v2 ) ≡ min v2 , ε−1 v1 , we can compute each of these by summing Let Πiτ
over profit form sales to each destiantion j: (t,t0 ]
Πiτ
XZ
=
j (−∞,τ ]
Πiτ
0
XZ
=
j
∞Z ∞ v1 ∞Z ∞
0
v1
(t,t0 ]
[p (v1 , v2 ) − v1 ] p (v1 , v2 )−ε Pjε−1 Xj Vjiτ (dv1 , dv2 ) (−∞,τ ]
[p (v1 , v2 ) − v1 ] p (v1 , v2 )−ε Pjε−1 Xj Vjiτ
(t,t0 ]
We will show below that Vjiτ (v1 , v2 ) =
λit0 −λit (−∞,τ ] (v1 , v2 ). λiτ Vjiτ
(t,t0 ]
Πiτ
=
(dv1 , dv2 )
It will follow immediately that
λit0 − λit (−∞,τ ] Πiτ λiτ
(t,t0 ]
(t,t0 ]
We now compute Vjiτ (v1 , v2 ). For each of the m managers in i, let Mi
(q) be the probability
that the no technique drawn between t and t0 delivers productivity better than q. Similarly, define 0
0
(t,t ] (t,t ] F˜i (q) ≡ Mi (q)m be the probability that none of the m managers drew a technique with 1 1 (t,t0 ] (t,t0 ] productivity better than q between t and t0 . Finally, let Fi (q) = limm→∞ F˜i m 1−β θ q .
We have
(t,t0 ]
V˜jiτ
i 0 0 ˜ (−∞,τ ] wk κjk M (−∞,τ ] wi κji m−1 M (−∞,t] wi κji M (t ,τ ] wi κji dM (t,t ] wi κji F k6=i k i i i i x x x x x v2 (v1 , v2 ) = i m−1 (t0 ,τ ] wi κji R v2 hQ 0] w κ w κ w κ w κ (−∞,τ ] (−∞,t] (t,t (−∞,τ ] i ji i i ji k jk ˜ +m Mi Mi Mi dMi k6=i Fk v2 v2 x x x v1 m
R ∞ hQ
0
(t,t ] The expression for V˜jiτ (v1 , v2 ) contains two terms.
The first represents the probability that
the best technique to serve j delivers marginal cost greater than v2 and was drawn by a man(t,t0 ] wi κji ager in i between t and t0 . For each of the m managers in i, dMi measures the x likelihood that the managers best draw between t and t0 delivered marginal cost x ∈ [v2 , ∞], hQ (t0 ,τ ] wi κji (−∞,t] wi κji ˜ (−∞,τ ] M is the probabiltiy that the manager had no better draws, and Mi k6=i Fk i x x is the probability that no other manager from any destination would be able to provide the good at marginal cost lower than x. The second terms represents the proability that the best technique to serve j delivers marginal cost between v1 and v2 and was drawn by a manager in i between t and t0 , and that no othe manager can delvier the variety with marginal cost lower than v2 .
60
wk κjk x
i
(−∞,t]
Using Mi (t,t0 ]
Mi
(t0 ,τ ]
(q) Mi
(t,t0 ]
(q) Mi
(−∞,τ ]
(q) = F˜i
(−∞,τ ]
(q), F˜i
(−∞,τ ]
(q) = Mi
(t,t0 ]
(q)m and F˜i
(q) =
(q)m , this can be written as
0
(t,t ] V˜jiτ (v1 , v2 ) =
R ∞ hQ v2
˜ (−∞,τ ] k Fk
wk κjk x
1 − 1−β
1 θ
1 − 1−β
v1 and m
(t,t0 ]
Vjiτ (v1 , v2 ) =
1 θ
i
x (t,t0 ] wi κji F˜i x 1 (t,t0 ] wi κji m (−∞,τ ] wi κji F˜i dF˜i x x (−∞,τ ] wi κji (t,t0 ] wi κji F˜i F˜i v2 x
R ∞ hQ v2
(−∞,τ ] k Fk
wk κjk x
(t,t0 ]
Finally, following the logic of Section ??, we have Fi (t,t0 ]
(t,t0 ]
Fi
wi κji x wi κji x
v2 and taking a limit as m → ∞ gives i dF (t,t0 ] wi κji i
x (t,t0 ] wi κji Fi x (t,t0 ] wi κji dFi x (t,t0 ] wi κji Fi x
R v2 hQ (−∞,τ ] wk κjk i + k Fk v2 v1
dFi
!
R v2 hQ (−∞,τ ] wk κjk i ˜ + k Fk v2 v1
Evaluating this at m
i dF˜ (t,t0 ] wi κji
−θ
(q) = e−(λit0 −λit )q , so that (−∞,τ ]
λit0 − λit dFi = (−∞,τ ] λiτ F i
wi κji x wi κji x
We thus have (t,t0 ]
Vjiτ (v1 , v2 ) =
λit0 − λit (−∞,τ ] Vjiτ (v1 , v2 ) λiτ
which completes the proof.
F
Multinationals
This section derives expressions for price indices, trade shares, and market clearing conditions. Across multinationals, let vi1 (s) and v2 (s) be the lowest and second lowest marginal costs of supplying good s to i. Then the price of good s in i is pi (s) = min
ε vi1 (s) , vi2 (s) ε−1
61
Define X
ϕi =
λk˜
˜ k
1 λk ϕi
πijk =
n X
!1−ρ wl κil δlk˜
−θ/[1−ρ]
l=1 n X
!1−ρ (wl κil δlk )
(wj κij δjk )−θ/[1−ρ]
−θ/[1−ρ]
−θ/[1−ρ] l=1 (wl κil δlk )
Pn
l=1
As with, Lemma 7, we begin with a lemma which will be useful intermediate step. Lemma 11 Suppose τ1 and τ2 satisfy τ1 < 1 and τ1 + τ2 < 1. Then Z s∈Sijk
˜ 1 , τ2 )πijk ϕτ1 +τ2 (wj κij δjk )τ1 θ qjk1 (s)τ1 θ pi (s)−τ2 θ ds = B(τ i
˜ 1 , τ2 ) is defined as in Lemma 7. where B(τ Proof. For v1 ≤ v2 , define V˜ijk (v1 , v2 ) to be the fraction of goods for which the lowest cost source for i is a multinational based in k producing in j, and for which that multinational is unable to supply the good at cost lower than v1 , and for which no other multinational can supply the good at cost lower than v2 . We can then express V˜ijk (v1 , v2 ) using F˜kj and Mkj to represent the derivatives of F˜k and Mk with respect the jth argument: V˜ijk (v1 , v2 ) = m
Z
∞
wl κil δlk Mkj x2
v2
Z
v2
+m v1
wl κil δlk Mkj x2
wl κil δlk x
Mk
l
wl κil δlk x
wl κil δlk x
Mk
l
m−1 Y l
wl κil δlk v2
F˜k˜
˜=k k6 m−1 Y
l
˜ =k k6
F˜k˜
wl κil δlk˜ x
dx l
wl κil δlk˜ v2
dx l
The two terms divide V˜ijk (v1 , v2 ) into instances where the lowest cost provider has cost less than v2 and between v1 and v2 respectively. To interpret the first term, note that for each of the m n o wl κil δlk multinationals based in k, the term wl κxil2 δlk Mkj is the probability that that particular x l
multinational can provide the good to i by producing in j at cost x(> v2 ) and cannot provide the n o m−1 Q n o wl κil δlk˜ wl κil δlk ˜ good at a lower cost by producing elsewhere, while Mk F is ˜ =k k ˜ k6 x x l
l
the probability that none of the other multinationals (the other m − 1 based in k or any based in
62
another country) can provide the good to i at cost lower than x. The second term is similar, except since x ∈ [v1 , v2 ], it uses the probability that none of the other multinationals can provide the good to i at cost lower than v2 . This can be written more concisely as V˜ijk (v1 , v2 ) = m
Z
∞
v1
wl κil δlk Mkj x2
wl κil δlk x
Mk
l
wl κil δlk max {x, v2 }
m−1 Y l
F˜k˜
˜ =k k6
wl κil δlk˜ max {x, v2 }
1
Note that since F˜k (q) m = Mk (q), we can differentiate each with respect to the jth argument to 1 F˜kj (q) m−1 m ˜ Fk (q) m
get Mkj (q) =
. We can therefore express V˜ijk as
∞
Z
V˜ijk (v1 , v2 ) =
F˜k
v1
n
wl κil δlk ˜ Fkj x2
n
wl κil δlk x
wl κil δlk x
o Q l
o 1−1/m
˜˜ ˜ Fk k n
F˜k
l
n
wl κil δlk˜ max{x,v2 }
wl κil δlk max{x,v2 }
o l
o 1/m dx l
− 1 θ − 1 θ Define Vijk (v1 , v2 ) ≡ V˜ijk m 1−β v1 , m 1−β v2 . As m grows large, this becomes
Z
∞
Vijk (v1 , v2 ) =
wl κil δlk Fkj x2
Fk
v1
With functional forms, Fk (q) = e
−λk
n
n
wl κil δlk x
wl κil δlk x
o
Y
l
Fk˜
o k
l
−θ/[1−ρ] 1−ρ n l=1 ql
P
wl κil δlk˜ max {x, v2 }
dx l
, the second term in the integrand can be
expressed as
Y
Fk˜
˜ k
We also have
wl κil δlk˜ max {x, v2 }
Fkj (q) Fk (q)
= λk θ
=
Y
l
e
−λk˜
Pn
l=1
wl κil δ ˜ lk max{x,v2 }
−θ/[1−ρ] !1−ρ
= e−ϕi max{x,v2 }
θ
˜ k
−θ/[1−ρ] 1−ρ −θ/[1−ρ]−1 n q qj , l=1 l
P
so that the first term of the integrand
is wl κil δlk Fkj x2
Fk
n
n
wl κil δlk x
wl κil δlk x
o
o l
=
n X wl κil δlk wl κil δlk −θ/[1−ρ] λk θ x2 x
!1−ρ−1
l=1
l θ−1
= θx
λk
n X
wj κij δjk x
!1−ρ−1 (wl κil δlk )
l=1
= ϕi πijk θxθ−1
63
−θ/[1−ρ]
(wj κij δjk )
θ − 1−ρ
−
θ −1 1−ρ
dx l
The measure Vijk can therefore be expressed as ∞
Z Vijk (v1 , v2 ) =
θ
e−ϕi max{x,v2 } ϕi πijk θxθ−1 dx
v1
with density d2 Vijk (v1 , v2 ) θ = πijk ϕi θv1θ−1 ϕθv2θ−1 e−ϕi v2 dv1 dv2 With this density, we can derive the desired expression for the integral. Z
τ1 θ
qjk1 (s)
−τ2 θ
pi (s)
Z
∞ Z v2
wj κij δjk v1
τ1 θ
min v2 ,
ε v1 ε−1
−τ2 θ
wj κij δjk v1
τ1 θ
min v2 ,
ε v1 ε−1
−τ2 θ
ds = 0
s∈Sijk
Z
0 ∞ Z v2
= 0
0
d2 Vijk (v1 , v2 ) dv1 dv2 dv1 dv2 θ
πijk ϕi θv1θ−1 ϕθv2θ−1 e−ϕi v2 dv1 dv2
Using the change of variables x2 = ϕi v2θ and x1 = ϕi v1θ Z qjk1 (s)
τ1 θ
−τ2 θ
pi (s)
τ1 θ
ds = πijk (wj κij δjk )
s∈Sijk
ϕiτ1 +τ2
Z 0
∞ Z x2 0
( x1−τ1 min x2 ,
ε ε−1
)−τ2
θ x1
˜ (τ1 , τ2 ) πijk ϕτ1 +τ2 (wj κij δjk )τ1 θ = B i
With this lemma in hand we can derive expressions for the trade share and price index. The share of i’s expenditure on goods produced in j by multinationals based in k is ε−1 ˜ 0, ε−1 πijk ϕ θ B i θ =P P R ε−1 = πijk 1−ε ε−1 p (s) ds ˜ ˜ i ˜ j,k Si˜j k˜ πi˜j k˜ ϕi θ ˜ B 0, θ ˜ j,k
R
Sijk
pi (s)1−ε ds
and the price level satisfies
Pi1−ε =
XZ ˜ ˜ j,k
Si˜j k˜
pi (s)1−ε ds =
X ˜ ˜ j,k
ε−1 ε−1 ˜ 0, ε − 1 π ˜˜ ϕ θ = B ˜ 0, ε − 1 ϕ θ B i ij k i θ θ
64
e−x2 dx1 dx2
The labor market clearing condition is
Lj wj Lj
XZ
κij δjk 1 X = ci (s) ds = wj κij δjk qjk1 (s) wj i,k s∈Sijk i,k Z X 1 Xi pi (s)−ε ds = wj κij δjk 1−ε Pi s∈Sijk qjk1 (s)
Z s∈Sijk
1 qjk1 (s)
pi (s)−ε
Xi ds Pi1−ε
i,k
Using Lemma 11, this is
wj L j
=
= =
1 ε ˜ − 1 , ε πijk ϕ− θ + θ (wj κij δjk )−1 B i θ θ wj κij δjk Xi ε−1 ˜ 0, ε−1 ϕ θ B i,k i θ 1 ε X ˜ B −θ, θ Xi πijk ˜ 0, ε−1 B θ i,k θ X Xi πijk θ+1
X
i,k
Source Distributions
Finally, we derive expressions for the source distributions. Before doing
that, it is useful to note the following relationship !ρ 1−ρ πijk
X l
πilk
!1−ρ
n X
−θ/[1−ρ]
1−ρ
(wj κij δjk ) 1 = λk (wl κil δlk )−θ/[1−ρ] Pn −θ/[1−ρ] ϕi (w κ δ ) l il lk l=1 l=1 ρ ! 1−ρ n X 1 −θ/[1−ρ] (wl κil δlk ) × λk ϕi l=1
=
λk (wj κij δjk )−θ ϕi
We first study learning from sellers, where learning is in proportion to expenditure. The source h i1−ε P R distribution is GSi (q) = j,k {s∈Sijk |qjk1
Ezra Oberfield
Federal Reserve Bank of Chicago
Princeton
June 1, 2015 Preliminary and Incomplete
