Appendix for "O shoring in a Ricardian World"

Transcription

1 Appendix for "O shoring in a Ricardian World" This Appendix presents the proofs of Propositions - 6 and the derivations of the results in Section IV. Proof of Proposition We want to show that Tm L m T m w m + m = T T w + w + m = Since T > Tm L m, it is enough to prove the inequality for m = : Thus, using w i = (T i =L i ) for i = ; ; m then we need to show that Tm L m! = T m Lm = (T m) = T m = T T T L T! = + L = L T T + L T We have T = L T T + L T Q.E.D. Proof of Proposition We have T = L T T + L T T = (T m) = : This implies that w () = ( + ) ~w () () T = ( + ) ( ) (L ) w () = T ( ) ( ) ( + ) (L ) + L L Obviously T ( ) (+) is decreasing, while () + L L is increasing in : To determine the sign of w() on [; ] we should then compare (L )

2 w() and w() with zero. Focusing rst on w(), using the de nition of, we get w() = ( + =L ) ( ) (L + ) < L + Turning to w (), note that w () = T ( ) T L = T L ( ( ) ) > () > ( ) = Thus, if ( ) =, then w () is always decreasing on [; ]: If > ( ) =, then w () is shaped like an inverted U on [; ]: Q.E.D. Proof of Proposition 3 Recall that P k T kc k. Thus, it is useful to use = T c + w + m where m is not a ected by. We know that c = = L ~, so This implies that T = ~ and w = T c + w = T L ( + ) + T (L ) T c + w = (T = ) ( + ) T (L ) T We need to compare ( + ) T (L ) with zero on T [; ). Obviously, = L >, while simple algebra reveals that =. Since f () <, then = implies that > for any [; [. This means that T c + w >, or > : But given P = = then this implies that P <. Q.E.D. Proof of Proposition 4 w We know that the sign of P is the same as the sign of w simple di erentiation and simpli cation reveals that w x( ) + + L=L = G(x; ) w x( + ) + P P = F (x; ) x (+) () x( + ) + L () + w P P : But

3 where x, = =L. Let x F () and x G () be de ned implicitly by F (x; ) = and G(x; ) =, respectively. The following lemma, whose proof is simple and therefore omitted, summarizes a number of properties of these functions: Lemma F (x; ) is decreasing in x, G(x; ) is increasing in x, + + L=L x F () = >, and x G () = > : ( ) Also, x F () < x G (), x F () >, x G () >. + Let x M () be de ned implicitly by G(x; ) = F (x; ). Such a solution necessarily exists since x F () < x G () and F (x; ) is decreasing in x and G(x; ) is increasing in x. Also, it is clear that < x F () < x M () < x G (). Since x > x M () implies G > F then it also implies that w =P is increasing. Similarly, x < x M () implies that w =P is decreasing. The following lemma (whose is proof is long and tedious and therefore provided in a separate online Appendix) is critical: Lemma x M () is increasing Let ^ be equal to x M () = : If ^; then x = x M () x M () for any : This implies that F (x) > G(x) (except the case when x = ^ and = ), so w =P is decreasing. This establishes the rst part of the proposition. To establish the second part, we need the following lemma: Lemma 3 For any > ^ = (x M ()) = we have x = < x M (()). Proof. The proof relies on showing that F ( ; ()) =, which implies that = x F (()). If this is true then x M (()) >, because since x F () < x M () for all then = x F (()) < x M (()), which establishes the result. But from the de nition of we see that = + and plugging this into F ( ; ) shows that F ( ; ()) =. This lemma implies that if > ^ then w =P is increasing for = and decreasing just before = (), with a unique point for which x M () = x at which G = F and hence (w =P ) =. This implies that the curve w =P as a function of in the interval [; ] is shaped like an inverted U. Proof of Proposition 5 The only thing left to show is that steady state P is decreasing in. It is su cient to show that mt = T t ct + twt is decreasing in. But mt = (=L F tg L ) r 'ct + r w 3

