Trade Without Scale Effects

Transcription

1 Trade Without Scale Effects Pedro Bento Texas A&M University May 2018 Abstract Across countries and over time, average incomes are related to population density, but not population keeping density fixed). At the same time, the number of firms in both manufacturing and services are essentially proportional to population but unrelated to density keeping population fixed). Why are these distinctions important? Moving from autarky to free trade is much more like increasing population while keeping density fixed. In this paper I extend a simple variety model by considering an economy made up of a large number of geographical markets of fixed area. Firms must incur a cost to access each market, and this cost is convex in the number of markets entered. Increasing both population and the number of markets has no effect on firms market entry decision, and so the number of firms per capita and average incomes remain unchanged. The implications of the model for trade are stark. If two identical countries open up to trade, there are no gains from trade. If two different countries open up to trade but no comparative advantage exists, there are no gains from trade. Countries only trade and gains from trade only exist) if comparative advantage exists. If comparative advantage exists, the gains from trade are lower than in a standard variety model. But if comparative advantage is ignored, the true gains from trade can be much higher than the gains calculated using a standard sufficient-statistic formula. I show these results are robust to other environments, such as a model of Ricardian trade and a model where consumers choose how many varieties to adopt. Keywords: international trade, intraindustry trade, firm size, productivity, markups, comparative advantage, love of variety, Ricardian trade, variety Engel curve, scale effects, increasing returns to scale. JEL codes: F1, O1, O4. I would like to thank everyone for helpful comments and suggestions. All remaining errors are my own. Texas A&M University, Department of Economics, 3056 Allen Building, 4228 TAMU, College Station, TX pbento@tamu.edu. 1

2 1 Introduction A common element across many models of trade for example Krugman, 1979 and Melitz, 2003) and growth Hsieh and Klenow, 2009) is increasing returns to labor at the aggregate level, or scale effects. More simply, these models predict that more populous economies should be richer and more productive. The mechanism, at the most basic level, is gains from variety. Given some fixed cost of producing a variety, a larger population can support more varieties, which increases income per capita and welfare. Krugman 1979) highlights the implication of this for trade. If two identical countries move from autarky to free trade, the gains to welfare are equivalent to the gains from doubling the population: twice the number of available varieties per consumer, and a factor increase in incomes of 2 1 σ 1, where σ describes the elasticity of substitution between varieties in a consumer s preferences. This paper is motivated by the data illustrated by Figure 1. This figure plots GDP per capita against population for 168 countries in the year The line running through the figure represents the OLS-estimated unconditional relationship between the two variables. This relationship is strongly negative and significant. For the sake of comparison, in Figure 2 I plot the relationship predicted by a simple variety model income as a function of population) on top of the data from Figure 1. The contrast is striking. Of course the unconditional negative relationship illustrated by Figure 1 need not be causal, and one could think of several omitted variables that might be driving this pattern. Rose 2006) conducts a much more serious search for scale effects using panel data, considering a host of country-specific variables and using several different estimators. He finds that the negative unconditional relationship between GDP per capita and population is not robust to controlling for other factors. But he also reports no evidence to support a positive relationship. What does seem to be positively associated with incomes and productivity is density. Ahlfeldt et al. 2015) document evidence of this in Berlin after World War II, and there are a large 1 This data is from Rose 2006). 2

3 GDP per Capita log scale, 000s $) LUX NOR USA ISL IRLDNK CHE AUT BEL AUSCAN FINHKG NLD PYF ITA FRA GBR DEUJPN NCL PRI SGP SWE MAC NZLISR ESP BRB BHS MLT BHR CYP SVN PRT GNQ KWT GRC CZE KOR OMN HUN KNA SVK SAUARG ATG MUS EST CRI HRV CHL POL TTO URY LTU MYSZAF MEX GRD BWA LVA RUS BRA DMA TON BLZ GAB BGR LCA NAM MKD PAN DOM TUN COL THA BIH ROM TUR VCT VEN WSM DZA CPV FJI IRN BLR GUY SWZ LBN JAM ALB JOR PRY SLV KAZPER GTM UKR TKM PHL ECULKAMAR EGY VUT NIC SYR IDN ARM HND AZE BOL LSO PNG ZWE SLBDJI GEO GINAGO CMR GHA VNM COM GMBMNG MRT HTIKHM SDN PAK TGO KGZ LAOSENCIVUZB BGD MDA NPL CAF UGA RWA COGBENBFA KEN GNB TCD MDG MOZ NGA ERITJKZMB NER MLIYEM ZAR ETH BDIMWI TZA SLE CHN Population log scale, millions) IND Figure 1: GDP per Capita and Population 168 Economies) GDP per Capita log scale) LUX NOR USA ISL IRLDNK CHE AUT BEL AUSCAN FINHKG NLD PYF ITA FRA GBR DEUJPN NCL PRI SGP SWE MAC NZLISR ESP BRB BHS MLT BHR CYP SVN PRT GNQ KWT GRC CZE KOR OMN HUN KNA SVK SAUARG ATG MUS EST CRI HRV CHL POL TTO URY LTU MYSZAF MEX GRD BWA LVA RUS BRA DMA TON BLZ GAB BGR LCA NAM MKD PAN DOM TUN COL THA BIH ROM TUR VCT VEN WSM DZA CPV FJI IRN BLR GUY SWZ LBN JAM ALB JOR PRY SLV KAZPER GTM UKR TKM PHL ECULKAMAR EGY VUT NIC SYR IDN ARM HND AZE BOL LSO PNG ZWE SLBDJI GEO GINAGO CMR GHA VNM COM GMBMNG MRT HTIKHM SDN PAK TGO KGZ LAOSENCIVUZB BGD MDA NPL CAF UGA RWA COGBENBFA KEN GNB TCD MDG MOZ NGA ERITJKZMB NER MLIYEM ZAR ETH BDIMWI TZA SLE CHN Population log scale, millions) IND Figure 2: GDP per Capita and Population Data vs. Model) 3

