APPENDICES * LABOR MARKET RIGIDITIES AND R&D-BASED GROWTH IN THE GLOBAL ECONOMY

Transcription

1 NDICS * for LBOR MRKT RIGIDITIS ND R&D-BSD GROWTH IN TH GLOBL CONOMY by Fuat Sener (Union College) December 4 aper published in the Journal of conomic Dynamics and Control, 3 (5), 6: * Not to be considered for publication. To be made available on the author s web site and also upon request from the author.

2 -1 ppendix (not to be considered for publication, to be made available on the author s web site and also upon request) 1. xistence and uniqueness of the steady-state equilibrium for the basic model with h h To prove the existence and uniqueness of the steady-state equilibrium, I follow a graphical approach. Recall that in the basic model, h h is assumed. To simplify notation let w* stand for the minimum wage w L *, and let subscript QM refer to steady-state equilibrium values. First, I analyze (n ) QM. Substituting for λb w L * from SS() into SS() using η (1 - η ) and n (1 - n ) yields: λ 1 φ (1 η ) λw * b φ(1 η ) ( ρ n)a k σ. n (.) (.1) (1 n ) b (1 n ) quation (.1) implicitly defines n in terms of the parameters of the model. Note that the LHS of (.1) is a constant, whereas the RHS is increasing in n for n (, 1). s n 1, the RHS ; and as n, the RHS INT, where INT (1 - η )(λw*b /b ) [(ρ - n)a k (φ(1 - η )/)]. Figure.1 plots the RHS and the LHS of (.1) with n on the horizontal axis. The necessary and sufficient condition for the existence and uniqueness of (n ) QM in the domain (, 1) is λ 1 φ λw * b φ(1 η ) > (1 η ) ( ρ n)a k σ. (.) b The next step is to investigate (u ) QM, which is implicitly defined by SS(U) as a function of n and the parameters of the model: λ 1 φ η [ λw * (1 u )] φη ( ρ n)a k σ n u (n ;.) (.3) n Note that the LHS of (.3) is a constant, whereas the RHS is decreasing in u for u (, 1). s u 1, the RHS, and as u, the RHS INT 1, where INT 1 η [λw*] [(ρ - n)ak (φ η /n )]/n. Figure. plots the RHS and the LHS of (.3) with u on the horizontal axis. Hence, given a unique (n ) QM (, 1) exists, the necessary and sufficient condition for the existence and uniqueness of (u ) QM in the domain (, 1) is λ 1 φ η [ λw*] φη < ( ρ n)a k. (.4) σ (n ) QM (n ) QM

3 - Figure.1. The existence and uniqueness of n LHS, RHS RHS λ 1 φ σ LHS INT (n ) QM 1 n Figure.. The existence and uniqueness of u LHS, RHS INT 1 RHS λ 1 φ σ LHS (u ) QM 1 u

4 -3 I now analyze (θ i ) QM for i,. Solving for w H from (7), Π (t) from (17), n from () and I from (3), and substituting the resulting expressions into (16) using (14), (15) and λb w L * gives: λ( ρ n)a ( λ 1) 1 b φ kw * (1 ( θ σ θ b θ ) ), θ (.) (.5) which implicitly defines θ as a function of the parameters of the model. Note that the LHS of (.5) is a constant, whereas the RHS is decreasing in θ for θ (, 1). s θ, the RHS ; and as θ 1, the RHS INT, where INT λ(ρ - n)a kw*(b /b ). Figure.3 plots the RHS and the LHS of (.5) with θ on the horizontal axis. The necessary and sufficient condition for the existence and uniqueness of (θ ) QM in the domain (, 1) is λ 1 b > λ( ρ - n)a kw * σ. (.6) b Solving for w H from (8), Π from (17), u from (1) and I from (3), and substituting the resulting expressions into (16), using (14) and (15) yields and λb w L * λ 1 1 n φ ( ρ n)a k θ σ θ (1 ( ) ) θ ( ) (n ;.) (.7) η which implicitly defines θ as an increasing function of n and the parameters of the model. Note that the LHS of (.7) is a constant, whereas the RHS is decreasing in θ for θ (, 1). s θ, the RHS ; and as θ 1, the RHS INT 3, where INT 3 (ρ - n)a kn /η. Figure.4 plots the RHS and the LHS of (.7) with θ on the horizontal axis. Hence, given a unique (n ) QM (, 1) exists, the necessary and sufficient condition for the existence and uniqueness of (θ ) QM in the domain (, 1) is λ 1 (n ) > ρ QM ( - n)a k. (.8) σ η Finally, I analyze the existences and uniqueness of relative wages. Let ω i w H i /w* represent the relative wage for i,. With (θ ) QM (, 1) determined, a unique and positive (ω ) QM can be instantly found from (7). To derive ω, solve for Π from (17), I from (3), and substitute these into (16) using (14) and (15). Dividing both sides of the resulting expression with w L * gives: ω w L * λ ( ρ n)a ( λ 1) k (1 ( θ φ η ) ) n. (.9)

