Offshoring in a one factor model: its DFS all over again!

Transcription

1 Offshoring in a one factor model: its DFS all over again! M.H. Davies a,b a Washington and Lee University, Lexington, VA 2445 b Center for Applied Macroeconomic Analysis, Australian National University, Canberra, ACT 2, Australia Abstract While the theoretical literature on offshoring is rich and complex, it may be generally summarized as a collection of special cases typified by partial equilibrium (goods or factor prices fixed), small numbers of goods, exogenously determined margins between offshored tasks, and fixed factor coeffi - cients. Using a Ricardian continuum framework, we present a simple general equilibrium model of offshoring with fully flexible goods and factor prices, a continuum of goods, endogenously determined margins of offshoring production and substitutability between stages of production. The model brings structure to the gallery of ambiguous results noted above, and allows focus of the effects of offshoring on prices, wages, production location, and welfare. F11 (Neoclassical Models of Trade) 1. Introduction and Motivation While rich and complex the theoretical literature on offshoring may be generally summarized as a collection of special cases typified by partial equilibrium: goods or factor prices fixed; small numbers of goods; exogenously determined margins between offshored tasks; and fixed factor coeffi cients. Although no single paper suffers from all of these deficiencies, each bears the burden of at least one of these limitations. The implications of offshoring in these models varies widely: a panoply of results. address: daviesm@wlu.edu (M.H. Davies) Preprint submitted to Elsevier September 27, 215

2 The key motivation of this paper is to solve a general equilibrium model of offshoring which, in contrast to the offshoring literature, has fully flexible goods and factor prices, a continuum of goods, endogenously determined margins of offshoring production and substitutability between stages of production. We present a simple general equilibrium model of offshoring that allows focus on the effects of offshoring on production location, wages, prices and welfare, and that brings some structure to the gallery of ambiguous results noted above. The Ricardian continuum framework (Dornbusch, Fischer, and Samuelson [2], henceforth DFS) is well suited to the analysis of North-South offshoring given its reliance on technological difference as a basis for trade. It is powerful and tractable framework with which one can analyze many issues of international trade, and may be viewed as the simplest general equilibrium trade structure. 1 We take two alternative and complementary approaches to modelling offshoring within this framework. The first is to specify a Ricardian continuum model of offshoring, and to compare the equilibrium to the pre-offshoring, or DFS, equilibrium. The examination of the differences between equilibria enable a determination of the effects of offshoring on wages, prices, welfare, and the location of production. The second approach is to model explicitly the impact of communication and coordination costs (g), one of the primary frictions that resist offshoring, on the offshoring equilibrium. This simplifies the analysis because the endogenous variables may be expressed as continuous functions of the exogenous variable g. This is distinct from the first approach in which the move from the pre-offshoring to the offshoring equilibria is a discrete change. A limitation of the Ricardian framework is that there is only one factor of production, and thus focus remains on the consequences of offshoring on aggregate welfare, rather than on the distribution of wages, or the skill premium. One interpretation of this analysis is that it is examining the impact of offshoring on average wages. Under this interpretation, an increase in average wages depicts a situation in which winners could compensate losers. We derive a number of results: (i) we show that offshoring causes the price level to fall by an amount equivalent to the gains from specialization. The 1 Simpler even that the two good Ricardian model since the specification of the supply side in the continuum model leads to a balance of trade schedule that is continous. This is as opposed to the two good set-up in which the balance of trade has points of discontinuity. 2

3 effect of offshoring on the price level is often overlooked in the popular debate about the benefits and costs of offshoring, and even within the offshoring literature; (ii) we show that although the price level falls, not all prices will fall if offshoring causes terms of trade effects; (iii) we determine a condition which quantifies the movement in the factoral terms of trade in terms of offshoring comparative advantage; (iv) we determine the impact of offshoring on welfare; and finally (v) we model explicitly the impact of transport and communications and coordinations costs on the arrangement of production, wages, prices and welfare. Briefly, we mention four studies which demonstrate the disintegration of the global production process. Hummels, Ishii, and Yi [6] note that vertical specialization 2 accounts for 3% of world exports, and one-third of the growth of world exports between 197 and 199. For the US, vertical specialization accounted for 22 percent of exports in 1997, and 3 percent of the growth in the export share of merchandise GDP between 1962 and 1997 (Hummels, Rapoport, and Yi, [7]; Hummels, Ishii and Yi, [6]). Yeats [11] finds that the share of transport and machinery components and parts is 3% of total OECD exports in the transport and machinery equipment sector. Feenstra [3], examining shares of US exports and imports by end-use categories over the period , finds that products are being imported into the US at an increasingly advanced stages of processing suggesting that US firms are substituting away from these processing activities at home. A central observation common amongst these papers, and others which examine the empirics of offshoring, is that Northern countries are substituting away from less sophisticated activities by offshoring them to the South, enabling specialization in more sophisticated activities. Hence, Northern resources are moving into the production of more sophisticated tasks, and Southern resources are being substituted for Northern factors of production is less sophisticated activities. This key observation is central to the modelling in this paper. The paper is organized as follows. Section 2 sets out and solves the model. Section 3 includes an examination of the effects of offshoring on wages, prices and welfare, and also the comparative static effects of pop- 2 Vertical specialization is closely related to offshoring and is defined by Yi [12] to occur when (i) goods are produced in multiple, sequential stages; (ii) two or more countries provide value added in the production sequence; and (iii) at least one country uses imported inputs in a stage of production, and exports some of the resulting production. 3