4 number of studies in the economic geography literature providing evidence of agglomeration effects at the city level. 2 This relationship between incomes and density is confirmed in the cross-country data of Rose 2006). In an OLS regression of GDP per capita on population, density, and openness, the estimated elasticities of GDP w.r.t. density and openness are both positive and significant, while the estimated elasticity of GDP w.r.t. population is not significantly different than zero. 3 I interpret these relationships as implying that incomes are increasing in density, keeping population fixed. But incomes are not increasing in population, keeping density fixed. Why is this distinction important? That an increase in density increases incomes is consistent with the intuition behind the assumption of scale effects in trade models. But in principle, opening up to trade is much closer to increasing population while keeping density fixed. Before turning to the implications of the above for the gains from trade, I first point to one last piece of related evidence from Bento and Restuccia 2017, 2018). Bento and Restuccia show that the average size of establishments across countries in both manufacturing and services are like average incomes) uncorrelated with population and sectoral employment. 4 Note that the inverse of average size is the number of establishments per person engaged. How is this evidence related to that of scale effects? Imagine that preferences are indeed described by a constant elasticity of substitution CES) aggregate of consumed varieties. Taken together, the data from Rose 2006) and Bento and Restuccia 2017, 2018) imply that increasing population while keeping density fixed increases the number of firms proportionally, but leaves the number of varieties available to each consumer unchanged. In this paper, I extend a simple variety model by considering an economy made up of a large 2 See Duranton and Puga 2014) for a partial survey of this literature. 3 In Rose 2006) openness is measured as the ratio of imports and exports over GDP, while density is population over area. Using data for 161 countries in the year 2000, the estimated elasticities standard errors) are ) for population, ) for openness, and ) for density, with the last two estimates significant at the 5% level. 4 Unlike incomes, the number of establishments per person from Bento and Restuccia 2017, 2018) is not significantly related to measures of density from Rose 2006), neither in manufacturing nor in services. 4

5 number of geographical markets of fixed area). Firms must incur a cost to access each market, and I assume firms face diseconomies of scope with respect to the number of markets they are active in. As a result the cost of entering a market is increasing in the number of markets entered. In reduced form, this cost looks much like the cost of acquiring customers in Arkolakis 2010). The main difference is that here the marginal cost per customer is decreasing in density i.e., the number of consumers per market), rather than the population of the entire economy. In autarky, an increase in population increases density, thereby increasing incomes - the usual scale effects. But if density is held constant if area increases with population), then incomes do not change. The number of firms increases proportionately with population, but with density fixed, the number of customers per firm stays constant. As a result, the number of varieties available to each consumer does not change. The implication of the model for trade is stark. If two identical countries open up to trade, there are no gains from trade. Further, if two different countries with respect to absolute advantage, population, etc...) open up to trade but no comparative advantage exists, there are no gains from trade. In the model, countries only trade and gains from trade only exist) if comparative advantage exists. The intuition behind these results is simple. If a firm s optimal number of customers is less than the total population which I assume to be the case for all firms), then opening up to trade without comparative advantage) does nothing to change the firm s decision. 5 As a result, each consumer is left with the same number of varieties as in autarky. With comparative advantage, each country at least partially) specializes in its relatively productive sector and exports to its trading partner. But the gains from trade are lower, as there are no gains from greater variety. What of intra-industry trade? The model suggests there will be no intra-industry trade, notwithstanding the voluminous evidence to the contrary. But the model and the evidence can be made consistent with each other if one maps model sectors to different groups of varieties within an industry in the data as in, for example, Holmes and Stevens, 2014). If German 5 This assumption of such an interior solution seems realistic. Even Walmart is not close to everyone in the U.S. 5

6 beers are distinct from American beers, and Germany and the U.S. each have a comparative advantage in their respective varieties, and consumers want both rather than just more American beers), then the model predicts intra-industry trade in beers. 6 What of the gains from intra-industry trade? If an economist mistakenly assumes that German and American beers are simply different varieties of the same good and that comparative advantage does not exist, then the usual gains from trade formula Arkolakis, Costinot, and Rodríguez-Clare, 2012) may underestimate the gains from trade. 7 I noted above that the average size of firms is independent of population in the data Bento and Restuccia, 2017, 2018). This fact is consistent with trade models in which markups are constant, like Melitz 2003). But it is inconsistent with models in which markups are variable and entry is endogenous, like Melitz and Ottaviano 2008). In these models, the larger number of firms associated with a larger population increases competition and lowers markups. As a result, the value of entry is reduced and the equilibrium number of firms increases less than proportionately with population. The implication is that on average, firms in larger countries should be larger. I extend my model to allow for endogenous markups through Cournot competition and show that a larger population does not affect average firm size, consistent with the data. The intuition is clear from the benchmark model. Since a larger population does nothing to affect the number of varieties available to each consumer, competition for any particular customer remains unchanged, leaving markups and average firm size also unchanged. If trade is driven by comparative advantage, the model generates outcomes broadly consistent with those highlighted in the trade literature. Only the most productive firms export, and more productive exporters sell more domestically and abroad, both in terms of the number of customers and sales per customer. Opening up to trade increases productivity investment for exporters but reduces it for domestic firms, and this difference is increasing with the size of the destination country. 6 More formally, the elasticity of substitution between the two groups of varieties American and German beers) must be lower than the elasticity between varieties within each group. 7 Levchenko and Zhang 2014) make a similar point. 6