5 -4 Figure.3. The existence and uniqueness of θ LHS, RHS RHS λ 1 σ LHS INT (θ ) QM 1 θ Figure.4. The existence and uniqueness of θ LHS, RHS RHS λ 1 σ LHS INT 3 (θ ) QM 1 θ

6 -5 With (θ ) QM (, 1) and (n ) QM (, 1), it follows from (.9) that (ω ) QM is unique and positive. For future use, one can simplify (.9). First, find an expression for n using (8) and (1) as n (θ ) ω η λw L */σ, then substitute this back into (.9) to obtain: ω λw L 1 * ( ρ n)a φσ(1 ( θ ( λ 1) k ( θ ) ) ω ( θ ;.) (.1) which defines the ω as an increasing function of θ and the parameters of the model. To sum up, there exists a unique-steady-state equilibrium in which (n ) QM, (u ) QM, (θ ) QM, and (θ ) QM (, 1); and (ω ) QM > and (ω ) QM > if and only if (.), (.4), (.6) and (.8) hold jointly. Since the parameters of the model enter the inequalities directly and indirectly via (n ) QM, one cannot determine sufficient conditions on parameters that would satisfy all the inequalities. However, the simulations reveal that for a large set of reasonable parameter values a unique steady-state equilibrium with the above properties exists.. roofs of comparative steady-state analysis.1. roof of roposition 1: Changes in the minimum wage rate ll derivatives are evaluated in the neighborhood of steady-state equilibrium. To simplify notation, I omit the subscripts QM. Totally differentiating (.1) gives (.1), totally differentiating (.3) implies: dn <. Given n n (w L *) by dw * du u dn u >. (.11) dw * 1443 n dw * { w * To analyze the variation in θ i, totally differentiate (.5) and (.7) to obtain: dθ dw * >, dθ θ dw * n 13 n 13 w * <. (.1) d(i n ) d(i n ) Using (.1) and (3), one can obtain < and >. In addition, using (.1) and (7), dw * dw * dω one can show that <. To analyze the change in ω, I totally differentiate (.1) and derive: dw * ω ω dw * { w * d ω dθ <. (.13) θ dw *

7 -6.. roof of roposition : Increased unemployment benefits When unemployment benefits is introduced, the relevant training arbitrage condition T() becomes θ σ(1 u (1 α))w L */w H, where α >. Using this to derive the steady-state equations implies that (.1), (.5), and (.9) remain the same, whereas (.3) and (.7) change as follows: λ 1 σ [ 1 u (1 α) ] a k(w * λ) n (1 u ) η φη ρ n ka n φ ) (1 u u (n ;)(.3) ) λ 1 a λσ kw θ L * (1 α)n α λw * θ ) η φη (1 ( θ ρ n ka n ) ) θ (n ;.) (.7) dn Observe that α does not enter (.1) and (.5); thus, dα dθ and, which, in turn, implies dα d(i n α ) dω α du by (3) and (7).Totally differentiating (.3) and (.7) one can obtain >, dα dθ dα d(i n ) >, respectively. Using these and (3), one can derive <. To determine the change in dα ω, totally differentiate (.9). This yields dω dα ω θ >. (.14) 13 θ α roof of roposition 3: Global and equi-proportionate technological change in R&D Totally differentiating (.1) gives dn <. Using (.3) one can obtain du u dn u {{ n { a >. (.15) To analyze the change in θ, I totally differentiate (.5) and derive dθ >, which, in turn, implies d(i n This implies: ) < by (3). To investigate the adjustment in θ, totally differentiate (.7), using a. a dθ sign d(a n sign ) sign (1 - ε(n, a )), (.16)

8 -7 where ε(n, a ) - ( n / a )(a /n ). Thus, dθ > if and only if ε(n, a ) 1. By (3), this, in turn, d(i n ) (I n ) (I n ) θ implies < 1443 a θ a if ε(n, a ) 1. To analyze the variation in ω, totally differentiate (.1). For the elastic case ε(n, a ) > 1, this yields: dω ω ω dθ <. (.17) { θ 13 1 a 3 For the inelastic case ε(n, a ) < 1, instead of (.1), consider ω σ(1 u )/θ, which is implied by (8). Totally differentiating this particular expression for ω implies: dω ω du ω dθ <. (.18) {{ θ 13 1 u 3 Thus, dω / is negative regardless of the value of ε(n, a )..4. roof of roposition 4: Changes in the distribution of global population Consider a change in η and note that dη - dη. For (.1) to hold, the ratio (1 η (1 n ) ) must remain constant. Thus d(1 n ) d(1 η ), which in turn implies (1 n ) (1 η ) dn (1 n ) dη (1 η ) > and dn d η <. Since η does not enter (.5), θ and therefore ω remain constant. With θ constant and dn /dη >, it follows from (3) that d(i n )/dη >. In urope, the effects depend on the variation in (n )* n /η. Totally differentiating (n )* with respect to η and appropriately substituting the merican levels using dη - dη, n (1 - n ) and η (1 - η ) gives d(n dη ) * η n η (1 η ). Hence, d(u dη ) d(u ) d(n ) * iff n η. (.19) d(n ) * dη Totally differentiating (.7) gives dθ dη θ ( n ) * 1443 d(n ) *, iff n η. (.) dη 1443

