Appendix A Market-Power Model of Business Groups Roert C. Feenstra Deng-Shing Huang Gary G. Hamilton Revised, Novemer 200 Journal of Eonomi Behavior and Organization, 5, 2003, 459-485. To solve for the equiliria, we will make use of the full-employment ondition, whih in written in the symmetri equilirium as: L GNk y GNβyφ GMk GM Gα N k N βy φ y x GM M k x x M x. (A) Using (5), (9), (4) and (8), this is simplified as, [ y( βη ( ) ) xσ α] βη ( ) Mk σ. L G N k M k ( ) Nk y x (A2) Another useful relation is the equality of GNP as total fator inome and the value of final sales, L GNqy Nqy. (A3) Using (8), (9), (2) and (4) this is written in the symmetri equilirium as, s L GNky s η y y Nk y η. (A4)
2 We should also indiate the values for internal and external sales of inputs, x and. External sales of eah input variety shown in (6) an e written as i, where i is sales to other groups and i is sales to downstream unaffiliated firms. Computing the derivative of unit-osts, the sales to other usiness groups are, G y ( ) jφj β Nj p G, j i M j M i,i j p j p M (G )y φ (G )M ( β) p N M p M i p i (A5a) where the seond line applies in the symmetri equilirium. The sales to unaffiliated firms are, i p yφ GM p ( β) M ( β) yjφ p G M i p N j N p p i (A5) M i where the seond line applies in the symmetri equilirium. Similarly, internal sales of eah produt variety are, x G M [ M M p p ] j, j i φ φ ( β)n ( β)n j [ M ( G ) M p M p ] y y j j j (A6)
3 where the seond line applies in the symmetri equilirium. Derivation of equation (8) in the text: When the usiness group hoose produt variety optimally, then, x ( σ ) k. (8) x Proof: We need to differentiate () with respet to M and set this equal to zero. Sine profits in () have een maximized with respet to the markup μ, it is legitimate to onsider a small hange dμ hosen to ensure that the hange in M has no effet of the final-goods prie q μ φ harged y group i. Speifially, we will hoose dμ suh that: dq μ dφ dμ φ 0 dμ / dφ ( μ / φ ). Suppose first that there are no external sales of the intermediate inputs y group i, 0. Sine the prie q is onstant, it follows that the demand y for eah final good of group i is also onstant, so that hange in M only affets osts. Then from () we have, d dm N [y ( μ dμ ) dφ y φ φ ] M k x φ x y N k x k x, (A7) M ( σ ) where the seond equality follows y using dμ / dφ ( μ / φ ), and the final equality follows from differentiating (2) and omparing the result to (A6). Then (8) follows y setting (A7) equal to zero, and using symmetry.
4 Now suppose that there are sales of the inputs outside the group. From (2), the osts to other groups depends on M p, hosen y group i. Sine profits in () are maximized with respet to p, we an onsider a small hange in p designed to keep this magnitude onstant, dm p ( σ)m p 0. (A8) We will onsider the optimal hoie of M with p adjusting as in (A8), whih ensures that the osts of the other groups and therefore their own final-goods pries are onstant. With q and q j all onstant, then demand y is also onstant. The hange in profits of group i is, d dm M d ( A8) (A8) dm (A8). Using () and the same steps as in (A7) and (A8), this is evaluated as, d dm (A8) x k ( σ ) x k ( σ ) x k ( σ ) x x x ( p ) M M ( p ) ( p ) ( σ ), p σ p p ( σ ) dx ~ (A8) dm (A8) (A9) where the seond equality follows y omputing / p from (A5a,) while keeping p M onstant in the denominator, and y omputing / dm from (A8). Setting (A9) equal to zero and using symmetry, we otain (8). QED