7 The model developed here suggests that gains from trade are driven solely by comparative advantage, rather than intra-industry trade. Given the results of Arkolakis, Costinot, and Rodríguez-Clare 2012), one might be tempted to dismiss this as relatively unimportant, since the gains from trade measured as a function of observed trade flows and trade elasticities) are identical in the Ricardian framework of Eaton and Kortum 2002). But I show that if varieties are perfectly substitutable within an industry that is, if one variety is as good as another), then there are still no gains from intra-industry trade. Within the framework of Bernard et al. 2003), the intuition for this result is the same as in the benchmark model. If firms must make a decision of how many markets to enter, their choice is unaffected by population keeping density fixed) or the freedom to trade. As a result, the expected productivity and price of the most productive seller in a given market is also unaffected by population or trade. Gains from trade can only be realized if there is a systematic comparative advantage across sectors among trading partners. In the benchmark model, scale effects are eliminated by imposing a costly decision on firms about how many markets and therefore customers) to serve. I also show that all of the same results can be obtained if consumers instead choose how many varieties to consume. Incorporating Li s 2018) partial equilibrium model of an Engel curve for varieties into a general equilibrium model, I show that consumers decisions about how many varieties to consume are unaffected by population. As a result, there are no scale effects and no gains from intra-industry trade. The counterfactual implications of scale effects in trade models has been relatively neglected in the trade literature. One notable exception is Ramondo, Rodríguez-Clare, and Saborío- Rodríguez 2016). 8 Motivated in part by Rose 2006), they introduce internal trade costs into the Ricardian model of Eaton and Kortum 2002). When population increases keeping density 8 In forcing their model trade economies to match incomes across countries, papers like Waugh 2010) have implicitly dealt with the lack of scale effects in the data by inferring low exogenous productivity that is, residual productivity not linked to population or trade) in larger countries. Ramondo, Rodríguez-Clare, and Saborío-Rodríguez 2016) make the same observation. 7

8 fixed), more potential producers means the most productive producer for each variety will tend to be even better. With internal trade frictions, this mechanism is dampened. As a result, scale effects are weakened, but not eliminated. The paper proceeds as follows. In the next section I present the benchmark model where firmlevel productivity is exogenous, markups are constant, and firms enter before fully realizing their productivity. In Section 3 I extend the model to allow for endogenous markups through Cournot competition. In Section 4 I make productivity endogenous by allowing for selection and a firm-level productivity investment. The following two sections show that the main results of the paper hold in a Ricardian model and in a model where consumers choose the number of varieties to consume. The last section concludes. 2 Benchmark Model 2.1 Autarky Consider a static two-sector economy where each person consumes a variety of imperfectly substitutable goods from each sector, each variety produced by a different firm. Labor is fixed equal to population) and is the only input used for both production and investment. Firms incur a fixed cost to enter and must also incur a cost to access consumers in different geographical markets. Firms are heterogeneous with respect to their productivity, and for now I assume firms do not learn their productivity until they enter. As a result free entry ensures the value of entry is driven to zero. I begin by describing each firm s cost of market access, before describing the rest of the environment in more detail. 8

9 2.1.1 Market Access Costs After a potential firm pays an entry cost and learns its productivity, it must decide how many markets to enter. I define a market as some fixed area of land, and I assume the number of customers per market what I call density) is constant across markets in an economy. Let this density be denoted by X. To access J markets, a firm must incur a cost of; w c L J, > 1, c L > 0, 1) where w denotes the real wage. Note that this cost is convex in the number of markets entered. I interpret this convexity as being the result of diseconomies of scope with respect to the number of markets a firm serves. I assume these diseconomies are internal to firm, rather than a reduced-form representation of variation in market characteristics. Indeed, I assume all markets within an economy are identical. Once a firm has access to a market, I assume it has access to all of that market s X consumers. It will be useful to transform this decision from one of how many markets to enter into one of how many customers to acquire. Let L i denote the number of customers acquired by firm i, where L i = J i X. Then the cost of acquiring L i customers is; w c L L i X. 2) I define a market as arbitrarily small, and treat this cost function as continuous in L i. Expressed in this way, the cost of acquiring customers is similar to that in Arkolakis 2010). An important difference is that this cost is decreasing in density, rather than population. 9

10 2.1.2 Environment There is a continuum of identical consumers of measure L, each supplying one unit of labor to firms. I assume consumers only value consumption, and are indifferent about how their labor is used. The consumption bundle consumed by each consumer C is a Cobb-Douglas aggregate of two sector-specific bundles, each of which is a CES constant elasticity of substitution) aggregate of each variety consumed; C s = Ns 0 C = C θ 1 1 C θ 2 2, 3) ) σ y σ 1 σ 1 σ ns dn, s {1, 2}, where N s is the number of varieties consumed from sector s by each consumer, σ is the elasticity of substitution between varieties indexed by n), y ns is the quantity of variety n consumed from sector s, and θ 1 + θ 2 = 1. Consumers take wages and prices as given, and I use C as the numéraire for the economy. There are a large number of potential firms, any of which can enter by paying a fixed cost equal to w c E, where w denotes the real wage. Once this cost is sunk firms learn their productivity z, drawn from some distribution with density fz). After entering and learning its productivity, a firm i in sector s must choose the number of consumers to sell to L is ) by incurring a cost as specified in function 2). Once firms have entered and chosen L is, they produce according to the following production function; y is = Z is l is, Z is AA s z is 4) where l denotes labor used in production for each customer, A is a productivity term common to all firms in the economy, and A s is an additional productivity term common to all firms within sector s. I emphasize that y is denotes output produced per customer. Note that all parameters and elasticities are assumed to be the same across sectors, but this assumption is irrelevant for all the qualitative results that follow. 10

