τ σ 1 (n 3 + m) + (n 1 + n 2 )τ σ 1 n 1, n 2, n 3, m 0 In this equilibrium, (A.1)-(A.4) are all equalities. Solving (A.1)-(A.

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1 Appendix 1 The Pre-PTA Equilibria The zero profit conditions for the national firms in countries 1, 2, 3 and multinational firms are: L 1 n 1 + m + n 2 + n 3 τ + L 2 τ 1 1 n 2 + m + n 1 + n 3 τ + τ 1 1 n 3 + m + n 1 + n 2 τ 1 + L 1 τ 1 n 1 + m + n 2 + n 3 τ + L 2 1 n 2 + m + n 1 + n 3 τ + τ 1 1 n 3 + m + n 1 + n 2 τ 1 + L 1 τ 1 n 1 + m + n 2 + n 3 τ + L 2 τ 1 1 n 2 + m + n 1 + n 3 τ + 1 n 3 + m + n 1 + n 2 τ 1 + L 1 n 1 + m + n 2 + n 3 τ 1 + L 2 n 2 + m + n 1 + n 3 τ 1 + n 3 + m + n 1 + n 2 τ A.1 A.2 A.3 A.4 n 1, n 2, n 3, m 0 A.1 When The Pre-PTA Equilibria Are Mixed Case I: n 1 > 0, n 2 > 0, n 3 > 0, m > 0 In this equilibrium, A.1-A.4 are all equalities. Solving A.1-A.3, we get: L 1 n 1 + m + n 2 + n 3 τ 1 = L 2 n 2 + m + n 1 + n 3 τ 1 = n 3 + m + n 1 + n 2 τ 1 A.5 We can derive the following from A.5: 2τ 1 < < 1 + τ 1 τ 1 A.6 Solving A.1-A.4, we get: L 1 n 1 + m + n 2 + n 3 τ 1 = L 2 n 2 + m + n 1 + n 3 τ 1 = n 3 + m + n 1 + n 2 τ 1 1

2 = 1 τ 1 A.7 Substituting A.7 in A.1, we get following condition for this equilibrium: = 3τ 1 1 τ 1 A.8 Case II: n 1 > 0, n 2 > 0, n 3 = 0, m > 0 In this case, A.1, A.2 and A.4 are equalities whereas A.3 is an inequality as there are no national firms in country 3. Subtracting A.2 from A.1, We get: L 1 n 1 + m + n 2 τ 1 = L 2 n 2 + m + n 1 τ 1 A.9 L1 L 2 τ 1 L 1 L 2 n 1 = L 2 L 1 τ 1 n 2 + L 2 L 1 τ 1 m A.10 Subtracting A.1 from A.4, we get: L 2 n 1 + m + n 2 τ 1 + m + n 1 + n 2 τ 1 = 2 1 τ 1 A.11 Substituting A.11 in A.1: L 1 n 1 + m + n 2 τ 1 = 1 3τ τ 1 1 τ 1 A.12 Substituting for n 1 from A.10 and solving for n 2 : n 2 = L 2 L 1 τ 1 F2 1 3τ τ τ 1 m 1 + τ 1 A.13 Subtracting A.2 from A.4: L 1 n 1 + m + n 2 τ 1 = 2 1 τ 1 m + n 1 + n 2 τ 1 A.14 2