9 -8 Using (3) and (.), one can obtain d(i dη n ) (I n ) (I n ) θ < if η n and ambiguous η η 1443 θ otherwise. To analyze ω, totally differentiate (.1) to obtain dω ω dη θ 13 dθ dη 13, iff n η. (.1)

10 B-1 ppendix B (not to be considered for publication, to be made available on the author s web site and also upon request) 1. Derivation of the steady-state equilibrium for the general model with h > and h > I first derive the SS() equation. Solving w H from Lemma (1) using T() and LM() gives w H. w H (,n ;.) n (b ) λ h η σ Substituting w H (,n ;.) from above into F() gives I ( λ 1) n (b ) h ( ρ n) a k. I (,n ;.) η σ Substituting I (,n ;.) from above into SM() and solving for θ yields λ n n (b ) ( 1) h ( θ η ) (, n ; b,a, ) 1 h ( ρ n)a k, θ (,n ;.) φη η σ where the signs above the variables/parameters stand for the signs of partial derivatives. Substituting the above into LM() implies θ (, n ; b,a, η b n ) η, SS() (,n ;.) where d/dn SS() > and hence the upward sloping SS() curve as shown in Figure B.1. For future use, note that for a given n, d/w L * SS(), d/ SS() <, d/dη SS() <, d/db SS() >. Second, I derive the SS(U) equation. Solving for w H from Lemma (1) gives: w H 1 b w L *. w H (;.) λ h Substituting w H (;.) from above into F() gives: I ( λ 1)h ( ρ n). I (;.) a k( λb w *) L Substituting I (;.) from above into SM() gives λ n ( 1)h h ( θ ) (, n ; b, η, w L *,a ) 1 ( ρ n)a k, θ (,n ;.) φη a k( λb w L*)

11 B- Finally, substitute the above into LM() to find: (1 u ) η θ (, n ; b, η, w L *,a n b ), SS(U) (,n, u ;.) where du /dn SS(U) < and hence the downward sloping SS(U) curve as illustrated in Figure B.1. For future use, note that for a given n, du /dw L * SS(U) <, du / SS(U) >, du /dη SS(U) >, du /db SS(U) <, du /d SS(U) >. Third, I derive the SS() equation. Combining Lemma (1) and T() gives (1 u ) θ /[σh [(/(λw L *)) b ]]. Substituting this into LM() immediately gives (θ ) n σh /[η [(/(λw L *b )) 1)]]. Finally combining this with the above θ (,n ;.) expression gives: n σh ( θ ) (, n ; b, η, w L *,a ), SS() (,n ;.) η 1 λw L * b where d/dn SS() >, which implies d/dn SS() < and hence the downward sloping SS() curve as illustrated in Figure B.1. For future use, note that for a given n, d/dw L * SS() >, d/ SS() <, d/dη SS() <, d/db SS() >. The steady-state equilibrium is fully illustrated in Figure B.1. It appears to be analytically infeasible to derive parametric conditions that guarantee the existence of equilibrium; however, numerical simulations suggest that for a wide range of reasonable parameters a unique steady-state equilibrium exists in which all endogenous variables attain nonnegative values in a relevant range.. Comparative Steady-State nalysis For the sake of conciseness, I provide a full account of the comparative steady-state analysis only for the case of a minimum wage hike. For the rest of the parameter changes, I outline the resulting shifts using Figure B.1. as a template and report the effects on the endogenous variables, which remain ambiguous in most cases..1. n increase in the minimum wage w L * (Figure B..) We know that for a given n, d/w L * SS() and d/dw L * SS() > ; thus, when w L * increases the SS() curve remains intact whereas the SS() curve shifts up. Hence, the equilibrium levels of and n both increase. To determine the change in θ, substitute for I (.) from above into SM() using n θ η /b (which is from LM()) to obtain:

12 B-3 [ ( θ ) ] φ θ θ (b ) 1 h ( λ 1) ( ρ n)a k h, θ (.) b σ which defines θ in terms of the parameters of the model. It follows from the above that the rise in triggers an increase in the equilibrium level of θ. The T() equation then implies that w H /w L decreases. Using I (.), it follows that the change in the equilibrium level of I n is ambiguous. I now turn to urope. Since du /d SS(U) >, the higher shifts the SS(U) curve to the left. On the other hand, since du /dw L * SS(U) < the higher w L * shifts the SS(U) line to the right. Consequently, the change in u remains ambiguous. To analyze the change in θ, first substitute (1 u ) θ /[σh [(/(λw L *)) b ]] into the LM() equation and obtain n (θ ) η ( λw L *b )/[σh cλw L *b ]. Substituting this expression for n into the SM() equation, one can obtain: (1 ( θ ( θ ) ) ) σφ w L *b 1 λw L *b ( ρ n)a 1 h k, θ (.) which defines θ in terms of the parameters of the model. It follows from the above that the rise in w L * and the endogenous increase in generates an ambiguous change in θ. With changes in both u and θ being ambiguous, the changes in w H /w * L and I n (as can be observed from the T() and I (.) equations) also remain indeterminate... n increase in unemployment benefit rate α (Figure B.3.) To study the effects of an increase in α, we need to rework the equations for urope. Using the T () equation and Lemma 1, one can derive an expression for (1 - u ). Substituting this (1 u ) expression and using θ (,n ;.) from above yields the new SS() curve as: 1 θ (, n σh ;.) b λw L * (1 α) α θ n b (, n ;.) η. SS () It is straightforward to show that d/dn SS () < and for a given, dn /dα SS () >. s a result an increase in α shifts the SS () curve up, leading to a rise in and n. Using the θ (.) equation, one can 1 Since the T() equation is not utilized in the derivation of θ (,n ;.), the introduction of α > does not change the θ (,n ;.) expression.

13 B-4 show that θ increases and w H /w L decreases. It follows from I (.) that the change in the equilibrium level of I n is ambiguous. Since du /d SS(U) > for a given n, the rise in shifts the SS(U) curve to the left. With n falling and SS(U) shifting to the left, it follows that u unambiguously increases. Using the θ (.) equation, one can show that the change in θ (.) is ambiguous. It then follows from the T() and I (.) equations that the changes in w H /w * L and I n also remain ambiguous..3. Global technological change in R&D in the form of a decline in both a and a (Figure B.4.) For a given n, d/ SS() < and d/ SS() < ; thus, a decline in both a and a shifts the SS() and SS() curves up. Consequently, increases whereas the change in n remains ambiguous. Since du /d SS(U) > for a given n, the rise in forces the SS(U) curve to the left. On the other hand, with du / SS(U) > for a given n, the decline in a forces the SS(U) curve to the right. The direction of the shift in the SS(U) curve remains indeterminate. With these findings, the changes in the rest of the endogenous variables also remain ambiguous..4. n increase in the population share of merica η (Figure B.5.) For a given n, d/dη SS() < and d/dη SS() < ; thus an increase in η shifts the SS() curve down and the SS() curve up. Consequently, n increases whereas the change in remains ambiguous. With du /d SS(U) > for a given n, the ambiguous change in forces the SS(U) curve in an ambiguous direction. On the other hand, with du /dη SS(U) >, the fall in η forces the SS(U) curve to the right. Obviously, the net impact on the SS(U) curve remains indeterminate. With these findings, the changes in the rest of the endogenous variables also remain ambiguous..5. Global sbtc in manufacturing in the form of a decline in both b and b (Figure B.6.) For a given n, d/db SS() > and d/db SS() > ; thus, a decline in both b and b shifts both the SS() and the SS() curves down. Consequently, decreases whereas the change in n remains ambiguous. Since du /d SS(U) > for a given n, the fall in forces the SS(U) curve to the right. On the other hand, with du /db SS(U) <, the fall in b forces the SS(U) curve to the left. Obviously, the direction of the shift in the SS(U) curve remains indeterminate. With these findings, the changes in the rest of the endogenous variables also remain ambiguous. Since the T() equation is not utilized in the derivation of the SS(U) equation, the introduction of α > does not affect the SS(U) curve.

14 B-5 Figure B.1. Steady-state equilibrium in the global conomy SS() u SS(U) SS() n n 1 n n Figure B.. n increase in the uropean minimum wage: w L * SS() u SS() SS() w L * n SS(U) w L * n n 1 n

15 B-6 Figure B.3. n increase in unemployment benefit rate: α SS() u SS(U) SS() α SS() n n n 1 n Figure B.4. Global technological change in R&D: a and a both SS() a SS() u SS() a SS() n SS(U) a n n 1 n

16 B-7 Figure B.5. n increase in the world population share of merica: η SS() SS() η SS() η u SS() n SS(U) η n n 1 n Figure B.6. Global skill-biased technological change in manufacturing: b and b both SS() SS() b u SS() SS() b n SS(U) b n n 1 n