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Redistributive policies and technology diffusion

  • Manuela Magalhães ORCID logo EMAIL logo and Tiago Neves Sequeira ORCID logo

Abstract

In this paper we examine the effects of redistributive policies in a transition economy in the presence of technology diffusion on labor and education decisions, and skill-premium. We set a micro-founded dynamic general equilibrium model with a skill-biased technology diffusion, elastic leisure/labor decisions, and investments in education. The economy is populated by two types of households – skilled and unskilled, which become skilled through investments in education. We highlight the importance of the general equilibrium effects of redistributive policies over the leisure/labor and education decisions and wages. Lump-sum transfers reduce investments in education, raising the share of unskilled individuals, decreasing their wage and, raising the skill-premium. Education subsidies raise investments in education, the skills supply, and unskilled wages and reduce the skill-premium during the slowdown of the technology diffusion.

JEL Classification: H23; J22; O33

Funding source: FCT

Award Identifier / Grant number: UID/ECO/04007/2013 (POCI-01-0145-FEDER-007659)

Funding statement: CEFAGE-UBI has financial support from FCT, Portugal, and FEDER/COMPETE 2020, through grant UID/ECO/04007/2013 (POCI-01-0145-FEDER-007659). Manuela Magalhães gratefully acknowledge financial support from FCT (via POCI, project number 24068/2005), from University of Warwick, from University of Alicante, and from the Spanish Ministry of Economics and Competition (ECO2012-36719).

Appendices

A Proofs

This appendix derives the Lemma 2 and Lemma 3 in the text. To do that, we find the FOCs of the households’ optimization problem defined by (8).

Proof of Lemma 2 and Lemma 3: The first-order conditions for the households’ inter-temporal optimization problem (8) are

(16)uctu=λtu,
(17)ucts=λts,
(18)ults=(1τt)wtsλts,
(19)ultu=wtuλtu,
(20)(ulteλust)=βaΓt+1,

where ΓVsVu and λ is the Lagrangian multiplier corresponding to the households’ budget contraint, which is introduced to the Bellman equation as defined by Lin and Arora (1991).

Substituting (16) into (20) we obtain Lemma 2. To obtain Lemma 3, we divide (18) by (19) and use the equations (16) and (17) to eliminate λu and λs. Then, we solve with respect to uctu and have

(21)uctu=(1τt)ultuultsSprtucts,whereSprt=wtswtu.

Next, we substitute (21) into Lemma 2 and obtain Lemma 3.

B Two-period model

This section derives the results for the two-period model described in Section 3.2. From the first-order conditions (16)–(20), we reduce the dimensionality of the problem to three equations

(22)ultuuctu=wtu,
(23)ultsucts=(1τt)wts,for
(24)(ulteuctust)=βa(ut+1sut+1u)β(1alt+1e)(ult+1euct+1ust+1).

Equation (24) is the Euler Equation (EE) for our model. To derive the EE, we use equation (16), equation (20) and the equation Γt=1βa(ult1eλust1). Then, rearranging we obtain the EE.

Next, we assume the log-log utility function described by (10) and rewrite the above equations (22)–(24) for T = 2,

(25)l1u=1l1e1+ηη1+ηΠ1+tr1+s1l1ew1u,
(26)l1s=11+ηη1+ηΠ1(1τ1)w1s,
(27)η1l1ul1e+s1c1u=βa(lnc2sc2u+ηln1l2s1l2u)+β(η1l2u),

where tr1=τ1γη1+ηm11m1. Note that we are also assuming that government carries out distributive policies only in period 1. Thus, as s2 = τ2 = 0, the optimal choice for l2e=0. Using the households’ budget constraints, we update the Euler equation (27),

(28)η1l1ul1e+s1η(1l1ul1e)w1u=βa((1+η)ln1l2s1l2u+ln((1τ2ϕ)w2sw2u)+ηln(1l2s1l2u))+β(η1l2u),

where

(29)w2sw2u=γα(n2(1m2)l2um2(1n2)l2s)1αΠ1=1αα((1n1)(1m1)w1ul1u+n1m1l1sw1s).