3 Substituting A.14 in A.1: m + n 1 + n 2 τ 1 = 1 + 3τ 1 1 τ 1 1 τ 1 A.15 Substituting for n 1 and solving for n 2 : n 2 = L 2 L 1 τ 1 F τ 1 1 τ 1 L 1 + L 2 τ 1 L 2 L 1 + L 2 τ 1 m A.16 Solving for m from A.13 and A τ 1 L 2 L 1 +L 2 τ 1 = L 2 L 1τ 1 A B A.17 Where 1 A = 1 F2 1 3τ τ τ 1 B = F τ 1 1 τ 1 L 1 +L 2 τ τ 1 L 2 L 1 +L 2 τ 1 = L 1 τ 1 L τ 1 L 1 +L 2 τ 1 < 0 The Left hand side of A.17 is always negative implying that the right hand side should be negative too. Setting right hand side equal to zero gives us the following condition for this equilibrium to exist: τ τ 1 1 3τ τ 1 1 τ 1 τ τ 2 1 < A.18 1 We can write A.9 as n 1 L 1 τ 1 L 2 + ml 1 L 2 + n 2 L 1 L 2 τ 1 = 0. If L 1 L 2 = L 1 τ 1 L 2 < 0 or L 2 L 1 τ 1 > 0 3

4 Solving A.13 and A.16 for m: L 2 L 1 τ 1 F2 1 3τ τ τ 1 n 2 L 2 L 1 τ 1 F τ 1 1 τ 1 L 1 + L 2 τ 1 n 2 L 2 L 2 A.19 A.20 Equating A.19 and A.20 and solving for n 2 : n 2 = L 2 F2 1 3τ τ 1 F τ 1 1 τ 1 A.21 Since the left hand side is positive, we set the right hand side to be positive and get: L τ 1 2 L τ 1 > A.22 We can, similarly, get two equations for n 1 and find the following condition: L τ 1 2 L τ 1 > A.23 Adding A.22 and A.23, we get the following condition for this equilibrium: 1 + 3τ τ τ 1 > A.24 Subtracting A.3 from A.4 and substituting A.9: L 1 n 1 + m + n 2 τ 1 > 1 τ 1 A.25 Substituting A.15 and A.25 in A.1: > 3τ 1 1 τ 1 A.26 Case III: n 1 = 0, n 2 = 0, n 3 > 0, m > 0 4

5 Subtracting A.3 from A.4: L 1 + L 2 m + n 3 τ 1 = 2 1 τ 1 A.27 Substituting A.27 in A.3: m + n 3 = 1 3τ τ 1 1 τ 1 A.28 Adding A.1 and A.2 and substitute A.27 and A.28: > 3τ 1 1 τ 1 A.29 Rewriting A.27 and A.28: 1 τ 1 1 3τ τ 1 n 3 L1 + L 2 1 τ 1 n 3 τ 1 2 A.30 A.31 Equating A.30 and A.31: n 3 = 1 3τ τ 1 L 1 + L 2 2 A.32 Since the left hand side is positive, the following condition should be satisfied: 2 1 3τ 1 1 τ 1 > A.33 Similarly, rewriting A.30 and A.31 in terms of m and noting that m > 0, we get: 2 τ 1 1 3τ 1 1 τ 1 < A.34 5

6 A.2 When The Pre-PTA Equilibria Are Pure Multinational Case I: n 1 = 0, n 2 = 0, n 3 = 0, m > 0 Since A.4 will be an equality in this case, we solve for m: L1 + L A.35 Substituting A.35 in A.3: L1 + L 2 τ 1 + L 1 + L 2 + < A τ 1 1 τ 1 Similarly, substituting A.35 in A.1 and A.2: < A.37 2L 1 L τ 1 L τ 1 < A.38 2L 2 L τ 1 L τ 1 < A.39 Adding A.38 and A.39: 1 + 3τ τ τ 1 < A.40 A.3 When The Pre-PTA Equilibria Are Exporting Case I: n 1 > 0, n 2 > 0, n 3 > 0, 0 Subtracting A.2 from A.1 and subtracting A.3 from A.1: L 1 n 1 + n 2 + n 3 τ 1 = L 2 n 2 + n 1 + n 3 τ 1 = n 3 + n 1 + n 2 τ 1 A.41 n 1 L 1 τ 1 + n 2 L 1 τ 1 τ 1 + n 3 L 1 τ 1 = 0 A.42 6