4 ulation and productivity shocks. Section 4 extends the model to include transport, and communication and coordination, costs and examines the effects of reductions of these frictions in a number of settings. Section 5 concludes. 2. Model Two countries, North (N) and South (S), are each endowed with a single productive factor (labor), in quantities L N and L S. There is a continuum of final goods j indexed on the interval [, 1]. The index j represents the technological sophistication of the production process, as in Young [13], with higher numbers indicating higher levels of sophistication Production Technology A final good j is produced by two tasks each made with a single factor of production, labor, according to Cobb-Douglas technology ( ) L i α ( ) y(j) = 1 (j) L ii 1 α 2 (j) α (, 1) a i (j) b ii (j) where L i k (j) is the factor input of task k {1, 2} for good j in location i,3 a i (j) is the unit labor requirement for task 1 in good j in country i, b ii (j) is the unit labor requirement for task 2 in good j in country ii, and where the production of each task may occur in either of two countries, i, ii {N, S}. The unit cost function of product j is: 4 cost(j; w i, w ii ) = ᾱ(w i a i (j)) α (w ii b ii (j)) 1 α where w i and w ii represent the wage rates in locations in i and ii where tasks 1 and 2 are produced. From the first order conditions of the cost minimization problem the ratio of labor in task 1 to task 2 is L i 1(j) /L ii 2 (j) = αw ii / (1 α) w i, and this is drawn in Figure 1. In the circumstance in which the both tasks are produced in the same location (i.e. i = ii) then L 1 (j) /L 2 (j) = α/ (1 α) and despite the opportunities for substitution in production between tasks 1 and 2, the optimal mix of L 1 to L 2 is fixed, 3 The use of the i and ii superscripts are to indicate that tasks could occur in the same, or different, locations. 4 where ᾱ = α α (1 α) α 1. 4

5 L 1 (j) L i 1 (j) α. w ii L ii 2 (j) = 1 α w i L 1 i (j) * w ii w i y(j) = 1 L 2 ii (j) * L 2 (j) Figure 1: Optimal ratio of labor in task 1 relative to task 2 and the technology is of the simple Ricardian form. In the pre-offshoring state of the world, technological and/or economic constraints impose that the production of tasks 1 and 2 for any good are integrated and thus produced in the same location. The pre-offshoring technology for the production of good j in country i may be written y i (j) = ᾱ 1 a i (j) α b i (j) α 1 L i (j) i = N, S j [, 1] (1) where L i (j) = L i 1(j) + L i 2 (j). 5 This expression exhibits the Ricardian form y i (j) = L i (j)/c i (j) where c i (j) = ᾱa i (j) α b i (j) 1 α is the unit labor requirement for good j in country i. All goods and tasks are produced under perfect competition, and hence unit cost and price are equal. We assume initially that there are no transport costs and that free trade equalizes goods and tasks prices across countries. Pre-offshoring, the price of good j in country i is given by p i (j) = w i ᾱa i (j) α b i (j) 1 α and the relative price of good j between the North and South is ( ) p N (j) p S (j) = wn a N α ( ) (j) b N 1 α (j) = wn c N (j) (2) w S a S (j) b S (j) w S c S (j) 5 Since L i 1 (j) /L i 2 (j) = α/ (1 α) then L i (j) = L i 1(j) + L i 2 (j) = L i 1(j)/α = L i 2(j)/ (1 α). 5

6 For ease of analysis we define A(j) = a N (j)/a S (j), B(j) = b N (j)/b S (j), and C (j) = c N (j)/c S (j), which are indices of relative unit labor requirements for tasks 1, tasks 2 and the integrated pre-offshoring production. Further, from (2) it is the case that C(j) = A(j) α B(j) 1 α, which represents the decomposition of the index of comparative advantage for pre-offshoring technology into the underlying indices for task 1 and task 2. We make the following assumption which follows since the North has a greater technological advantage in more sophisticated activities. Assumption 1 A(j) and B (j) are continuous and decreasing in j. A natural outcome of Assumption 1 it that C(j) = c N (j)/c S (j) is continuous and strictly decreasing in j. Providing that A(j) is suffi ciently strongly negatively sloped, B(j) could be increasing, or non-monotonic, in j, while ensuring that C (j) is decreasing in j. However, imposing the assumption above on the B(j) schedule eliminates the possibility that a country is the best producer of tasks over segments of continuum that are not contiguous, preventing taxonomy in this analysis Consumption Households consume a continuum of final goods indexed by the variable j over interval [, 1] and share identical homothetic preferences ln U = 1 φ (j) ln x(j)dj where x(j) is the consumption of final good j and φ (j) is the expenditure share on good j, where Φ (j) = j φ (j) dj, and Φ (1) = 1.6 All households spend a constant fraction of income, φ (j), on good j. Without loss of generality, prices p(j) are normalized such that world expenditure is 1. This normalization sets expenditure as the numeraire against which prices are measured, and has no effect on the evolution of real magnitudes such as output volumes or relative prices. Thus, the world demand for good j is x(j) = φ (j) /p (j) Pre-offshoring Model In the pre-offshoring equilibrium the cut-off good, m, is defined ω = C(m) (3) 6 This could also be viewed as a third stage of production, in which final goods are assembled into a single non-traded good for consumption, as in Yi [12]. 6

