Appendix: Proofs

Proof of Lemma 1. Facing the budget constraint P

j=A,B (unemployed) consumer income I^nh comprises his share of profits and wage income (un-employment benefits). The goods demand functions of the single consumer are Y_ij^nh = (Pij/P)^−ηI^nh/(P G), whereP ≡(_G¹ P

j=A,B

P_G/2

i=1 P_ij^1−η)^1/(1−η)denotes the aggregate price index. The demand function Y_ij^d for the producer of good i in country j is Y_ij^d = (Pij/P)^−η(1/G)(P

h=A,B

P_L

n=1 I^nh/P). World real income in terms of the aggregate good isY ≡P revenue function of each firm (in terms of the aggregate good) can be written as R_ij = R^ij(N_ij, Y) =p_ijY_ij = [aN_ij^β + (1−a)K_ij^β]^κ/β(Y /G)^1−κ, where K_ij and G are suppressed as arguments of the revenue function. Marginal revenue with respect to employment is

R_N^ij_ij =κ ε^YN_ij Y_ij^κ

ε^YN_ij is the elasticity of output with respect to employment, with 0 < ε^YN_ij <1. Further-more,R^ij_Nij,Y = (1−κ)Y⁻¹R^ij_Nij >0, andR^ij_Nij,Nij =N_ij⁻¹£

(κ−β)ε^YN_ij −(1−β)¤

R^ij_Nij <0, where the negative sign in the latter derivative results because (κ−β)ε^YN_ij <1−β for all permissible values ofβ,κ andε^YN_ij . The concavity of the revenue function with respect to employment guarantees that the optimal employment level is found by the first-order con-ditionR^ij_Nij −wij = 0. This equation implicitly determines Nij as a function N^ij(wij, Y) withN_w^ij_ij = 1/R^ij_Nij,Nij <0 andN_Y^ij =−R^ij_Nij,Y/R^ij_Nij,Nij >0.

Proof of Lemma 3. As a first step one has to insert eq. (A.1) in the first-order condition R^ij_Nij −w_ij = 0. Since N_j = (G/2)N_ij, K_j = (G/2)K_ij and N_j = (1−u_j)L, the inverse labor demand function of countryj =A, B is

w_j =n^j(u_j, Y)≡κ a[(1−u_j)L]^β−1

12The derivation of the firm’s goods demand function under monopolistic competition in the goods market follows a standard approach in the macroeconomics literature. See, for instance, Weitzman (1985), Blanchard and Kiyotaki (1987), and Dutt and Sen (1997).

with the partial derivatives n^j_uj = (wj/(1−uj))£

(1−β)−(κ−β)ε^YN_j ¤

> 0 and n^j_Y = (1−κ)(wj/Y)> 0, where ε^YN_j = ε^YN_j (Nj) ≡a(Nj/Yj)^β and j =A, B. n^j_uj >0 since the expression in brackets is positive for all permissible values ofβ and κ.

Proof of Lemma 4. The bargained real wage is found by maximizing the Nash product maxwij It has to be noted that

w_ijN^ij(·)

R^ij(·)−wijN^ij(·) = R_N^ij_ij(·)N^ij(·)/R^ij(·)

1−R^ij_Nij(·)N^ij(·)/R^ij(·) = κε^YN_ij (N^ij(wij, Y))

1−κε^YN_ij (N^ij(wij, Y)) (A.5) is the elasticity of profits with respect to the real wage. The last expression in eq. (A.5) follows from inserting the terms forR^ij andR^ij_Nij derived in the Proof of Lemma 2. Taking account of eq. (A.5) when solving forw_ij in eq. (A.4) leads to Lemma 4. The second-order condition for a maximum of the Nash product requires that

χ_j∂[w_ij/(w_ij −z_j)] is negative. The first term in this expression is negative. Furthermore,

ε^{N W}_ij = 1 It is assumed that σ < 1 which implies β < 0. As a consequence, sign¡

∂ε^{N W}_ij /∂N_ij¢

One must also bear in mind that due to Lemma 2, N_w^ij_ij < 0. It follows that all terms in eq. (A.6) are negative. As a result, the second-order condition for a maximum of the Nash product is fulfilled if it is assumed that the elasticity of substitution between labor and capital is less than one.

