Estimating Trade in a Regional Input-Output Table Johannes Bröcker & Johannes Burmeister September 3, 2015 1/24
Outline Introduction Theoretical Framework Prerequisites CHARM GRETA Empirical Test Finland China Additional Tests Summary 2/24
Motivation Survey methods for setting up regional I-O tables are nice, but costly (Hewings, 1985). Thus, there is a need for cheap non-survey methods, less accurate but better than nothing. A typical procedure distributes output and final demand by industry according to indicators such as employment and income. This yields region r s output in industry j, x j r and region r s final demand D i r for goods from industry i. 3/24
Motivation (ctd.) Furthermore, technical input coefficients a ij (input per unit output from all sources) in the region are assumed to be the same as in the nation. Total domestic use for goods from industry i in region r is thus y i r = j a ij x j r + D i r. How to estimate regional exports e i r and imports m i r? This reduces to determining the domestic flow t i r as, by definition, e i r = x i r t i r and m i r = y i r t i r. 4/24
Prerequisites Minimal Requirements Consistency requires (r is region, N is nation) t i r min (x i r, y i r ) t i r t i N t i r 0. (A) (B) (C) 5/24
CHARM Cross-Hauling Adjusted Regionalization Method (CHARM) The industry standard for solving the problem seems to be CHARM (Kronenberg, 2009). In its original form it reads v s = x s y s + h s (x s + y s ) with v s := e s + m s, which is supposed to hold for s = r (region) as well as s = N (nation). 1 This can be rewritten as a trade equation [appendix 1], t s = min(x s, y s ) h s (x s + y s )/2. (1) 1 Industry subscript is omitted up to slide 13. 6/24
CHARM Cross-Hauling Adjusted Regionalization Method (CHARM) (ctd.) One obtains h N by solving (1) for h N with national data, i.e. with s = N. Assuming h r = h N, (1) with s = r delivers t r. CHARM fulfills (A) [appendix 1] but violates (B) and (C) [appendix 2]. 7/24
GRETA Charming GRETA (Gravity Regionalisation of Trade Approach) GRETA is theory based fulfilling (A) to (C), and (almost) as cheap as CHARM. GRETA s trade equation reads t s = (x s + y s + f s ) 2 with internal trade barrier f s > 0. (x s + y s + f s ) 2 4 x s y s (2) 8/24
GRETA Charming GRETA (ctd.) t s is the equilirbium solution T ss of a 2-regions (region s and rest of world w) CES trade model, with value flows as follows [appendix 3]: ( ) pk τ 1 σ kl T kl = α k y l. q l Notation: α k p k q l τ kl σ measure of product variety at origin k price at origin k CES composite price at destination l price mark-up due to trade barriers between k and l elasticity of substitution between varieties 9/24
GRETA Charming GRETA (ctd.) CES trade matrix: origin destination s w s w ( ) α psτ 1 σ ( ) ss s q ys s α psτ 1 σ sw s q yw w α w ( pw τ ws q s ) 1 σ ys α w ( pw τ ww q w ) 1 σ yw With f s := y w ( τ 2 sw τ ss τ ww ) 1 σ Two assumptions: τ sw = τ ws (symmetry) Take the limiting solution for y s /y w 0 (small open region) 10/24
GRETA Charming GRETA (ctd.) One obtains f N by solving (2) for f N with national data, i.e. s = N. Assuming f r = f N, (2) with s = r delivers t r. But one can do better, see extension on slide 14. 11/24
GRETA Properties of GRETA s trade equation 0 t min (x, y) [appendix 3] m t x = m+e+f 0 [appendix 4] t y = e m+e+f 0 [appendix 4] t t f = m+e+f 0 [appendix 4] f = 0 t = min (x, y) [appendix 5] f lim t = 0 [appendix 5] t (λx, λy, f ) λt (x, y, f ) for any λ 1. [appendix 6] If f N f r then t r t N (note:f N f r is sufficient, not necessary) [appendix 7] For x, y, f > 0, all relations also hold with strict inequalities replacing weak inequalities. 12/24