5 Lemma Suppose that N 0. Then eah group will sell inputs to the other groups if and only if, G > σ σ, (9) in whih ase the optimal pries are given y: Proof: p G. p [ σ s ( )] G x (20) Let p and q denote the pries hosen y group i, with p j and q j the pries of the other groups j,,g, j i. As p is inreased, this will raise the osts to the other groups and therefore inrease q j μ j φ j, holding fixed the optimal markups μ j. Sine profits in () have een maximized with respet to the markup μ, it is legitimate to onsider a small hange dμ to ensure that the hange in the pries of final goods y all groups are equal. Speifially, we will hoose dμ suh that: dq dμ φ dq μ ( φ / p ). (A0) j j j The left of this expression if the hange in the prie of final goods for group i, due to a small hange in its markup, and on the right is the hange in the final goods prie of another group j, due to a hange in the prie p of intermediate inputs sold y group i (ut holding the markup of group j fixed at its optimal level). By ensuring that all final goods pries hange y the same amount, this ensures that the relative outputs and market shares of all final
6 goods are unhanged. This will simplify the alulation of the hange in group i profits due to the omned hange (dμ, ) satisfying (A0). Using symmetry of the initial equilirium, we divide (A0) y q μ φ, and q j j j rewrite this expression as, dq l q dq j l φj l. (A0 ) q p φ j j Thus, with a rise in p leading to equi-proportional inreases in q, q j, and φ j, and total onsumer expenditure fixed at L, these prie inreases must e mathed with equiproportional redutions in final goods output y and y j. Thus, dy (A0) y dq, i,,g. (A) q (A0) Notie that this implies that q y is onstant under (A0), as is φ j y j. The total hange in profits for group i is, d (A0) p d dq (A0) dq (A0) G j, j i d dq j (A0) dq j (A0). Using (), this is evaluated as, d (A0) M ( p ) p N φ dy (A0). (A2) In the first term of (A2), the elastiity ( x ~ / p )( p / ) (holding φ j y j onstant) as [ σ ( σ) ] group i, given y: s x is omputed from (A5a) ~ x, where s x is the intermediate market share of
7 s x M M p G M p j, j i j j j M p j. This expression appears as (2) under symmetry. In the seond term of (A2), we note that equi-proportional inreases in q and redution in y has the effet of holding q y onstant in profits, and simply reduing y y the amount given y the last term in (A0 ). To evaluate this term, differentiate (2) to ompute, φ p j p y M j M (G )N, ( β) M G M M p i i φ,i j j p i (A3) where the seond line applies with symmetry, with given y (A5a). Then omning (A)-(A3) and using symmetry, we otain, d (A0) M p p [ σ s ( σ ) ] x ( ) G. (A4) Setting (A4) 0, we see that there is a finite solution for p if G > σ/(σ-), and this solution is (20). Otherwise, expression (A4) remains positive for all positive values of p, so the usiness group optimally hooses p. QED
8 With these preliminary results, and heneforth using symmetry of the equilirium, we have the following haraterizations of the V-group equiliria (Proposition ) and the D- group equiliria (Proposition 2): Proposition Assume N 0. Then the V-group equiliria an take one of two forms. Either: (a) the usiness groups do not sell inputs to eah other ( 0 ), and the numer of groups is given y the unique positive solution to, G 2 Δαη Δα G Δ L ( η ) L ( ) 0, (A5) provided that G < σ/(σ-), where Δ ( σ )/[( η )( β)] ; or, () the usiness groups do sell to eah other ( > 0 ), while the numer of groups is given y any positive solution to, ~ G 2 Δαη ( η ) Δ ~ α ~ G ( Δ ) 0, (A6) L L provided that G > σ/(σ-), where Δ ~ [( σ/ fp ( )) ] / [( η )( β)], and, p p G fp ( ) ( ) ( ) ( ) σ. (A7) ( G ) p
9 Proof: (a) Sine N 0 then s y /G, so that s y /(-s y ) /(G-). Setting Π 0, and using 0 with (2) and (4) we otain, N k ( G ) M k ( G ) α. (A8) y x Using N 0, (A2) and (A4) are simplified as, [ ( ( ) ) ] L G N k βη M k σ α M k σ, (A2 ) y x x L GNky η G. (A4 ) Condition (8) eomes x (σ-)k x sine 0, so that using (4) and (A6), ( σ ) k x ( η )( β) k y N ( M Mp ), (A9) where p σ/(σ-). The aove are 4 equations in 4 unknowns- G, N, M and M. Setting (A2 ) (A4 ) to eliminate L, and using (A8) repeatedly to onvert terms involving N to involve M instead, we an derive the quadrati equation, G 2 G ( σ ) ( β)( η ) Mk M k α M k x ( β)( η ) x x x σ α M k 0 (A20) Using (A9) we an solve for (M /M ) as,