There are two special cases in which we can obtain the analytical effects of redistributive policies for this economy with technology diffusion and endogenous human capital accumulation. First, when the technology is skill-neutral, α = 1. Second, when technology is skill-biased but profits are not distributed by household, α ≠ 1 and Π = 0.

B.1 Skill-neutral technology diffusion, α = 1

If α = 1, the Spr1 = Spr2 = γ and Π1 = Π2 = 0 [see equation (29)]. Forwarding the equations (25) and (26) by one period and substituting Π2=l2e=0, we have that l2s=11+η and l2u=11+η, given that tr2 = 0. To obtain the hours invested in education l1e, we use equations (25)–(27) again. From equation (26) we have that l1s=η1+η for the skill-neutral technology because Π = 0. And from the firms’ optimization problem we know that wtu=1wts=γ, for α = 1, ∀ t = {1, 2}. Thus, updating equation (25) yields

(30)l1u=11+η(1ηs1l1el1e)τ1ϕγη(1+η)2m11m1.

Substituting (29) into (28), and after some mathematical manipulation, we obtain the optimal choice for l1e,

(31)l1e=1l1uη(1+s1)βalnγ+β(1+η).

Next, substituting (30) into (31) and rearranging, we obtain the hours invested in education by unskilled workers for the skill-neutral technology diffusion,

(32)l1e=11s1(1+τ1ϕγ(1+η)m11m1(1+η)(1+s1)βalnγ+β(1+η)).

The previous equation gives the optimal choice for hours invested in education as a function of taxes and education subsidies. Substituting l1e into (30), we obtain the optimal choice for the unskilled labor,

(33)l1u=11+η(1(1+ηs1)(11s1(1+τ1ϕγ(1+η)m11m1(1+η)(1+s1)βalnγ+β(1+η))))τ1ϕγη(1+η)2m11m1.

Therefore, we can explore the general equilibrium effects of redistributive policies in labor supplied to production and education activities. To do that we compute the total derivatives with respect to s1 and tr1=τ1γη1+ηm11m1 and obtain the global effects of each redistributive policy.

(34)dl1udtr1=1η>0,
(35)dl1edtr1=1η(1+η)<0,
(36)dl1eds1=2βaln(γ)+β(2+2η)3η3(2s11)2β(aln(γ)+1+η)<0,
(37)dl1uds1=2(βaln(γ)+(β12s1+2s12)η+β3/2)(2s11)2β(aln(γ)+1+η)>0.

The sign of the previous effects with respect to subsidies s1 is found for the calibration used in this study. Still the signs for the effects of tr are valid for any calibration.

B.2 Skill-biased technology diffusion (α ≠ 1) and no profits are distributed by households

If no profits are distributed and α ≠ 1; replacing the fiscal revenue by tr1=τϕw1sl1sm1/(1m1), equations 2527 become

(38)l1u=1l1e1+ηη1+η(τ1ϕl1sw1sm1)/(1m1)+s1l1ew1u
(39)l1s=11+η,
(40)η1l1ul1e+s1c1u=βa(lnc2sc2u)+β(1+η),

As l2s=l2u and τ2 = 0, the term lnc2sc2u=lnSpr2=ln(γα(n21n2)1α(1m2m2)1α). Substituting this result into equation (40) and using equation (22) yields

(41)η1l1ul1e+s1w1u(1l1ul1e)=βaln(γα(n21n2)1α(1m2m2)1α)+β(1+η).

To investigate the redistributive policy effects, we consider the two policies separately:

B.2.1 Lump-sum transfers (s1 = 0)

Updating equations (38) and (41) for this case, we have

(42)l1u=1l1e1+ηη(1+η)2m11m1w1sw1uτ1,
(43)η1l1ul1e=βaln(γα(n21n2)1α(1m2m2)1α)+β(1+η).

