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Externalities and fiscal policy in a Lucas-type model
Economics Letters 88 (2005) 251 – 259 www.elsevier.com/locate/econbase
Externalities and fiscal policy in a Lucas-type model Manuel A. Go´mezT Department of Applied Economics, Facultad de Ciencias Econo´micas, Campus de Elvin˜a, 15071, University of A Corun˜a, Spain Received 30 April 2004; received in revised form 22 February 2005; accepted 22 March 2005 Available online 23 May 2005
Abstract This paper devises a fiscal policy capable of decentralizing the optimal growth path in a Lucas-type model when average human capital has an external effect in the goods sector and average learning time has an external effect in human capital accumulation. D 2005 Elsevier B.V. All rights reserved. Keywords: Human capital; Optimal policy; Externalities JEL classification: O41
1. Introduction Several authors have considered externalities in endogenous growth models with human capital accumulation (e.g., Lucas, 1988; Chamley, 1993; Benhabib and Perli, 1994). In this case, optimal growth paths and competitive equilibrium paths may not coincide, although an adequate government intervention can correct this market failure. Actually, Garcı´a-Castrillo and Sanso (2000) and Go´mez (2003) devise fiscal policies that are capable of decentralizing the optimal equilibrium in Lucas’ (1988) model when average human capital has an external effect in goods production. This paper devises a fiscal policy by means of which the optimal equilibrium can be decentralized in a Lucas-type model. We extend the analyses of Garcı´a-Castrillo and Sanso (2000) and Go´mez (2003) to T Tel.: +34 981 167000; fax: +34 981 167070. E-mail address: [email protected]. 0165-1765/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2005.03.001
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allow also for the average learning time to have an external effect on human capital accumulation. This issue is interesting since these external effects are also plausible (e.g., Chamley, 1993; Benhabib and Perli, 1994) and one may conjecture that the optimal policy will depend on the class of externalities considered. We assume diminishing point-in-time returns in human capital accumulation, because they have been shown to be essential for capturing the characteristic life-cycle phases of human capital accumulation (e.g., Rosen, 1976). The optimal policy proposed by Garcı´a-Castrillo and Sanso (2000) must rely on lump-sum taxation at least in the transitional phase. More realistically, we assume that the government cannot resort to lump-sum taxes. Interestingly enough, we find that a similar policy to that devised by Go´mez (2003), conveniently modified to correct for an externality of average learning time as well, can decentralize the optimal equilibrium. Section 2 describes the market economy and Section 3 the centralized economy. Section 4 devises the optimal policy. Section 5 concludes.
2. The decentralized economy Consider an economy inhabited by a constant population, normalized to one, of identical infinitelylived agents who derive utility from consumption, c, according to Z l eqt c1r 1 =ð1 rÞdt qN0; rN0: ð1Þ 0
Time endowment is normalized to unity, which can be allocated to work, u, or learning, 1 u. Human capital, h, is accumulated according to h˙ ¼ Bð1 uÞv ð1 uÞx ah
BN0; vN0; xz0; v þ xV1;
ð2Þ
where (1 u)a expresses externalities associated to average learning time. The government taxes capital income at a rate s k , labor income at a rate s h and subsidizes investment in education at a rate s h . The sole cost of education is foregone earnings, w(1u)h, a fraction s h of which is therefore financed by the government. The agent’s budget constraint is k˙ ¼ ð1 sk Þrk þ ð1 sh Þwuh c þ sh wð1 uÞh;
ð3Þ
where r is the interest rate, and w, the wage rate. We shall express (3) equivalently as k˙ ¼ ð1 sk Þrk þ ð1 sˆ h Þwuh c þ sh wh;
ð4Þ
where sˆ h ¼ sh þ sh :
ð5Þ
Output, y, is produced with Cobb–Douglas technology: y ¼ Ak b ðuhÞ1b hwa
AN0; 0bbb1; wz0;
where h a expresses externalities associated to average human capital. Profit maximization by competitive firms implies that r = by/k and w = (1 b)y / (uh).