7 where ω = w S /w N is the South s relative wage. The South has a comparative advantage in, and produces, goods on [, m) and the North produces higher indexed goods on [m, 1]. 7 Prices are determined by p(j) = w S c S (j) for j [, m) (4) = w N c N (j) for j [m, 1] Since the share of labor is one then w i L i (j) = p(j)y(j) = φ (j) and the derived demand for labor in the production of good j in country i is L i (j) = φ (j) /w i for good j. 8 The labor market clearing condition in the South may be written w S L S = Φ (m) ( w S L S + w N L N) which when rearranged gives the pre-offshoring balance of trade BOT (m) = Φ (m) L N (5) 1 Φ (m) L S which determines the relative wage that will clear labor markets in both countries, or equivalently that ensures that trade is balanced, for cut-off good m Pre-offshoring Equilibrium The equilibrium relative wage ω and the good m are solved using (3) and (5), and the solution is at intersection of the C(j) and BOT (j) loci, which are analogous to A(z) and B(z,L /L) in Figure 1 in DFS (p.825). At the equilibrium production is located effi ciently and trade is balanced Offshoring Model Initially offshoring is economically infeasible or the fragmentation of the integrated production process into separate tasks is technically infeasible. Technological progress makes the spatial separation of tasks possible and the production technology may be represented as follows: y 1 (j) = L 1 (j)/a(j), y 2 (j) = L 2 (j)/b(j) and y(j) = y 1 (j) α y 2 (j) 1 α for α (, 1), where y k (j) is the output of task k for good j. Offshoring allows firms the opportunity to exploit differences in relative factor prices and technology to reduce costs by 7 Throughout this paper we maintain the assumption that the cut-off good / task is produced in the North. This has no effect on the solution of the model 8 Note that the normalization of prices of final goods such that p(j)y(j) = 1 means that p(j)y(j) = φ (j) j. 7

8 substituting foreign for domestic factors. Once offshoring becomes feasible the C(j) schedule, which is a geometric average of A(j) and B(j), disintegrates into A(j) and B(j), which guide the location of tasks 1 and 2. For a given ω, the boundaries of offshoring for tasks 1 and 2 are at z and v, where ω = A(z) (6) ω = B(v) (7) There are two possible arrangements of production when tasks are located in different countries: (North, South) and (South, North). 9 The second arrangement is ruled out by the specification of relative advantage below. Definition 1. A country is defined as having a relative advantage in the production of task 1 (task 2) if given any ω, z and v, where z and v are defined by ω = A(z) and ω = B(v) then it is the case that z < v ( z > v). Applying Definition 1, 1 the assumption below follows naturally. Assumption 2 The North has a relative advantage in the production of task A justification of this assumption is that for any good j, task 1 is more technologically sophisticated than task 2. For (N orth, South), the offshoring production technology for good j is y o (j) = ( L N 1 (j) a N (j) ) α ( ) L S 1 α 2 (j) b S (j) with unit cost ᾱ(w N a N (j)) α (w S b S (j)) 1 α. 12 Since there are two tasks and given Assumption 2, there are three possible production patterns for any 9 Notation is (location of task 1, location of task 2). 1 Taylor [1] (p. 233) makes a similar definition. 11 Since at any given w = A (z) = B (v) then z < v, and given Assumption 1 it follows directly that A(j) < B(j) for all j [, 1]. 12 The model is this paper is not strictly a model of North-South trade, as defined by Flam and Helpman [4], Stokey [9], and Matsuyama [8]. A model of North-South trade, by their definition, is one in which the North has an absolute advantage in all goods, which ensures that in equilibrium the North s relative wage always exceeds unity. A natural 8

9 ω ω 1 B(j) C(j) A(j) z m v j Figure 2: Offshoring allocation of tasks good: (South, South), (N orth, South), and (N orth, N orth). The allocation of production is illustrated in Figure 2 in which the comparative advantage loci for tasks 1 and 2 have been projected onto the same axis. The South produces tasks 1 over [, z) and tasks 2 over [, v) with the corresponding production intervals in the North being [z, 1] and [v, 1]. Thus goods on [, z) are produced entirely in the South, goods on [z, v) have offshored tasks with task 1 produced in the North and task 2 in the South, and goods on [v, 1] are produced entirely in the North. The price of good j, p (j), is given by p(j) = w S c S (j) for all j [, z) p(j) = ᾱ ( w N a N (j) ) α ( w S b S (j) ) 1 α for all j [z, v) (8) p(j) = w N c N (j) for all j [v, 1] Since the task-shares of task 1 and task 2 are α and (1 α) then the derived extension of this assumption in a model with offshoring is that the North has an absolute advantage in the production of all tasks. In this model such an assumption would impose that B (j) < 1 for all j. Given Assumption 3, it would follw that A (j) < 1 for all j. We do not make this assumption and the relative wage in equilibrium is not restricted to a value below unity. 9

10 demands for labor for task 1 and task 2 in the production of good j in country i are L i 1 (j) = αφ (j) /w i and L i 2(j) = (1 α) φ (j) /w i. 13 From the South s labor market equilibrium, balance of trade is determined as BOT (z, v) = αφ (z) + (1 α) Φ (v) L N (9) 1 αφ (z) (1 α) Φ (v) L S where BOT (z, v) indicates the relative wage at which trade is balanced. Equations (6), (7), and (9) specify the offshoring equilibrium, which is 3 equations and 3 unknowns, w, z, and v Transformation of Offshoring Equilibrium The following transformation of the offshoring model allows it to be simplified into a form that is isomorphic to the Ricardian continuum model. Defining the variable µ = αz + (1 α) v as the task-share weighted average of the cut-off tasks, then the effi cient task location conditions, (6) and (7), may be expressed as follows. Lemma 2. The supply side of the offshoring model may be represented by ω = D (µ) (1) Proof. Inverting the effi cient production location conditions, (6) and (7), gives z = a (ω) and v = b (ω), and we define µ = αa (ω) + (1 α) b (ω). Given that a (ω) and b (ω) < then this may be rewritten µ = d (ω) where d (ω) < since µ is a convex combination of a (.) and b (.). Inverting this gives ω = D (µ), where D (.) <. Given that z and v are uniquely determined at each ω then the task share weight average µ = αz + (1 α) v is also uniquely determined. Defining Φ (µ) = αφ (z)+(1 α) Φ (v) as the share of world income spent on Southern tasks, then the offshoring balance of trade (9) may be expressed 14 BOT (µ) = Φ (µ) L N (11) 1 Φ (µ) L S 13 This follows since w i L i 1 (j) = αp(j)y(j) = αφ (j) and w i L i 2(j) = (1 α)φ (j). 14 where Φ (µ) = α z φ (j) dj + (1 α) v φ (j) dj 1