Abstract We provide a tractable theory of innovation and diffusion of technologies to explore the role of international trade and foreign direct investment (FDI). We model innovation and diffusion as a process involving the combination of new ideas with insights from other industries or countries. We provide conditions under which each country’s equilibrium frontier of knowledge converges to a Frechet distribution, and derive a system of differential equations describing the evolution of the scale parameters of these distributions, i.e., countries’ stocks of knowledge. In particular, the growth of a country’s stock of knowledge depends only on the its trade and FDI shares and the stocks of knowledge of its trading partners. We use this framework to quantify the dynamic gains from trade in the short and long run. We explore the model’s potential to account for cross-sectional TFP differences, long-run changes in TFP, and post-war growth miracles. ∗
1
Economic miracles are characterized by protracted growth in productivity, per-capita income, and increases in trade and FDI flows. The experiences of Japan and South Korea in the postwar period and the recent performance of China are prominent examples. These experiences suggest an important role played by openness in the process of development.1 Yet quantitative trade models relying on standard static mechanisms imply relatively small gains from openness, and therefore cannot account for growth miracles.2 These findings call for alternative channels through which openness can affect development. In this paper we present and analyze a model of an alternative mechanism: the impact of openness on the creation and diffusion of best practices across countries. We model innovation and diffusion as a process involving the combination of new ideas with insights from other industries and countries. Insights occur randomly due to local interactions among domestic producers. In our theory openness affects the creation and diffusion of ideas by determining the distribution from which domestic producers draw their insights. Our theory is flexible enough to incorporate different channels through which ideas may diffuse across countries. We focus on two main channels: (i) insights are drawn from those that sell goods to a country, (ii) insights are drawn from technologies used domestically, whether foreign or domestically owned. In our model, openness to trade and Foreign Direct Investment (FDI) affects the quality of the insights drawn by domestic producers by selecting different sellers to a country and/or affecting the technologies used to produce domestically. We use the model to explore several questions. First, we study how barriers to trade and FDI alter the learning process. At the micro level, the insights one draws depend on local interactions. At the aggregate level, the growth of a country’s stock of knowledge depends on its trade and FDI shares and the stocks of knowledge of its trading partners, and that of foreign firms operating domestically. Starting from autarky, opening up to trade and FDI results in a higher temporary growth rate, and permanent higher level, of the stock of knowledge, as the producers are exposed to more productive ideas. Nevertheless, the case of costless trade does not necessarily result in the highest short-run growth rate of the stock of knowledge. For example, a country can change the growth rate of its stock of knowledge by altering the composition of its trading partners. If 1 See Feyrer (2009a,b) for recent estimates of the impact of trade on income, and a review of the empirical literature. See also the discussion in Lucas (2009b). 2 See Connolly and Yi (2009) for a quantification of the role of trade on Korean’s growth miracle. Atkeson and Burstein (2010) also find relatively small effects in a model with innovation.
2
learning from sellers is important, a country could increase the growth rate of its stock of knowledge by tilting imports toward its higher-wage trade partners; imports from higher-wage countries tend to be produced at higher productivity, as the high wage must be overcome with a low unit labor requirement. In this case, the growth rate of the stock of knowledge is maximized when import costs perfectly offset the wage differences among its trading partners. However, this generically conflicts with maximizing the static gains from trade. We next use our model to quantify the dynamic gains from openness, studying in particular how opening to trade and FDI shape the diffusion of ideas. In a world that is generally open, if a single closed country opens to trade, it will experience an instantaneous jump in real income, a mechanism that has been well-studied in the trade literature. Following that jump, this country’s stock of knowledge will gradually improve as the liberalization leads to an improvement in the composition of insights drawn by its domestic producers. Here, the speed of convergence depends on the nature of learning process. If insights are drawn from goods that are sold to the country, then convergence will be faster, as opening to trade allows producers to draw insight from the relatively productive foreign producers. In contrast, if insights are drawn from technologies that are used locally, the country’s stock of knowledge grows more slowly. In that case, a trade liberalization leads to better selection of the domestic producers, but those domestic producers have low productivity relative to foreign firms. Opening to FDI also tends to lead to faster convergence than opening to trade. Opening to FDI provides more immediate access to the ideas of foreign producers. Over time, the exposure to more productive foreign multinationals producing domestically leads to faster growth of the stock of knowledge. In the case of opening to FDI, the stock of knowledge grows faster when learning is from technologies used locally. We also study whether trade and FDI are substitutes or complements in the diffusion of ideas. For both the static and dynamic gains from opening, this depends crucially on the correlation of multinationals’ productivities across potential production locations. When this correlation is high, trade and FDI are substitutes in increasing the speed of learning and raising real incomes. Finally, we specify a quantitative version of the model that includes non-traded goods and intermediate inputs, and equipped labor with capital and education. Using cross-country data on trade flows and GDP, we study whether the model can account for the post-war growth of South Korea. We first vary a parameter that modulates the strength of diffusion. We find that the role 3
of diffusion is largest for intermediate values of this parameter; When the Literature Review
Our work builds on a large literature modeling innovation and diffusion of
technologies as a stochastic process, starting from the earlier work of Jovanovic and Rob (1989), Jovanovic and MacDonald (1994), Kortum (1997), and recent contributions by Alvarez et al. (2008) and Luttmer (2012).3 We are particularly related to recent applications of these frameworks to study the connection between trade and the diffusion of ideas (Lucas, 2009a; Alvarez et al., 2013; Perla et al., 2013; Sampson, 2014). Our theory captures the models in Kortum (1997) and Alvarez et al. (2008, 2013) as special cases. When the contribution of insights to the development of new technologies is zero, β = 0 in our notation, our framework simplifies to a version of Kortum (1997) with exogenous search intensity. As in that paper, ours is a model with semi-endogenous growth. When insights from domestic sellers are the only input to the development of new technologies, β = 1, our framework simplifies to the model in Alvarez et al. (2008, 2013) with stochastic arrival of ideas. Beyond analyzing the intermediate cases, β ∈ (0, 1), the behavior of the model is qualitatively different from either of the two special cases β = 0 or β = 1. With β = 0, there is no diffusion of ideas and thus no dynamic gains from trade. With β = 1, changes in trade costs alter a country’s growth rate, and the equilibrium frontier of knowledge is closer to a logistic distribution. More importantly, when β = 1 and trade barriers are finite, changes in trade barriers have no impact on the tail of the distribution of productivity, and therefore, the model has a more limited success in providing a quantitative theory of the level and transitional dynamics of productivity. With β < 1, the frontier of knowledge converges to a Frechet distribution. This allows us use the machinery of Eaton and Kortum (2002), Bernard et al. (2003), and Alvarez and Lucas (2007) which have been remarkably successful as quantitative trade models. We therefore believe that studying the intermediate case of β ∈ (0, 1) is a step toward a quantitative model of the cross-country diffusion of ideas. Finally, we are able to nest alternative sources of insights, e.g., learning from those who sell goods to a country, learning from those that produce within a country, and study the role of both trade and FDI in determining the distribution of insights. Eaton and Kortum (1999) also build a model of the diffusion of ideas across countries in which 3
Lucas and Moll (2014) and Perla and Tonetti (2014) extends these models by studying the case with endogenous search effort, a dimension that we abstract from.
