行政院國家科學委員會專題研究計畫 成果報告
異質生產者與內生彈性下之空間經濟學
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 99-2410-H-004-223- 執 行 期 間 ： 99 年 08 月 01 日至 100 年 10 月 31 日 執 行 單 位 ： 國立政治大學國際貿易學系 計 畫 主 持 人 ： 徐則謙 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 公 開 資 訊 ： 本計畫可公開查詢中 華 民 國 101 年 02 月 10 日
中 文 摘 要 ： 傳統之經濟地理學著重在鏈結所產生的向心力上。我以一簡 單模型指出當市場結構 内生時，這些力量可被經濟成長或國際貿易所弱化。因此， 如同某些實證文獻所示， 在形成人類經濟活動分部不均的成因上，自然的地理條件比 經濟的力量來得重要。 中文關鍵詞： 經濟地理、內生市場結構、異質生產者、國際貿易
英 文 摘 要 ： Conventional economic geographic models concentrate on linkages to generate
centripetal forces. I propose a tractable model to show that the forces can
be weakened by economic growth or trade once we endogenize the market structure.
It then shows that the first nature might be more important than the economic
forces to generate agglomeration as observed by some empirical studies.
英文關鍵詞： Spatial Economy, Endogenize Market Structure, Heterogeneous Producers, International Trade 表
A Spatial Economy with Heterogeneous Firms
and Endogenous Elasticity
Tsechien Hsu
Department of International Business
National Chengchi University
∗January 31, 2012
Abstract
Conventional economic geographic models concentrate on linkages to generate centripetal forces. I propose a tractable model to show that the forces can be weakened by economic growth or trade once we endogenize the market structure. It then shows that the first na-ture might be more important than the economic forces to generate agglomeration as observed by some empirical studies.
1
Introduction
Recently, spatial economics has extended its foundation to consider the em-pirical evidence showing that firms are heterogeneous in productivity. How-ever, in those models, they either assume that the spatial concentration in one location has already been there (Nocke, 2006; Baldwain and Okubo, 2006) or the elasticity of demand is constant in the conventional way (Baldwain and Okubo, 2006). The purpose of this research is going to show that the as-sumption that the elasticity is constant might be nocuous when we consider the possibility of economic growth or international trade. In particular, we are going to see that the first nature is more important in determining the geographic distribution of economic activities, while the economic behaviors are not strong enough to generate centripetal forces as observed in empir-ical studies (Davis and Weinstein, 2002). On the other hand, the lack of consideration on firm heterogeneity, such as Ottaviano, Tabuchi, and Thisse (2002), ignores the rich information on how trade affect the entry and exit and moving decisions of firms depending on their productivity.
In this project, the economic centripetal forces are conventional forward and backward linkages when there is a transportation cost. Larger mar-kets provide higher demand for producers (backward linkages) and provide cheaper goods for workers (forward linkages). Contrary to Nocke (2006) who assumes that the size of demand in two locations are fixed and therefore high-productivity firms tend to near large-size of location because of their comparative advantage in dealing with the fierce competition there and the benefit from procuring larger demand, when we endogenize the size of lo-cations, the forward and backward linkages make all manufacturing activity concentrated in one location. There is no need to compare one location with the other. Specifically, Nocke (2006) implicitly assumes that the demand cannot flow from one location to the other and ignores the important eco-nomic behavior that workers have incentive to move toward a location with higher real reward. By adding this possibility manufactures cannot locate in a small-size location since the real wage they offer is obviously lower than that in the large-size location. This is the first result different from the existing literature I propose.
As pointed out by Fujita, Krugman, and Venables (2000), the centripetal forces are sensitive to the elasticity of demand of varieties. As the elasticity increases, the range of the transportation cost to sustain the core-periphery structure shrinks. It is intuitive since fierce competition forces producers to close to their demand even though the transportation cost is negligible. Once we endogenize the elasticity as in Hsu (2009), we are able to see how the distribution of economic activities change as some interesting exogenous
variables change. For instance, as the population grows, larger number of varieties can be provided and the level of competition is higher. In Hsu (2009), it implies that the elasticity is higher. The higher elasticity generates centrifugal forces. As firms are heterogeneous in productivity, it is interesting to analyze which firm move first as the economy is less concentrated. It might also be an explanation that why in the modern world, economic activities have the tendency to move out of the cities, for instance, the moving of some manufacturing activities toward the suburban or even rural areas. The explanation is better than the traditional view of decreasing transportation costs, because it is well known that the decrease of the transportation cost might actually strengthen the centripetal forces.