7 n 2 L 2 τ 1 + n 1 L 2 τ 1 τ 1 + n 3 L 2 τ 1 = 0 A.43 Add C.42 and C.43: n 1+ n 2 L 1 + L 2 τ τ 1 = n 3 2L3 τ 1 L 1 + L 2 A.44 Since n 1, n 2 and n 3 are all positive, the other two terms in the expression must be of the same sign. 2 2 τ 1 L 1 + L 2 < 0 = L 1 + L 2 > 2τ 1 L 1 + L 2 τ τ 1 < 0 = L 1 + L 2 < 1 + τ 1 τ 1 = 2τ 1 < L 1 + L 2 < 1 + τ 1 τ 1 A.45 Subtracting A.3 from A.4: L 1 1 τ 1 n 1 + n 2 + n 3 τ 1 + L 2 1 τ 1 n 2 + n 1 + n 3 τ 1 < 2 A.46 Using A.46: Similarly, we can find: L 1 n 1 + n 2 + n 3 τ 1 < L 2 n 2 + n 1 + n 3 τ 1 < n 3 + n 1 + n 2 τ 1 < 1 τ 1 1 τ 1 1 τ 1 A.47 A.48 A.49 2 We can show that expression in brackets on the right has to be negative. Let us suppose it is positive. 2 τ 1 L 1 + L 2 > 0 = τ 1 + τ 1 L 1 + L 2 > 0 = 1 + τ 1 L 1 + L 2 > 0 = 1 + τ 1 L 1 + L 2 τ 1 > 0. This leads to a contradiction. Futhermore, let us now suppose 2 τ 1 L 1 + L 2 = 0 = 1 + τ 1 L 1 + L 2 τ 1 which is again a contradiction implying 2 τ 1 L 1 + L 2 < 0. 7

8 Substituting A.47,A.48 and A.49 in A.1: < 3τ 1 1 τ 1 A.50 Case II: n 1 > 0, n 2 > 0, n 3 = 0, 0 Subtracting A.2 from A.1: L 1 n 1 + n 2 τ 1 = L 2 n 2 + n 1 τ 1 A.51 Rearranging terms: n 1 = L1 L 2 τ 1 L 2 L 1 τ 1 n 2 A.52 Subtracting A.3 and A.4 from A.2 and using A.51: L3 1 τ 1 n 1 + n 2 τ 1 < A.53 Using A.52 and A.53 to solve for n 1 and n 2 : n 2 > n 1 > L3 L 2 L 1 τ 1 L 1 + L 2 τ 1 L3 L 1 L 2 τ 1 L 1 + L 2 τ 1 A.54 A.55 Substituting n 1 and n 2 and A.51 in A.2, we get the following condition for this equilibrium: τ 1 1 2τ 1 1 τ 1 > A.56 Subtracting A.4 from A.1: L 2 n 2 + n 1 τ 1 + n 2 + n 1 τ 1 < 2 1 τ 1 A.57 8

9 Substituting this back in A.1: L 1 n 1 + n 2 τ 1 > 1 τ τ 1 1 τ 1 A.58 Solving for n 1 and n 2 and using A.52 n 2 < L 2 L 1 τ τ 1 1 τ τ 1 n 1 < L 1 L 2 τ τ 1 1 τ τ 1 A.59 A.60 Putting these back in A.1, we get another condition for this equilibrium: τ τ 1 1 3τ τ 1 1 τ 1 τ τ 2 1 < A.61 Substituting for n 1 in A.1 by using A.52 and solving for n 2 : n 2 = L 2 L 1 τ 1 1 τ 1 τ τ 1 + τ 1 L 1 + L 2 L 1 + L 2 n 1 = L 1 L 2 τ 1 1 τ 1 τ τ 1 + τ 1 L 1 + L 2 L 1 + L 2 A.62 A.63 Substituting these values in A.3 and simplifying, we get: > 1 + τ 1 1 τ 1 A.64 Case III: n 1 = 0, n 2 = 0, n 3 > 0, 0 From A.3, we get: n 3 = L1 + L A.65 Substituting this in A.1, A.2 and A.4: 2 τ 1 1 3τ 1 1 τ 1 > A.66 9