4
the distribution of productivities in each country is Frechet, and where the evolution of the scale parameter of the Frechet distribution in each country is governed by a system of differential equations. In their work insights are drawn from the distribution of potential producers in each country, according to exogenous diffusion rates which are estimated to be country-pair specific, although countries are assumed to be in autarky otherwise. Therefore, changes in trade and FDI costs do not affect the diffusion of ideas. The model shares some features with Oberfield (2013) which models the formation of supply chains and the economy’s input-output architecture. In that model, entrepreneurs discover methods of producing their goods using other entrepreneurs’ goods as inputs.4
1
Technology Diffusion with a General Source Distribution
We begin with a description of technology diffusion in a single country given a general source distribution. The source distribution describes the set of insights that domestic producers might access. In the specific examples that we explore later in the paper, the source distribution will depend on the profiles of productivity across all countries in the world, but in this section we take it to be a general function satisfying weak tail properties. Given the assumption on the source distribution, we show that the equilibrium distribution of productivity in a given economy is Frechet, and derive a differential equation describing the evolution of the scale parameter of this distribution. We consider an economy with a continuum of goods s ∈ [0, 1]. For each good, there are m producers. We will later study an environment in which the producers engage in Bertrand competition, so that (barring ties) at most one of these producers will actively produce. A producer is characterized by her productivity, q. A producer of good s with productivity q has access to a labor-only, linear technology y(s) = ql(s),
(1)
where l(s) is the labor input and y(s) is output of good s. The state of technology in the economy is described by the function Mt (q), the fraction of producers with knowledge no greater than q. We 4 Here, the evolution of the distribution of marginal costs depends on a differential equation summarizing the history of insights that were drawn. In Oberfield (2013), the distribution of marginal costs is the solution to a fixed point problem, as each producer’s marginal cost depends on her potential suppliers’ marginal costs.
5
call Mt the distribution of knowledge at t. The economy’s productivity depends on the frontier of knowledge. The frontier of knowledge is characterized by the function F˜t (q) ≡ Mt (q)m . F˜t (q) is the probability that none of the m producers of a good have productivity better than q. We now turn to a description of the dynamics of the distribution of knowledge. We model diffusion as a process involving the random interaction among producers of different goods or countries. We assume each producer draws insights from others stochastically at rate αt . However there is randomness in the adaptation of that insight. More formally, when an insight arrives to a producer with productivity q, the producer learns an idea with random productivity zq 0β and adopts the idea if zq 0β > q. The productivity of the idea has two components. There is an insight ˜ t (q 0 ). The second drawn from another producer, q 0 , which is drawn from the source distribution G component z is drawn from an exogenous distribution with CDF H(z). We refer to H(z) as the exogenous distribution of ideas.5 This process captures the fact that interactions with more productive individuals tend to lead to more useful insights, but it also allows for randomness in the adaptation of others’ techniques to alternative uses. The latter is captured by the random variable z. An alternative interpretation of the model is that z represents an innovator’s “original” random idea, which is combined with random insights obtained from other technologies.6 ˜ t (q 0 ), and the Given the distribution of knowledge at time t, Mt (q), the source distribution, G exogenous distribution of ideas, H(z), the distribution of knowledge at time t + ∆ is Z Mt+∆ (q) = Mt (q) (1 − αt ∆) + αt ∆
∞
˜ t (x) H q/xβ dG
0
The first term on the right hand side is the distribution of knowledge at time t, which gives the fraction of producers with productivity less than q. The second term is the probability that a ˜ t (q) and H(z) are exogenous. The distinction between these distriFrom the perspective of this section, both G butions will become clear once we consider specific examples of source distributions, in which the source distribution will be an endogenous function of the countries’ frontiers of knowledge. 6 If β = 0 our framework simplifies to a version of the model in Kortum (1997) with exogenous search intensity. The framework also nests the model of diffusion in Alvarez et al. (2008) with stochastic arrival of ideas if β = 1, H ˜ t = F˜t . is degenerate, and G 5
6
producer did not have an insight between time t and t + ∆ that raised her productivity above q. This can happen if no insight arrived in an interval of time ∆, an event with probability 1 − αt ∆, or if at least one insight arrived but none resulted in a technique with productivity greater than q, R∞ ˜ t (x). an event that occurs with probability 0 H q/xβ dG Rearranging and taking the limit as ∆ → 0 we obtain Mt+∆ (q) − Mt (q) d ln Mt (q) = lim = −αt ∆→0 dt ∆Mt (q)
Z
∞h
i ˜ t (x). 1 − H q/xβ dG
0
With this, we can derive an equation describing the frontier of knowledge. Since F˜t (q) = Mt (q)m , the change in the frontier of knowledge evolves as: d ln F˜t (q) = −mαt dt
Z
∞h
i ˜ t (x). 1 − H q/xβ dG
0
To gain tractability, we make the following assumption about the exogenous component of ideas.
Assumption 1 The exogenous distribution has a Pareto right tail with exponent θ, so that
lim
z→∞
1 − H(z) = 1. z −θ
We thus assume that the right tail of the exogenous distribution of ideas is regularly varying. The restriction that the limit is equal to 1 rather than some other positive number is without loss of generality; we can always choose units so that the limit is one. We also assume that the strength of diffusion, β is strictly less than one. Assumption 2 β ∈ [0, 1). ˜ t has a sufficiently For this section we make one additional assumption: the source distribution G thin tail.7 ˜ t (q)] = 0. Assumption 3 At each t, limq→∞ q βθ [1 − G 7
In later sections when we endogenize the source distribution, this assumption will be replaced by an analogous assumption on the right tail of the initial distribution of knowledge, limq→∞ q βθ [1 − M0 (q)] = 0 and the restriction that β < 1.
7
We will study economies where the number of producers for each good is large. As such, it will be convenient to study how the frontier of knowledge evolves when normalized by the number of 1 1 ˜ t m (1−β)θ q producers for each good. Define Ft (q) ≡ F˜t m (1−β)θ q and Gt (q) ≡ G Proposition 1 Suppose that Assumption 1, Assumption 2 and Assumption 3 hold. Then in the limit as m → ∞, the frontier of knowledge evolves as: d ln Ft (q) = −αt q −θ dt
Z
∞
xβθ dGt (x)
0
Motivated by the previous proposition, we define λt ≡
Rt
−∞ ατ
R∞ 0
xβθ dGτ (x)dτ . With this,
one can show that the economy’s frontier of knowledge converges asymptotically to a Frechet distribution. 1/θ
−θ
Corollary 2 Suppose that limt→∞ λt = ∞. Then limt→∞ Ft (λt q) = e−q . 1/θ
−θ
Proof. Solving the differential equation gives Ft (q) = F0 (q)e−(λt −λ0 )q . Evaluating this at λt q 1/θ
1/θ
−1 −θ q
gives Ft (λt q) = F0 (λt q)e−(λt −λ0 )λt e−q
1/θ
. This implies that, asymptotically, limt→∞ Ft (λt q) =
−θ
Thus, the distribution of productivities in this economy is asymptotically Frechet and the dynamics of the scale parameter is governed by the differential equation λ˙ t = αt
Z
∞
xβθ dGt (x).
