A Factor-Endowment Theory of Endogenous Firm Heterogeneity

Transcription

1 A Factor-Endowment Theory of Endogenous Firm Heterogeneity Taiji Furusawa Hitotsubashi University Hitoshi Sato Institute of Developing Economies This Draft: June 27 Abstract In the model where the choice of technology by firms endogenously determine productivity differences, we investigate the link between factor endowment and the productivity both in the firm and industry levels. We find among others that firms in capitalabundant countries tend to adopt new advanced technologies more in their production processes, and that opening to international trade increases the average adoption rate of new technologies although firms with intermediate productivity levels adopt less of new technologies than in autarky. Preliminary and incomplete Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo The Institute of Developing Economies (IDE), Wakaba, Mihama, Chiba hitoshi

2 1 Introduction Productivity varies across firms even within the same industry. Some firms are more efficient than others since they hire more-productive workers or they adopt more-advanced production technology. Indeed, Bartel and Lichtenberg (1987) show the evidences that educated workers have comparative advantage in implementing new technology. Doms et al. (1997) also find that the plants that adopt new automation technology tend to have more skilled workers. Even if new advanced production technology is in the public domain, some firms choose not to adopt it perhaps due to the lack of human capital that is essential to the technology adoption. This linkage between the difference in the intrinsic productivity of firms, partly caused by the difference in the amount of human capital, and the adoption rate of new technology explains the coexistence of capital-intensive advanced technology and labor-intensive old technology within the same industry. Anecdotal evidences suggest that even the same firm often uses different technology depending on the plant location. Firms tend to adopt capitalintensive advanced technology in capital-abundant developed countries while they tend to use labor-intensive old technology in labor-abundant less-developed countries. Firms in lessdeveloped countries use less-advanced technology on average, not just because of its lack of capacity to absorb new technology but perhaps because that less-developed countries tend to be labor-abundant so firms that operate there deliberately adopt old technology to save capital. Melitz (23) introduces productivity difference across firms within the same industry and builds a general equilibrium model in which less productive firms sell only domestically while productive firms sell both domestically and internationally. He also investigates how trade liberalization affects firms with different productivity levels differently. In his model, different productivity levels are assigned to ex ante symmetric firms exogenously; and once assigned, productivity of a firm remains unchanged. In reality, however, efficiency gap between productive firms and unproductive ones tend to expand with technological improvement. In our model, we partially endogenize firm-specific productivity levels by allowing firms to choose either labor-intensive old technology or capital-intensive new technology. Laborintensive old technology is standardized so that all firms can utilize it in the same manner. Capital-intensive new technology, on the other hand, requires cognitive skills that 1

3 vary across firms, so the difference in productivity arises once they adopt new production technologies. Our model specification is rich enough so that efficiency gap between inherently-productive firms and inherently-unproductive firms expands with technological improvement; inherently-productive firms become more efficient as the technology advances while inherently-unproductive firms stop adopting new technologies once the technological frontier reaches a certain level. In the model where the choice of technology by firms endogenously determine productivity differences, we investigate the link between factor endowment and the productivity both in the firm and industry levels. We find among others that firms in capital-abundant countries tend to adopt new advanced technologies more in their production processes, and that opening to international trade increases the average adoption rate of new technologies although firms with intermediate productivity levels adopt less of new technologies than in autarky. Bernard et al. (27) also investigate the link between factor endowment and the industry-wide productivity distribution across firms. However, they assume as Melitz (23) that the productivity of each firm is exogenously given. 2 The Model This section lays out the model that contains two sectors, two factors, and a continuum of potentially heterogenous firms. One sector produces a homogenous good that serves as the numeraire in the model. The other sector is characterized by a continuum of differentiated goods à la Melitz (23). However, firms heterogeneity stems from each firm choice of production technologies. 2.1 Preferences Consider an economy populated with E consumers that have identical preferences as follows: U = x + lnc, (1) where x is consumption of the homogeneous good and C is an index of consumption of the differentiated goods. C is further defined by [ σ/(σ 1) C = x(i) di] (σ 1)/σ, (2) i Ω where x(i) is the consumption for a differentiated commodity i and Ω denotes the set of available goods. The varieties of differentiated goods are substitutable and an elasticity of 2

4 substitution between any two varieties is equal to σ > 1. As is well known, these preferences yield the following iso-elastic demand function for differentiated good i of x(i) = Ep(i) σ, (3) P 1 σ where p(i) represents the price of differentiated commodity i and P the price index such that 2.2 Production Technology [ 1/(1 σ) P = p(i) di] 1 σ. (4) i Ω We assumed that production of the homogenous good requires labor only. Any firm can produce the homogenous good with 1/w units of labor per unit of output, which ensures that the equilibrium wage rate is equal to w. Production of differentiated products requires a continuum of intermediate goods of which the set is the interval [,1]. Specifically, the following Cobb-Douglas is employed for the production of each variety of differentiated goods: where y(j) is input of the intermediate good j. [ 1 ] x(i) = exp ln y(j)dj, (5) The intermediate goods are produced by labor and capital. There potentially exist two types of technologies for each intermediate good so that firms producing differentiated products choose which technology they adopt for each intermediate good. The first technology is a labor-intensive technology in which the production of intermediate good j involves a constant marginal requirement l(j) units of labor. We assume that this production technology is well known to the economy and always available to any firms. The second-type technology contrasts with the first-type in several respects. First, the second type is capital-intensive: it saves labor but requires more capital relative to the first-type technology. Second, the extent to which firms can efficiently use the second-type technology is positively correlated to their intrinsic productivity levels (or skills). Third, the availability of the second-type technology is limited only to some intermediate goods. This is a heuristic way to introduce a technological frontier into the economy: an expansion of the range of intermediate goods for which the second-type technology is available implies that new technologies and physical capital are complements with skilled labor. As suggested by 3