We finally consider how trade affects the centripetal and centrifugal forces. The idea can be demonstrated by figure 1. Each circle is a location. The circles filled with black are locations in the foreign country. The figure is managed to show that a foreign location has equal distance to each home location (measured by the length of lines linked locations) but one is more distant than the other. We assume that labor cannot move internationally, a not so unreasonable assumption. To see the forces clearly, we also assume that producers in one country cannot choose production location in the other country, an unnecessary assumption which we do not assume in the formal analysis to have the result. It is clearly to see that trade must increase the level of competition in the home country and generate centrifugal forces. However, in the foreign country, there is a tendency for firms to locate to the lower circle because it has the shortest distant to each of the location in the home country and at the same time there is also centrifugal forces because the competition is fiercer. At this moment it can not be clearly shown which force is dominant; however it is clear that there is a tendency that the economic activities are more dispersed distributed in the home country. Notice that this result cannot be generated by the conventional model because of the constant elasticity assumption.
The result then strengthen the point of view of Sachs and Jared Diamond that the first nature (the initial natural geographic distribution) is much more important than an accidental historic economic activities. We see that the agglomeration is not so easily sustained as the population grows. We also see that the geographic distribution of locations in both countries will affect the distribution of activities under trade. It might disperse activities or con-centrate them in a location depending on the initial geographic distribution instead of the initial economic activities.
2
Methodology
In this section, I extend Hsu (2009) to consider distribution of activities between two locations. Suppose there are two countries and each of them has two locations. Location is indexed by j ∈ {1, 2, 3, 4}. Index 1 and 2 are locations in the home country and rest of them are locations in the foreign country. There are two sectors, agriculture (a) and manufacturing (m). Each sector uses farmer and labor as the specific input and they are inelastically supplied (Laand Lm). It is assumed that farmers are tied to the land and therefore immobile; however, labors are perfectly mobile between locations. There is an ice-berg transportation cost τj,j0 > 1 when transporting
manufacturing goods between two locations. There is no cost for transporting agricultural goods and no transportation cost for the manufacturing goods within a location. It is also assumed that farmers are evenly distributed between two locations to avoid any centripetal forces simply generated by the initial uneven distribution of demand. The location index is omitted if it is not necessary.
2.1
Demand
Consumers have identical indirect utility functions. The utility function of an agent living in a location is
V =X k
βkln Uk(λI, {Pk(ω)}) , (1)
where k ∈ {a, m} is the sector index, Uk is the indirect subutility of sector k in a location, 0 < βk < 1 ∀ k and P βk = 1. I is the total income, Pk(ω) is the price of variety ω in sector k in a location and ω ∈ Ωk, where Ωk is the set of goods available in sector k in this location. λ is the fraction of income in this location. Total income is chosen as a numeraire.
The subutilities of the two sectors, by applying a continuous version of Feenstra (2003) comprise a continuous number of varieties. The varieties are determined endogenously and homothetically enter the subutilities:
ln Uk= ln λβk− αk− Z ω∈Ωk αk(ω) ln Pk(ω)dω (2) − 1 2 Z ω∈Ωk Z ω0∈Ω k γk(ω, ω0) ln Pk(ω) ln Pk(ω0)dωdω0.
The parameters of the function are chosen as follows so that the utility is homogeneous of degree one in prices, and varieties enter the utility function
symmetrically: αk= α0+ e Ω − Ωk 2γΩkΩe , αk(ω) = 1 Ωk , γk(ω, ω0) = − γ(Ωk− R ω0=ωdω 0) dωΩk when ω = ω0, γk(ω, ω0) = γ Ωk ∀ ω 6= ω0,
where γ > 0 affects the elasticity of demand, as will be clear later. eΩ is the total number of potential varieties in this economy, which I treat as fixed and large. It should be noted that the definition of γk(ω, ω0) is not a normal function. Mathematically, it is called a delta function. Its integral at ω0 = ω equals −γ:
Z ω0=ω
γ (·) dω0 = −γ.
Therefore, when ω is a continuous measure, γ (ω, ω0) equals Ωγ
k when ω
0 6= ω; otherwise it jumps to ∞ but equals −γ when it is integrated at the point ω = ω0.
The market share of each variety can be obtained easily by differentiating equation (2) with respect to the log prices. This gives us:
sk(ω) = λβkzk, (3) where
zk = hk− γ ln Pk(ω) (4) is the market share of product ω in sector k. hk is defined as:
hk = 1 Ωk + γln Pk, (5) where ln Pk = Z ω∈Ωk ln Pk(ω) Ωk dω (6)
is the average log price level in sector k. hk is decreasing with Ωk and is increasing with the average price of the industry; therefore, it is the inverse of the level of competition in that sector. A firm controls more demand in a sector when the level of competition is lower in that sector. Notice that total income is normalized to one so that sk also represents the total revenue of the firm.