(2)
0
We call λt the stock of knowledge. In the rest of the paper we analyze alternative models for the source distribution Gt . A simple example that illustrates basic features of more general cases is Gt (q) = Ft (q). This corresponds to the case in which diffusion opportunities are randomly drawn from the set of domestic best practices across all goods. In a closed economy this set equals the set of domestic producers and sellers. In this case equation (2) becomes λ˙ t = αt Γ(1 − β)λβt
where Γ(u) =
R∞ 0
xu−1 e−x dx is the Gamma function. Growth in the long-run is obtained in this 8
framework if the arrival rate of insight grows over time, αt = α0 eγt . In this case, the scale of the Frechet distribution λt grows asymptotically at the rate γ/(1 − β), and per-capita GDP grows at ˆ t = λt eγ/(1−β)t the rate γ/[(1−β)θ]. In general, the evolution of the de-trended stock of knowledge λ can be summarized in terms of the de-trended arrival of ideas α ˆ t = αt eγt ˆ˙ t = α ˆβ − λ ˆ t Γ(1 − β)λ t
γ ˆ λt , 1−β
and on a balanced growth path on which α ˆ is constant, the de-trended stock of ideas is 1 1−β α ˆ (1 − β) ˆ= λ Γ(1 − β) . γ
In the model that follows, potential producers engage in Bertrand competition. In that environment, an important object is the joint distribution of the productivities of best and second best producers of a good. We denote the CDF of this joint distribution as F˜t12 (q1 , q2 ), which, for q1 ≥ q2 , equals8 F˜t12 (q1 , q2 ) = Mt (q2 )m + m [Mt (q2 ) − Mt (q1 )] Mt (q2 )m−1 . Since the frontier of knowledge at t satisfies F˜t (q) = Mt (q)m , the joint distribution can be written as F˜t12 (q1 , q2 )
= 1+m
1/m F˜t (q1 )/F˜t (q2 ) − 1 F˜t (q2 ),
q1 ≥ q2 .
1 1 Normalizing this joint distribution by the number of producers, Ft12 (q1 , q2 ) ≡ F˜t12 m (1−β)θ q1 , m (1−β)θ q2 , we have that for large m,
Ft12 (q1 , q2 ) = [1 + log Ft (q1 ) − log Ft (q2 )] Ft (q2 ),
2
q1 ≥ q2 .
International Trade
We first consider a world where n economies interact through trade, and ideas diffuse through the contact of domestic producers with those who sell goods to the country as well as with those that 8 Intuitively, there are two ways the best and second best productivities can be no greater than q1 and q2 respectively. Either none of the productivities are greater than q2 , or one of the m draws is between q1 and q2 and none of the remaining m − 1 are greater than q2 .
9
produce within the country. Given the results from the previous section, the static trade theory is given by the standard Ricardian model in Eaton and Kortum (2002), Bernard et al. (2003), and Alvarez and Lucas (2007), which we briefly introduce before deriving the equations which characterize the evolution of the profile of the distribution of productivities of countries in the world economy. In each country, consumers have identical preferences over a continuum of goods. We use ci (s) to denote the consumption of a representative household in i of good s ∈ [0, 1]. Utility is given by u(Ci ), where the the consumption aggregate is Z Ci =
1
ci (s)
ε−1 ε
ε/(ε−1) ds
0
so goods enter symmetrically and exchangeably. We assume that ε − 1 < θ, which guarantees the price level is finite. Let pi (s) be the price of good s in i, so that i’s ideal price index is i 1 hR 1−ε 1 . Letting Xi denote i’s total expenditure, i’s consumption of good s is Pi = 0 pi (s)1−ε ds ci (s) =
pi (s)−ε Xi . Pi1−ε
In each country, individual goods can be manufactured by many producers, each using a laboronly, linear technology (1). As discussed in the previous section, provided countries share the same exogenous distribution of ideas H(z), the frontier of productivity in each country is described by a −θ
Frechet distribution with curvature θ and a country-specific scale λi , Fi (q) = e−λi q . Transportation costs are given by the standard “iceberg” assumption, where κij denotes the units that must be shipped from country j to deliver a unit of the good in country i, with κii = 1 and κij ≥ 1. We now briefly present the basic equations that summarize the static trade equilibrium given the vector of scale parameters λ = (λ1 , ..., λn ). Because the expressions for price indices, trade shares, and profit are identical to Bernard et al. (2003), we relegate the derivation of these expressions to Appendix B. Given the isoelastic demand, if a producer had no direct competitors, it would set a price with a markup of
ε ε−1
over marginal cost. Producers engage in Bertrand competition. This means that
lowest cost provider of a good to a country will either use this markup or, if necessary, set a limit price to just undercut the next-lowest-cost provider of the good. Let wi denote the wage in country i. For a producer with productivity q in country j, the cost of
10
providing one unit of the good in country i is
wj κij q .
The price of good s in country i is determined
as follows. Suppose that country j’s best and second best producers of good s have productivities qj1 (s) and qj2 (s). The country that can provide good s to i at the lowest cost is given by
arg min j
wj κij qj1 (s)
If the lowest-cost-provider of good s for i is a producer from country k, the price of good s in i is pi (s) = min
wj κij ε wk κik wk κik , , min ε − 1 qk1 (s) qk2 (s) j6=k qj1 (s)
That is, the price is either the monopolist’s price or else it equals the cost of the next-lowest-cost provider of the good; the latter is either the second best producer of good s in country k or the best producer in one of the other countries. In Appendix B, we show that, in equilibrium, i’s price index is
Pi = B
X
λj (wj κij )−θ
−1/θ
j
where B is a constant.9 Let Sij ⊆ [0, 1] be the set of goods for which a producer in j is the lowest-cost-provider for country i. Let πij denote the share of country i’s expenditure that is spent on goods from country R j so that πij = s∈Sij (pi (s)/Pi )1−ε ds. In Appendix B, we show that the expenditure share is λj (wj κij )−θ πij = Pn . −θ k=1 λk (wk κik ) A static equilibrium is given by a profile of wages w = (w1 , ..., wn ) such that labor market clears in all countries. The static equilibrium will depend on whether trade is balanced and where profit from producers is spent. For now, we take the each country’s expenditure as given and solve for the equilibrium as a function of these expenditures. Labor in j is used to produce goods for all destinations. To deliver one unit of good s ∈ Sij to 9
B
1−ε
=
1−
ε−1 θ
−θ −θ ε ε 1 − ε−1 + ε−1 Γ 1−
11
ε−1 θ
.
i, the producer in j uses κij /qj1 (s) units of labor. Thus the labor market clearing constraint for country j is XZ
Lj =
κij ci (s)ds. qj1 (s)
s∈Sij
i
Similarly, the total profit earned by producers in j can be written as
Πj =
wj κij ci (s)ds. pi (s) − qj1 (s) s∈Sij
XZ i
In Appendix B, we show that these can be expressed as
wj Lj =
θ X πij Xi θ+1 i
and
Πj =
1 X πij Xi θ+1 i
Under the natural assumption that trade is balanced and that all profit from domestic producers is spent domestically, then Xi = wi Li + Πi and the labor market clearing conditions can be expressed as X
wj Lj =
πij wi Li
i
As a simple benchmark, it is useful to consider the case with costless trade, κij = 1, all j, and countries of equal size Li = Lj , all j 6= i. In this case, relative wages are wiF T = wiF0 T
λi λ i0
1 1+θ
and the relative expenditure shares are FT πij FT πij 0
=
λj λj 0
1 1+θ
.
(3)
Given the static equilibria, we next solve for the evolution of the profile of scale parameters λ =
12
(λ1 , ..., λn ) by specializing (2) for alternative assumptions about source distributions. We consider source distributions that encompass two cases: (i) domestic producers learn from sellers to the country, (ii) domestic producers learn from other producers in the country.
2.1
Learning from Sellers
Following the framework introduced in Section 1, we model the evolution of technologies as the outcome of a process where technology managers combine “own ideas” with random insights from technologies in other sectors or countries. We first consider the case in which insights are drawn from sellers to the country. In particular, we assume that insights are randomly drawn from the distribution of sellers’ productivity in proportion to the expenditure on each good.10 In this case, the source distribution is given by the expenditure weighted distribution of productivity of sellers
Gi (q) = GSi (q) ≡
XZ s∈Sij |qj (s) 0 arbitrarily. Since limz→∞ 1−H(z) z −θ 1−H(z) z −θ
1−H(z) z −θ
1−H(z) z −θ
≤K
= 1, there is a z ∗ such that z > z ∗ implies
< 1 + δ. For z < z ∗ , we have that z θ [1 − H(z)] ≤ (z ∗ )θ [1 − H(z)] ≤ (z ∗ )θ . Thus for any z, n o ≤ K ≡ max 1 + δ, (z ∗ )θ
Claim 6 Suppose that Assumption 1 and Assumption 3 hold. Then in the limit as m → ∞, the frontier of knowledge evolves as: d ln Ft (q) = −αt q −θ dt
Z
∞
xβθ dGt (x)
0
1
1 − (1−β)θ
Proof. Evaluating the law of motion at m (1−β)θ q and using the change of variables w = m
x
we get 1 ∂ = −mαt ln F˜t m (1−β)θ q ∂t
Z
∞
0
Z
∞
= −mαt 0
1
"
m (1−β)θ q 1−H xβ
!#
1 (1−β)θ
˜ t (x) dG
1 q ˜ (1−β)θ m d G m w 1 − H t β 1 m (1−β)θ w
1 1 ˜ t m (1−β)θ q which have the From above, we have that Ft (q) ≡ F˜t m (1−β)θ q , and Gt ≡ G following derivatives 1 1 ˜ 0t m (1−β)θ q G0t (q) = m (1−β)θ G 1 1 Ft0 (q) = m (1−β)θ F˜t0 m (1−β)θ q 1 ˜t m (1−β)θ q ∂ F ∂Ft (q) = ∂t ∂t The equation becomes ∂ ln Ft (q) = −mαt ∂t
Z 0
∞
1 (1−β)θ
q m 1 − H β dGt (w) 1 m (1−β)θ w 46
This can be rearranged as ∂ ln Ft (q) = −αt q −θ ∂t
Z 0
∞
"
1 − H m1/θ qw−β −θ m1/θ qw−β
# wβθ dGt (w)
We want to take a limit as m → ∞. To do this, we must show that we can take the limit inside the integral. By Lemma 5, there is a K < ∞ such that for any z, 1−H(z) ≤ K. Further, given the z −θ R∞ ˜ t (w) is finite. Thus we can take the limit assumptions on the tail of Gt , the integral 0 Kwβθ dG inside the integral using the dominated convergence theorem to get ∂ ln Ft (q) = −αt q −θ ∂t
B
Z
∞
wβθ dGt (w)
0
Trade
B.1
Equilibrium
This section derives expressions for price indices, trade shares, and market clearing conditions that determine equilibrium wages. The total expenditure in i is Xi . Throughout this section, we h i −θ maintain that Fi12 (q1 , q2 ) = 1 + λi q1−θ − λi q2−θ e−λi q . For a variety s ∈ Sij (produced in j and exported to i) that is produced with producw κ
tivity q, the equilibrium price in i is pi (s) = jq ij , the expenditure on consumption in i is 1−ε 1−ε pi (s) pi (s) 1 X , consumption is Xi , and the labor used in j to produce variety s i Pi Pi pi (s) 1−ε κ /qj1 (s) pi (s) Xi for i is ijpi (s) Pi Define πij ≡
λj (wj κij )−θ P −θ . k λk (wk κik )
We will eventually show this is the share of i’s total expenditure
that is spent on goods from j. We begin with a lemma which will be useful in deriving a number of results. Lemma 7 Suppose τ1 and τ2 satisfy τ1 < 1 and τ1 + τ2 < 1. Then # τ2
"
Z qj1 (s)
τ1 θ
−τ2 θ
pi (s)
˜ 1 , τ2 ) ds = B(τ
s∈Sij
X k
47
λk (wk κik )
−θ
πij
λj πij
τ1
˜ 1 , τ2 ) ≡ where B(τ
1−
τ2 1−τ1
+
τ2 1−τ1
ε ε−1
−θ(1−τ1 )
Γ (1 − τ1 − τ2 )
Proof. We begin by defining the measure Fij to satisfy q2
Z Fij (q1 , q2 ) = 0
Y
Fk12
k6=j
wk κik x wk κik x , wj κij wj κij
Fj12 (dx, x)+
q1
Z
Y
Fk12
q2 k6=j
wk κik q2 wk κik q2 , wj κij wj κij
Fj12 (dx, q2 ) (16)
Fij (q1 , q2 ) is the fraction of varieties that i purchases from j with productivity no greater than q1 and second best provider of the good to i has marginal cost no smaller than
wj κij q2 .