5 the literature of skill-biased technological change (SBTC) such as Autor et al. (1998), Doms et al. (1997), and Bresnahan et al. (22), these specifications are empirically consistent. Specifically, in order to simplify the algebra, we assume that the first and the second technologies are extreme in terms of factor intensity so that the second technology requires a constant marginal requirement k(j)/ϕ units of capital where ϕ > denotes firm-specific productivity levels. Without loss of generality, we order intermediate goods j by the extent to which the second technology replaces labor with capital. Defining the following function λ(ϕ,j) that measures the capital required to save labor by one unit: λ(ϕ,j) = k(j) ϕl(j), (6) we order j such that λ(ϕ,j) is non-decreasing in j, that is, an intermediate good with higher j has a higher capital cost of labor reduction. Note that the second technology is more efficient for firms with high productivity levels in the sense that they would save the same amount of labor with fewer capital than firms with low productivity levels. We introduce the technological frontier to the model by defining a z [,1] such that capital-intensive technologies are known for all intermediate goods for j [,z] while they have not yet discovered for the remaining intermediate goods j (z, 1]. Thus, while firms can choose whether or not to adopt the second-type technology for intermediate j [,z], they have to use the first-type technology for the remaining. A rise in z means that the second-type technology becomes available for more intermediate goods with increasing λ. This specification implies that newly discovered technologies are more capital intensive than those discovered before. Although z is exogenously given in the base model, we will later discuss an endogenized technological frontier as an extension of the model. Imposing these specifications, we will refer to the first-type technology as the old technology and the second-type technology as the new technology. Firms adopt the new technology as long as they can reduce the production costs. Supposing that a firm adopts the new technology for intermediate goods in [,m] where m z, the unit variable cost function is given by [ m ĉ(ϕ,m,r,w) = exp ln rk(j)dj + 1 m ] ln wl(j)dj m ln ϕdj, (7) where r is the rental rate of capital. The partial derivative of c with respect to m, ĉ m (ϕ,m,r,w) = c(ϕ,m,r,w)[ln rk(m) ln wl(m) ln ϕ], (8) 4

6 implies that as long as rk(m)/wl(m) ϕ holds, ceteris paribus, the firm can reduce the unit variable cost by expanding the adoption of the new technologies if they have been discovered. Otherwise, the unit variable cost is increasing in m. Using the expression in equation (6), this condition can be rewritten as follows: ĉ m (ϕ,m,r,w) λ(ϕ,m) w r, (9) which also reveals that normalizing l(j) to one for all j does not alter the analysis in essential manners so that we will henceforth simplify λ such that λ(ϕ,j) = k(j)/ϕ. The extent to which firms adopt the new technology, m(ϕ,r,w), is given by z, if rk(z)/w ϕ m(ϕ,r,w,z) = k 1 (ϕw/r), if rk()/w < ϕ < rk(z)/w, otherwise. Consequently, the unit variable costs for firms with productivity level ϕ are [ ] m(ϕ,r,w,z) c(ϕ,r,w,z) = exp ln rk(j)dj + (1 m(ϕ,r,w,z))ln w m(ϕ,r,w,z)ln ϕ. The upper panel of Figure 1 illustrates three examples of λ with different productivity levels for a given w/r. λ is nondecreasing in j by construction. Firms with productivity ϕ 1 have the highest productivity level among the three and λ(ϕ 1,j) < ω for the entire range of intermediate goods, which implies that the unit variable cost is monotonically decreasing in z as is shown as c(ϕ 1,r,w,z) in the lower panel of the figure. Such firms are willing to adopt the new technologies for all intermediate goods if they are available. Thus, the optimal technology adoption is z. Firms with ϕ 2 are in an intermediate case; by adopting the new technologies up to k 1 (ϕw/r), the firm can reduce its production cost. However, it does not have incentive to adopt the new technologies for intermediate goods beyond this since adopting such technologies increases the production cost. Hence, the unit variable cost c(ϕ 2,r,w,z) for z k 1 (ϕw/r). Finally, firms with ϕ 3 are in the opposite situation to those with ϕ 1. Adopting new technologies is unprofitable for any intermediate goods so that such firms stick to the old technologies even though the new technologies become available. The corresponding unit variable cost is c(ϕ 3,r,w,z) = w in the figure. To simplifying the notation, we henceforth supress the parameter z from the arguments of the functions m, c, and others, except in the comparative statics analysis with respect to z. (1) (11) 5

7 As depicted in Figure 2, firms are categorized into three cohorts according to their intrinsic productivity levels. Adopting the old technology for all intermediate, the least productive firms with ϕ < rk()/w are homogenous in the production cost even though their intrinsic productivity levels vary. Firms with mid-range productivity levels of rk()/w < ϕ < rk(z)/w choose the extent of technological adoption as interior solutions. It follows that technological progress (i.e. a rise in z) does not directly affect this group s technological adoption. Rather, this group may change the technological adoption through general equilibrium ramifications, such as changes in the rental rate r. In contrast, the most productive firms with ϕ rk(z)/w are directly affected by technological progress since their technological choices are corner solutions. Their technological choices are independent from changes in factor prices. Facing the iso-elastic demand function in (3), a firm with productivity level ϕ sets the price at p(c(ϕ, r, w)) = σc(ϕ, r, w)/(σ 1) and produces x(ϕ,r,w) = E [ ] p(c(ϕ,r,w)) σ (12) P P units of variety i. Assuming that each differentiated good requires the same amount of labor input f > as a fixed cost, the profit, net of the fixed cost wf, is given by π(ϕ,r,w) = E [ ] p(c(ϕ,r,w)) 1 σ wf. (13) σ P As is standard in the Dixit-Stiglitz-type monopolistic competition, the ratios of any two firms outputs x(ϕ,r,w), and revenues R(ϕ,r,w) depend only on the ratio of their unit variable costs: i.e., x(ϕ 1,r,w) x(ϕ 2,r,w) = [ ] c(ϕ1,r,w) σ, c(ϕ 2,r,w) R(ϕ 1,r,w) R(ϕ 2,r,w) = [ ] c(ϕ1,r,w) 1 σ. (14) c(ϕ 2,r,w) 2.3 Entry and Exit The design of differentiated goods requires an initial investment f e > measured in units of labor. After investing in f e units of labor, each firm independently draws its intrinsic productivity level ϕ from the common probability distribution on (, ) having the cumulaive distribution function H with continuous density function h. that is a random variable following a common distribution h(ϕ) with support (, ). Since the the profit π(c(ϕ,r,w)) is strictly increasing in ϕ for ϕ > rk()/w, there exist two cases of the marginal firms earning zero profit. The first case is that the marginal 6