10 < 2τ 1 A.67 B The Post-PTA Equilibria The zero profit conditions for the national firms based in the integrated region,the national firms in country 3 and the multinational firms are: n I + m + n + τ 1 3 τ 1 n 3 + m + n Iτ 1 + τ 1 n I + m + n + 3 τ 1 n 3 + m + n Iτ 1 + n I + m + n + 3 τ 1 n 3 + m + n Iτ B.1 B.2 B.3 n I, n 3, m 0 B.1 When The Post-PTA Equilibria Are Mixed Case I: n I > 0, n 3 > 0, m > 0 Subtracting B.1 from B.3 and B.2 from B.3 n 3 + m + n Iτ 1 = 1 τ 1 B.4 n I + m + n 3 τ 1 = 1 τ 1 B.5 Substitute B4 and B.5 in B.1: = 2τ 1 1 τ 1 B.6 10

11 From B.4 and B.5, we get: n 3 + m + n Iτ 1 = n I + m + n 3 τ 1 B.7 = τ 1 < < 1 τ 1 B.8 Case II: n I > 0, n 3 = 0, m > 0 Subtracting B.1 from B.3: m + n I τ 1 = 1 τ 1 B.9 Substituting B.9 in B.1: n I + m 1 τ τ 1 1 τ 1 B.10 Rewriting B.10 as follows: 1 τ 1 1 τ τ 1 n I B.11 From B.9, we get: L3 1 τ 1 n Iτ 1 B.12 Solving for n I and m from B.11 and B.12: n I = 1 τ τ 1 L 3 L3 τ 1 1 τ τ 1 B.13 B.14 For n I and m to be greater than zero, the following conditions must be met: < 1 2τ 1 1 τ 1 B.15 11

12 > τ 1 1 2τ 1 1 τ 1 B.16 Substituting B.9 and B.10 in B.2, we get another condition for this equilibrium > 2τ 1 1 τ 1 B.17 Case III: n I = 0, n 3 > 0, m > 0 Subtracting B.2 from B.3: m + n 3 τ 1 = 1 τ 1 B.18 Substituting this back in to B.2: n 3 + m 1 τ τ 1 1 τ 1 B.19 Rewriting B.18 and B.19 as follows LI 1 τ 1 n 3 τ 1 1 τ 1 1 τ τ 1 n 3 B.20 B.21 Solving for n 3 and m n 3 = 1 τ τ 1 LI τ 1 1 τ τ 1 B.22 B.23 For n and m to be greater than zero, the following conditions must be met: 3 < 1 2τ 1 1 τ 1 B.24 12

13 > τ 1 1 2τ 1 1 τ 1 B.25 Substituting B.18 and B.19 in B.1, we get another condition for this equilibrium: > B.2 When The Post-PTA Equilibria Are Multinational Case I: n I = 0, n 3 = 0, m > 0 2τ 1 1 τ 1 B.26 Solving for m from B.3: LI B.27 Substituting B.27 in B.1 and B.2: < 1 2τ 1 1 τ 1 B.28 < 1 2τ 1 1 τ 1 B.29 B.3 When The Post-PTA Equilibria Are Exporting Case I: n I > 0, n 3 = 0, 0 Solving for n I directly from B.1: n I = LI + + B.30 Substituting B.30 in B.2 and B.3, we get the following two conditions for this equilibrium: Case II: n I = 0, n 3 > 0, 0 < τ 1 1 2τ 1 1 τ 1 < 1 τ 1 B.31 B.32 13