There are
two terms in the sum. The first term integrates over goods where j’s lowest-cost producer has productivity no greater than q2 , and the second over goods where j’s lowest cost producer has productivity between q1 and q2 . The corresponding density
∂2 ∂q1 ∂q2 Fij (q1 , q2 )
will be useful because
it is the measure of firms in j with productivity q that are the lowest cost providers to i and for which the next-lowest-cost provider has marginal cost wj κij /q2 . We first show that i − 1 λ q−θ h j Fij (q1 , q2 ) = πij + λj q2−θ − q1−θ e πij 2
The first term of equation (16) can be written as Z
q2
0
Y
Fk12
k6=j
wk κik x wk κik x , wj κij wj κij
Fj12 (dx, x) =
q2 − P k6=j λk
Z
e
wk κik wj κij
−θ
x−θ
−θ
θλj x−θ−1 e−λj x dx
0
−θ
=
λj (wj κij ) e P −θ λ (w κ ) k k ik k
−
P
k λk
wk κik wj κij
−θ
q2−θ
λ
− π j q2−θ
= πij e
ij
The second term is Z
q1
Y
q2 k6=i
Fk12
wk κik q2 wk κik q2 , wj κij wj κij
Fj12 (dx, q2 )
−
P
= e
k6=j
λk
wk κik wj κij
−θ
q2−θ
Z
q1
q2 −
P
= e
k
λ
λk
− π j q2−θ
= e
48
ij
wk κik wj κij
−θ
q2−θ
h i λj q2−θ − q1−θ i
h λj q2−θ − q1−θ
−θ
θλj x−θ−1 e−λj q2 dx
Together, these give the expression for Fij , so the joint density is − 1 λ q−θ 1 ∂2 j Fij (q1 , q2 ) = θλj q1−θ−1 θλj q2−θ−1 e πij 2 ∂q1 ∂q2 πij We next turn to the integral a markup of
ε ε−1
R s∈Sij
qj1 (s)θτ1 pi (s)−θτ2 ds. Since the price of good s is set at either
over marginal cost or at the cost of the next lowest cost provider, this integral
equals ∞Z ∞
Z 0
q2 ∞Z ∞
Z = 0
q2
q1θτ1
wj κij ε wj κij , q2 ε − 1 q1
−θτ2
wj κij ε wj κij , q2 ε − 1 q1
−θτ2
min
q1θτ1 min
Using the change of variables x1 =
(wj κij )−θτ2 πij
R ∞ R x2
˜ 1 , τ2 ) ≡ Define B(τ
0
0
λj πij
τ1 +τ2 Z
qj1 (s)
∞ Z x2
0
1 x−τ 1
Z
λj −θ πij q1
min x2 ,
θτ1
−θτ2
pi (s)
0
ε ε−1
∂ 2 Fij (q1 , q2 ) dq1 dq2 ∂q1 ∂q2 − 1 λ q−θ 1 j θλj q1−θ−1 θλj q2−θ−1 e πij 2 dq1 dq2 πij λj −θ πij q2
and x2 =
( x1−τ1 min x2 , −τ2
θ
x1
, this becomes
ε ε−1
λj (wj κij )−θ P −θ , k λk (wk κik )
ds = B(τ1 , τ2 ) (wj κij )
we have (wj κij )−θτ2
49
λj πij
e−x2 dx1 dx2
x1
e−x2 dx1 dx2 , so that the integral is
−θτ2
s∈Sij
Using πij =
)−τ2
θ
τ 2
=
hP
πij
λj πij
τ1 +τ2
−θ k λk (wk κik )
iτ2
. Finally we com-
˜ 1 , τ2 ): plete the proof by providing an expression for B(τ
˜ (τ1 , τ2 ) = B
Z
∞ Z x2
0
Z
0
1 min x2 , x−τ 1
∞ Z x2
= (
0
Z
(
−θ ε x2 ε−1
)
1−τ1 ∞ x2
−
ε ε−1
−θ
1−
ε ε−1
)−τ2
θ
−θ(1−τ1 ) Z
e−x2 dx1 dx2
x1 Z
−θ
ε ( ε−1 )
∞Z
+ 0
x2
0
1 x−τ 1
x2 2 −x2 dx2 x−τ 2 e
∞
+ ε ε−1
ε ε−1
−θτ2 Z
ε ε−1
∞
Z 0
∞
)−τ2
θ
e−x2 dx1 dx2
x1 ε ε−1
0
−θ(1−τ1 )
1 −τ2 −x2 e dx2 + x1−τ 2 1 − τ1 1 − τ1 − τ2 0 −θ(1−τ1 ) −θ(1−τ1 ) ε ε 1 − ε−1 ε−1 = + Γ (2 − τ1 − τ2 ) 1 − τ1 1 − τ1 − τ2 ( −θ(1−τ1 ) ) τ2 τ2 ε 1− + Γ (1 − τ1 − τ2 ) = 1 − τ1 1 − τ1 ε − 1
=
(
1−τ1
1 − τ1
0
ε ε−1
1 −τ2 −x2 dx1 dx2 x−τ 1 x2 e
=
−θ
1−τ1 −τ2 x2
1 − τ1 − τ2
x21−τ1 −τ2 e−x2 dx2
where the final equality uses the fact that for any x, Γ(x + 1) = xΓ(x). We first use this lemma to provide expressions for the price index in i and the share of i’s expenditure on goods from j. Claim 8 The price index for i satisfies #− 1 1 "X θ 1−ε−1 ε − 1 ˜ 0, Pi = B λk (wk κik )−θ θ k
where B ≡
1−
ε−1 θ
−θ −θ ε ε 1 − ε−1 + ε−1 Γ 1−
ε−1 θ
. πij =
λ (w κ )−θ P j j ij −θ k λk (wk κik )
is the share
of i’s expenditure on goods from j. Proof. The price aggregate of goods provided to i by j is
R s∈Sij
pi (s)1−ε ds. Using Lemma 7, this
equals # ε−1 "X θ ε − 1 −θ 1−ε ˜ pi (s) ds = B 0, λk (wk κik ) πij θ s∈Sij
Z
k
50
e−x2 dx2
The price index for i therefore satisfies
Pi1−ε =
# ε−1 "X θ ε − 1 ˜ 0, λk (wk κik )−θ pi (s)1−ε ds = B θ s∈Sij
XZ j
k
and i’s expenditure share on goods from j is R s∈Sij
pi (s)1−ε ds Pi1−ε
= πij
We next turn to the market clearing conditions. Claim 9 Country j’s expenditure on labor is
θ θ+1
P
i πij Xi .
i Proof. i’s consumption of good s is pi (s)−ε PX1−ε . If j is the lowest-cost provider to i, then j’s i
κ
expenditure on labor per unit delivered is wj qj1ij(s) . The total expenditure on labor in j to produce R w κij i goods for i is then s∈Sij qj1j (s) pi (s)−ε PX1−ε ds. Using Lemma 7, the total expenditure on labor in j i
is thus XZ i
s∈Sij
Z X wj κij Xi Xi pi (s)−ε 1−ε ds = qj1 (s)−1 pi (s)−ε ds wj κij 1−ε qj1 (s) Pi P s∈Sij i i " #ε X − 1 θ X θ λj 1 ε X i −θ ˜ − , = B wj κij 1−ε λk (wk κik ) πij θ θ πij Pi i k
˜ −1, ε = The result follows from B θ θ
C
θ ˜ θ+1 B
0, ε−1 and θ
wj κij Pi1−ε
hP
−θ k λk (wk κik )
iε θ
λj πij
− 1 θ
˜ 0, ε−1 =B θ
Source Distributions
This appendix derives expressions for the source distributions under various specifications. We begin by describing learning from sellers.