Proof of Lemma 5. It holds thatm^ij_µij <0, i.e. mij and µij are negatively related. It

Proof of Lemma 6. Taking account of Lemma 4, eqs. (1) and (2), and bearing in mind that in equilibriumw_ij =w_j and µ^ij =µ^j, one obtains

F(w_j, u_j, Y, φ_j, χ_j,ρ˜_j,˜b_j) = 0, F(·)≡w_j{(1−ρ˜_j)−[µ^j(w_j, Y, φ_j, χ_j)θ^j(u_j)]⁻¹} −˜b_j. (A.11) If ˜b_j > 0 (˜b_j = 0), the term in brackets must be positive (zero) in order to guarantee that w_j > 0. Hence, it must hold that (1−ρ˜_j)µ^j(w_j, Y, φ_j, χ_j)θ^j(u_j) ≥ 1. Eq. (A.11) implicitly defines w_j as a function of the other variables if ∂F(·)/∂w_j 6= 0. This is the case for ˜ρ_j >0 and ˜b_j ≥0 as well as for ˜ρ_j = 0 and ˜b_j >0. One obtains

w_u^j_j =−w_jµ^jθ_u^j_jΨ⁻¹_j <0, w^j_Y =−w_jθ^jµ^j_YΨ⁻¹_j >0, w^j_ρ˜j =w_j(µ^jθ^j)²Ψ⁻¹_j >0 w^j_φj =−w_jθ^jµ^j_φjΨ⁻¹_j <0, w_χ^j_j =−w_jθ^jµ^j_χjΨ⁻¹_j >0, w^j_˜b

j = (µ^jθ^j)²Ψ⁻¹_j >0, (A.12) where Ψ_j ≡µ^jθ^j[(1−ρ˜_j)µ^jθ^j −1] +w_jθ^jµ^j_wj >0.

Proof of Lemma 7. Since national prices p_j (in terms of the aggregate good) may differ, aggregate output has to be written as Y = p_AY_A+p_BY_B. Inserting the inverse goods demand function of each country and the national version of the CES production function into this equation and bearing in mind thatN_j = (1−u_j)L, one obtains:

Y =y(u_A, u_B)≡2^κ−1κ

However, the sign of (∂w_j/∂u_j)|_LDj is not immediately obvious. It holds that

∂wj

The positive sign arises because (1−β)>(κ−β)ε^YN_j + (1−κ)ε^YN_j , where the right-hand eq. (A.2) implicitly defines the labor demand functionN^j(w_j, Y). Applying the implicit function rule leads to

where the last equality follows from the proof of Lemma 3. Hence,

w^j_Y = n^j_Y

Proof of Proposition 4. Using the notation of Assumption 4, in country A the vari-ablex_Achanges, wherex_A ∈X_A ={φ_A, χ_A,ρ˜_A,˜b_A}. Bearing in mind eq. (3), the following

where the Jacobi matrixJ is defined as J≡

Ã(n^A_uA +n^A_Yy_uA)−(w^A_uA +w_Y^Ay_uA) (n^A_Y −w_Y^A)y_uB

(n^B_Y −w^B_Y)y_uA (n^B_uB +n^B_Yy_uB)−(w^B_uB +w^B_Yy_uB)

!

. (A.18)

Step 1: Determination of the sign of |J|. One obtains

|J|=£

−(n^A_uA+n^A_Yy_uA)w_u^B_B −(n^B_uB +n^B_Yy_uB)w_u^A_A + (w^A_uA +w^A_Yy_uA)w^B_uB +w_Y^ByuB(w_u^A_A −n^A_uA)−n^B_uBw_Y^AyuA

¤+n^B_uB(n^A_uA+n^A_YyuA) +n^A_uAn^B_YyuB. (A.19) Recalling the signs of the derivatives from Lemma 3, Lemma 6, Lemma 7, Proposi-tion 1 and ProposiProposi-tion 2, it becomes clear that all terms in brackets are positive. Hence, n^B_uB(n^A_uA +n^A_YyuA) +n^A_uAn^B_YyuB >0 is a sufficient condition for |J|>0. Using the

Step 2: Solution of the system in eq. (A.17). One obtains

∂u_A

∂x_A = 1

|J|w_x^A_A£

(n^B_uB +n^B_Yy_uB)−(w^B_uB +w^B_Yy_uB)¤

. (A.23)

From Propositions 1 and 2 it follows that the expression in brackets is positive. Hence, sign (∂u_A/∂x_A) = sign(w^A_xA). It also holds that

Hence, sign (∂u_B/∂x_A) = sign(w_x^A_A). The comparative-static effects on real wages can be determined by considering the labor demand equations. One obtains for country A:

∂wA

∂x_A = (n^A_uA +n^A_YyuA)∂uA

∂x_A +n^A_YyuB

∂uB

∂x_A. (A.25)

Due to Proposition 1,n^A_uA+n^A_YyuA >0. Furthermore,n^A_Y >0 andyuB <0 due to Lemma 3 and Lemma 7, respectively. According to the results above,∂uA/∂xAand ∂uB/∂xA have the same sign. The sign of∂w_A/∂x_Atherefore is not immediately obvious. Taking account of the solutions for the change in unemployment leads to:

∂w_A and Lemma 7, respectively. Therefore, sign (∂w_A/∂x_A) = sign¡

w^A_xA¢

. The change in real wages in country B is given by

∂w_B

From the signs of the partial derivatives in Lemma 3 and Lemma 6 it immediately follows thatn^B_uBw_Y^B−n^B_Yw^B_uB >0. Since yuA <0, sign (∂wB/∂xA) = −sign¡

w^A_x¢

. The respective sign of w_x^A_A can be taken from Lemma 6. Then Proposition 4 follows.