8 firms productivity level ϕ is above rk()/w. Firms with productivity levels below ϕ earn negative profits and immediately exit from the market. Thus, the zero-profit condition for the marginal firms is given by E σ [ p(ϕ ],r,w) 1 σ = wf. (15) P The second case is that even the least productive firms with ϕ k()/z may enter the market, earning zero profit. In such a case, there does not exist cutoff productivity levels. Although it is not difficult to handle these two cases, we concentrate on the case where ϕ is large enough that m(ϕ,r,w) >, which is ensured, for example, by assuming k() =. Letting N e be a mass of new entrant firms and N a mass of surviving firms, the entry-exit dynamics are simply [1 H(ϕ )]N e = N, (16) where H(ϕ) is a cumulative distribution of productivity parameter ϕ. Let c(ϕ,r,w) denote the unit variable cost for the average firms such that [ 1/(1 σ) c(ϕ,r,w) = c(ϕ,r,w) 1 σ dµ ϕ (ϕ)]. (17) The price index P in equation (4) can be written as follows: [ 1/(1 σ) P = p(c(ϕ,r,w)) 1 σ Ndµ ϕ (ϕ)] = N 1/(1 σ) p( c(ϕ,r,w)), (18) where p( c(ϕ,r,w)) = [ p(c(ϕ,r,w)) 1 σ dµ ϕ (ϕ) ] 1/(1 σ) and µϕ (ϕ) is the conditional distribution of h(ϕ) such that h(ϕ) µ ϕ (ϕ) = 1 H(ϕ if ϕ ϕ ) (19) otherwise. The price p( c(ϕ,r,w)) is an average price in the sense that the price index P can be replicated by N homogenous firms with price p( c(ϕ,r,w)). In fact, as is shown in the appendix, p( c(ϕ,r,w)) is a weighted average of the prices of differentiated commodities. Since the revenue earned by firms with the average unit variable cost is simply E/N, the zero-profit condition in equation (15) can be expressed by E σn [ c(ϕ ],r,w) 1 σ c(ϕ = wf. (2),r,w) 7

9 The free entry condition ensures that the ex ante expected profit equals the entry cost wf e. Noting that the profit of the average firms, net of the fixed cost wf, is simply π = E/(σN) wf, the free entry condition is expressed as [ ] E [1 H(ϕ )] σn wf = wf e. (21) Using equation (2), the profit of the average firm, π = E/(σN) wf can be written such that π(ϕ,r,w) = wfφ(ϕ,r,w), (22) where φ(ϕ,r,w) [ c(ϕ,r,w)/c(ϕ,r,w)] 1 σ 1. Substituting this expression into the free-entry condition in (21), we obtain Φ(ϕ,r,w) [1 H(ϕ )] φ(ϕ,r,w) = F, (23) where F f e /f. Equation (23) restates the free-entry condition in terms of the price of capital r and the cutoff productivity level ϕ. 2.4 Factor Market By Shephard s lemma, capital requirements for the unit production of a differentiated product are c(ϕ, r, w)m(ϕ, r, w)/r (see the Appendix). Thus, the full-employment condition is given by c(ϕ,r,w)m(ϕ,r,w) x(ϕ,r,w)ndµ ϕ (ϕ) = K, (24) r where K denotes the capital endowment. Using equation (18) and the pricing rule, we can rewrite the full-employment condition as follows: αe m(ϕ,r,w) = rk, (25) where α (σ 1)/σ represents the share of income that distributed to the intermediate goods and m(ϕ,r,w) is the average level of technological adoption such that [ ] c(ϕ,r,w) 1 σ m(ϕ,r,w) = m(ϕ,r,w) c(ϕ dµ ϕ (ϕ), (26),r,w) where the relative revenues of differentiated goods are used as the weights (see (14)). Recalling that the production technology for the final good is a Cobb-Douglas type and the new technology employs capital only, equation (25) implies that the average technological adoption m also means the income share distributed to the capital. 1 1 αe(1 m) is distributed labor force employed in the old technology. The aggregate profit (1 α)e is distributed labor force used for the fixed cost f and the entry investment f e. 8

10 Since m is a summary index of the technology adoption, it is central to understand how m is affected by changes in cutoff productivity levels ϕ, the rental rate r, and the wage rate w (i.e., the productivity of the nuemeraire good). The following lemma states the results. Lemma 1. Given that all surviving firms adopt the new technologies for some intermediate goods, the average technological adoption m is increasing in ϕ and w and decreasing in r. Namely, m ϕ (ϕ,r,w) >, m r (ϕ,r,w) <, m w (ϕ,r,w) >, for ϕ < αez K k(z) w αez and r < K. Otherwise, m is constant at z. Proof. See the Appendix. The proof is straightforward but tedious so that we relegate it to the Appendix. As shown in the appendix, if the cutoff productivity level is beyond the above threshold value, all firms adopt the new technologies for all intermediate goods. As a result, the average productivity level is pinned down at z and becomes insensitive to ϕ,r, and w. Otherwise, there are the two types of firms: firms that partially adopt the new technologies and firms that fully adopt them. In what follows, we discuss this case. Intuitively, an increase in the cutoff productivity level ϕ forces firms with the lowest productivity levels to exit from the market so that the average level of technological adoption rises. The responses of the average technological adoption to changes factor prices also seem quite initiative. When the rental rate is high, firms decline to adopt the new technologies, which reduces the average adoption level. The result of a higher wage rate should be opposite. In addition to this substitution effect, a rise in the rental rate lowers the average adoption level through changing each firm s weight in the average technological adoption (recall that m is a weighted average of firms technological adoption levels). Since firms with higher productivity levels intensively use capital relative to those with lower productively levels, the impact of a higher rental rate on revenues is greater for firms with higher productivity levels than those with lower productivity levels. The revenues earned by firms with technological adoption levels higher (lower) than the average adoption level decline more (less) than the average revenue. As a result, the impact of firms with higher adoption levels on the average 9

11 technological adoption reduces while that of firms with lower adoption levels increase, which leads to a decline of the average technological adoption level. The free-entry condition in (23) and the full-employment condition in (25) simultaneously solve two endogenous variables ϕ and r. Substituting ϕ and r into equation (22) generates the average profit π. Since the mass of surviving firms can be written by N = the model can solve all endogenous variables, ϕ,r, and N. 3 Properties of Equilibrium E σ( π + wf), (27) This section solve the model and examine several comparative statistics. 3.1 Equilibrium We start to solve the model with clarifying properties of the free-entry condition in (23) and the full-employment condition in (25). It is rather straightforward to establish the followings: Lemma 2. The free-entry condition in equation (23) is represented by a down-sloping schedule in a (r,ϕ ) space. The full-employment condition in equation (25) is an upwardsloping schedule with an upper-bound rental rate. Proof. See the Appendix. Relegating the proof to the appendix, we here state intuitions of the lemma. When the entry to the market becomes more difficult (a higher ϕ ), the average profit π (this is also ex-ante expected profit) must rise in order to make firms incur the initial investment f e. This is enabled by a lower rental rate that raises the profits earned by inframarginal firms through both reducing the production costs and encouraging the technology adoption. Thus, the free-entry (FE) schedule is down-sloping in the (r,ϕ ) space. With respect to the full-employment (FM) schedule, a higher capital price lowers the demand for capital by discouraging the technology adoption and rasing the production costs. In order to compensate this demand decline, the active firms must become more productive on average. This implies that surviving firms should be more selective so that a higher ϕ is needed. From Lemma 2, we state the following proposition: 1