51
−1
.
C.1
Learning from Sellers
Here we characterize the learning process when insights are drawn from sellers in proportion to the expenditure on each seller’s good. Consider a variety that can be produced in j at productivity q. Since the share of i’s expenditure on good s is (pi (s)/Pi )1−ε , the source distribution is
Gi (q) =
XZ j
(pi (s)/Pi )1−ε ds
{s∈Sij |qj1 (s)≤q}
The change in i’s stock of knowledge depends on Z
∞ βθ
q dGi (q) =
XZ
0
qj1 (s)βθ (pi (s)/Pi )1−ε ds
s∈Sij
j
Using Lemma 7, this is Z
∞
q βθ dGi (q) =
0
j
=
C.1.1
X ˜ B ˜ B
# ε−1 "X β θ λj ε − 1 −θ ˜ λk (wk κik ) πij 1−ε B β, θ πij Pi k β X β, ε−1 λj θ πij ε−1 πij 0, θ 1
(17)
j
Alternative Weights of Sellers
Here we explore two alternative processes by which individuals can learn from sellers. In the first case, individuals are equally likely to learn from all active sellers, independently of how much of the seller’s variety they consume. In the second case, insights are drawn from sellers in proportion to consumption of each sellers’ goods. In each case, the speed of learning is the same as our baseline (equation (17)) up to a constant. Learning from All Active Sellers Equally If producers are equally likely to learn from all active sellers, the source distribution is P R Gi (q) =
j {s∈Sij |qj1 (s)≤q} ds
P R
j s∈Sij
The change in i’s stock of knowledge depends on
R∞ 0
52
ds P R
q βθ dG
i (q)
=
j
s∈S
P Rij j
qj1 (s)βθ ds
s∈Sij
ds
. Using Lemma 7,
this is Z
β β X ˜ λj λj B(β, 0) X = Γ(1 − β) πij πij ˜ 0) πij πij B(0,
∞ βθ
q dGi (q) = 0
j
j
Learning from Sellers in Proportion to Consumption i . If producers learn in proportion to consumpi’s consumption of goods s is ci (s) = pi (s)−ε PX1−ε i
tion, then the source distribution is −ε Xi ds j {s∈Sij |qj1 (s)≤q} pi (s) P 1−ε
P R
i
Gi (q) =
−ε Xi ds j {s∈Sij } pi (s) Pi1−ε
P R
The change in i’s stock of knowledge depends on Z
P R
∞ βθ
q dGi (q) = 0
j s∈Sij
P R
qj1 (s)βθ pi (s)−ε ds
j s∈Sij
pi (s)−ε ds
Using Lemma 7, this is
Z
iε β hP P ˜ λj −θ θ ε λ (w κ ) π B β, ij πij k ik k k j θ iε hP P ˜ −θ θ ε πij k λk (wk κik ) j B 0, θ ˜ β, ε X B λj β θ π ij ˜ 0, ε πij B θ
∞
q βθ dGi (q) =
0
=
j
C.2
Learning from Producers
Here we characterize the learning process when insights are drawn from domestic producers in proportion to labor used in production. For each s ∈ Sij , the fraction of j’s labor used to produce the good is
1 κij Lj qj1 (s) ci (s)
i with ci (s) = pi (s)−ε PX1−ε . Summing over all destinations, the source i
distribution would then be
Gj (q) =
XZ i
s∈Sij |qj1 (s)≤q
53
1 κij Xi pi (s)−ε 1−ε ds Lj qj1 (s) Pi
The change in j’s stock of knowledge depends on ∞
Z
βθ
q dGj (q) = 0
XZ i
q βθ
s∈Sij
1 κij Xi pi (s)−ε 1−ε ds Lj qj1 (s) Pi
Using Lemma 7, this is Z
∞
0
#ε "X β− 1 θ X κij Xi θ λj 1 ε −θ βθ ˜ B β − , λ (w κ ) π q dGj (q) = ij k k ik Lj Pi1−ε θ θ πij j k
Using the expressions for Pi and πij from above, this becomes ∞
Z 0
C.2.1
β ˜ β − 1, ε B λj 1 X θ θ q dGj (q) = πij Xi ε−1 ˜ 0, wj Lj πij B θ βθ
j
Alternative Weights of Producers
Here we briefly describe the alternative learning process in which insights are equally likely to be dawn from all active domestic producers. We consider only the case in which trade costs satisfy the triangle inequality κjk < κji κik , ∀i, j, k such that i 6= j 6= k 6= i. We will show that, in this case, all producers that export also sell domestically. This greatly simplifies characterizing the learning process. Towards a contradiction, suppose there is a variety s such that i exports to j and k exports to wi κji qi (s)
i. This means that
≤
wk κjk qk (s)
and
wk κik qk (s)
≤
wi κii qi (s) .
Since κii = 1, these imply that κji κik ≤ κjk ,
a violation of the triangle inequality and thus a contradiction. R
In this case, the source distribution is Gi (q) =
s∈Sii |qi1 ≤q R s∈Sii
ds
ds
. The change in i’s stock of knowledge
depends on Z
R
∞ βθ
q dGi (q) = 0
s∈Sii
q βθ ds
R s∈Sii
ds
Using Lemma 7, this is
Z
∞ βθ
q dGi (q) = 0
˜ B(β, 0)πij
λj πij
˜ 0)πij B(0,
54
β = Γ(1 − β)πij
λj πij
β
D
Simple Examples
D.1
Symmetric Countries
If countries are symmetric, there are two possible values of πij :
πii = πij
=
1 1 + (n − 1)κ−θ κ−θ , 1 + (n − 1)κ−θ
i 6= j
Normalizing the wage to unity, the price level is − 1 1 θ P = Bλ− θ 1 + (n − 1)κ−θ
The de-trended scale parameter on a balance growth path is # 1 ε−1 −θ(1−β) 1−β Γ 1 − β − α 1 + (n − 1)κ θ ˆ λ(κ) = (1 − β) 1−β γ Γ 1 − ε−1 (1 + (n − 1)κ−θ ) θ "
Using this expression, per-capita income, yi = wi /Pi , is 1 1 θ ε − 1 − 1−ε 1 −θ y(κ) = Γ 1 − λ θ 1 + (n − 1)κ θ 1 # θ(1−β) 1 " 1 α Γ 1 − β − ε−1 ε − 1 − 1−ε −θ(1−β) θ(1−β) θ (1 − β) 1 + (n − 1)κ = Γ 1− θ γ Γ 1 − ε−1 θ The de-trended stock of knowledge and per-capita income relative to costless trade are ˆ λ(κ) ˆ λ(1) y(κ) y(1)
" =
1 + (n − 1)κ−θ(1−β)
#
1−β
1 1−β
n
β − 1−β
(1 + (n − 1)κ−θ ) 1 1 1 + (n − 1)κ−θ θ λ(κ) θ = n λ(1)
In particular, per-capita income in autarky relative to the case with costless trade β 1 y(∞) − θ(1−β) = |{z} n− θ n | {z } y(1)
static
55
dynamic
D.2
A Small Open Economy
Consider a small open economy. The economy is small in the sense that actions in the economy have no impact on other countries’ expenditures, price levels, wages, or stocks of knowledge.