11 The model with offshoring takes a form that is structurally identical to DFS, with two equations (1) and (11), and two unknowns, ω and µ. This allows a solution of the offshoring equilibrium ω and µ, with a diagrammatic representation similar to Figure 1 in DFS (p.825) with appropriate relabelling. Given the unique relationship between µ and z and v, then this equilibrium determines ω, z and v which ensure trade balance and effi cient location of production for both tasks. The index µ may be thought of as an index of the technological sophistication of production, since it is a task-share weighted average of z and v. Since the models are isomorphic then if in the offshoring equilibrium µ > m it follows that offshoring causes ω to increase. 3. Wages, Prices and Welfare 3.1. What happens to wages? We solve for a condition which determines the effect of offshoring on the relative wage. In the offshoring equilibrium the North expands its production of tasks 1 and contracts its production of tasks 2 relative to the initial equilibrium (and in the South the opposite) however the impact of offshoring on the relative wage cannot be established by qualitative analysis alone. Thus far we have modelled offshoring as discrete shock, a movement between the pre-offshoring and offshoring states. To facilitate this analysis we introduce communication and coordination costs, g, which are frictions that resists offshoring, making it costly. This simplifies the analysis by making the endogenous variables in the analysis, ω, z and v, continuous functions of the exogenous g. Communication and coordination costs of the iceberg variety fall on the production of both tasks when a good is produced using offshoring technology. This reflects the resource cost of arranging spatially separated tasks, where a fraction 1 g of each unit of a task melts leaving only g < 1 units. The costs could fall asymmetrically on tasks, and could also vary across goods, however we assume for simplicity that g 1 (j) = g 2 (j) = g for all goods j. 15 The impact of communication and coordination costs is to increase the effective unit labor requirement of task 1 in the North to a N (j)/g, 15 This allows a specification of the pre-offshoring equilibrium which is independent ( ) α ( ) 1 α of communication and coordination costs, since C(j) = g 1 α A(j) g 1 1 α B(j) = A(j) α B(j) 1 α is independent of g. 11

12 and task 2 in the South to b S (j)/g, when these tasks are offshored. When goods are produced using offshoring technology the production function is y o (j) = g ( ) α ( ) 1 α L1 (j) L2 (j) a N (j) b S (j) The cost to produce good j using offshoring technology is c O (j) = ᾱ ( w N a N (j ) ) α ( w S b S (j) ) 1 α g The task 1 cut-off is determined by comparing the cost using offshoring technology to the cost producing the good entirely in the South, where z is determined by c O (z)/c S (z) = (A (z) /ω) α /g = 1, and Similarly, the task 2 boundary v is determined by ω = g 1 α A(z) (12) ω = g 1 1 α B(v) (13) From (12) and (13) it is apparent that at given ω a fall in communication and coordination costs (an increase in g) decreases z and increases v, increasing the range of goods over which offshoring occurs. To determine the balance of trade we again look at labor market equilibrium in the South which leads to an expression for the balance of trade which is identical to (9). Although communication and coordination costs reduce the range of goods over which offshoring occurs, because goods that remain integrated in production are still traded then the balance of trade is unaltered. Given (12) and (13), the cut-off tasks z and v may be written z(ω(g)) and v(ω(g)) and substituting these expressions into the balance of trade allows the representation of the system as one equation and one unknown ω (g) = From (14) the following proposition is established. Φ (µ (ω (g))) 1 Φ (µ (ω (g))).ln (14) L S Proposition 3. A reduction in communication and coordination costs (an increase in g) leads: 16 Under this specification, the impact of a decrease in communication and coordination costs is equivalent to a Hicks-neutral technological improvement. 12

13 ( ) dv i. to an increase in task trade: dz > ; dg dg and ii. the South s terms of trade, ω, to change according to Proof. The proof is in the Appendix What happens to prices? ˆω if φ (z) φ (v) A (z) (v) A (z) B B (v) Based on the pre-offshoring and offshoring location of task production, the goods continuum is divided into four segments. Goods on the interval [, z) are produced entirely in the South in both the pre-offshoring and offshoring states, and prices will change in the same proportion as the change in the wage of the South: p 1 (j)/p (j) = w1 S /w S, where subscripts and 1 indicate the pre-offshoring and offshoring states. Similarly the goods on the interval [v, 1] are produced entirely in the North in both the pre-offshoring and offshoring situations and p 1 (j)/p (j) = w1 N /w N. Goods on the interval [z, m), produced pre-offshoring in the South, are now produced using offshoring technology with task 1 from the North and task 2 from the South. The opportunity to split the integrated production process into two tasks and offshore allows Southern firms to substitute towards cheaper Northern task 1. Taking the ratio of the offshoring to pre-offshoring prices gives ( ) ( ) p 1 (j) w S α p (j) = 1 A (j) for j [z, m) (15) w S ω 1 Since the North has a comparative advantage in task 1 on [z, 1] then A (j) /ω 1 1 on [z, m). Goods on the interval [m, v), produced pre-offshoring in the North, are also produced using offshoring technology and the ratio of offshoring to pre-offshoring prices for these goods is ( p 1 (j) w N p (j) = 1 w N ) ( ) 1 α ω1 for j [m, v) (16) B (j) Since the South has a comparative advantage in task 2 on [, v) it follows that ω 1 /B (j) < 1 on [m, v), and from (16) the greatest relative fall in prices occurs for the pre-offshoring cut-off good m. The intuition is as follows: from Figure 2, moving from good v (which is produced entirely in the North) 13