Solving this non-linear system of two equations and rearranging, we have

(44)(1+η)(ηη1+ηl1e+η1+ηm11m1γα(n1(1m1)m1(1m1))(1α)τ1)(z2(1α)(1m1)al1e)=0,

where z=βaln(γα(n21n2)1α)+(1α)(12m1)+β(1+η), and the previous equation (44) is valid for values of m close to 1, because we approximate the function ln1m2m212m2. Next, by the Implicit Function Theorem, we can obtain dl1edtr1

(45)dl1edtr1<0,

for the calibration assumed in this paper.

B.2.2 Education subsidies (s1 ≠ 0)

Again, updating equations (41) and (38) for this case, we obtain the following expressions:

(46)l1u=(z2(1m1)al1e)(1l1e)ηz2(1m1)al1e,
(47)w1u=α1n11m1((1n1)1α(1m1)α(l1u)α+γαn11α(m1l1s)α)1αα(l1u)α1,
(48)11+η=l1u+11+ηl1e+x(lu)α+η1+ηs1l1ew1u,

where x=η21+η(η1+η)1αm11m1γα(n11n11m1m1)1α. Equation (48) implicitly defines l1e as a function of income taxes and (τ1) and education subsidies (s1). By the Implicit Function Theorem, we can find how education subsidies affect the investment in education:

(49)dl1eds10.

C Computation of the transition dynamics

Consider that there is a skill-biased technology diffusion and the economy is therefore in transition between two steady states. In this economy, households (skilled and unskilled) consume, invest in education, and supply labor. The investment in education allows the unskilled households to acquire skills and become skilled. From t = 0 to t = 40, i.e. during the period of the technology diffusion, there is an increase of the tax rate τ. This tax rate over the skilled workers’ wage can be distributed by unskilled workers as a lump sum transfer or as a constant subsidy per hours invested in education. To solve the two dimensional and dynamic optimization problem defined by (8), and analyze how redistributive polices affect the unskilled and skilled workers’ decisions, we develop the following algorithm:

  1. In each t, nt, τt and VtI(,t+1) for I={s,u} are known.

  2. For t = T, assume VtI(,t+1)=0 for I={s,u} and guess mT.

  3. For t = T, solve the discrete nonlinear programming problem defined by (8) using an hybrid combination of a genetic algorithm (GA) and a sequential quadratic programming (SQP) algorithm. The nonlinear equations are incorporated in both algorithms by using a merit function that consists of the objective function as well as the nonlinear constraints through penalty functions.

  4. Check whether the solution obtained in the segmented market and in step 3 verify the condition wtu<wts. If so, store all values, otherwise proceed with the optimization in the unsegmented market and store all values.

  5. With values stored for t = T, solve by backward induction the problem defined by (8) as defined in step 3 for t = T − 1, …, 0.

  6. Repeat steps 2–4 until the m0 obtained by backward solution coincides with the m0 obtained for the initial steady-state, that is, when n is constant and there is no technology diffusion.

  7. Verify that the solution is not sensitive to T.

D Comparison with laissez-faire economy

Figure 7: Impulse response functions to a permanent lump-sum transfers increase and laissez-faire economy variable values.Notes: Orange line: laissez-faire. Blue line: redistributive economy.
Figure 7:

Impulse response functions to a permanent lump-sum transfers increase and laissez-faire economy variable values.

Notes: Orange line: laissez-faire. Blue line: redistributive economy.

Figure 8: Impulse response functions to a permanent education subsidy increase and laissez-faire economy variable values.Notes: Orange line: laissez-faire. Blue line: redistributive economy.
Figure 8:

Impulse response functions to a permanent education subsidy increase and laissez-faire economy variable values.

Notes: Orange line: laissez-faire. Blue line: redistributive economy.

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Published Online: 2018-05-18

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