D.2.1
Steady State Gains from Trade
D.2.2
Speed of Convergence
We use the notation x ˇ to denote log-deviation from of x from its steady state (or BGP) value. To derive the speed of convergence, we want expressions for how the trade shares and wages change over time. The trade shares and market clearing condition for i are
πij rji wi Li
λj (wj κij )−θ Pn −θ k=1 λk (wk κik ) Xj πji = wj Lj X = Xj πji
=
j
For a small open economy, we have ˇ i − θw π ˇii = (1 − πii ) λ ˇi ˇ i − θw j= 6 i : π ˇij = −πii λ ˇi ˇ i − θw i 6= j : π ˇji = λ ˇi ˇ i − θw j 6= i : rˇji = π ˇji − w ˇi = λ ˇi − w ˇi
The change in the wage can be found from linearizing the labor market clearing condition: X ˇ i − θw ˇ i − θw ˇi w ˇi = rii w ˇi + (1 − πii ) λ ˇi + rji λ j6=i
w ˇi (1 − πii ) w ˇi w ˇi
ˇ i − θw ˇ i − θw = πii w ˇi − πii λ ˇi + λ ˇi ˇ i − θw = (1 − πii )2 λ ˇi ˇ i − θw = (1 + πii ) λ ˇi
56
This last equation can be expressed in two ways: ˇ i − θw λ ˇi =
w ˇi =
ˇi λ 1 + θ(1 + πii )
(1 + πii ) ˇ λi 1 + θ(1 + πii )
Plugging these back into the shares, we have (1 − πii ) ˇ λi 1 + θ (1 + πii ) πii ˇi j 6= i : π ˇij = − λ 1 + θ(1 + πii ) 1 ˇi i 6= j : π ˇji = λ 1 + θ (1 + πii ) 1 (1 + πii ) ˇ −πii ˇi j= 6 i : rˇji = π ˇji − w ˇi = − λi = λ 1 + θ (1 + πii ) 1 + θ (1 + πii ) 1 + θ (1 + πii ) π ˇii =
ˇi. We now proceed to characterizing transition dynamics for the stock of knowledge, λ 1−β ˆ β πij λj P 1−β ˆ β . π k ik λk
Learning from Sellers Let ΩSij ≡
The change in the the deviation of i’s stock of
knowledge from the BGP is ˇi ∂λ ∂t
=
ˆi 1 ∂λ BS α ˆ i X 1−β ˆ β γ = πij λj − ˆ i ∂t ˆi 1−β λ λ j
Log-linearizing around the steady state gives ˇi ∂λ ∂t
≈
= =
BS α ˆ i X 1−β ˆ β ˇj − λ ˇi πij λj (1 − β)ˇ πij + β λ ˆi λ j P 1−β ˆ β ˇj − λ ˇi πij + β λ γ j πij λj (1 − β)ˇ P 1−β ˆ β 1−β j πij λj γ X S ˇj − λ ˇi Ωij (1 − β) π ˇij + β λ 1−β j
ˇi 1 − β ∂λ γ ∂t
=
X
ˇj − λ ˇi ΩSij (1 − β) π ˇij + β λ
j
57
ˇ j = 0, π For a small open economy, we have λ ˇii =
ˇi (1−πii )λ 1+θ(1+πii ) ,
and π ˇij =
ˇi −πii λ 1+θ(1+πii )
for j 6= i. The
law of motion can be written as ˇi 1 − β ∂λ γ ∂t
X ˇ i + (1 − β) ˇi = ΩSii (1 − β) π ˇii + β λ ΩSij π ˇij − λ j6=i
= = = = ˇi ∂λ ∂t
=
X ˇi ˇi (1 − πii ) λ −πii λ ˇ i + (1 − β) ˇi (1 − β) ΩSij + βλ −λ 1 + θ (1 + πii ) 1 + θ (1 + πii ) j6=i −πii (1 − π ) ii S S ˇi + β + (1 − β) (1 − Ωii ) −1 λ Ωii (1 − β) 1 + θ (1 + πii ) 1 + θ (1 + πii ) ΩSii − πii S ˇ − βΩii λi − 1 − (1 − β) 1 + θ (1 + πii ) ΩSii − πii S ˇi + β 1 − Ωii λ − (1 − β) − (1 − β) 1 + θ (1 + πii ) β ΩSii − πii S ˇi −γ 1 − + 1 − Ωii λ 1 + θ (1 + πii ) 1 − β
ΩSii
Finally, we can use this to get at the speed of convergence for real income: 1 1 ˇ (1 − πii ) ˇ i = Aλ ˇi 1− λ λi − π ˇii = θ θ 1 + θ (1 + πii ) ˇi dλ ΩSii − πii β d ˇi w ˇi − Pˇi = A = −γ 1 − + 1 − ΩSii Aλ dt dt 1 + θ (1 + πii ) 1 − β ΩSii − πii β S = −γ 1 − + 1 − Ωii w ˇi − Pˇi 1 + θ (1 + πii ) 1 − β w ˇi − Pˇi =
Learning from Producers Let ΩPij ≡
ˆ i /πji )β rji (λ P β. ˆ k rki (λi /πki )
The change in the the deviation of i’s
stock of knowledge from the BGP is ˇi ∂λ ∂t
=
β ˆi 1 ∂λ BP α ˆi X ˆ γ = rji λi /πji − ˆ ˆ 1−β λi ∂t λi j
58
Log-linearizing around the steady state gives ˇi ∂λ ∂t
≈ =
β BP α ˆi X ˆ ˇi rˇji − β π ˇij − (1 − β)λ rji λi /πji ˆi λ j γ X P ˇi Ωij rˇji − β π ˇij − (1 − β) λ 1−β j
ˇi 1 − β ∂λ γ ∂t
=
X
ˇi ΩPij [ˇ rji − β π ˇij ] − (1 − β) λ
j
Using the expressions for π ˇii , π ˇji and rˇji , along with πii = rii , the law of motion can be written as ˇi 1 − β ∂λ γ ∂t
= ΩPii [ˇ rii − β π ˇii ] +
= =
E
ˇi ΩPij [ˇ rji − β π ˇij ] − (1 − β) λ
j6=i
=
ˇi ∂λ ∂t
X
=
X −πii 1 (1 − πii ) ˇ P ˇ ˇ ˇi λi + Ωij λi − β λi − (1 − β) λ (1 − β) 1 + θ (1 + πii ) 1 + θ (1 + πii ) 1 + θ (1 + πii ) j6=i ( ) P Ωii (1 − β) (1 − πii ) + 1 − ΩPii (−πii − β) ˇi − (1 − β) λ 1 + θ (1 + πii ) ) ( ΩPii (1 − β) (1 − πii ) − πii 1 − ΩPii (1 − β) − β 1 − ΩPii (1 + πii ) ˇi − (1 − β) λ 1 + θ (1 + πii ) ( ) 1 − ΩPii (1 + πii ) ˇ ΩPii − πii β −γ 1 − + λi 1 + θ (1 + πii ) 1 − β 1 + θ (1 + πii )
ΩPii
Research
This section proves the following claim: Claim 10 If Πiτ is total flow of profit earned by entrepreneurs in i at time τ , then the flow of profit earned in i at time τ from ideas generated between t and t0 (with t < t0 < τ ) is: λit0 − λit Πiτ λiτ 0
(t,t ] Proof. For v1 ≤ v2 , let V˜jiτ (v1 , v2 ) be the probability that at time τ , the lowest cost technique
to provide a variety to j was discovered by a manager in i between times t and t0 , that the marginal cost of that lowest-cost technique is no lower than v1 , and that marginal cost of the (t,t0 ]
next-lowest-cost supplier is not lower than v2 . Just as in Section ??, we will define Vjiτ (v1 , v2 ) = − 1 − 1 (t,t0 ] limm→∞ V˜jiτ m (1−β)θ v1 , m (1−β)θ v2 . 59
(t,t0 ]
be profit from all techniques drawn in i between t and t0 . Thus total profit in i at n o−ε (−∞,τ ] ε τ is Πiτ . Defining p (v1 , v2 ) ≡ min v2 , ε−1 v1 , we can compute each of these by summing Let Πiτ
over profit form sales to each destiantion j: (t,t0 ]
Πiτ
XZ
=
j (−∞,τ ]
Πiτ
0
XZ
=
j
∞Z ∞ v1 ∞Z ∞
0
v1
(t,t0 ]
[p (v1 , v2 ) − v1 ] p (v1 , v2 )−ε Pjε−1 Xj Vjiτ (dv1 , dv2 ) (−∞,τ ]
[p (v1 , v2 ) − v1 ] p (v1 , v2 )−ε Pjε−1 Xj Vjiτ
(t,t0 ]
We will show below that Vjiτ (v1 , v2 ) =
λit0 −λit (−∞,τ ] (v1 , v2 ). λiτ Vjiτ
(t,t0 ]
Πiτ
=
(dv1 , dv2 )
It will follow immediately that
λit0 − λit (−∞,τ ] Πiτ λiτ
(t,t0 ]
(t,t0 ]
We now compute Vjiτ (v1 , v2 ). For each of the m managers in i, let Mi
(q) be the probability
that the no technique drawn between t and t0 delivers productivity better than q. Similarly, define 0
0
(t,t ] (t,t ] F˜i (q) ≡ Mi (q)m be the probability that none of the m managers drew a technique with 1 1 (t,t0 ] (t,t0 ] productivity better than q between t and t0 . Finally, let Fi (q) = limm→∞ F˜i m 1−β θ q .
We have
(t,t0 ]
V˜jiτ
i 0 0 ˜ (−∞,τ ] wk κjk M (−∞,τ ] wi κji m−1 M (−∞,t] wi κji M (t ,τ ] wi κji dM (t,t ] wi κji F k6=i k i i i i x x x x x v2 (v1 , v2 ) = i m−1 (t0 ,τ ] wi κji R v2 hQ 0] w κ w κ w κ w κ (−∞,τ ] (−∞,t] (t,t (−∞,τ ] i ji i i ji k jk ˜ +m Mi Mi Mi dMi k6=i Fk v2 v2 x x x v1 m
R ∞ hQ
0
(t,t ] The expression for V˜jiτ (v1 , v2 ) contains two terms.
The first represents the probability that
the best technique to serve j delivers marginal cost greater than v2 and was drawn by a man(t,t0 ] wi κji ager in i between t and t0 . For each of the m managers in i, dMi measures the x likelihood that the managers best draw between t and t0 delivered marginal cost x ∈ [v2 , ∞], hQ (t0 ,τ ] wi κji (−∞,t] wi κji ˜ (−∞,τ ] M is the probabiltiy that the manager had no better draws, and Mi k6=i Fk i x x is the probability that no other manager from any destination would be able to provide the good at marginal cost lower than x. The second terms represents the proability that the best technique to serve j delivers marginal cost between v1 and v2 and was drawn by a manager in i between t and t0 , and that no othe manager can delvier the variety with marginal cost lower than v2 .
60
wk κjk x
i
(−∞,t]
Using Mi (t,t0 ]
Mi
(t0 ,τ ]
(q) Mi
(t,t0 ]
(q) Mi
(−∞,τ ]
(q) = F˜i
(−∞,τ ]
(q), F˜i
(−∞,τ ]
(q) = Mi
(t,t0 ]
(q)m and F˜i
(q) =
(q)m , this can be written as
0
(t,t ] V˜jiτ (v1 , v2 ) =
R ∞ hQ v2
˜ (−∞,τ ] k Fk
wk κjk x
1 − 1−β
1 θ
1 − 1−β
v1 and m
(t,t0 ]
Vjiτ (v1 , v2 ) =
1 θ
i
x (t,t0 ] wi κji F˜i x 1 (t,t0 ] wi κji m (−∞,τ ] wi κji F˜i dF˜i x x (−∞,τ ] wi κji (t,t0 ] wi κji F˜i F˜i v2 x
R ∞ hQ v2
(−∞,τ ] k Fk
wk κjk x
(t,t0 ]
Finally, following the logic of Section ??, we have Fi (t,t0 ]
(t,t0 ]
Fi
wi κji x wi κji x
v2 and taking a limit as m → ∞ gives i dF (t,t0 ] wi κji i
x (t,t0 ] wi κji Fi x (t,t0 ] wi κji dFi x (t,t0 ] wi κji Fi x
R v2 hQ (−∞,τ ] wk κjk i + k Fk v2 v1
dFi
!