12 Proposition 1. There exists a unique equilibrium referenced by the vector (r,ϕ ) that simultaneously solve the free-entry condition in (23) and the full-employment condition in (25). Proof. See Appendix. Figure 3 depicts the determination of equilibrium values of r and ϕ. It is possible that the FE schedule (the free-entry condition) cuts through the FM schedule (the fullemployment condition) at a vertical part of the FM schedule. However, in this case, all surviving firms are homogenous in terms of technological adoption. 2 For making things interesting, we will fucus on the case in which firms are heterogenous in technological adoption. Once r and ϕ are determined, the characteristics of the average surviving firms such as the average unit variable cost c (equation (17)), the average profit π (equation (22)), and the average technology adoption m (equation (26)) are identified, which means that the firms distribution including the mass of surviving firms is also determined. Notice that a change in the size of the economy (i.e. a proportional change in E and K) does not alter both the free-entry (FE) schedule and the full-employment (FM) schedule. This means that the profile of the average firm is independent of the size of the economy: an increase in the country size simply make the distribution of firms thick (a rise in N), preserving the shape of the distribution of firms the same as before. Hence, the size independency, a standard property of the monopolistic competition model with the CES preferences, holds in this model from the perspective of the average firm. 3.2 Comparative Statics We perform several comparative statics analysis. Capital Endowment Consider an increase in the capital endowment K (i.e., an increase in the capital endowment per capita K/E). K shows up only in the full-employment condition. Thus, while the free-entry (FE) schedule is intact, the full-employment (FM) schedule shifts leftward. As a result, the rental rate r declines and the cutoff productivity level ϕ rises: firms with the lowest productivity levels will exit from the market even though a lower 2 Thus, this case is essentially the same as the one examined by Melitz (23). 11

13 rental rate stemming from the capital expansion reduces their production costs and facilitates the adoption of the new technologies that are not used before the capital expansion. The key mechanism here is the complementarity between capital and intrinsic productivity of firms: the capital expansion is more beneficial to firms with higher productivity levels than those with lower productivity levels. As a result, some least productive firms have to exit from the market due to their lack of competitiveness. 3 Although not all firms cannot benefit from the capital expansion, the aggregate productivity and the average technological adoption level increase. Fixed Cost Consider a decrease in the fixed cost f. An example of this exercise is a relaxed regulation of entry of start-up firms. Interestingly, Djankov et al. (22) reveals the regulation of entry of start-up firms significantly vary across countries. For example, they document that while the cost of necessary process for starting operation in New Zealand is virtually nothing (three steps procedures that take three days), that in France is about 14 per cent of GDP per capita (15 steps that takes 53 days). 4 In the model, a decrease in f (i.e. an increase in F) shifts the free-entry (FE) schedule downward since all operating firms become more profitable for any given r and the cutoff productivity level, hence, must decline (recall that the cutoff profit level is π(ϕ,r) = f). As a result, both the cutoff productivity level ϕ and the rental rate r fall. The model s interpretation is quite intuitive. Lowering the entry cost, for example, invites less productive firms entry (a lower ϕ ). However, since the tail of firm distribution is stretched, the average technological adoption decreases, which leads to a lower rental rate. 5 (With respect to changes in welfare, see the appendix.) These results are recorded as follows. Proposition 2. An increase in the capital endowment facilitates the adoption of the new technologies and the least productive firms exit from the market (a rise in the cutoff productivity level ϕ ). As a result, the aggregate productivity rises. A decrease in the fixed cost f raises profitability of operating firms and encourages more firms entry into the market. However, the average technological adoption level falls, leading 3 This is somehow similar to the Rybcznski effect by which a increase in capital leads to an expansion of the capital-intensive sector and a shrink of the less-capital intensive sector in the standard Heckscher-Ohlin model. 4 Djankov et al. (22) study 85 countries and find that the cost-to-per-capita-gdp ratio decreases uniformly with GDP per capita. 5 Although each operating firm adopts the new technologies more, the increase in the number of low productivity firms lowers the average technological adoption. The intensive margin of the technological adoption is dampened by the (negative) extensive margin of the technological adoption. 12

14 to a lower rental rate. Proof. See the Appendix. Technological Progress An expansion of the technological frontier (i.e. an increase in z) brings about more ambiguous results. First, although the new technology becomes available in more intermediated goods, only firms with ϕ > rk(z) (i.e. firms choosing the full adoption of the new technologies) adopt the new technologies. Thus, in such a situation, the entry to the market becomes difficult, which leads to an upward shift of the free-entry (FE) schedule. The facts that the most productive firms uses the new technologies more and that the least productivity firms will retire from the market raise the average technological adoption, which leads to a rise in the capital price. The expansion of the technological frontier also affects the full-employment condition. Since the group of the most productive firms increases the demand for capital, the rental rate goes up (for any cutoff productivity levels) and the full-employment schedule shifts rightward. This change in the factor market raises the rental rate further and dampens the favorable effect of the technological progress on the most productive firms (recall the complementarity between skill and capital). Therefore, it is not clear whether or not the cutoff productivity rises in a new equilibrium (see Figure 5). Productivity in the Other Sector Consider now a rise in w, namely, an increase in the productivity of the homogenous good sector. Facing a decline in the relative price of capital (r/w), firms with partially adopting the new technologies (i.e. productivity level below rk(z)/w < ϕ) adopt the new technologies more. The unit variable costs of firms with lower productivity levels increase more than the unit variable cost of the average firms since low productivity firms have high labor intensity. Thus, the average profit π increases, which leads to an upward shift of the free-entry (FE) schedule. At the same time, as in an expansion of the technological frontier, the capital market becomes tight and the fullemployment (FM) schedule shifts rightward. Thus, in total, while the rental rate rises, it is not clear if the cutoff productivity rises. The impact of the technological progress in the other sector on r and ϕ is similar to a rise in z. Also, in both cases, the average technological adoption rises. However, benefitted firms are opposite since while the most productive firms expand technological adoption in the case of a higher z, the mid-range and low productivity firms expand technological adoption in the case of a higher w. This 13