11 2.1.3 Equilibrium I focus on the decentralized equilibrium of the economy in which producers in each product market take the real wage w as given and behave as monopolistic competitors. Each entrant chooses how many consumers to sell to, and potential firms enter until the value of doing so is exactly zero. Each consumer takes income Y, the number of varieties available N s, and the price of each variety P ns as given, and chooses a quantity of each available variety to maximize consumption, subject to a budget constraint. The first order conditions for a given consumer s problem imply a fixed fraction of income spent on goods from each sector; P s C s = θ s Y, 5) where P s is the unit price of C s, and the following inverted demand function for each variety consumed by the consumer; P ns = θ s Y C σ 1 σ s ) y 1 σ ns. 6) Operating profits for firm i in sector s per customer are; π is = P is y is wy is Z is = θ s Y C σ 1 σ s ) y σ 1 σ is wy is Z is. 7) Given Z is and aggregate prices, a firm is will demand productive labor such that operating profits are maximized. The optimal decision of each firm depends only on its sector and its productivity, so optimal output and profits per customer) can be expressed as; ) σ σ 1 θ s Y yz, s) = σ wc σ 1 σ s ) σ 1 σ 1 θ s Y πz, s) = w σc σ 1 σ s where Z should be understood to also be a function of the firm s sector. ) σ Z σ, 8) ) σ Z σ 1, 9) 11

12 Profit per customer πz, s) is the marginal benefit to a firm with productivity Z in sector s of acquiring an additional customer. Each firm chooses its number of customers LZ, s) by equating this marginal benefit to its marginal cost, resulting in the following first order condition; wc L LZ, s) X = LZ, s)πz, s). 10) The number of customers acquired by a firm is increasing in Z, as is output per customer. This suggests that the most productive firms might choose to sell to all customers, especially if population is small and/or density is very high. I ignore this possibility here. Given equation 10), the value of a producer in sector s with productivity Z can now be expressed as; V Z, s) = LZ, s)πz, s) wc L LZ, s) X or ) 1 V Z, s) = LZ, s)πz, s). 11) Let NZ, s) denote the mass of firms with productivity Z in sector s catering to a particular customer. Taking into account that income Y is equal to the wage w, equations 3) and 8) can be used to express C s in the following way; ) σ 1 C s = θ s σ z ) 1 Z σ 1 NZ, s)dz σ 1, 12) and operating profits πz, s) can be rewritten as; πz, s) = wθ s Z σ 1 σ ). 13) z Zσ 1 NZ, s)dz Using equations 11) and 13), V Z, s) can now be expressed as; ) 1 θ s LZ, s)z σ 1 V Z, s) = w σ ). 14) z Zσ 1 NZ, s)dz 12

13 I now turn to NZ, s). The number of firms entering with productivity Z in sector s is simply Z the number of firms entering sector s multiplied by f AA s ). Let Ms) denote the number of firms entering. Then the number of firms with productivity Z catering to each consumer NZ, s) is; ) Z LZ, s) NZ, s) = Ms) f AA s L. 15) With this expression in hand, I can now use the free entry condition to solve for the number of firms in each sector. Free entry ensures that the expected value of entering is equal to zero. Since productivity is only realized after entry, this implies that E z V Z, s) = wc E, or; ) 1 θs Ms) = c E σ L. 16) Notice that the number of firms scales up proportionately with population. This implies that the average size of firms does not depend on population, an implication that is consistent with the evidence documented in Bento and Restuccia 2017, 2018) and Fernandes, Klenow, Meleshchuk, Pierola, and Rodríguez-Clare 2015). All other outcomes of interest in the economy can be expressed as functions of Ms). Income as well as output and consumption) per capita is equal to; ) σ 1 Y = AA θ 1 1 A θ 2 2 θ θ 1 1 θ θ 2 2 σ X c L ) 1 σ 1) M1) θ 1 M2) θ 2 L ) 1 z σ 1) σ 1) 1 fz)dz. 17) Like average firm size, income per capita is independent of population if density X is held constant. The number of sector-s varieties available to each consumer Ns) is equal to; Ns) = Ms) L LZ, s)fz)dz = X θs σc L ) 1 Ms) L ) 1 σ 1 z 1 fz)dz ) 1. 18) σ 1) z 1 fz)dz Ns) is also independent of population holding density fixed), but increasing in density. The denser the population, the more customers acquired by each firm. As a result consumers have 13

14 more varieties available, even though the total number of varieties in the economy remains unchanged. Finally, the price of C 1 relative to the price of C 2 is equal to; P 1 P 2 = A 2 A 1 ) 1 M2) σ 1). 19) M1) 2.2 Trade or the Lack Thereof) I now consider the above country opening up to trade with another country. I impose a balanced trade condition, I assume all costs are incurred at home, and I further assume that the cost of acquiring customers for a firm i is; w c L L i + L i ) X, 20) where L i denotes foreign customers, and density is assumed to be the same in both economies. I assume customer acquisition costs depend only on home wages for simplicity, but this assumption does not affect any qualitative results. 9 I start by considering the possibility of trade without comparative advantage, then move on to the case of trade with comparative advantage No Comparative Advantage Imagine trade becomes possible between the economy described above and another economy with the same relative prices P 1 /P 2. Then the gains from trade are exactly zero. This can be seen most intuitively for two identical countries. From equation 16), a doubling of population does nothing to the number of firms per capita. As a result there is no change in the number 9 This holds as long as the fraction of costs incurred at home paying the home wage) does not depend on the fraction of customers that are domestic, as in Arkolakis 2010). 14