M.A. Go´mez / Economics Letters 88 (2005) 251–259
253
The government runs a balanced-budget and cannot resort to lump-sum taxation: sk rk þ sh wuh ¼ sh wð1 uÞh; or using (5), sk rk þ sˆ h wuh ¼ sh wh:
ð6Þ
Let J be the current-value Hamiltonian of the agent’s problem, J ¼ c1r 1 =ð1 rÞ þ kðð1 sk Þrk þ ð1 sˆ h Þwuh c þ sh whÞ þ lBð1 uÞv ð1 uÞx a h: The first-order conditions are cr ¼ k;
ð7aÞ
kð1 sˆ h Þwh ¼ lvBð1 uÞv1 ð1 uÞx a h;
ð7bÞ
k˙ =k ¼ q ð1 sk Þr;
ð7cÞ
ˆ h Þwu þ sh wÞ; l˙ =l ¼ q Bð1 uÞv ð1 uÞx a ðk=lÞðð1 s
ð7dÞ
lim kkeqt ¼ lim lheqt ¼ 0:
ð7eÞ
tYl
tYl
Hereafter, let c z = ˙z / z denote the growth rate of the variable z. In what follows, the equilibrium conditions h = h a and 1 u = (1 u)a will be imposed. From (7a) and (7c), we get cc ¼ ð ð1 sk Þby=k qÞ=r:
ð8Þ
Using (6), (4) can be expressed as ck ¼ y=k c=k:
ð9Þ
Log-differentiating (7b) yields ck s˙ˆ h =ð1 sˆ h Þ þ cy cu ¼ cl þ ð1 v xÞcu u=ð1 uÞ þ ch :
ð10Þ
From (7b) and (7d), we get cl ¼ q Bð1 uÞvþx Bð1 uÞvþx1 ðvu þ vsh =ð1 sˆ h ÞÞ:
ð11Þ
The following system characterizes the dynamics of the economy in terms of the variables u, x = kh (1b+w)/(b1) and q = c / k, that are constant in the steady state: cq ¼ ðð1 sk Þb rÞAxb1 u1b =r þ q q=r;
ð12aÞ
cx ¼ Axb1 u1b ð1 b þ wÞBð1 uÞvþx =ð1 bÞ q;
ð12bÞ
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cu ð1 uÞ=ðbð1 uÞ þ ð1 v xÞuÞ vsh s˙ˆ h vþx1 b1 1b b sk Ax u q þ Bð1 uÞ ð1 b þ wÞð1 uÞ þ vu þ : 1 sˆ h 1 sˆ h ð12cÞ Eq. (12a) is obtained from (8) and (9), and (12b) from (2) and (9). Substituting c l from (11), c k from (7c) and c y from c y = bc k + (1 b)c u + (1 b + w)c h in (10), and then replacing c k from (9) and c h = B(1 u)v+x in the ensuing expression, we get (12c).
3. The centrally planned economy The central planner takes all the relevant information into account. He maximizes (1) subject to k˙ = Ak b u 1b h 1b+x c and h˙ = B(1 u)v+x h. Let J¯ be the current-value Hamiltonian of the planner’s problem, J¯ ¼ c1r 1 =ð1 rÞ þ k¯ Ak b u1b h1bþw c þ l¯ Bð1 uÞvþx h: The first-order conditions are cr ¼ k¯ ;
ð13aÞ
k¯ ð1 bÞAk b ub h1bþw ¼ l¯ ðv þ xÞBð1 uÞvþx1 h;
ð13bÞ
k˙¯ =k¯ ¼ q bAk b1 u1b h1bþw ;
ð13cÞ
l˙¯ =l¯ ¼ q Bð1 uÞvþx k¯ =l¯ ð1 b þ wÞAk b u1b hbþw ;
ð13dÞ
lim k¯ keqt ¼ lim l¯ heqt ¼ 0:
ð13eÞ
t Yl
t Yl
Proceeding as in the case of the market economy, we arrive at cq ¼ ðb rÞAxb1 u1b =r þ q q=r;
ð14aÞ
cx ¼ Axb1 u1b Bð1 b þ wÞð1 uÞvþx =ð1 bÞ q;
ð14bÞ
cu ¼
ð1 b þ wÞBð1 uÞvþx ðð1 bÞð1 uÞ þ ðv þ xÞuÞ ð1 bÞbð1 uÞq : ð1 bÞðbð1 uÞ þ ð1 v xÞuÞ
ð14cÞ
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Equating the growth rates to zero yields the steady state. From (14a) and (14b), we get x4 ¼
Að1 bÞbu41b Bð1 b þ wÞrð1 u4Þvþx þ ð1 bÞq
1=ð1bÞ ð15aÞ
;
q4 ¼ Bð1 b þ wÞðr bÞð1 u4Þvþx =ðð1 bÞbÞ þ q=b:
ð15bÞ
Substituting (15b) into (14c) and simplifying yields Bð1 b þ wÞð1 u4Þvþx1 ðð1 rÞð1 u4Þ þ ðv þ xÞu4Þ ¼ ð1 bÞq:
ð15cÞ
Now, we analyze when (15c) has a solution u* a (0,1). From (15c), we define PðuÞ ¼ Bð1 b þ wÞð1 uÞvþx1 ðð1 rÞð1 uÞ þ ðv þ xÞuÞ ð1 bÞq; which is defined at least in [0,1). P is twice continuously differentiable and monotonically increasing because PVðuÞ ¼ Bð1 uÞvþx2 ð1 b þ wÞðv þ xÞðrð1 uÞ þ uð1 v xÞÞN0;
8uað0; 1Þ:
If v þ x b 1; lim PðuÞ ¼ þl, and so, (15c) has a solution u* a (0,1) if and only if u Y1
Pð0Þ ¼ Bð1 b þ wÞð1 rÞ ð1 bÞqb0; or equivalently, qð1 bÞNBð1 b þ wÞð1 rÞ:
ð16aÞ
If v þ x ¼ 1; lim PðuÞ ¼ Pð1Þ ¼ Bð1 b þ wÞ ð1 bÞq, and so, (15c) has a solution u* a (0,1) if uY1 and only if Pð0Þ ¼ Bð1 b þ wÞð1 rÞ ð1 bÞqb0; Pð1Þ ¼ Bð1 b þ wÞ ð1 bÞqN0; or equivalently, Bð1 b þ wÞNqð1 bÞNBð1 b þ wÞð1 rÞ:
ð16bÞ
As (15b) and (15c) imply q* N 0 and x* N 0 if u* a (0,1), the steady state is feasible. We can also state the following proposition (see the proof in Appendix A). Proposition 1. The steady state of the optimal-growth problem is locally saddle-path stable.