15 observation is recorded in the following proportion: Proposition 3. An expansion of the technological frontier (a rise in z) increases the rental rate but the effect on the cutoff productivity level is ambiguous. An increase in the productivity in the homogenous good sector has the similar effects on the rental rate and the cutoff productivity level. However, an expansion of z benefits the most productive firms while a productivity improvement in the homogeneous good sector may benefit less productive firms. Proof. See the Appendix. 4 Open Economy Analysis This section extend the basic mode to an open economy setting where n + 1 symmetric countries freely trade. As is discussed above, opening up free trade without trade costs does not alter firm distribution. Thus, following Melitz (23), we introduce an icebergtype trade cost and an additional fixed cost for exporting. To simplifying the notation, we normalize w to one and suppress w from the arguments of the functions. 4.1 Set-up Assuming an iceberg-type trade cost τ > 1, each firm s pricing rule is given by p(c(ϕ, r)) = c(ϕ, r)/α for the domestic market and p(τ c(ϕ, r)) = τ p(c(ϕ, r)) for the foreign market. Denoting the revenue earned from the domestic market by R(ϕ,r)(= E(p(c(ϕ,r))/P) 1 σ ), the revenue from the foreign market is given by R x (ϕ,r) = E(τp(ϕ,r)/P) 1 σ = τ 1 σ R(ϕ,r). Thus, if a firm does not export, its revenue is R(ϕ,r) while if it exports, its revenue is R(ϕ,r) + nr x (ϕ,r) = (1 + nτ 1 σ )R(ϕ,r). The profits earned from the domestic market and a foreign market are π d (ϕ,r) = R(ϕ,r) σ f, π x (ϕ,r) = R x(ϕ,r) σ f x. (28) Let ϕ x is a cutoff productivity level with which firms can earn zero profit from exporting. It is easily checked that τ σ 1 f x > f is necessary for ϕ x > ϕ. We assume that this condition is satisfied for any τ > 1 (i.e. f x > f is sufficient). With this assumption, the ex-ante conditional probability that a firm can export is given by p x = [1 H(ϕ x)]/[1 H(ϕ )]. The mass of exporting firms is expressed by N x = p x N. The total mass of varieties that consumers can enjoy is given by N t = N + nn x = (1 + np x )N. 14

16 Similar to equation (17), the average unit variable cost of exporting firms is [ 1/(1 σ) c(ϕ 1 x,r) = 1 H(ϕ x ) c(ϕ,r) dh(ϕ)] 1 σ. (29) Hence, the average price of exporting firms is τp( c(ϕ x,r)). We have two zero-profit conditions: ϕ x π d (ϕ,r) =, π x (ϕ x,r) =. (3) Using these zero-profit conditions, the average profits from the home market and the foreign market are where φ(ϕ,r) = [ c(ϕ,r)/c(ϕ,r)] 1 σ 1. π d = fφ(ϕ,r), π x = f x φ(ϕ x,r), (31) Also, the two zero-profit conditions suggest that [ ] c(ϕ τ 1 σ x,r) 1 σ c(ϕ = f [ ] x,r) f c(ϕ x,r) = c(ϕ,r) f 1/(σ 1). (32) τ That is, ϕ x can be expressed as a function of ϕ and r. Now the free-entry condition can be expressed by Φ(ϕ,r) [1 H(ϕ )][φ(ϕ,r) + p x n(f x /f)φ(ϕ x,r)] = F. (33) The full-employment condition is obtained in a similar manner to the base model: αen N t ϕ [ ] c(ϕ,r) 1 σ c t (ϕ m(ϕ,r),r) + αenn x N t h(ϕ) 1 H(ϕ ) dϕ ϕ x [ τc(ϕ,r) c t (ϕ,r) ] 1 σ m(ϕ,r) f x h(ϕ) 1 H(ϕ dϕ = rk, (34) x) where c t (ϕ,r), the average unit variable cost of all firms in the market, is given by a weighted average of c and c x : i.e., c t (ϕ,r) 1 σ = N N t c(ϕ,r) 1 σ + nn x N t c x (ϕ x,r) 1 σ = N N t [ c(ϕ,r) 1 σ + np x c x (ϕ x,r) 1 σ]. (35) With a little arrangement, we can rewrite the full-employment condition as follows: [ 1 1 H(ϕ αe ) ϕ c(ϕ,r) 1 σ m(ϕ,r)h(ϕ)dϕ + npx ] 1 H(ϕ x ) ϕ τc(ϕ,r)1 σ m(ϕ,r)h(ϕ)dϕ x c(ϕ,r) 1 σ + np x c x (ϕ x,r) 1 σ = rk, (36) 15

17 which implies that the inside of the square brackets is an open economy version of the average technology adoption, m t. Note that when n = (i.e. autarky), it becomes identical to the average technology adoption m in equation (25). Therefore, we have the open economy version of the full-employment condition similar to that in the closed economy: αe m t (ϕ,r) = rk, (37) where m t (ϕ,r) is defined by the inside of the square brackets in equation (36). 4.2 Equilibrium Having the free-entry condition in (33) and the full-employment condition in (37), we are now prepared to determine the equilibrium in the open economy. First, notice that the basic properties of the free-entry condition and the full-employment condition are unchanged from the base model. The free-entry condition is a down-sloping schedule while the fullemployment condition is an upward-sloping schedule in a (r,ϕ ) space. Therefore, the two schedules can determine a unique set of ϕ and r. What would happen to the equilibrium values of ϕ and r by opening up trade? In order to see this, it is enough to see the impact of an increasing in n, the number of foreign countries, on equilibrium values of ϕ and r. The results are summarized in the following proposition. Proposition 4. Free trade with identical economies raises the rental rate relative to autarky. However, it is ambiguous whether or not the entry to the market becomes difficult relative to autarky. Proof. See the Appendix. The effect of opening up international trade is similar to that of an expansion of the technological frontier. Given r, the ex-ante expected profits φ(ϕ,r) + p x n(f x /f)φ(ϕ x,r) increases, which shifts the free-entry schedule in (33) up. The rental rate and the cutoff productivity level go up. As in Melitz (23), some productive firms only access to the foreign markets while some less productive firms exit from the domestic market. However, since exporting firms are more capital intensive than the (total) average firm, the capital market becomes tight, which raises the capital price furthermore. The impact of a tight capital market is more severe to high productivity firms due to the skill-capital complementarity. Thus, this capital market effect mitigates the firm selection effect stemming from opening up trade. Hence, in total, it is not clear whether or not the cutoff productivity level rises. 16