15 of varieties per consumer, from equation 18). Combined with the fact that the distribution of productivity across firms remains constant, by construction here, this implies that output per capita is therefore unaffected equation 17). With perfectly frictionless trade, trade volumes are indeterminate, as firms are indifferent about where they sell their goods. But with epsilon trade costs, no trade will occur. To see this, notice that nothing changes with respect to each firm s marginal decision of how many customers to acquire equations 10 and 13). In a model where firms could freely access all consumers, each consumer would automatically have access to more varieties with trade as with a higher population), and so incomes would increase. The lack of gains from trade is not dependent on the foreign country being identical. As long as relative prices are the same in autarky, no firm has an incentive to acquire customers in the foreign country. For example, assume the home country has a higher productivity A across all firms. As a result, wages will be lower in the foreign country. One might think firms at home might want to take advantage of the lower cost of acquiring customers in the foreign country due to lower wages). But these lower costs are exactly offset by lower profits abroad due to lower incomes), making trade unprofitable Comparative Advantage Now assume that there exists comparative advantage across countries. Let stars denote foreign variables. For simplicity, assume that the two countries are identical except for sectoral productivities A s. In particular, let A 1 = A 2 > A 1 = A 2. Assume further that consumers put equal weight on each sector, θ 1 = θ 2 = 1/2. Under these assumptions relative prices will equalize across sectors and countries, and each country will completely specialize in its more productive sector. Equation 16) shows that the total number of firms in each sector will not 10 If density differs across the two potential trading partners, then firms in the low-density country will have an incentive to acquire customers in the high-density country, as the marginal cost of acquiring a customer is lower and the profit from selling to a wealthier customer is higher. But balanced trade precludes this, since no firm in the high-density country has an incentive to export. 15

16 change. Equation 17) therefore shows the gains from trade for each country will simply be; Y trade Y autarky = A1 A 2 ) 1/2, 21) equivalent to productivity increasing in the less productive sector to that of the more productive sector. If firms could freely access all consumers the standard model), then the gains from trade in this example would increase by a factor of 2 1 σ 1. If customers were free, then the number of varieties available to each consumer would double, generating the usual gains from variety. Are the gains from trade therefore lower than has been estimated in the literature? Not necessarily. If researchers ignore comparative advantage, then the gains from trade in an analogous standard model where customers are freely acquired) would be miscalculated as 2 1 σ 1. If A1 /A 2 is high enough, the true gains from trade driven by comparative advantage) could easily be higher. Levchenko and Zhang 2014) make a related point about ignoring comparative advantage within the context of a model with scale effects. 3 Variable Markups and Firm Size In the model above, the average size of firms L/M) is independent of population. This same implication is generated by models with scale effects like for example) Melitz 2003), and is consistent with the evidence on firm size in Bento and Restuccia 2017, 2018). But in models with variable markups like Melitz and Ottaviano, 2008, for example) this is not the case. As population increases, the number of firms increases less than proportionately. More firms lowers average markups, which dampens profits, lowering the value of starting a firm. As a result these models imply that firms in more populous countries should be larger on average, which is counterfactual. In this section I show how the model presented above can be extended to 16

17 rationalize how average markups and firm size can both be endogenous yet remain independent of population. To keep things simple, I assume away heterogeneity in productivity across firms within a sector. The productivity of each firm in sector s is therefore AA s. To endogenize markups I now assume a finite number of firms in each sector, engaging in Cournot competition. With Ns) identical firms competing, the markup over marginal cost of each firm will be; MUs) = ) ) σ Ns), 22) σ 1 Ns) 1 which is decreasing in Ns). Optimal output and profits per customer) will be equal to; ys) = θ s Z s Ns)MUs), 23) and πs) = wθ sz s Ns) MUs) 1 MUs) ). 24) Each firm chooses its number of customers Ls) as before, by equating the marginal cost and benefit of acquiring a customer; wc L Ls) X = Ls)πs). 25) As before, the number of varieties per consumer within each sector Ns) can be expressed as a function of the number of firms Ms); Ns) = Ms) Ls) L. 26) 17

18 Using the free entry condition, I can now solve for the number of firms Ms); ) ) 1 MUs) 1 Ms) = θ s L. 27) MUs) Customers per firm Ls) and varieties per consumer Ns) are; Ls) = Xc 1 L ) 1, 28) 1 and Relative prices are; Ns) = Xc 1 L P 1 P 2 = A 2 A 1 1 N2) N1) ) 1 MUs) 1 MUs) ). 29) ) ) MU1), 30) MU2) and income per capita is; ) θ1 ) θ2 θ1 A 1 θ2 A 2 Y = A N1) θ 1 N2) ) θ 1 2 σ 1. 31) MU1) MU2) Equation 29) makes clear that varieties per consumer Ns) does not depend on population L, and so neither do markups MUs). A larger population generates more firms Ms) 27), but firms do not change the number of customers they cater to Ls) 28). As before, incomes are constant 31). As in Section no trade will occur with another country if relative prices are the same. But again, trade will occur if there exists comparative advantage. Under the same assumptions as in Section 2.2.2, where both countries are identical except that A 1 = A 2 > A 1 = A 2, trade will occur with the home country specializing in sector 1 and the foreign country specializing in sector 2. Markups do not change. 18

19 4 Endogenous Productivity 4.1 Autarky In this section I endogenize firm-level productivity and show that the main results of Section 2 remain unchanged. Here I again adopt the assumptions from Section 2, with heterogeneous productivity across potential firms and monopolistic competition constant markups), but I extend the model in two ways. First, I allow each firm i to choose its productivity A i s) by making the following investment; w c As A γ i s)zσ 1 1), γ > 1, cas > 0, 32) where Z i s) is now equal to z i AA i s). 11 Cross-sector differences in As) will now be driven by differences in c A. Second, I now assume potential entrants know their initial productivity z before entry. As a result, there will be a sector-specific productivity threshold z s) below which potential entrants choose not to enter. I further assume that the mass of potential entrants with productivity z is proportional to population, so that fz) describes the mass of potential firms with productivity z per capita. 12 Finally, I assume fz) = αz α min/z α+1, i.e., z follows a Pareto distribution with lower bound z min and scale parameter α. Equations 8) through 14) still apply, except that integrals must condition on Z > Ẑs) = ẑs)aas). I repeat the relevant equations here for convenience. 11 The presence of z σ 1 1) in the cost function for As) simplifies the model by ensuring every firm within a country-sector chooses the same As). In a dynamic setting, like Atkeson and Burstein 2010), this would ensure that firm growth is independent of initial firm size. 12 I could alternatively assume that each firm is owned by an entrepreneur who chooses to start a firm rather than work, and that entrepreneurial productivity is heterogeneous across the population. I choose not to do this simply to keep the model as close to that of Section 2 as possible. 19