0 M σ M σ ( η )( β) ( σ ) Nky Mky ( η )( β) ( G ) ( σ ) α Mkx, where the seond equality follows from (A8). Sustituting this into (A20), we otain a rather long quadrati equation for G, whih has the following two solutions: (i) M (ii) M 0 and G ( σ ) ( β)( η ) σ α Mk x, or, k > x σ ( σ ) > kx 0 and G σ β η α ( )( ) M k x. These solutions are viale equiliria provided that 0, so that the group finds it optimal to not sell its inputs externally, as assumed aove. From the aove Lemma, we know that 0 if and only if G < σ/(σ-). This immediately rules out the solution in (ii) whenever k x is lose to k x. The solution in (i) is viale provided that G < σ/(σ-). To simplify the expression in (i), we an sustitute (A8) into (A4 ) to solve for, α M k x α L α. G[ η( G ) ) (A2) Sustituting (A2) into (i), we otain the quadrati equation in part (a). Part () is proved along with Proposition 2, elow. QED
Proposition 2 Assume N 0. Then in the D-group equilirium unaffiliated upstream firms are profitale (M > 0 ), and the usiness groups also sell to eah other ( > 0 ), while the numer of groups is given y: whih implies, [ ( G ) p ] k G x σ (A22) k σ x σ Proof: σ σ p ( σ ) k x G (p ). (A23) σ k x Now we suppose that G > σ/(σ-) so that the group sells externally. Setting Π 0 and using (4), we otain: Nky ~ xm ( p ) Mk x α. (A24) ( G ) The full-employment onditions (A2 ) and (A4 ) ontinue to hold. Condition (8) is ( ) x ~ x σ k, so that using (A6) and (A5a) we otain, x ( ) ( σ ) ( G ) ~ k x G x p x Sustituting this into (A24) we otain, x [ p ]. N k y ( G ) ( ) M k f p α, x (A24 ) where f(p ) is defined in (A7).
2 Then setting (A2 ) (A4 ) to eliminate L, and repeatedly sustituting (A24 ) to replae terms involving N with those involving M, we otain the quadrati equation, G 2 G σ fp ( ) ( β)( η ) Mk M k x x α fp ( ) M k σ ( β)( η ) x α fp ( ) 0. M k x (A25) When M 0, then G is solved as: G σ fp ( ) ( β)( η ). α fp ( ). M k x (A25 ) When M > 0, we first rewrite the expression ( ) σ k x ~ x using (A6) and (A5a) as, x ( σ ) k x y ( β)( η ) [( ) ] [ M M( G ) p Mp ] k N G p, where p σ/(σ-). It follows using (A24 ) and (A7) that we an solve for M as, Mp M [ ( G ) p ] ( σ ) α M k ( G ) f( p ) ( )( ) fp ( ) x η β σ. (A26) Notie that when M 0 in (A26), we again solve for G as in (A25 ). When M > 0, we sustitute (A26) into (A25) to otain a rather long quadrati equation in G. One solution (for M 0) is (A25 ), and the other solution (for M > 0) is given y (A23). In order to solve ompletely for all endogenous variales, we an make use of (A4 ) together with (A24 ) to otain,
3 α M k x αfp ( ) L α. G[ η( G ) ) (A27) Sustituting (A27) into (A25 ), we otain (A6). Then there are five unknowns p, s x, G, M, and M and five equations to solve for them (20), (2), (A6)-(A7) or (A23), (A26) and (A27). QED Propositions and 2 provide us with the equations used to ompute the V-group and D-group equiliria. Finally, we turn to the ase of U-groups, in whih ase N > 0. We first need to derive the pries harges y these groups for the external sale of intermediate inputs, assume that the usiness group annot disriminate in its sales to other groups or to downstream unaffiliated firms. Then the optimal prie for external sales of the intermediate input is: Derivation of Priing Equation for U-groups: With M 0 and N > 0, the optimal prie p will satisfy: θ sy p G s y p ( ) ( θ) σ ( σ) θsx G ( λ ) (A28) where θ is the fration of external sales ~ x that are sold to other groups, and, ( G ) p λ. (A29) Gp