18 5 Concluding Remarks We have set up an endogenously-heterogeneous-firm general equilibrium model and showed, among others, that firms in capital-abundant countries tend to adopt new advanced technologies more in their production processes and that opening to international trade increases the average adoption rate of new technologies although firms with intermediate productivity levels adopt less of new technologies than in autarky. We have also shown that the technological improvement, represented by an increase in the technology frontier beyond which only old technology is available, raises the rental rate so firms with high productivity levels adopt more of new technology than before while those with intermediate productivity levels adopt less. Technological improvement is brought to the firms through costly innovation. The incentive to innovate new technology rises with the capital endowment, for example, since it would raise the wage-rental ratio. Endogenizing the innovation process is the next step of research. 17

19 References Autor, D. H., Katz, L. F., and Kruger, A. B. (1998). Computing Inequality: Have Computers Changed the Labor Market? Quarterly Journal of Economics, 113(4): Bartel, A. and Lichtenberg, F. (1987). The Comparative Advantage of Educated Workers in Implementing New Technology. Review of Economics and Statistics, 69(1):1 11. Bernard, A. B., Redding, S. J., and Schott, P. K. (27). Comparative Advantage and Heterogenous Firms. Review of Economic Studies, 74(1): Bresnahan, T. F., Brynjolfsson, E., and Hitt, L. M. (22). Information Technology, Workplace Organization, and The Demand for Skilled Labor: Firm-Level Evidence. Quarterly Journal of Economics, 117(1): Djankov, S., La Porta, R., Lopez-Silanes, F., and Shleifer, A. (22). The Regulation of Entry. Quarterly Journal of Economics, 117(1):1 37. Doms, M., Dunne, T., and Troske, K. R. (1997). Workers, Wages, and Technology. Quarterly Journal of Economics, 112(1): Melitz, M. J. (23). The Impact of Trade on Intra Industry Reallocations and Aggregate Industry Productivity. Econometrica, 71(6):

20 A Appendix A.1 The Average Price From p(ϕ,r,w) 1 σ = p(ϕ,r,w) 1 σ dµ ϕ (ϕ), p(ϕ,r,w) = = = [ p(ϕ,r,w) p(ϕ,r,w) p(ϕ,r,w) ] σ dµ ϕ (ϕ) [ ] c(ϕ,r,w) σ p(ϕ,r,w) c(ϕ dµ ϕ (ϕ),r,w) [ x(ϕ,r,w) p(ϕ,r,w) x(ϕ,r,w) ] dµ ϕ (ϕ) (equation (14) is used) (A.1) where x(ϕ,r)(= [E/P][ p(ϕ,r)/p] σ ) is the output of firms with price p(ϕ,r). Hence, p is the average price weighted by the relative outputs of differentiated products. A.2 The Unit Capital Requirement Denoting the partial derivative of c with respect to r as c r, we obtain [ ] m(ϕ,r,w) c r (ϕ,r,w) = c(ϕ,r,w) r 1 dj + m r (ϕ,r,w)[ln rk(m) lnw ln ϕ], (A.2) where m r is the partial derivative of m(ϕ,r,w) with respect to r. For firms with ϕ rk(z)/w, m r =. For firms with rk()/w < ϕ < rk(z)/w, ln rk(m) lnw ln ϕ =. Thus, the unit capital requirement is given by c(ϕ, r, w)m(ϕ, r, w)/r. A.3 Proof of Lemma 1 As is shown in (1), once ϕ passes through rk(z)/w, all incumbent firms fully adopt the new technologies, which leads to m = z. If this is the case, from the full-employment condition, the rental rate is r = αez K, (A.3) and this is the upper bound of the rental rate. Substituting this rental rate into rk(z)/w, we obtain the threshold value of ϕ as follows: ϕ = αez K k(z) w. (A.4) Now suppose that this is not the case. The partial derivative of m(ϕ,r,w) with respect 19

21 to ϕ is given by [ m ϕ c(ϕ, ) 1 σ h(ϕ )m(ϕ, )c(ϕ, ) 1 σ m(ϕ, )c(ϕ, ) 1 σ h(ϕ )c(ϕ, ) 1 σ] dh(ϕ) ϕ (ϕ,r,w) = [ ] 2 ϕ c(ϕ, ) 1 σ dh(ϕ) = c(ϕ, ) 1 σ h(ϕ ) ϕ [m(ϕ, ) m(ϕ, )] c(ϕ, ) 1 σ dh(ϕ) [ ] 2 ϕ c(ϕ, ) 1 σ dh(ϕ) = h(ϕ )[ m(ϕ,r,w) m(ϕ [,r,w)] c(ϕ ],r,w) 1 σ 1 H(ϕ ) c(ϕ.,r,w) (A.5) Since m(ϕ, ) is nondecreasing in ϕ, it follows that m(ϕ, ) m(ϕ, ) on the support of h. Therefore, m/ ϕ > is obtained. The partial derivative of m(ϕ,r) with respect to r is given by m(ϕ,r,w) r = + 1 σ r m(ϕ,r,w) r [ [ ] c(ϕ,r,w) 1 σ c(ϕ dµ ϕ (ϕ),r,w) {m(ϕ,r,w)} 2 [ c(ϕ,r,w) c(ϕ,r,w) ] 1 σ dµ ϕ (ϕ) { m(ϕ,r,w)} 2 ]. (A.6) It is straightforward to see that the first term in the right-hand side (RHS) is negative since m(ϕ,r,w), if rk(z)/w ϕ; = r k(m( )) rk (m( )) <, if rk()/w < ϕ < rk(z)/w. (A.7) In order to see that the second term is non-positive, notice that the inside of the square brackets is a weighted variance of m(ϕ, r, w). Using the relative revenues as weights, the variance of m(ϕ, r, w) is Var(m) = [ ] [m(ϕ,r,w) m(ϕ,r,w)] 2 c(ϕ,r,w) 1 σ dµ ϕ (ϕ). c(ϕ,r,w) With some algebraic manipulations, we can easily see that [ ] [m(ϕ,r) m(ϕ,r)] 2 c(ϕ,r) 1 σ dµ ϕ (ϕ) c(ϕ, r) = {m(ϕ,r)} 2 [ c(ϕ,r) c(ϕ,r) ] 1 σ dµ ϕ (ϕ) { m(ϕ,r)} 2, (A.8) with equality when all firms are in the same technology adoption level. We now have m [ ] m c(ϕ,r,w) 1 σ r (ϕ,r,w) = r (ϕ,r,w) c(ϕ dµ ϕ (ϕ) + 1 σ Var(m) <.,r,w) r (A.9) 2