The number of firms is continuous, so each firm treats the average price as given as it changes the price. The elasticity of demand can therefore be obtained as: εk(ω) = 1 − d ln sk(ω) d ln Pk(ω) (7) = 1 + γ zk(ω) .
The assumption that γ > 0 guarantees that the elasticity is greater than one. Because zk(ω) is a function of the average price and the number of producers in sector k, the elasticity is not constant, contrary to the conventional setting in trade models under monopolistic competition (Helpman and Krugman, 1985). It can be observed that equation (7) increases with Ωk and decreases with ln Pk, meaning that when the market is more competitive, the demand faced by each firm is more elastic. It is clear to see that the market structure is endogenous, for instance, if producers are charging the same price; then, as the number of producers goes to ∞, the elasticity also goes to ∞. The market structure is perfect competition. The elasticity also increases as γ increases.
2.2
Production
It is assumed that there is a large number of potential entrants at each point in time. As shown by Figure 2, at each point in time potential entrants first decide which location to enter. They then pay a fixed cost to discover their productivity. The fixed cost is a fixed amount of specific input and is financed by issuing stocks. Productivity is drawn from a distribution func-tion G(A), where A ∈0, A is the productivity level. Finally, the potential entrants decide whether to produce after discovering their productivity level. To produce one unit of output, producers have to use A(ω)1 units of the input; therefore, the higher A(ω) is, the higher the productivity of producer ω. If they do not produce, they simply exit. The equilibrium of those decisions at each point in time can be solved by backward induction. Only the stationary equilibrium is considered. The market structure in each sector is monopolis-tic competition. We assume that the fixed entry cost is zero for agricultural sector; therefore at equilibrium the market structure in the agricultural sec-tor endogenously degenerates to perfect competition. It is not a necessary assumption but reduce the complexity.
Suppose that producers know their productivity level. Let ck(j, j0) be the marginal cost in sector k when the good is transported from location j to
location J0. Clearly, ck(j, j0) = τj,j0ck(j, j). Because each firm produces a
unique variety, it has monopoly power and the price it charges is marked up over its marginal cost. We therefore have:
Pk(ω) = ck(j, j0) A(ω) 1 − 1 εk(ω) −1 . (8)
Therefore, for a given level of competition, hk, and factor price, w, equations (4), (7) and (8) determine the equilibrium price charged by firm ω in a location.
2.3
Entry, Exit and Market Selection Decisions
The profit generated by firm ω in sector k in region j is:
πkj(ω) = skj(ω) εkj(ω) = λjβk z2 kj(ω) zkj(ω) + γ , (9)
It is clear that zkj(ω) is a function of A(ω). A producer produces if and only if the total profit is greater than or equal to zero which implies the following condition to determine the productivity level of the marginal producers
max {zkj(A) : ∀j ∈ {1, 2, 3, 4}} = 0. (10) Using the terminology of Melitz (2003), this is called the ZCP (zero cutoff point) condition.
Because a potential entrant has to pay a fixed cost f ck(j, j) before discov-ering his/her productivity level, the entrant will pay the fixed cost so long as it is lower than the expected profit. Free entry implies that at equilibrium:
max {πkj : ∀j ∈ {1, 2, 3, 4}} = f ck(j, j), (11) where πkj is the long-run expected profit of a potential entrant when it pro-duced at location j. It is clear that a producer will produces at location j if and only if the expected profit is the highest there.
2.4
Factor Market Equilibrium
The last endogenous variable to be determined is the equilibrium factor price. Recall that the fixed cost is financed by issuing stocks. It is assumed that there is a complete financial market; therefore, stockholders can diversify their portfolio well. This implies that at each point in time the dividend
of a unit of well-diversified stock portfolio is equal to the average profit of producers. This average profit level is a constant at the stationary equilib-rium, which has the same risk-free feature as the payment received by factors hired for production. Under the condition that there is no friction in the fac-tor market, each facfac-tor owner’s compensation, whether it is in the form of wages (when he/she produces output) or in the form of stock (when his/her endowment comprises the fixed cost), is the same at equilibrium.
Because of the free entry condition, the expected net profit is zero for all potential entrants. This implies that on average, existing producers have zero net profit. All expenditure is therefore equal to factor income. This implies that the nominal wage is
βk Lk
(12)
in each sector.