4 Using c = = N w and w = ( = N ) then! =( ) mt = (=L F = tg L ) r ' N + r ( = N ) w t = ( N =L F tg L ) ('r w + r ) But plugging in from the equations (4) and (5) we get that which is increasing in. Q.E.D. 'r w + r = 'w + Proof of Proposition 6 I rst show that x = =w is increasing in. From (9) we get ( = N ) = z ( + x) = z (z + ( ) ), where z ( ) w. Since ( ) is increasing in then z must be decreasing in. In turn, this implies that x must be increasing in. Now, recall that r is determined as the solution of r = r ( + ( r ) =w ). Both the LHS and the RHS are linear functions in r, with the LHS increasing and the RHS decreasing. An increase in moves the RHS schedule upward because =w increases with, while the LHS schedule remains the same. This implies that r increases. In the text, before stating proposition 6 I also stated that ( r ) =w is increasing in. To see this, note that since r is increasing in alpha, then the RHS of r = r ( + ( r ) =w ) must be increasing in, so ( r ) =w is increasing in. Finally, to prove that r is decreasing in, from (5) I need to show that ( r ) is increasing in. But we know that ( r ) =w is increasing in while is constant and w is increasing. This implies that ( r ) must be increasing in. Q.E.D. Equilibrium with o shoring costs: countries Let C(w ; ; ) F (; ) + Z w Integration and simpli cation yields xdf (x; = )dx + w ( F (w ; = )) C(w ; ; ) = ( =) exp( w = ) + ( =) exp( ) Letting s(w ; ; ) F (w = ; ) and (w ; ; ) = or (w ; ; )s(w ; ; ), (w ; ; ) (=) exp( w = ) w = exp( w = )+(=) exp( ) then an equilibrium for < m is determined by the following equations: 4

5 and C(w ; ; ) = =( T s(w ; ; )) () = () L (w ; ; ) =( s(w ; ; )) Totally di erentiating above yields equation () yields dc d d = (T = ) ( s) ds d As! then s!, dw = (T = ) ds d ds d d = (= ) exp( w = ) w =w exp( w = ) + (w = ) exp( w = ) When! then ds=d! w =. Noting that w! (T = ) as!, we see d Simple di erentiation = exp( w = ) exp( ) + w exp( w = ) exp( ) Since both the numerator and denominators converge to as! then we can use L Hopital s Theorem to nd that lim w = lim! w ( ) (w = ) exp( w = ) + exp( ) w = ( =) = ( =) h i (w = ) ( ) w! This implies that lim! dw =d > if and only if (w = ) > =( ). 5

6 To show that <, note that this is equivalent to ( + w = ) exp( w = ) < ( + ) exp( ) x) is clearly de- This inequality holds since the function f(x) ( + x) exp( creasing. Equilibrium with o shoring costs: 3 countries I rst characterize an equilibrium in which w > w 3 >. In this case the distribution of c is 8 >< Pr(C c ) = >: if c < F (c ; = ) if c [ ; w 3 [ F (c ; ') if c [w 3 ; w [ if c w where ' = + =w 3, and unit cost of the common input in country is c (w ; ; w 3 ) F (; ) + Z w3 df (x; = ) +w 3 (F (w 3 ; ') F (w 3 ; = )) + The equilibrium condition for country is Z w w 3 df (x; ') + w ( F (w ; ')) T ( c (w ; ; w 3 ) = s where s = exp( 'w ) is the total share of services o shored. This share is distributed between countries and 3 as follows: =w s = F (w 3 ; = ) + [F (w ; ') F (w 3 ; ')] ' and s 3 = F (w 3 ; = + =w 3 ) F (w 3 ; = ) + [F (w ; ') =w3 F (w 3 ; ')] ' Note that s = s + s 3. On the other hand, we have that 8 < if c 3 < Pr(C 3 c 3 ) = F (c 3 ; = ) if c 3 [ ; w 3 [ : if c 3 w 3 and c 3 (w ; ; w 3 ) F (; ) + Z w3 xdf (x; = ) + w 3 ( F (w 3 ; = ) 6