4 Proof: Let p and q denote the pries hosen y group i, with p j and q j the pries of the other groups j,,g, j i. As p is inreased, this will raise the osts to the other groups and therefore inrease q j μ j φ j, holding fixed the optimal markups μ j. Sine profits in () have een maximized with respet to the markup μ, it is legitimate to onsider a small hange dμ to ensure that the market shares of all other usiness groups are held onstant. Imposing symmetry on the pries q j q of all other groups j,,g, j i, as well as on the numer of final produts N j N for j,,g, and the pries of downstream unaffiliated firms q j q for j,,n, then the market share of group j i is written from () as, η N q s. ( ) y η η η [ N q (G )N q N q ] Totally differentiating this expression, we find that s y is onstant provided that, where, d ln q d ln q s y d ln q (G ), (A30) s y s y η N q s (A3) y η η η [ N q (G )N q N q ] is the omned market share of all downstream unaffiliated firms. Thus, we will evaluate the total hange in profits for group i due to a small hange, assuming that the downstream pries of other groups q and unaffiliated firms q are
5 adjusted holding their markups μ and μ fixed at their optimal level. Further, we assume that the markup for group i, μ, is adjusted so that dq satisfies (A30), i.e. the market shares of all other usiness groups j,,g, j i, are onstant. Beause total onsumer expenditure equals L from (7 ), when the market shares for the other groups s y are fixed, then so is expenditure on their produts, so that q y μ φ y are oth fixed: the inrease in q due to the rising input prie p will e mathed y an equi-proportional redution in y. The same is not true for group i, however: the inrease in q satisfying (A30) will e mathed y an redution in y that need not e in the same proportion. Then the total hange in profits for group i is, d ln p d ln q d ln y q yn ( q φ ) (A32) yn ( A30) (A30) p ~ x M p ln ~ x p ln p q y N d ln q ( q φ ) d ln y yn ( A30) (A30) The first terms on the right are simply the partial effet on profits of hanging the input prie p, whih is evaluated using its elastiity of demand. The other two terms are the effet on profits of hanging q aording to (A30), and the indued effet of all prie hanges in q and q on y. In order to evaluate the indued effet on y, we write this from (7) as:
6 y q L, (7 ) η η η [N q (G )N q N q ] η where we have made use of symmetry: q j q for all other groups j,,g, j i, N j N for j,,g, and q j q for j,,n. Totally differentiating (7 ) and using (A30) we find that, d ln y d ln q ηs y d ln q d ln q (A33) s (A30) y (A30) (A30). ( A30) From the optimal priing rule (2), and using symmetry so s y s y, we have that, s y η ( q φ ) q φ. (A34) s y Sustituting (A30) and (A33) into (A32), and making use of (A34) along with Gs y s y, we otain: dπ p ~ x M p ln ~ x p ln p d ln q s y d ln q d ln q y φ N. (A35) s ( A30) y (A30) (A30) To simplify this expression further, use (2), (5) and the onstant optimal markups, along with (A5) to ompute that, d ln q (A30) ln φ pm (A36) ln p (G )y φn,
7 d ln q (A30) ln φ pm, (A37) ln p yφn eah of whih an e sustituted into (A35). Finally, to ompute ln ~ x / ln p ) we use (A5) to otain: ( ln ~ x ln p { σ ( σ) [ θsx ( θ) / G] }, (A38) where ( / ) θ ~ x is the share of a usiness group s external sales of intermediate inputs this is sold to other groups. Notie that the derivative in (A38) is omputed while holding φ y onstant in the numerator of (A5a), as disussed aove. We also treat φ y onstant in the numerator of (A5) when omputing this elastiity. Sustituting (A36) (A38) into (A35), and setting the latter equal to zero, we otain: p (G p θ s y Nk ( θ) ) s y Nk ( θ) σ ( σ) θs x G y y s s y y. (A39) To simplify this further, note that (-θ)/θ is the supply from eah usiness group to all downstream unaffiliated firms, relative to all other usiness groups. This must equal the demand from downstream unaffiliated firms relative to that from usiness groups, given y ( ~ x / ~ x ) λ( N k / N k ), using (A5), (9), (4) and (A29). Thus, y y θ λnky θ ( G ) N k Sustituting (A40) into (A39), we otain (A28). QED y. (A40)