14 p 1 (j)/p (j) 1 z z * m v l 1 j Figure 3: Relative movement of goods prices when South s relative wage rises. towards good m, the North begins to offshore task 2 to the South. It is giving up a task which it is relatively ineffi cient at producing. This relative ineffi ciency increases in the approach to, and is at its highest at, m, where ω 1 /B (m) is smallest for all j [m, v). Thus the greatest relative cost savings due to offshoring occurs at this point. Similar intuition can be applied when moving from z towards m, with the South offshoring task 1 to the North, and the ratio A (j) /ω 1 falling as m is approached. If the terms of trade move in favour of the South, then prices on [.z) increase in proportion w1 S /w S, and from (15), moving from z towards m the ratio A (j) /ω is falling and there is some z above which prices fall relative to their pre-offshoring level where p 1 (z ) /p (z ) = The prices of goods on [v, 1] fall in proportion to w1 N /w N and from (16) prices on [m, v) fall by relatively more, with the relative fall increasing when moving towards m from v as ω 1 /B (j) decreases. The relative movement in prices may be seen in Figure 3. Looking at Figure 4, when the factoral terms of trade move against the South, moving from v towards m the ratio ω 1 /B(j) falls, and prices of goods below v will fall relative to their pre-offshoring level, where p 1 (v ) /p (v ) = The prices 17 z is determined by A(z ) = ( ) w S 1 α /w1 N 18 v solves B(v ) = ( w N 1 ) α 1 α ( w S 1 ( w S 1 ) 1 α α ) / ( w N ) 1 1 α. 14

15 p 1 (j)/p (j) 1 z m v * v 1 j Figure 4: Relative movement of goods prices when South s relative wage falls of goods on [, z) fall in proportion to w1 S /w S and, from (15), the prices on [z, m) fall by relatively more than this. If offshoring does not cause a change in the factoral terms of trade then the prices of goods on (z, v) will fall relative to their pre-offshoring levels, however when offshoring causes changes in the factoral terms of trade the prices of some good will rise. 19 The following proposition determines the impact of offshoring on the price level. Proposition 4. In the move from the pre-offshoring to the offshoring equilibrium the price level falls by an amount equivalent to the gains from trade due to offshoring, GF T O, where ln P 1 ln P = GF T O (17) and GF T O = α m φ (j) (ln A (z) ln A (j)) dj+(1 α) v φ (j) (ln B (j) ln B (v)) dj. z m Proof. The proof is in the Appendix. The intuition of this result is straight-forward. The two terms inside the parentheses, which are unambiguously positive since A(z) A(j) on [z, m) and B(j) > B(v) on [m, v), represent the sums of excess comparative advantage due to increased specialization in production of tasks 1 and 2 19 unless z and v 1. 15

16 as a result of offshoring. Offshoring unambiguously decreases the price level. A analogous result may be derived by examining the effect of a fall in communication and coordination costs on the price level in the offshoring equilibrium. Lemma 5. A fall in communication and coordination costs (an increase in g) leads to a fall in the price level, where d ln P dg = 1 (Φ (v) Φ (z)) < g Proof. The proof is in the Appendix. A reduction in communication and coordination costs, which is an increase in g, is equivalent to Hicks-neutral technological progress in offshoring technology. This increases v and decreases z. Although at the margin the prices p(v) and p(z) are equal under both the integrated and offshored technology, the fall in communication and coordination costs reduces the cost of producing the inframarginal goods on (z, v). Following Grossman and Rossi-Hansberg [5], the term 1 (Φ (v) Φ (z)) is defined as the productivity g effect. Corollary 6. In the move from the pre-offshoring to the offshoring equilibrium the increase in world real income, defined as an expenditure weighted index of the output of all goods, ln Y i = φ (j) ln y i (j) dj, is equivalent in measure to the gains from trade due to offshoring, ln Y 1 ln Y = GF T O Proof. The proof is in the Appendix. The increase in world output is equivalent to the gains from specialization due to the increased trade from offshoring Welfare We begin this section with the following Lemma. Lemma 7. The utility per capita is the real wage. 2 2 This Lemma is attrbuted to Peter Neary, and comes from his Oxford lecture notes on the Ricardian Continuum Model. 16

17 Proof. The proof is in the Appendix. Lemma 8. The welfare gains per capita from offshoring in the North may be expressed ( ) ( ) w N w N ln ln = GF T O Φ (m) (ln ω 1 ln ω ) P 1 P and in the South ( ) ( ) w S w S ln ln P P 1 = GF T O + (1 Φ (m)) (ln ω 1 ln ω ) Proof. The proof is in the Appendix. The gains from trade due to offshoring are unambiguously positive. The terms of trade effect is the product of the North s share of expenditure on imports from the South and the change in the North s factoral terms of trade. This term is positive or negative depending on the direction of the movement in the factoral terms of trade, and is opposed between countries World Welfare To calculate the change in world welfare, we take a population weighted average of the welfare per capita changes in each country. Given that the population share of the South is λ = L S / (L N + L S ), the change in world welfare per capita as a result of offshoring is ( ( ) ( ) ) ( ( ) ( ) ) w S w S w N w N λ ln ln + (1 λ) ln ln (18) P 1 P P 1 P = GF T O + (λ Φ (m)) (ln ω 1 ln ω ) The change in world welfare is the sum of the gains from trade due to offshoring and the net terms of trade effect. Since a terms of trade gain for one country is a loss to the other, the change in the terms of trade is weighted by a term which reflects the initial inequality in per capita incomes determined by the difference between the South s population share and its initial expenditure share, (λ Φ (m)). From (5) the sign of (λ Φ (m)) depends of the magnitude of ω relative to 1 since λ Φ (m) = L S L S + L ω L S N L N + ω L when ω S 1 17