22 The proof of m/ w > is similar to that of m/ w <. The partial derivative of m with respect to w is given by m [ ] c(ϕ, ) 1 σ w (ϕ,r,w) = w m(ϕ,r,w) c(ϕ dµ ϕ (ϕ), ) [ + 1 σ [ ] c(ϕ, ) 1 σ m(ϕ, ){1 m(ϕ, )} w c(ϕ dµ ϕ (ϕ) m( ){1 m( )}]. (A.1), ) The first term in the RHS is positive since m r (ϕ,r,w) =, if rk(z)/w ϕ; ϕ) rk (m( )) >, if rk()/w < ϕ < rk(z)/w. (A.11) The inside of the squire brackets in the second term is the covariance of m and 1 m. Obviously, Cov(m, 1 m) <. Therefore, m [ ] m c(ϕ, ) 1 σ w (ϕ,r,w) = w (ϕ,r,w) c(ϕ dµ ϕ (ϕ) + 1 σ Cov(m,1 m) > (A.12), ) w A.4 Proof of Lemma 2 Totally differentiating the free-entry condition in equation (23), dϕ dr = Φ r, (A.13) Φ ϕ where Φ r and Φ ϕ are the partial derivatives of Φ with respect to r and ϕ, respectively: Φ r = (1 [ σ)m(ϕ, ) m(ϕ, ) ] rc(ϕ, ) ϕ m(ϕ, ) c(ϕ, )1 σ dh(ϕ) c(ϕ, ) 1 σ dh(ϕ) <, (A.14) ϕ and Φ ϕ = (1 σ)m(ϕ,r) ϕ c(ϕ,r) 1 σ ϕ c(ϕ,r) 1 σ dh(ϕ) < (A.15) Thus, the slope of the free-entry condition schedule (FE schedule) becomes [ dϕ [ ] dr = m(ϕ, r) c(ϕ,r) 1 σ ϕ r m(ϕ,r) c(ϕ dµ ϕ (ϕ) 1],r) [ = ϕ m(ϕ ],r) r m(ϕ,r) 1, (A.16) with equality whenever m(ϕ,r) = m(ϕ,r) where all firms are homogeneous in terms of productivity levels. Then, totally differentiating the full-employment condition in equation (25), we obtain 21

23 dϕ dr = αe m r(ϕ,r,w) K αe m ϕ (ϕ,r,w) >, (A.17) where m ϕ (ϕ,r) and m r (ϕ,r) are the partial derivatives of m with respect to ϕ and r, respectively. Lemma 1 states that m ϕ (ϕ,r,w) > and m r (ϕ,r) <. Thus, dϕ /dr > is immediate. A.5 Proof of Proposition 1 We show that lim r [ φ(ϕ ],r,w) = for given ϕ and w. For this purpose, it is sufficient to see that lim c(ϕ, ) r c(ϕ, ) =. Without loss of generality, we can set w = 1. Recalling that c(ϕ,r) [ ] c(ϕ,r) 1 σ c(ϕ,r) = c(ϕ dµ ϕ (ϕ), (A.18),r) we compute c(ϕ,r)/c(ϕ,r), using equation (11). This is [ ] c(ϕ, r) m(ϕ,r) c(ϕ,r) = exp ln r ln k(j)dj m(ϕ,r)ln ϕ + m(ϕ,r)ln ϕ. (A.19) Since and m(ϕ,r) [ c(ϕ, r) c(ϕ,r) < exp ln r lim exp r we can conclude that lim r [ c(ϕ,r) c(ϕ,r) [ ln r ] m(ϕ,r) m(ϕ,r) m(ϕ,r) m(ϕ,r) ln k(j)dj ln k(j)dj ] ] (A.2) =, (A.21) =, which implies that lim r [ c(ϕ,r) c(ϕ,r) ] =. Therefore, lim r φ(ϕ,r) =. It immediately follows that when r goes to, ϕ must approaches infinity. This fact along with lemma 2 proves that the FE schedule and the FM schedule have at most one intersection. A.6 Welfare Welfare per consumer is given by W = 1/P. With a little arrangement, we have W = N 1/(σ 1) α c(ϕ,r) [ ] E 1/(σ 1) α = σf c(ϕ,r). (A.22) Therefore, welfare per consumer is negatively correlated to the unit production cost of the cutoff firm. For example, as is shown in the main text, if K increases, ϕ goes up and r falls. Therefore, the welfare per consumer unambiguously increases. In the case of a decrease in 22

24 the fixed cost f, it can be shown that there is no change in W. Suppose that f changes from f to f < f. The ratio of corresponding welfare levels is where (14) is used. W W = = [ ] σf 1/(σ 1) c(ϕ,r ) σf c(ϕ,r) [ R(ϕ,r) R(ϕ,r ) ] 1/(σ 1) [ R(ϕ,r) R(ϕ,r ) ] 1/(1 σ) = 1, (A.23) A.7 Proof of Proposition 2 Consider the partial derivative of φ with respect to w, φ w. [ c(ϕ φ w (ϕ ] [, ) σ c(ϕ ], ) w,r,w) = (1 σ) c(ϕ, ) c(ϕ, ) w c(ϕ, ) c(ϕ, ) {c(ϕ, )} 2 = σ 1 [ c(ϕ ], ) 1 σ w c(ϕ [ m(ϕ, ) m(ϕ, )] >., ) (A.24) Thus, responding to an increase in w, the FE schedule shifts upwards. In order to obtain the results of comparative statics, totally differentiating the FE schedule in (23) and the FM schedule in (25), we obtain where Φ ϕ dϕ + Φ r dr + Φ w dw = df, αe m ϕ dϕ + (αe m r K)dr + αe m w dw = rdk α mde. Φ ϕ = (1 σ)m(ϕ, ) ϕ c(ϕ, ) 1 σ c(ϕ, ) 1 σ dh(ϕ) < ϕ Φ r = (1 [ σ)m(ϕ, ) m(ϕ, ) rc(ϕ, ) 1 σ ϕ m(ϕ, ) c(ϕ, )1 σ dh(ϕ) Φ w = σ 1 [ c(ϕ ], ) 1 σ w c(ϕ [ m(ϕ, ) m(ϕ, )] >, ) m ϕ = h(ϕ )[ m(ϕ, ) m(ϕ [, )] c(ϕ ], ) 1 σ 1 H(ϕ ) c(ϕ >, ) m r = m w = m(ϕ, ) r m(ϕ, ) w ϕ ] c(ϕ, ) 1 σ dh(ϕ) < [ ] c(ϕ, ) 1 σ c(ϕ dµ ϕ (ϕ) + 1 σ Var( m) <, ) r [ ] c(ϕ, ) 1 σ c(ϕ dµ ϕ (ϕ) + 1 σ Cov(m,1 m) >., ) w (A.25) (A.26) Then, [ ] [ ][ ] dϕ 1 αe mr K Φ = r df Φ w dw. dr Φ ϕ (αe m r K) αeφ r m ϕ αe m ϕ Φ ϕ rdk α mde αe m w dw (A.27) 23