8 In the denominator of (A28), s x is still given y (2) ut with M 0, and is interpreted as the share of total demand for intermediates y a group (inluding internal demand) oming from one other group. We ould analogously define the share of total demand for intermediates y a unaffiliated downstream firm supplied y one group, whih is simply (/G). Thus, the weighted average [ θs ( θ) / G] appearing in the denominator of (A28) an e interpreted as the share of total demand for intermediates supplied y one group, so that the entire denominator is simply the elastiity of demand for the inputs of a group. If the numerator were unity, then (A28) would e a onventional Lerner priing formula. Instead the numerator exeeds unity, refleting the fat that when a group sells an input, it will give ompeting firms a ost advantage, therey lowering profits in the final goods market. Aordingly, the usiness group harges a higher prie for its inputs than would a firm that is not vertially-integrated aross oth markets. With the pries given y (A28), the U-group equilirium is haraterized y: x Proposition 3 (U-Group Equiliria) Assume M 0. Then in the U-group equilirium the usiness groups sell inputs to unaffiliated downstream firms (N > 0) and to eah other ( > 0 ), while the numer of groups is given y any positive solution to: ~ αδg L 2 s s y y s η G s y y ~ ~ αδη Nk Δ L Nk y y Nk Nk y y 0, (A4) provided that G > σ/(σ-), where Δ ~ [( σ/ g(p )) ] / [( η )( β)] and,
9 g(p p [( G ) λ( Nky / Nky)] ) ( σ )( p ). (A42) p [( G ) λ( Nky / Nky)] Proof: Using Π 0 and (2), we otain: s s y y Nky ~ xm p Mkx ( ) α. (A43) Using ~ x ~ x ~ x from (A5) with M 0, together with x from (A6) and x ~ x ( σ ) k from (8), we an derive, x [ ] ( ) ( ) M ~ x p Mkx g p, where g(p ) is defined in (A42). Sustituting this into (A43) we otain, s s y y Nky Mkxg(p) α. (A43 ) Setting (A2) (A4) with M 0, and sustituting (A43 ) to replae terms involving M with those involving N, to otain: s G G s y α N k y y ( σ/ g(p )) Nk Nk ( η )( β) y y. (A44) In order to fully determine the equilirium, we also need to solve for (N /N ). We will make use of the relations, η η φky y q φ s y φk y y q φ η s. (A45) y
20 The first equality of (A45) follows from (9) and (4); the seond equality from the CES demand system; and the third equality from the priing formulas (8) and (2). Making use of the first and last expressions we otain, η η η φ sy ( ) ky ( ) φ η s y k y (A46) and sustituting this into the last equality of (A45) we have, q q s s η y y ( η ) y ( η ) k k y. (A47) The shares s y and s y of one usiness group and all unaffiliated firms are related y Gs y s y. It follows that (s y /s y ) equals, s y Gs y N N q q η η N N s η s y y, (A48) where the first equality follows from the CES demand system, and the seond from (A47). Rewriting (A48) we otain, Nk Nk y y s s y y sy G ( ) s. η y (A49) To simplify this further, note that λ in (A29) equals ( σ )/( β) φ φ, using (2) and (5) with M 0. It follows from (A46) that, where, s s y y ηλ δ ( ky ky) / η, (A50a)
2 Sustituting (A50) into (A49), we otain, ( β)( η ) δ. (A50) ησ ( ) Nk N k y y λ / k k k k G η λ / ( ). (A5) ( y y) ( y y) δ η δ η To solve for the level of N and M, we an make use of the full-employment onditions (A2) and (A4) together with (9) and (4) to derive, and, M N L k y s Gη s y y Nk Nk η y y ( G ( Nky Nky) ), (A52) L ( βη ( )) / Gkx G G sy sy Nky Nky kx α. (A53) σ η ( /( )) η( / ) σ Sustituting (A52) into (A44) we otain the quadrati equation (A4). The equilirium is now defined y six variales G, p, s x, θ, λ and (N /N ) with six equations given y (2), (A28), (A29), (A40), (A4)-(A42) and (A5). QED