7 and s 3 ( ; w 3 ) = F (w 3 ; = ) The equilibrium conditions for countries and 3 are c i (w ; ; w 3 ) = T i = L e i for i = ; 3, with c (w ; ; w 3 ) =. To derive L e 3, note that country uses =( s) of every service, and if the o shoring cost of a service in country 3 is 3 then it takes 3 =( s) units of labor to produce =( s) units of a service delivered in country. The expectation of 3 for services o shored by country to country 3 is Z w =w3 3 (w ; ; w 3 ) F (w 3 ; ') F (w 3 ; = ) + (=w 3 ) xdf (x; ') ' This implies that 3 3 s units of labor are left in country 3 for nal good production. But this country will o shore s 3 services to country, so it will use This implies that 3 s e = s 3 w 3 of each service for domestic production. 3 s s 3 On the other hand, country export services to countries and 3. The labor it takes to export services to country is s, where is the expectation of for services o shored by country to country, and is given by Z w3 (w ; ; w 3 ) ( exp( )) + (= ) xdf (x; = ) w Z w =w +(= ) xdf (x; ') ' Country also exports services to country 3, and by the reasoning above, we know that it takes 3 3 s s 3 units of labor to do so, with 3 representing the expectation of for services o shored by country 3 to country, given by w 3 3 (w ; ; w 3 ) ( exp( )) + (= ) Z w3 xdf (x; = ) Thus, we nd that el = L s 3 3 s s 3 If the previous system yields wages that do not respect w > w 3 > then it is not an equilibrium. Other possible equilibrium con gurations have 7

8 w > = w 3, w = w 3 >, and w = = w 3. The last case entails full o shoring. Such an equilibrium satis es T m L m = T ( s) = L s s = s3 s L s3 s 3 T 3 ( s 3 ) s3 s with s = s +s 3. For every s 3 these equations determine s and s 3, so these variables are not uniquely pinned down in equilibrium: there is no uniqueness because of the absence of o shoring costs for the services that are traded, but all equilibria entail the same wages. If one can nd a solution with s < F (; ) and s ; s 3 ; s 3 < F (; ), then this solution corresponds to an equilibrium with full o shoring. Since F (; ) and F (; ) both converge to as! then necessarily there is some critical value of such that for higher values of the equilibrium entails full o shoring. I now establish the equilibrium conditions when wages entail w > = w 3 and w = w 3 >. If w > = w 3 then countries and 3 are integrated and their wage should be the same as the one we would would get in a two country system, with + T 3 = w 3 = 3 L + s but with countries and 3 drawing their o shoring cost from F (; ), and with s = F (w =3 ; ) and Z w=3 = F (; ) + xdf (x; ) This equilibrium has and 3 and s 3 such that and 3 L s 3 3 s 3 s T 3 with the restriction that 3 = s 3 F (; ) and s 3 A s 3 s 3 Z w=3 ; 3 F (; ) + (=) xdf (x; ) Although there are 3 equations for 3 unknowns (s, s 3 and s 3 ), these equations are linearly dependent, so they determine only two unknowns. A A 8

9 The rst term on the RHS is the measure of services for which or 3 are equal to, while the second term is the o shoring cost for services with i ]; w =w 3 ]. Now consider the case with w = w 3 >. This entails (T + T 3 ) ( s) w = w 3 = w 3 = + and where s = F (w 3 = ; ) and L s A ( + ) Z w3= = F (; ) + xdf (x; ) For this to be an equilibrium, we need that 3 = s 3 F (; ), have s 3 = s = F (w 3 = ; ) and and the equations of the full system. Z w3= 3 = = F (; ) + xdf (x; ) For the long run equilibrium, we have c i (w ; ; w 3 ) = i = N w i for i = ; ; 3, with c (w ; ; w 3 ) =. This equation for i = can be solved directly to yield = ( = N ), as in (8). Plugging this into the equation for i = 3 yields an equation that can be solved for the equilibrium wage in country 3, w 3. We can easily check that w 3 is increasing in. Finally, plugging the solution w 3 () into the equation for i = yields the equilibrium w (). I have not been able to show that w () is increasing. 9