Finally, workers work at location j if and only if
ln Ukj ≥ ln Ukj0 ∀j 6= j0. (13)
This complete the description of the model.
3
Contribution and Further Proceeding
Equilibrium is then simultaneously determined by equations (4), (5), (6), (7), (8), (9), (10), (11), (12), and (13). The equilibrium structure of the model has been discussed in Hsu (2009). I simply extend the model to consider the decisions on production and working locations. Most of the result has been derived and the rest of the work is to simulate to confirm the intution obtained by mathematical derivation and to write the draft of the paper.
References
Baldwain, R., and T. Okubo (2006): “Heterogeneous Firms, Agglomer-ation and Economic Geography: Spatial Selection and Sorting,” Journal of Economic Geography, 6, 323–346.
Davis, D. R., and D. E. Weinstein (2002): “Bones, Bombs, and Break Points: The Geography of Economic Activity,” American Economic Re-view, 92, 1269–1289.
Feenstra, R. C. (2003): “A Homothetic Utility Function for Monopolis-tic Competition Models, Without Constant Price ElasMonopolis-ticity,” Economics Letters, 78(1), 79–86.
Fujita, M., P. Krugman, and A. J. Venables (2000): The Spatial Economy: Cities, Regions and International Trade. MIT.
Helpman, E., and P. R. Krugman (1985): Market Structure and Foreign Trade. The MIT Press.
Hsu, T. (2009): “A Proportional Theory with Heterogeneous Producers and Endogenous Elasticity,” .
Melitz, M. J. (2003): “The Impact of Trade on Intra-industry Realloca-tions and Aggregate Industry Productivity,” Econometrica, 71(6), 1695– 1725.
Nocke, V. (2006): “A Gap for Me: Enterpreneurs and Entry,” Journal of the European Economic Association, 4, 929–956.
Ottaviano, G., T. Tabuchi, and J.-F. Thisse (2002): “Agglomeration and Trade Revisited,” International Economic Review, 43(2), 409–35.
Figure 1: International Geographic Distribution Choose which location to enter Draw productivity level from G(·) Produce or exit Invest f amount of intermediate inputs financed by issuing stocks
出席國際學術會議心得報告
計畫編號 NSC 99-2410-H-004-223 計畫名稱 異 質 生 產 者 與 內 生 彈 性 下 之 空 間 經 濟 學 出國人員姓名 服務機關及職稱 徐則謙 助理教授 國立政治大學國貿系會議時間地點 Seoul, Korea, August 11-13, 2011
會議名稱 Asia Meeting of Econometric Society
發表論文題目 A Proportional Theory of Heterogeneous Firms and Endogenous Elasticity
一、參加會議經過 會議開始於八月十一日。除發表文章外，每日到場聆聽他人的研究成果，收獲甚豐。 二、與會心得 該研討會為國際知名的大型研討會。與會者大都是在經濟學領域上聲譽有佳的學 者，當中亦不乏年輕精進者。能與之討論現今國際經濟學發展的狀況獲益良多。 三、論文摘要
This paper develops a Heckscher--Ohlin model containing heterogeneous producers with dynamic entry, exit and export decisions, and an endogenous market structure. Because of the endogenous markups, and contrary to the conventional export-driven mechanism, competition increases aggregate productivity by reallocating factors and changing the entry and exit decisions of producers. Therefore, trade can have a profound impact on variables in import-competing sectors rather than exporting sectors. This more accurately reflects our intuitive expectations and empirical observation. A Heckscher--Ohlin model with homogeneous producers and perfect competition is then a special case of this model, which offers the possibility for further theoretical extension into fields employing the traditional Heckscher--Ohlin model.
四、被接受證明
Dear Professor Tsechien Hsu:
Thank you for submitting your paper to the Asian Meeting of the Econometric Society.
We would like to inform you that the decision on your submission has been made, which can be found in the Conference Maker.
Thank you again submitting your paper. Sincerely,
國科會補助計畫衍生研發成果推廣資料表
日期:2012/02/10國科會補助計畫
計畫名稱: 異質生產者與內生彈性下之空間經濟學 計畫主持人: 徐則謙 計畫編號: 99-2410-H-004-223- 學門領域: 國際經濟學無研發成果推廣資料
99 年度專題研究計畫研究成果彙整表
計畫主持人：徐則謙 計畫編號： 99-2410-H-004-223-計畫名稱：異質生產者與內生彈性下之空間經濟學 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 1 0 0% 研究報告/技術報告 1 1 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 3 3 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次其他成果