18 The population share, λ, is an exogenous parameter which ceteris paribus uniquely determines relative wage, ω, and the cut-off good, m, in the preoffshoring state of the world. As the population share of the South, λ, decreases the relative wage, ω, increases. Defining λ as the population share which yields a pre-offshoring relative wage of unity, ω = 1, we identify three cases: (i) λ = λ (ω = 1) : per capita utility is equal in the North and the South, and Φ (m) = λ. The terms of trade effects cancel out as they are exactly equal and opposite in magnitude. (ii) λ < λ (ω > 1) : the South s share of world expenditure exceeds it s population share (Φ (m) > λ) and if the South s relative wage falls then, the net terms of trade effect is positive as the North s gain per worker exceeds the South s loss and, world welfare rises unambiguously. If the South s relative wage increases then the terms of trade gain (per capita) in the South is less than the offsetting loss in the North, and the net terms of trade effect is negative. (iii) λ > λ (ω < 1) : the South s share of world expenditure is less than it s population share (Φ (m) < λ) and if ω increases due to offshoring state then the net terms of trade effect is positive because the North s loss is exceeded by the South s gain. If ω falls then the net terms of trade effect is negative (North s gain is outweighed by the South s loss). From (ii) and (iii) it is clear that any movement in the terms of trade that increases inequality between the North and the South will have a negative impact on world welfare, offsetting some part of the gains from trade due to offshoring. This is a result of the specification of the utility function, which is maximized when income per capita is equalized across Northern and Southern workers. Measuring the Net Terms of Trade Effect:. In the following analysis we show that the net terms of trade effect is positive for the period 1985 to 21. Using information on world population and income shares to calibrate the size of the net terms of trade effect, according to Cypher [1] (p. 18) in 1985 λ 1985 =.817 and Φ (m 1985 ) =.22 and thus (λ 1985 Φ (m 1985 )) =.597. This implies, from the pre-offshoring balance of trade, that ω 1985 = Φ (m 1985) (1 λ 1985 ) λ 1985 (1 Φ (m 1985 )) =.63 18

19 The offshoring balance of trade is ω = Φ (µ) L N / (1 Φ (µ)) L S where Φ (µ) is the South s share of world expenditure. From the same source λ 21 =.837 and Φ (µ 21 ) =.33. Thus ω 21 = Φ(µ 21 )(1 λ 21) λ 21 =.85. Taking 1985 (1 Φ(µ 21 )) as the point at which offshoring began to accelerate (Yi, [12]), the net terms of trade effect may be calculated as (λ 1985 Φ (m 1985 )) (ln ω 21 ln ω 1985 ) =.13 which is positive since the terms of trade have moved in favour of the poorer South. 21 In conclusion, when the terms of trade move to reduce inequality in per capita incomes between then North and the South, as they have here, then world welfare rises unambiguously. When the terms of trade move to increase world inequality, the net terms of trade effect offsets some of the gains from trade due to offshoring Comparative Statics: Population,Productivity and Demand Shocks We mention briefly, and describe intuitively, the effects of population and productivity shocks on the offshoring equilibrium. Given the structure of the model, the analysis is identical to the corresponding shocks in DFS, although the interpretation of results is altered. Population shock: An expansion of the North s labor supply ceteris paribus shifts the BOT (µ) curve upward, as in DFS Figure 2 (p.826), leading to a rise in the ω and an fall in µ, and thus a fall in the task 1 and 2 extensive margins, z and v. The terms of trade move against the North, and the range of tasks 1 and 2 that the North performs increases. Productivity shocks: A uniform global productivity shock reduces the unit labor requirements of all tasks in the North and the South in the same proportion leaving A (j) and B (j) schedules unchanged and thus the D (µ) schedule unaffected. The balance of trade is also unaffected because expenditure shares are constant and any improvement in unit labor requirements, and thus fall in prices, is met with an increase in consumption that 21 This is not a true ceteris paribus experiment because other factors have influenced the evolution of income shares. 19

20 leaves the labor allocated to any task or good unchanged, and the BOT (µ) schedule unmoved. Thus, as in the pre-offshoring model, a uniform global productivity improvement has no effect on the offshoring equilibrium. A uniform improvement in Northern technology shifts the D (µ) schedule inward, as both the A (j) and B (j) schedules shift to the left. The balance of trade is unaffected as expenditure shares are fixed, and thus the North s relative wage increases, and the extensive margins of tasks 1 and 2 expand for the North, and z and v fall. Since a uniform Northern productivity improvement has the same impact as a negative uniform Southern productivity shock of the same proportion, then a uniform Southern productivity improvement leads to an increase in the South s terms of trade and an increase in z and v. Demand shifts: A demand shift towards more sophisticated goods leads to a rightward shift of the balance of trade schedule, BOT (µ), 22 and a fall in South s terms of trade as demand shifts towards goods, and the underlying tasks, in which the North has a comparative advantage. The South s extensive margins for tasks 1 and 2 expand, with a corresponding contraction in the tasks performed by Northern labor. 4. Transport costs When transport costs are positive it is necessary to specify whether the production process places a constraint on the order of tasks, in particular if the execution of a task requires the physical presence of the output of another task in order for that task to be completed. With zero transport costs this can be ignored because the output of a task can be transported costlessly across borders multiple times for the application of further tasks without any implication for the extensive margins of task trade. One can imagine two alternative settings. In the first setting, tasks are sequential and the output of one task must be physically present in order for the next task to be executed. The alternative is that tasks are non-sequential and may be performed independently of each other, and assembled in any location into final goods. We differentiate the first setting as sequential tasks from the second which is non-sequential tasks. Of course, when there are 22 A demand shift towards more sophisticated goods means that Φ (j) decreases at any given j. 2