Proof of Corollary 1. i) Consider the case of an adverse labor market shock in coun-try A. Due to Proposition 4, uA and uB increase. Since the stock of capital is fixed, it immediately follows that YA, YB and hence also Y decline. For the impact on relative prices, consider the first-order condition for firm’s labor demand in the proof of Lemma 2, which can be written as κ pij ∂Yij

∂Nij = wij. An adverse labor market shock in country A leads to a decline in employment in countryB, which implies an increase in the marginal product of labor. Since wiB declines for all i, it can be concluded that piB and hence pB must have decreased. Moreover, pA must have increased since it is not possible that in both countries prices decline relative to the aggregate price level. From the national goods demand functions follows that YA/YB = (pB/pA)^1−κ1 , where 1/(1−κ) =η denotes the elasticity of substitution between goods. Since pB/pA declines, it follows that YA/YB

declines. As a result, YA declines by more than YB. ii) For a positive labor market shock the same reasoning applies, but the signs have to be reversed, i.e. uA and uB decline,YA

and YB increase, pA decreases, pB increases, pA/pB declines, and YA/YB increases, which implies thatYA increases by more than YB.

Proof of Proposition 5. According to the Proposition it must hold that |n^B_Yy_uA| =

|w_Y^By_uA|, implying w^B_Y − n^B_Y = 0 if ˜b_j = 0. From the wage-setting equation in the proof of Lemma 6 it is evident that with ˜b_j = 0 and ˜ρ_j = ρ_j one obtains (1 −ρ_j)− [µ^j(w_j, Y, φ_j, χ_j)θ^j(u_j)]⁻¹ = 0. Hence, (1−ρ_j)µ^jθ^j = 1. It then follows from eq. (A.14)

thatw^j_Y =−µ^j_Y/µ^j_wj =n^j_Y, where the last equality follows from the proof of Proposition 3.

As a result, w_Y^j −n^j_Y = 0, for j =A, B.

Proof of Proposition 6. Bearing Proposition 5 in mind, it follows from eq. (A.24) that ∂u_B/∂x_A = 0. Due to eq. (A.23) it also follows that ∂u_A/∂x_A = |J|⁻¹w_x^A(n^B_uB − w_u^B_B). Since n^B_uB > 0 and w^B_uB < 0, sign (∂u_A/∂x_A) = sign¡

w^A_xA¢

. Eq. (A.25) becomes

∂w_A/∂x_A = (n^A_uA +n^A_Yy_uA)(∂u_A/∂x_A). Due to Proposition 1, (n^A_uA +n^A_Yy_uA) > 0. As a result, sign (∂w_A/∂x_A) = sign¡

w_x^A_A¢

. Furthermore, ∂w_B/∂x_A =n^B_Yy_uA(∂u_A/∂x_A), where n^B_Yy_uA <0 because of Lemma 3 and Lemma 7. Hence, sign (∂w_B/∂x_A) =−sign¡

w_x^A_A¢ . Proof of Proposition 7. It has to be shown that |n^B_Yy_uA|> |w_Y^By_uA|, implying w_Y^B− n^B_Y < 0. Since ˜ρ_j = 0, the expression Ψ_j defined in the proof of Lemma 6 simplifies to Ψe_j ≡µ^jθ^j(µ^jθ^j−1) +w_jθ^jµ^j_wj >0. It then follows from eq. (A.12) that

w^j_Y =−w_jθ^jµ^j_Y

Ψe_j =− µ^j_Y/µ^j_wj

©µ^j(µ^jθ^j −1)/¡ w_jµ^jwj

¢ª+ 1. (A.26)

Along the lines of the proof of Proposition 3, it is easily seen that w_Y^j can be written as w_Y^j = n^j_Y/{[µ^j(µ^jθ^j −1)/(w_jµ^j_wj)] + 1}. Since the denominator is greater than one, it follows thatw^j_Y −n^j_Y <0.

Proof of Proposition 9. According to Proposition 2, the slope of the wage-setting curve is w_u^j_j+w^j_Yyuj. Shifts of the wage-setting curve due to the aggregate income effect are given as w^j_Yyuk for j, k = A, B and j 6= k. The expression Ψj, defined in the proof of Lemma 6, is smaller than the expression Ψej, which has been defined in the proof of Proposition 7. Hence, if Ψj in Lemma 6 is replaced byΨej, it is immediately evident that

¯¯w_u^j_j|FRB

¯¯<¯

¯w_u^j_j|two-tier

¯¯ and w^j_Y|FRB< w_Y^j|two-tier. From this the Proposition follows.

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Im Dokument The Impact of the Unemployment Benefit System on International Spillover Effects∗ (Seite 23-32)