20 Output per customer is; yz, s) = θ s Z σ ẑs) Zσ 1 NZ, s)dz), 33) operating profits per customer are; πz, s) = wθ s Z σ 1 ). 34) σ ẑs) Zσ 1 NZ, s)dz each firm s first order condition with respect to its number of customers is; wc L LZ, s) X = LZ, s)πz, s), 35) and each firm chooses its productivity As) by equating the marginal costs and benefits. ) wc As A γ i s)zσ 1 1) σ 1 = LZ, s)πz, s), 36) γ The value of a firm in sector s with productivity Z is now; ) γ σ + 1) γ V Z, s) = LZ, s)πz, s). 37) γ The number of firms entering with productivity Z in sector s is ; MZ, s) = L f ) Z, 38) AAs) if z > ẑs), and zero otherwise. The number of firms with productivity Z catering to each consumer NZ, s) is; LZ, s) NZ, s) = MZ, s) L. 39) 20

21 The threshold ẑs) must be such that V Ẑ, s) = wc E; ) γ σ + 1) γ σ 1 θ s Ẑ LẐ, s) c E = γσ ẑs) Zσ 1 LZ, s)fz)dz), or, using equations 34) through 36); ẑs) σ 1 1) = c E γσ z σ 1 1) fz)dz. θ s [γ σ + 1) γ] ẑs) If z is Pareto distributed, the above can be easily solved for ẑs); ) α ẑs) = z min αc E γσ 1) θ s [α 1) σ 1)][γ σ + 1) γ], 40) with the following expression for LZ, s); LZ, s) = X c E γ c L [γ σ + 1) γ] ) 1 ) σ 1 z 1. 41) ẑs) The number of firms entering each sector is; Ms) = ) α zmin L, 42) ẑs) the number of varieties from sector s consumed per capita is; Ns) = X α 1) σ 1) α 1) σ + 1 ) γ σ + 1) γ c E γ ) 1 θ s σc 1/ L, 43) and sector-specific productivity in sector s is; As) = ) 1 γ c E σ 1) c As [γ σ + 1) γ]ẑs) σ 1, 44) 1) 21

22 with relative productivity across sectors equal to; A1) A2) = ) 1 ca2 c A1 γ θ 1 θ 2 ) σ 1) αγ 1). 45) Income per capita is equal to; ) 1 Y = θ θ 1 1 θ θ 2 2 ) σ σ 1 X c L σ σ 1) γ σ + 1) γ c E γ ) 1 σ 1) AA θ 1 1)A θ 2 2)ẑ1) θ1ẑ2) θ 2. 46) Notice that as in Section 2, varieties available per consumer equation 43), customers per firm 41), average firm size 42), and average income 46) are all independent of population L, while all but firm size are increasing in density X. 4.2 Trade or the Lack Thereof) No Comparative Advantage As in Section 2 there is no trade here without comparative advantage. To see this, imagine all parameters are the same across countries except that average productivity is higher in the home country i.e., A > A ). Start from the autarky equilibrium and consider the incentives facing a home firm if trade were suddenly possible and free. From equation 20), the marginal cost of acquiring a foreign customer is; wc L LZ, s) + L Z, s)) 1 X, which is equal to πz, s) in the autarky equilibrium. From equations 34) and 46), the marginal benefit of acquiring a foreign customer is; πz, s) w w A = πz, s). A 22

23 With the marginal benefit and cost of acquiring a foreign customer the same as in autarky, there is no incentive for a firm to acquire foreign customers and therefore no trade. Note that the above holds for all Z including Ẑ, implying that any firm unwilling to enter in autarky will similarly be unwilling to enter when trade is possible Comparative Advantage Now assume there exists comparative advantage, driven by symmetric differences in the investment cost of productivity, so c A1 = c A2 < c A1 = c A2. Also, let population differ in the two countries, so L L. 13 Relative prices will again equalize across countries with free trade. Unlike in Section 2.2.2, however, each country will continue to produce domestically in its relatively unproductive sector. To solve the model it is useful to first treat the two countries as one large country, solve for productivity cutoffs and aggregate outcomes, and then work backwards to solve for countryspecific outcomes. Equations 33) through 39) still hold except that Ẑs) will reflect a different ẑs) and As) in each country. Since firms in both countries are indifferent between domestic and foreign customers, consumers in both countries will have the same number of varieties available from each sector. Let LZ, s) denote the total number of customers foreign and domestic) acquired by a firm. Then the number of varieties available to a consumer from firms with productivity Z in sector s is; NZ, s) = ) ) ) ) L Z L Z L + L LZ, s)f + AAs) L + L LZ, s)f, 47) AA s) where L ) = 0 for all Z < Ẑ. Ẑs) can be solved for using the following condition, derived 13 Note that with A = A and θ 1 = θ 2 = 1/2 in both countries, wages are equal across countries both in autarky and with trade. 23