21 more than two tasks it is possible that the production of a good might involve both sequential and non-sequential tasks. Suppose that a good is produced from two sequential tasks, and task 1 must be performed before task 2. If the North wants to offshore task 2 then the North s output of task 1 must be sent to the South for combination with task 2 before shipment of the completed good to Northern (or Southern) consumers. On the other hand with non-sequential tasks, if the North is offshoring task 2, then the South can export the output of task 2 to the North for combination with task 1. Under zero transport costs whether tasks are sequential or non-sequential has no implication for the cost of production Transport Cost with Sequestial Tasks In this section we examine sequential tasks, which seems the most natural case. 23 With transport costs, it is necessary to distinguish the cost of a good by both its location and where it was produced. Hence c i p(j) is defined as the cost of good j in country i when it is produced in location p = {N, S, O}, where O indicates offshoring technology. We now determine the extensive margins for tasks. When does the South start to purchase task 1 from the North? The cost to the South of producing good j themselves is c S S (j) = ᾱws a S (j) α b S (j) 1 α while the cost of producing good j using task 1 from the North is c S O (j) = t α ᾱ ( w N a N (j) ) α ( w S b S (j) ) 1 α. Defining zs, the cut-offgood, by c S O (z S)/c S S (z S) = 1 then ω = ( 1 t ) A(z S ) (19) When does the North start to purchase task 1 from the South? By similar analysis, z N, the cut-off good for the North s task 1 is ( ) 1 ω = A(z N ) (2) t Since (19) and (2) are identical then z S and z N coincide and z S = z N = z. The relative costs are identical at each of these margins because the North 23 For example, the iphone is built from a sequential application of tasks, and may cross the same border multiple times as it is built. 21

22 is exporting the output of task 1 to the South for assembly with task 2 and then importing the completed final good from the South, so that the output of task 1 is imported as part of the final good. When does the North start to purchase task 2 from the South? When tasks are sequential, if the North is offshoring task 2 to the South it must send the output of task 1 to the South so that task 2 production activities can occur. It re-imports the output of task 1 when it imports the final good from the South. Comparing of c N N (j) = ᾱwn a N (j) α b N (j) 1 α and c N O = t (1+α) ᾱ ( w N a N (j) ) α ( w S b S (j) ) 1 α the cut-off task, vn, is determined by ω = t 1+α 1 α B(vN ) Note that in the expression for c N O (j) the term t (1+α) reflects that task 1 which has share α crosses the border twice, while task 2 with share (1 α) crosses once. When does the South start to purchase task 2 from the North? As j rises, the South, which is initially purchasing task 1 from the North and assembling it for its own consumption will begin to buy the entire good from the North, and v S is defined by ( ) 1 ω = B(v S ) t Figure 5 indicates the arrangement of production for given ω, and Figure 6 indicates the specific allocation of tasks. As can be seen, since v N < v S due to transport costs, then task 2 is non-traded over the interval [v N, v S ) Balance of Trade From the labor market equilibrium in the South, the balance of trade is 24 ω = Φ (µ N) θ Φ (µ N ).LN L S (21) We define the following terms which are used throughout the remainder of the analysis: µ S = αz S + (1 α)v S and µ N = αz N + (1 α)v N. These 24 See the Appendix for a derviation of this expression. 22

23 ω ω 1 (1/t)B(k) t (1+α)/(1 α) B(k) (1/t) A(j) z v N v S j,k Figure 5: Transport costs positive, tasks sequential S produces 1 and 2 for N & S N produces 1 for N & S z S produces 2 S produces 1 for N & S v N N produces 1 for N & S, 2 for N 2 for S only v S N produces 1 and 2 for N & S Figure 6: Transport costs positive, tasks sequential: arrangement of production 23

24 terms account for the fact that when there are transport costs there will be non-traded tasks. It follows that Φ (µ N ) is the North s import share while 1 Φ (µ S ) is the South s import share. Because there are non-traded tasks the share of world output that is traded is less than unity, where θ = 1 (Φ (µ S ) Φ (µ N )) is the share of world income that is traded Effect of Fall in Transport Costs on Price Level and Welfare The price level in the South now differs from that in the North because of transport costs. The effect of a fall in transport costs on the aggregate price level in the South is d ln P S = 1 ( ) 1 θ dt t (1 Φ (µ S)) + (1 Φ (µ S )) ˆω (22) θ which is the sum of two terms. The first term is the fall in the price level as resources are released from transportation of tasks/goods. This term is weighted by the South s import share 25 because the South only bears these costs on imports. The second term is the effect of the change in the terms of trade on the price level as a result of a change in t. In contrast to the case of communication and coordination costs only, the effects on the aggregate price level of opposing movement in w N and w S do not exactly cancel out because there are non-traded tasks. Rearranging (22) gives the effect on welfare per capita in the South of a fall in transport costs ( ) w d ln S P S = 1 dt t (1 Φ (µ S)) + (1 Φ (µ S )) ˆω which is the sum of the import share weighted transport cost term, identical to that in the expression directly above, and the terms of trade term, which is the change in the terms of trade weighted by the import share. Similarly for the North, d ln P N = 1 ( ) 1 θ dt t (Φ (µ N) + 2α (Φ (v N ) Φ (z))) Φ (µ N ) ˆω (23) θ 25 The import share is defined as the share of income that is spent on imported tasks/goods. 24