R v2 hQ (−∞,τ ] wk κjk i ˜ + k Fk v2 v1
Evaluating this at m
i dF˜ (t,t0 ] wi κji
−θ
(q) = e−(λit0 −λit )q , so that (−∞,τ ]
λit0 − λit dFi = (−∞,τ ] λiτ F i
wi κji x wi κji x
We thus have (t,t0 ]
Vjiτ (v1 , v2 ) =
λit0 − λit (−∞,τ ] Vjiτ (v1 , v2 ) λiτ
which completes the proof.
F
Multinationals
This section derives expressions for price indices, trade shares, and market clearing conditions. Across multinationals, let vi1 (s) and v2 (s) be the lowest and second lowest marginal costs of supplying good s to i. Then the price of good s in i is pi (s) = min
ε vi1 (s) , vi2 (s) ε−1
61
Define X
ϕi =
λk˜
˜ k
1 λk ϕi
πijk =
n X
!1−ρ wl κil δlk˜
−θ/[1−ρ]
l=1 n X
!1−ρ (wl κil δlk )
(wj κij δjk )−θ/[1−ρ]
−θ/[1−ρ]
−θ/[1−ρ] l=1 (wl κil δlk )
Pn
l=1
As with, Lemma 7, we begin with a lemma which will be useful intermediate step. Lemma 11 Suppose τ1 and τ2 satisfy τ1 < 1 and τ1 + τ2 < 1. Then Z s∈Sijk
˜ 1 , τ2 )πijk ϕτ1 +τ2 (wj κij δjk )τ1 θ qjk1 (s)τ1 θ pi (s)−τ2 θ ds = B(τ i
˜ 1 , τ2 ) is defined as in Lemma 7. where B(τ Proof. For v1 ≤ v2 , define V˜ijk (v1 , v2 ) to be the fraction of goods for which the lowest cost source for i is a multinational based in k producing in j, and for which that multinational is unable to supply the good at cost lower than v1 , and for which no other multinational can supply the good at cost lower than v2 . We can then express V˜ijk (v1 , v2 ) using F˜kj and Mkj to represent the derivatives of F˜k and Mk with respect the jth argument: V˜ijk (v1 , v2 ) = m
Z
∞
wl κil δlk Mkj x2
v2
Z
v2
+m v1
wl κil δlk Mkj x2
wl κil δlk x
Mk
l
wl κil δlk x
wl κil δlk x
Mk
l
m−1 Y l
wl κil δlk v2
F˜k˜
˜=k k6 m−1 Y
l
˜ =k k6
F˜k˜
wl κil δlk˜ x
dx l
wl κil δlk˜ v2
dx l
The two terms divide V˜ijk (v1 , v2 ) into instances where the lowest cost provider has cost less than v2 and between v1 and v2 respectively. To interpret the first term, note that for each of the m n o wl κil δlk multinationals based in k, the term wl κxil2 δlk Mkj is the probability that that particular x l
multinational can provide the good to i by producing in j at cost x(> v2 ) and cannot provide the n o m−1 Q n o wl κil δlk˜ wl κil δlk ˜ good at a lower cost by producing elsewhere, while Mk F is ˜ =k k ˜ k6 x x l
l
the probability that none of the other multinationals (the other m − 1 based in k or any based in
62
another country) can provide the good to i at cost lower than x. The second term is similar, except since x ∈ [v1 , v2 ], it uses the probability that none of the other multinationals can provide the good to i at cost lower than v2 . This can be written more concisely as V˜ijk (v1 , v2 ) = m
Z
∞
v1
wl κil δlk Mkj x2
wl κil δlk x
Mk
l
wl κil δlk max {x, v2 }
m−1 Y l
F˜k˜
˜ =k k6
wl κil δlk˜ max {x, v2 }
1
Note that since F˜k (q) m = Mk (q), we can differentiate each with respect to the jth argument to 1 F˜kj (q) m−1 m ˜ Fk (q) m
get Mkj (q) =
. We can therefore express V˜ijk as
∞
Z
V˜ijk (v1 , v2 ) =
F˜k
v1
n
wl κil δlk ˜ Fkj x2
n
wl κil δlk x
wl κil δlk x
o Q l
o 1−1/m
˜˜ ˜ Fk k n
F˜k
l
n
wl κil δlk˜ max{x,v2 }
wl κil δlk max{x,v2 }
o l
o 1/m dx l
− 1 θ − 1 θ Define Vijk (v1 , v2 ) ≡ V˜ijk m 1−β v1 , m 1−β v2 . As m grows large, this becomes
Z
∞
Vijk (v1 , v2 ) =
wl κil δlk Fkj x2
Fk
v1
With functional forms, Fk (q) = e
−λk
n
n
wl κil δlk x
wl κil δlk x
o
Y
l
Fk˜
o k
l
−θ/[1−ρ] 1−ρ n l=1 ql
P
wl κil δlk˜ max {x, v2 }
dx l
, the second term in the integrand can be
expressed as
Y
Fk˜
˜ k
We also have
wl κil δlk˜ max {x, v2 }
Fkj (q) Fk (q)
= λk θ
=
Y
l
e
−λk˜
Pn
l=1
wl κil δ ˜ lk max{x,v2 }
−θ/[1−ρ] !1−ρ
= e−ϕi max{x,v2 }
θ
˜ k
−θ/[1−ρ] 1−ρ −θ/[1−ρ]−1 n q qj , l=1 l
P
so that the first term of the integrand
is wl κil δlk Fkj x2
Fk
n
n
wl κil δlk x
wl κil δlk x
o
o l
=
n X wl κil δlk wl κil δlk −θ/[1−ρ] λk θ x2 x
!1−ρ−1
l=1
l θ−1
= θx
λk
n X
wj κij δjk x
!1−ρ−1 (wl κil δlk )
l=1
= ϕi πijk θxθ−1
63
−θ/[1−ρ]
(wj κij δjk )
θ − 1−ρ
−
θ −1 1−ρ
dx l
The measure Vijk can therefore be expressed as ∞
Z Vijk (v1 , v2 ) =
θ
e−ϕi max{x,v2 } ϕi πijk θxθ−1 dx
v1
with density d2 Vijk (v1 , v2 ) θ = πijk ϕi θv1θ−1 ϕθv2θ−1 e−ϕi v2 dv1 dv2 With this density, we can derive the desired expression for the integral. Z
τ1 θ
qjk1 (s)
−τ2 θ
pi (s)
Z
∞ Z v2
wj κij δjk v1
τ1 θ
min v2 ,
ε v1 ε−1
−τ2 θ
wj κij δjk v1
τ1 θ
min v2 ,
ε v1 ε−1
−τ2 θ
ds = 0
s∈Sijk
Z
0 ∞ Z v2
= 0
0
d2 Vijk (v1 , v2 ) dv1 dv2 dv1 dv2 θ
πijk ϕi θv1θ−1 ϕθv2θ−1 e−ϕi v2 dv1 dv2
Using the change of variables x2 = ϕi v2θ and x1 = ϕi v1θ Z qjk1 (s)
τ1 θ
−τ2 θ
pi (s)
τ1 θ
ds = πijk (wj κij δjk )
s∈Sijk
ϕiτ1 +τ2
Z 0
∞ Z x2 0
( x1−τ1 min x2 ,
ε ε−1
)−τ2
θ x1
˜ (τ1 , τ2 ) πijk ϕτ1 +τ2 (wj κij δjk )τ1 θ = B i
With this lemma in hand we can derive expressions for the trade share and price index. The share of i’s expenditure on goods produced in j by multinationals based in k is ε−1 ˜ 0, ε−1 πijk ϕ θ B i θ =P P R ε−1 = πijk 1−ε ε−1 p (s) ds ˜ ˜ i ˜ j,k Si˜j k˜ πi˜j k˜ ϕi θ ˜ B 0, θ ˜ j,k
R
Sijk
pi (s)1−ε ds
and the price level satisfies
Pi1−ε =
XZ ˜ ˜ j,k
Si˜j k˜
pi (s)1−ε ds =
X ˜ ˜ j,k
ε−1 ε−1 ˜ 0, ε − 1 π ˜˜ ϕ θ = B ˜ 0, ε − 1 ϕ θ B i ij k i θ θ
64
e−x2 dx1 dx2
The labor market clearing condition is
Lj wj Lj
XZ
κij δjk 1 X = ci (s) ds = wj κij δjk qjk1 (s) wj i,k s∈Sijk i,k Z X 1 Xi pi (s)−ε ds = wj κij δjk 1−ε Pi s∈Sijk qjk1 (s)
Z s∈Sijk
1 qjk1 (s)
pi (s)−ε
Xi ds Pi1−ε
i,k
Using Lemma 11, this is
wj L j
=
= =
1 ε ˜ − 1 , ε πijk ϕ− θ + θ (wj κij δjk )−1 B i θ θ wj κij δjk Xi ε−1 ˜ 0, ε−1 ϕ θ B i,k i θ 1 ε X ˜ B −θ, θ Xi πijk ˜ 0, ε−1 B θ i,k θ X Xi πijk θ+1
X
i,k
Source Distributions
Finally, we derive expressions for the source distributions. Before doing
that, it is useful to note the following relationship !ρ 1−ρ πijk
X l
πilk
!1−ρ
n X
−θ/[1−ρ]
1−ρ
(wj κij δjk ) 1 = λk (wl κil δlk )−θ/[1−ρ] Pn −θ/[1−ρ] ϕi (w κ δ ) l il lk l=1 l=1 ρ ! 1−ρ n X 1 −θ/[1−ρ] (wl κil δlk ) × λk ϕi l=1
=
λk (wj κij δjk )−θ ϕi
We first study learning from sellers, where learning is in proportion to expenditure. The source h i1−ε P R distribution is GSi (q) = j,k {s∈Sijk |qjk1