14 Solving for n 3 from B.2: n 3 = LI + + B.33 Substituting B.33 in B.1 and B.3, we get the following two conditions for this equilibrium: Case III: n I > 0, n 3 > 0, 0 < τ 1 1 2τ 1 1 τ 1 < τ 1 B.34 B.35 Subtracting B.1 and B.2 from B.3: Substituting B.36 and B.37 in B.1: n I + n 3 τ 1 n I τ 1 + n 3 < 1 τ 1 B.36 < 1 τ 1 B.37 < 2τ 1 1 τ 1 B.38 Subtracting B.2 from B.1: n I + n 3 τ 1 = n I τ 1 + n 3 B.39 Rewriting B.39 as follows: n 3 = L3 τ 1 τ 1 n I B.40 For n I and n to be greater than zero, the following condition must be met: 3 τ 1 < < 1 τ 1 B.41 14

15 C The Derivation of the Utility Functions C.1 The Pre-PTA Utility Function We derive the utility function in the case when n 1, n 2, n 3, m > 0. Other cases can be derived from this general case by making appropriate substitutions. The utility function of a country i in the pre-pta regime can be given as follows: U i = Q i Y1 i C.1 i, j, k = 1, 2, 3 i j k Where Q i = 1 q π π and π = 1 i and Q i can be written as Q i = n i + mq π ii + n jq π ji + n kq π ki 1 π C.2 The quantity of a variety produced in country j and sold in country i and the quantity of a variety produced in country i and sold in country i are: q ji = q ii = n i + mp 1 ii n i + mp 1 ii p ji + n j p 1 ji p ii + n j p 1 ji i j k + n k p 1 L i ki + n k p 1 L i ki C.3 C.4 Using p ij = p ik = p ii /τ = c 1 τ to substitute for the prices above, we can rewrite Q i: n i + m 1 c π + n j Q i = n i + m 1 c 1 + nj c 1 τ c 1 τ π + n k 1 + nk c 1 τ c 1 τ π 1 π L i π 1 π C.5 15

16 Simplifying C.5: Q i = n i + m + n j τ 1 + n k τ L i 1 c C.6 Since, the utility function is of Cobb-Douglas type, we can write the demand for the numeraire good Y = 1 L i. Substituting for Y and Q i in the utility function, we can write the the utility of a representative individual in country i in the pre-pta case as follows: U Pr e i = C n i + m + n j τ 1 + n k τ 1 1 C.7 n i, m, n j, n k 0 Where C = ; c i, j, k = 1, 2, 3 and i j k. C.2 The Post-PTA Utility Function We derive the utility function in the case when n I, n 3, m > 0. The utility function of a country i in the post-pta regime is: U i = Q i Y1 i C.8 i = I, 3 i j Where Q i = 1 q π π and π = 1 i and Q i can be written as Q i = n i + mq π ii + n kq π ki 1 π C.9 The quantity of a variety produced in country j and sold in country i and the quantity of a variety produced in country i and sold in country i are: q ji = p ji n i + mp 1 ii + n k p 1 L i ki C.10 16

17 Using p ij = p ii /τ = c 1 q ii = p ii n i + mp 1 ii + n k p 1 L i ki τ to substitute for the prices, we can rewrite Q i: n i + m 1 c π + n c π j 1 τ Q i = n i + m 1 c 1 + c 1 π L i π nj 1 τ 1 π C.11 C.12 Simplifying C.12: Q i = n i + m + n j τ L i 1 c C.13 Writing Y = 1 L i and substituting for Y and Q i in the utility function, we can write the the utility function of a representative individual in country i in the post-pta case as follows: U Post i = C n i + m + n j τ 1 1 C.14 where C = c and ni, m, n j 0 17

GS/ECON 5010 Answers to Assignment 3 W2005

GS/ECON 5010 Answers to Assignment 3 W2005 GS/ECON 500 Answers to Assignment 3 W005 Q. What are the market price, and aggregate quantity sold, in long run equilibrium in a perfectly competitive market f which the demand function has the equation