GRETA Extension: f r f N Let τ rw = τ Nw, but τ rr τ NN. Then, for industry i, f i r = f i N ( τ i NN τ i rr ) 1 σ i Let τ i ss = exp (γ i d ss ), with distance parameter γ i and average internal distance d ss of region s. Then, log f i r = log f i N + β i (d rr d NN ) with β i = γ i ( σ i 1 ), β i > 0 if σ i > 1.. 13/24
GRETA Extension: f r f N (ctd.) To make it cheap, one may approximate d rr d NN by Fr F N (up to a multiplicative constant), F s denoting the area of the region or nation. β i is the distance parameter of a standard gravity equation. A simple estimate is obtained from the cross-sectional regression with subscript c denoting countries. log f i c = A i + β i F c + u i c, (3) 14/24
15/24 Extension: f r f N (ctd.) The following table shows half-life distances log (2)/ ˆβ i in km (squareroot of area), estimated regressing (3) with GTAP 8 data: Sector hld in km Sector hld in km Sector hld in km Paddy rice 933 Sugar 888 Trade 626 Wheat 393 Food products nec 1279 Communication 863 Cereal grains nec 707 Beverages and tobacco 1138 Financial services 507 Vegetables, fruit, nuts 602 Textiles 779 Insurance 621 Oil seeds 563 Wearing apparel 1703 Business services 488 Sugar cane, sugar beet 1013 Leather products 730 Recreational services 743 Plant-based fibers 350 Wood products 886 Public Administration 574 Crops nec 415 Paper, publishing 575 Bovine cattle 1260 Petroleum, coal 524 Animal products nec 618 Chemicals 666 Raw milk 399 Mineral products nec 869 Wool 330 Ferrous metals 976 Forestry 714 Metals nec 593 Fishing 508 Metal products 614 Coal 272 Motor vehicles 609 Oil 261 Transport equipm. 447 Gas 266 Electronic equipment 601 Minerals nec 400 Machinery nec 544 Bovine meat products 594 Manufactures nec 641 Meat products nec 811 Electricity 1044 Vegetable oils and fats 758 Gas 402 Dairy products 1251 Water 1086 Processed rice 638 Construction 459
Finland The State of Uusimaa 2002 Survey CHARM GRETA in mn. e Intra-trade i Exports i Imports i t i r 56,281 (100%) e i r 34,078 (100%) m i r 24,619 (100%) Trade 66,405 (118%) 23,954 (70%) 14,495 (59%) 64,366* (114%) 25,993* (76%) 16,534* (67%) 16/24
Billion Euro 17/24 15 BENCHMARK CHARM GRETA Uusimaa intra trade flows 10 Agriculture Forestry Fishing Mining Food Textiles Wood Paper Coke Minerals Metals Machinery Computers Vehicles Furniture Electricity Construction Trade Hotels Transport Finance Real estate Public amdin Education Health Sewage 5 0
Finland The State of Uusimaa 2002 (ctd.) For comparison, we calculate column sum totals µ j of the Leontief inverse with regional input coefficients rr ij as r ij r ( t = ar ij i ) r yr i with RHS from survey data except for t i r. t i r from survey data t i r from CHARM estimates t i r from GRETA estimates 18/24
The State of Uusimaa 2002 (ctd.) Survey CHARM GRETA With: Regional Output Multiplier µ j Mean 1.38 1.57 1.52* ν 1-13.43 10.31* ν 2-49.17 37.71* ν 3-12.01 8.73* ν 4-15.32 11.66* ν 5-13.43 10.31* SD - 0.072 0.054* SD = ( 100 ν 1 = n ν 2 = 100 ν 3 = 100 ν 4 = 100 ) j ( ˆµ j µ j ) /µ j, ( ) ˆµ µ / ( µ 1), j ( 1 ν 5 = n n j j ( ) q j µ j ˆ µ j /µ j, ) ( ˆµ j µ j ) 2 / j ˆµ j µ j /µ j, j µ 2 j, [ ( ) 1 ) } 2 ] 0.5. {( ˆµ j µ j /µ j ν 5
China The Province of Hubei 2005 in bn. yuan Intra-trade i Exports i Imports i Survey CHARM GRETA t i r 1,922 (100%) e i r 279 (100%) m i r 282 (100%) Trade 1,998 (104%) 206 (73%) 220 (74%) 1,940* (101%) 265* (94%) 279* (94%) 20/24
Hubei intra trade flows 200 BENCHMARK CHARM GRETA 150 bn yuan 100 50 0 Agriculture Coal Oil and gas Metals Textiles Clothing Wood Oth. Mining Food and Tobacco Chemicals Oth. minerals Metal smelting Fabricated Metals Paper Oil, coke & nuclear General equipment Transportation Eq. Electrical Machinery Computers Instruments Arts and crafts Waste Electricity Gas Transport Post Information Trade Water Construction Hotels Finance R&D Technology Environment Real estate Leasing HH services Education Health Culture Public mgmt 21/24