24 using equations 34), 35), and 37); Ẑs) A ) σ 1 1) ) c E γσ = θ s [γ σ + 1) γ] L ) + L or L As)z) σ 1 ) 1) fz)dz + L A s)z) σ 1 1) fz)dz, Ẑs) Ẑs) AAs) AA s) ) As) Ẑs) α = z min A) α α L + A s) α L αc E γσ 1) L + L θ s [α 1) σ 1)][γ σ + 1) γ], 48) with the following country-sector-specific cutoffs for z; ẑs) = Ẑs) AAs) = ẑ s) A s) As). 49) Opening up to trade encourages entry lowers ẑ) in each country s relatively productive sector, while discouraging entry in its less-productive sector. This change in the number of producers at home is greater when the home population is smaller and when the foreign population is larger. Firm productivity As) can be solved for using equations 34), 35), 36), 47), and 48), and remains the same function of ẑs) as in autarky; A γ s) = c E σ 1) c As [γ σ + 1) γ]ẑs) σ 1 1), 50) A1) A 1) = A 2) A2) = Relative productivity across sectors is equal to; ca2 c A1 ) 1 γ σ+1) γ. A1) A2) = ) 1 ca2 c A1 γ θ 1 θ 2 ) σ 1) αγ 1) L + L c A1 /c A2 ) L + L c A2 /c A1 ) α 1) ) 1 γ γ σ+1) γ α 1) γ σ+1) γ. 51) 24

25 Comparing equations 45) and 51), it is clear that opening up to trade increases firm-level productivity As) in the relatively productive sector with low c A ) but decreases productivity in the unproductive sector. Further, the productivity gap between the two sectors is wider when the foreign population is larger. The total number of customers acquired by a firm with productivity Z in sector s can be solved for using equations 34), 35), 47), and 48); LZ, s) = X c E γ c L [γ σ + 1) γ] ) 1 Z Ẑs) ) σ ) Comparing equations 41) and 52), it is clear that firms in the relatively productive sector increase their number of customers with trade, while firms in the unproductive sector contract. Combining equations 47) and 52), the number of varieties available to each consumer in each sector remains the same as in autarky 43). With perfectly free trade, where a particular firm sells its goods is indeterminate. I therefore again assume some epsilon iceberg cost incurred by exporters, so that all firms in a country s less productive sector sell domestically. 14 To find the fraction of customers served by domestic firms that are foreign in sector 1), first note that all firms with productivity Z serve the same total number of customers LZ, 1). The fraction of total customers that are foreign is L / L + L ). The fraction of customers served by domestic firms that are foreign is therefore; ) [ 1 L + L L L M ] Z, 1), 53) MZ, 1) or L L + L ) [ 1 ) ] A A α+1 1), 54) AA1) since M Z,1) MZ,1) = L L ) α+1 A A 1) AA1) from equation 42). Clearly the fraction of customers that are 14 That all firms in a country s less productive sector only sell domestically is the result of both the epsilon exporting cost and the assumption that z follows a Pareto distribution. 25

26 foreign is both constant across exporters and increasing in the foreign population. And given that LZ, s) for exporters is also increasing in the foreign population equations 48 and 52), it is clear that the absolute number of foreign customers per exporter L Z, s) is higher when the foreign population is higher. This is consistent with the evidence in Arkolakis 2010). The domestic trade share expenditure on domestic goods as a share of total expenditure) in the exporting sector is one. The domestic trade share in the importing sector is; L L + L A s) As) ) α. 55) The larger the trading partner, the lower the share of expenditure going to domestic producers. Finally, average incomes Y ) with trade will be larger than those in autarky by a factor of; Y trade Y autarky = [ L + L Ω] θ 1 [ L + L Ω 1 ] θ 2 L + L ) γ σ+1) γ γ 1), 56) Ω ca1 c A2 ) 1 γ σ+1) γ. In the special case θ 1 = θ 2 = 1/2, the gains are highest when both countries are equally populated. 5 Trade in a Ricardian Model Without Scale Effects In this section I sketch out a Ricardian model to provide intuition for why both scale effects and gains from intra-industry trade disappear when market access costs are imposed on firms. Again consider an economy composed of many markets, but now the economy resembles that developed in Bernard et al. 2003). The utility of each consumer is still described by equation 3), but now the number of goods in each sector is fixed and equal to 1. For each good there 26

27 exists many potential producers in the economy with productivity Zs) = AA s z, where z is firm-specific, A s is common across potential producers in sector s, and A is common across all potential producers. I assume each z is drawn from some distribution, and that the mass of potential producers for each good scales up proportionally with population. Producers still exhibit constant returns to scale in production, and demand for each good is still described by equation 6). Since potential producers for each good produce identical varieties, the most productive producer in a market will capture that market. Denote the k th highest productivity for a good among potential producers present in a market by z [k]. Then the best producer in a market will charge a price equal to; { w P [1] = min, Z [2] ) } σ w. 57) σ 1 Z [1] Profits per customer for the best firm, relative to the wage, are increasing in Z [1], and the probability that Z = Z [1] is increasing in Z. I make the following additional assumptions. First, potential producers must incur a fixed labor cost to enter. Second, producers choose how many markets to enter, but not which markets or alternatively, they enter without knowing anything about other entrants). Under these assumptions potential firms with higher Z will choose to enter more markets, and there will exist a productivity threshold Ẑ such that all potential producers with Z > Ẑ will enter. Now consider a doubling of the population, along with a doubling of the number of markets. As in the models in the previous sections, entry and market decisions remain unchanged, resulting in a doubling of the number of producers, no change in the expected values of Z [1] and Z [2] in each market, and therefore no change in welfare. Opening up to trade with an identical country remains equivalent to a doubling of the population, keeping density fixed. Trade, and gains from trade, require that comparative advantage exist between the two countries. 27