25 Φ ϕ (αe m r K) αeφ r m ϕ > is immediate. Hence, for the first result (an increase in K), we have [ ] [ ] dϕ 1 Φr rdk =, (A.28) dr Φ ϕ (αe m r K) Φ r m ϕ Φ ϕ rdk which follows that dϕ /dk > and dr/dk <. For the second result (i.e., an increase in F = f e /f), we have [ ] dϕ 1 = dr Φ ϕ (αe m r K) Φ r αe m ϕ [ ] (αe mr K)dF. (A.29) αe m ϕ df Therefore, dϕ /df < and dr/df <. Finally, with respect to an increase in w, we have [ ] [ ] dϕ 1 αe( ( mr K/(αE))Φ = w + Φ r m w )dw. (A.3) dr Φ ϕ (αe m r K) Φ r αe m ϕ αe( m ϕ Φ w Φ ϕ m w )dw Therefore, while the sign of dϕ /dw is unclear, the capita price goes up, dr/dw >. From the full-employment condition, m/r should be constant. The average technological adoption, hence, increases. A.8 Proof of Proposition 3 In order to simplify the notation, we here set w = 1. The average adoption level m can be written as rk(z) [ ] c(ϕ,r) 1 σ [ ] c(ϕ,r) 1 σ m(ϕ,r;z) = m(ϕ, r) c(ϕ dµ ϕ (ϕ) + z,r) rk(z) c(ϕ dµ ϕ (ϕ).,r) (A.31) Thus, the partial derivative with respect to z, m z, is given by m z (ϕ,r;z) = rk (z)[m(rk(z),r) z] [ ] c(rk(z),r) 1 σ c(ϕ,r) + [1 + z(1 σ)(ln rk(z) ln ϕ)] rk(z) [ ] c(ϕ,r) 1 σ c(ϕ dµ ϕ (ϕ). (A.32),r) Noting that m(rk(z),r) = z and ln rk(z) ln ϕ < for the relevant support of ϕ, m z becomes [ ] c(ϕ,r) 1 σ m z (ϕ,r;z) = [1 + z(1 σ)(ln rk(z) ln ϕ)] c(ϕ dµ ϕ (ϕ) >. (A.33),r) rk(z) Then, consider the partial derivative of Φ(ϕ,r;z) with respect to z. Denoting it by Φ z, [ ] c(ϕ,r) 1 σ Φ z (ϕ,r;z) = (1 σ) rk(z) c(ϕ [ln rk(z) ln ϕ]dh(ϕ) >. (A.34),r) 24

26 Thus, the comparative statics with respect to the technology frontier z is given by [ ] [ ][ ] dϕ 1 αe mr K Φ = r Φz dz dr Φ ϕ (αe m r K) Φ r αe m ϕ αe m ϕ Φ ϕ αe m z dz = 1 [ ] ( Φz (αe m r K) + Φ r αe m z )dz, (αe m ϕ Φ z Φ ϕ αe m z )dz (A.35) where Φ ϕ (αe m r K) Φ r αe m ϕ. Since m ϕ >,Φ z >,Φ ϕ <, and m z >, αe m ϕ Φ z Φ ϕ αe m z > holds, which implies that dr/dz >. However, Φ r αe m z is negative while Φ z (αe m r K) is positive. Thus, in general, the sign of dϕ /dz is not clear. A.9 Proof of Proposition 4 It is immediate to see that Φ in equation (33) is increasing in n. The partial derivative of m t with respect to n is [ m t n = 1 p x c 1 σ 1 H(ϕ τc(ϕ,r) 1 σ m(ϕ,r)h(ϕ)dϕ p x c 1 σ ] x x) 1 H(ϕ c(ϕ,r) 1 σ m(ϕ,r)h(ϕ)dϕ ) = p x c 1 σ c 1 σ x ϕ x [ m x (ϕ x,r) m(ϕ,r)], ϕ (A.36) where [ c 1 σ + np x c x 1 σ ] 2. Since m x (ϕ x,r) m(ϕ,r) >, it follows that m t / n >. Thus, the comparative statics with respect to the number of trading countries is [ ] [ ][ ] dϕ 1 αe mtr K Φ = r Φn dn dr Φ ϕ (αe m tr K) αeφ r m tϕ 1 = Φ ϕ (αe m tr K) Φ r m tϕ αe m tϕ Φ ϕ m tn dn [ ] [ Φn (αe m tr K) + Φ r m tn ]dn. [αe m tϕ Φ ϕ Φ ϕ m tn ]dn (A.37) 25

27 w/r λ(ϕ 3, ) λ(ϕ 2, ) (w/r) λ(ϕ 1, ) 1 j c w c(ϕ 3,r,w, ) c(ϕ 2,r,w, ) c(ϕ 1,r,w, ) m(ϕ 2,r,w, ) z 1 z Figure 1: Technology Adoption and the Unit Variable Costs I 26

28 m z no adoption partial adoption full adoption ϕ c w rk()/w rk(z)/w ϕ Figure 2: Technology Adoption and the Unit Variable Costs II 27

29 ϕ FM αezk(z) wk ϕ FE r αez K r Figure 3: The Equilibrium Cutoff and Capital Price ϕ FM FE r Figure 4: An Increase in the Capital Endowment 28

30 ϕ FM FE r Figure 5: An Expansion of the Technological Frontier 29