25 Rearranging (23) leads to ( ) w d ln N P N = 1 dt t (Φ (µ N) + 2α (Φ (v N ) Φ (z))) Φ (µ N ) ˆω The coeffi cient on the first term in both of the expressions above, Φ (µ N ) + 2α (Φ (v N ) Φ (z)), is the North s import share which exceeds Φ (µ N ) = αφ (z) + (1 α) Φ (v N ), which is the share of Northern income spent on tasks/goods from the South. This is because the North is re-importing tasks 1 on [z, v N ) sent to the South for assembly with tasks 2 into final goods. These tasks cross the border twice and incur transport costs on each crossing; first prior to combination with tasks 2 as they are sent to the South, and second as completed goods Transport, and Communication and Coordination, Costs with Non-Sequential Tasks The objective of this section is to demonstrate the effect on production location of declining communication and coordination costs when there are positive transport costs. This reveals how these costs interact in their effects on task boundaries and the location of production. To constrast with the section above, we examine the case of non-sequential tasks. Initially there is no offshoring because communication and coordination costs are at the prohibitive level, g. In the presence of communication and coordination, and transport, costs the task boundaries are determined as: 26 ω = g 1 α ta(zn ) ω = g 1 1 α tb(vn ) (24) ω = g 1 1 α t A(z S) ω = g α t B(v S) (25) Although it is simple to calculate the balance of trade here, to focus on the supply side we abstract from the effects of changes in g and t on the demand side and assume that the relative wage remains fixed. The prohibitive communication and coordination costs for the North, gn, is determined by setting z N = v N = m N and substituting into (24) it follows that ( 1 g N ) 1 α ta (mn ) = (g N) 1 1 α tb (mn ) g N = ( ) α(1 α) A (mn ) B (m N ) 26 See the Appendix for the calculation of these boundaries. 25

26 Similarly the prohibitive communication and coordination costs for the South, g S, is determined by setting z S = v S = m S and substituting into (25) g S = ( ) α(1 α) A (ms ) B (m S ) For simplicity we assume that A(j) and B(j) are correlated so that B (j) = σa (j) 27 which simplifies the analysis by imposing a common prohibitive level g = gn = g S = σ α(1 α). Initially communication and coordination costs are at the prohibitive level, g, in both countries and the pre-offshoring situation is depicted in Figure 7 with the dashed lines labelled (1/t)C (j) and tc (j) guiding goods production location, similar to Figure 3 in DFS (p. 83). Goods in the range [m N, m S ] are non-traded, with the North producing goods on [m N, 1] but exporting to the South only over [m S, 1], and the South producing goods on [, m S ) and exporting goods on [, m N ) to the North. A global fall in communication and coordination costs means that, there is a uniform increase in g from g and, offshoring will erupt on both the m N and m S margins and the North will begin to export task 1 and the South task 2, as seen in Figure 7. Goods on [v N, z S ] remain non-traded, while the South exports final goods on [, z N ) to the North, and the North exports final goods on [v S, 1] to the South. The North exports tasks 1 on [z S, v S ) which are combined with tasks 2 from the South to produce goods for Southern consumption. The South exports tasks 2 on [z N, v N ) which are combined with tasks 1 from the North to produce goods for Northern consumers. The specific allocation of tasks is given in Figure 8. As g increases further eventually v N will exceed z S and all goods will involve some traded tasks. The critical level of g at which z N = v S is (t 2 σ) α(1 α). 28 We briefly mention two cases of falling transport costs in the presence of communication and coordination costs. When communication and coordination costs are at the prohibitive level then a fall in transport costs leads to an increase in m N and a decrease in m S and a fall in the range of goods that are non-traded, however there is no offshoring. In the case where communication and coordination costs are below the prohibitive level so that there is offshoring, then the pattern of production is as described in Figure 7 and, a 27 The definition of relative advantage imposes that σ > This is calculated using (24) and (25) and setting v N = z S. 26

27 (1/t)C(j) ω tc(j) ω 1 g (1/(1 α) (1/t)B(j) g (1/α) (1/t)A(j) g (1/(1 α) tb(j) g (1/α) (1/t)A(j) z N m N v N z S m S v S Figure 7: Transport costs positive: falling communication and coordination costs S produces 1 and 2 for N & S z N N produces 1 for N only S produces v N 2 for N & S, 1 for S Non traded goods 1 z S N produces 1 for N & S, 2 for N S produces v S 2 for S only N produces 1 and 2 for N & S Figure 8: Allocation of tasks with transport and communication and coordination costs 27

28 fall in transport costs causes z N and v N to rise and z S and v S to fall. This leads to an increase in the range of traded goods as the intervals [, z N ) and [v s, 1] expand, and a contraction in the range of non-traded goods [v N, z S ). The range of offshored goods changes little as z N and v N, and z S and v S, move together. When B (j) = σa (j), and if z N and v N, and z S and v S, are close such that A (z N ) = B (v N ) and A (z S ) = B (v S ) then v N z N and v S z S are unchanged. One final point is that when both transport costs and communication and coordination costs fall this causes opposing movement on z N and v S and reinforcing movement on v N and z S. As these frictions fall v N is increasing and z S is decreasing and, as transport costs and communication and coordination costs approach zero, they approach v S and z N. 5. Conclusion With a simple modification we incorporate offshoring into the general equilibrium structure of the Ricardian continuum model of DFS. This allows an examination of the implications of offshoring in a setting with fully flexible goods and factor prices, endogenously determined margins of offshoring production, substitutability between tasks, and a continuum of goods. We examine the effects of offshoring on the location of task production, prices and the price level, wages, and welfare when trade is free and also in the presence of frictions that resist offshoring: transport, and communication and coordination, costs. A transformation of the offshoring model, which has three equations and three unknowns, allows it to be simplified into a form that is isomorphic to the Ricardian continuum model, which elucidates the analysis of equilibrium. The North specializes in tasks 1 and the South in tasks 2, as determined by relative advantage. Offshoring causes the price level to fall by an amount equivalent to the gains from trade, a benefit of offshoring that is often overlooked in the literature (both popular and academic), and if relative wages change as a result of offshoring then not all prices necessarily fall. The change in country welfare is the sum of the gains from increased specialization due to offshoring, which is always positive, and a terms of trade term. The change in world welfare is the sum of the gains from specialization and the net terms of trade effect, the sign of which is dependent upon whether the relative wage moves to reduce or increase inequality. 28