28 6 Trade When Consumers Choose Varieties In this section I extend Li 2018) to show that the results of this paper hold when consumers, rather than firms, determine how many varieties to consume. I begin by describing the environment, then characterize the autarky equilibrium, and then discuss trade. 6.1 Environment I continue to define aggregate and sectoral consumption as in equation 3), but now I assume the utility of a consumer is equal to; UC, N 1, N 2 ) = C c N N 1 + N 2 ), ) 58) where c N > 0 and > 1. As in Li 2018), consumers face a trade-off between the number σ 1 of varieties consumed and the quantity consumed of each variety. I depart from Li by making the simplifying assumption that consumers choose the number of varieties to consume, but not which varieties to consume i.e., they consume a random sample of available varieties in each sector). Firms face the same decisions as in Section 2, except that they need not incur any cost to access customers. As in Section 2, potential firms must incur an entry cost before realizing their productivity. 6.2 Equilibrium Following Li 2018) I separate the consumer s problem into two stages. In the second stage the consumer takes the number of varieties consumed as given and chooses a quantity of each 28

29 available variety to maximize consumption, subject to a budget constraint. In the first stage, the consumer chooses how many varieties to consume. The second stage of the consumer s problem is standard, resulting in the following inverted demand function for good n in sector s; where P 1 σ ns is the mean value of P 1 σ ns P ns P s = P s = θs w P s N s P 1 σ ns ) 1 σ y 1 σ ns, 59) ) 1 1 σ, 60) across all varieties consumed by the consumer. The price of aggregate consumption is again normalized to 1, so the following must hold; P = 1 = P θ 1 1 P θ 2 2 θ θ 1 1 θ θ 2 2, 61) and the consumer spends a constant fraction of income on varieties from each sector; P s C s = θ s w. 62) In the first stage of the consumer s problem, the consumer chooses how many varieties N s to consume from each sector. Given equations 58), 59), and 60), the consumer chooses N 1 and N 2 to maximize the following objective function; wθ θ 1 1 θ θ 2 2 N 1 Pn1 1 σ ) θ 1 σ 1 N 2 P 1 σ n2 ) θ 2 ) σ 1 c N N 1 + N 2. 63) The resulting optimal variety choices are characterized by the following equation; N s = ) 1 θ s w. 64) c N σ 1) 29

30 As in Li 2018), higher incomes encourage consumers to adopt more varieties. 15 I now turn to the firm s problem. For a firm in sector s with productivity Z, operating profits per customer are; πz, s) = P Z, s)yz, s) wyz, s) Z θs w = P 1 σ s ) 1 σ yz, s) σ 1 σ wyz, s). 65) Z Maximizing profits with respect to output results in the following optimal output and profits per customer; ) σ σ 1 θ s Z σ yz, s) =, 66) σ w σ 1 Ps 1 σ πz, s) = w σ 1)σ 1 θ s Z σ 1, 67) σ σ w σ 1 Ps 1 σ where Z should again be understood to also be a function of the firm s sector. Denote the number of firms in sector s by M s. Then the number of customers per firm in sector s is equal to; L N s M s. 68) Free entry ensures that potential firms enter until the value of entry is equal to zero; L N s M s E Z [πz, s)] w c E = 0. 69) Combining equations 59),??), 8), 9), and??), the number of firms in each sector can be expressed as; M s = θ s σc E L. 70) Notice that the number of firms scales up proportionately with population. This implies that the average size of firms does not depend on population, again consistent with the evidence documented in Bento and Restuccia 2017, 2018). 15 Note that this result depends on how the cost of adopting varieties is specified. If consumers had to incur a labor cost, rather than a direct utility cost, then higher incomes would not affect the optimal variety choice. 30

31 I can now solve for aggregate consumption using equations 4), 63), and 64); C = w = AA θ 1 ) 1 A θ 2 ψ 2 σ 1) θ θ 1 1 θ θ 2 2 σ c ψn ψ ) 1 ψ ) +ψ ψ E[z σ 1 ] ψ ψ 71) where ψ 1. Utility is equal to; σ 1 ) ψ U = C, 72) and relative prices are equal to; P 1 P 2 = A 2 A 1 θ2 θ 1 ) ψ. 73) Like average firm size and the number of varieties per consumer, real income per capita w) and utility are independent of population. If the aggregate productivity term A was increasing in density, then equations 64), 70), 71), and 72) show that higher density would be associated with higher incomes, welfare, and varieties consumed per consumer, but no difference in average firm size. 6.3 Trade or the Lack Thereof) No Comparative Advantage Again, there is no trade here without comparative advantage. Assume that consumers must incur some epsilon cost to access foreign varieties, and can then choose to consume a sample of domestic varieties, foreign varieties, or both. If trade opens up between two countries with identical relative prices, then no consumer has an incentive to consume foreign varieties. 31

32 6.3.2 Comparative Advantage If relative sectoral productivities differ across two countries, then consumers in each country will have an incentive to consume varieties from the relatively productive sector in the foreign country. Assume consumers put equal weight on each sector, θ 1 = θ 2 = 1/2, and sectoral productivities are symmetrically different, A 1 = A 2 > A 1 = A 2. For simplicity, assume that the two countries are otherwise identical. Under these assumptions relative prices will equalize across sectors and countries, and each country will completely specialize in its more productive sector. Equation 70) shows that the total number of firms in each sector will not change, but the higher incomes generated by trade encourage consumers to consume more varieties in each sector 64). 16 Equations 71) and 72) show that incomes and welfare will increase by a factor of; w trade w autarky = U trade U autarky = A1 A 2 ) 2 ψ). 74) 7 Conclusion Empirically, more populous countries seem to have proportionally more firms but no higher incomes than smaller countries. In this paper I interpret this to mean that consumers in larger countries do not systematically enjoy more varieties than those in smaller countries. I extend a simple variety model of trade by imposing a market access cost for firms, convex with respect to the number of markets served. I show this simple model generates no scale effects with respect to population, and no gains from intra-industry trade. Trade and gains from trade only occur when comparative advantage exists, and the gains from trade through comparative advantage are lower than in a standard model because trade need not increase the number of varieties adopted by consumers. On the other hand, the true gains from trade can be higher than those 16 If adopting varieties required consumers to incur a labor cost, then higher incomes would not encourage more variety adoption. 32