The Province of Hubei 2005 (ctd.) Survey CHARM GRETA With: Regional Output Multiplier µ j Mean 1.88 1.93 1.87* ν 1 2.3-0.92* ν 2 5.36-1.97* ν 3 3.73 0.38* ν 4 4.45 3.85* ν 5 2.98 2.57* SD 0.029 0.027* SD = ( 100 ν 1 = n ν 2 = 100 ν 3 = 100 ν 4 = 100 ) j ( ˆµ j µ j ) /µ j, ( ) ˆµ µ / ( µ 1), j ( 1 ν 5 = n n j j ( ) q j µ j ˆ µ j /µ j, ) ( ˆµ j µ j ) 2 / j ˆµ j µ j /µ j, j µ 2 j, [ ( ) 1 ) } 2 ] 0.5. {( ˆµ j µ j /µ j ν 5
Additional Tests Further Data: Multi-regional I-O Data for Japan 2005 Regional I-O table for Catalonia, Spain 23/24
Conclusion GRETA represents a cheap and easy to use recipe for regionalizing national I-O tables. GRETA is theory based. GRETA fulfills minimal consistency requirements. GRETA outperforms the industry standard. 24/24
24/24 Appendix First to CHARM: 1. By definition, Thus (x + y v)/2 = (x e + y m)/2 = (t + t)/2 = t. t = [x + y x y h(x + y)]/2 = min(x, y) h(x + y)/2, where we made use of x + y x y = 2 min(x, y). (A1) From t N min(x N, y N ) follows h N 0. With h r = h N we thus obtain t r min(x r, y r ).
24/24 2. First, x N = y N = 10 and t N = 0 implies t r = 1/2 < 0. Second, x N = 10, y N = 20, t N = 0 and x r = y r = 1 implies t r = 1/3 > t N. (Note that t N = 0 is sufficient, not necessary for the counterexamples to hold.) Then to GRETA: For ease of notation define z := (x + y + f )/2 and S := z 2 xy, such that the GRATA trade equation (2) reads t = z S. 3. The GRETA trade equation (2) is derived as the equilibrium solution t = T ss of a 2-regions CES trade model (see slide 9) with value flows in the trade matrix as follows: α s ( ) ps τ 1 σ ( ) ss q ys s α ps τ 1 σ sw s q yw w ) 1 σ yw α w ( pw τ ws q s ) 1 σ ys α w ( pw τ ww q w.
24/24 Define and ( ) 1 σ ps τss a := α s yw τ 1 σ ww, q w ( ) y 1 σ s pw τss b := α w τ 1 σ ww, yw q s ( ) τ 2 1 σ sw f := y w. τ ss τ ww ( ) Then, since α pw τ 1 σ ww w q w 1, the trade matrix reduces to ( ) ab a f b. f y w Summing over the first row and column of the trade matrix, respectively,
24/24 leads to the equation system t + a f t + b f = x = y with t = ab, by definition. Summing up and multiplying both equations, respectively, results in 2t + (a + b) f = x + y t 2 + t(a + b) f + tf = xy. By rearranging the first equation to (a + b) f = x + y 2t and plugging this into the second one, we obtain rewritten as t 2 + t(x + y 2t) + tf = xy, t(x + y + f ) xy = t 2. Solving for t leads to the two possible solutions t 1,2 = z ± z 2 xy. (A2)
24/24 For the expression under the square-root in (A2) we get z 2 xy (x + y) 2 /4 xy = (x 2 2xy + y 2 )/4 = (x y) 2 /4 0. Both solutions are thus real. As z (x + y)/2 min(x, y) the larger solution (the one with + ) yields t > min(x, y) if f > 0 or x = y. Hence, the larger solution cannot apply. For the smaller solution we obtain t 0, because S z. Furthermore, t min(x, y) is immediate from t = min(x, y) for f = 0 (see item 5) and t f 0, t f = 1 2 z 2S 0. 4. t x = 1 2 z y 2S = S z + y m = 2S m + e + f 0. Similar operations yield t y and t f. We disregard the case m + e + f = 0 that can only occur if f = 0 and x = y, as is immediate from item 5.
24/24 5. For f = 0 we obtain t = (x + y)/2 (x + y) 2 /4 xy = (x + y)/2 (x 2 2xy + y 2 )/4 = (x + y)/2 x y /2 = min(x, y), where the last equality is (A1). As to the other end, let t and t be solutions for f and f f and given x and y. Hence, t t due to t f 0. Thus xy t = x + y + f t xy x + y + f t. Taking the limit for f yields t 0. 6. If λ 1, then t(λx, λy, f ) t(λx, λy, λf ) = λt(x, y, f ). The inequality follows from t f 0, the equality is obvious from (2).
7. Obvious from item 4. 24/24