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4. The optimal policy This section devises a fiscal policy capable of decentralizing the optimal equilibrium. First, note that (12b) and (14b) coincide. Comparing (12a) with (14a), they coincide if s k = 0. Equating (12c) and (14c), with s k = 0, yields ðð1 bÞx þ wðv þ xÞÞð1 sˆ h Þu ð1 uÞ1vx sˆ˙ h þ sh ¼ : ð17Þ ð1 bÞv Bv We guess a constant optimal sˆh , i.e., sˆ˙h = 0. Hence, (17) reduces to sh ¼
ð1 b þ wÞx þ wv ð1 sˆ Þu: ð1 bÞv
Solving (6) and (18), with s k = 0, we obtain w x wx þ ; sˆ h ¼ 1 b þ w v þ x ð1 b þ wÞðv þ xÞ which is constant as guessed, and 0 b sˆh b 1. The optimal subsidy is then w x wx þ sh ¼ u; 1 b þ w v þ x ð1 b þ wÞðv þ xÞ which satisfies 0 b s h b 1. Now, (5), (19) and (20a) yield w x wx þ ð1 uÞ; sh ¼ sˆ h sh ¼ 1 b þ w v þ x ð1 b þ wÞðv þ xÞ
ð18Þ
ð19Þ
ð20aÞ
ð20bÞ
which satisfies 0 b s h b 1. These results are consistent with the Pigouvian tax intuition that a subsidy directed at the source of a positive externality will correct the market failure. The following proposition summarizes these findings. Proposition 2. The optimal equilibrium can be decentralized if capital income is not taxed and investment in human capital is subsidized at a rate sh ¼
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ u:
The subsidy can be financed by taxing labor income at a rate sh ¼
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ ð 1 uÞ :
Lump-sum taxation is not required to balance the government budget. Comparing (7a) and (13a), decentralization of the optimal equilibrium requires k = k¯. Hence, c k = ck¯ which, using (7c) and (13c), yields s k = 0. Log-differentiating (7b) and (13b), recalling that a constant sˆh was guessed, decentralization also requires c l = cl¯ ; i.e., using (7d) and (13d), ðk=lÞð1 sˆ h þ sh =uÞð1 bÞAk b u1b hbþw ¼ k¯ =l¯ ð1 b þ wÞAk b u1b hbþw : ð21Þ
M.A. Go´mez / Economics Letters 88 (2005) 251–259
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Hence, the optimal policy must be set so that private and social marginal products of human capital in the goods sector, measured in terms of human capital as numeraire, coincide. This can be accomplished by setting s h according to (18), which, thus, corrects for the effects of two divergences: the difference between the private and the social marginal product of human capital in goods production because of the externality associated to average human capital, and the difference ¯ ¯ between the imputed real price of physical capital in the centralized economy, k / l , and that in the market economy, k / l, because of the externality associated to average learning time (see (7b) and (13b)). The government’s budget constraint (6), with s k = 0, provides the additional condition required to determine sˆh and s h . The optimal policy just derived is similar to that devised by Go´mez (2003), conveniently adapted to account for the externality of average learning time as well. Eqs. (20a) and (20b) comprise three terms. The first term, wu / (1 b + w) and w(1 u) / (1 b + w), respectively, depends on the relative size of the externality associated to average human capital, and fully corrects for this external effect in the absence of externalities in the educational sector (x = 0). This is the optimal policy derived by Go´mez (2003). The second term, xu / (v + x) and x(1 u) / (v + x), respectively, depends on the relative size of the externality associated to average learning time, and fully corrects for this external effect in absence of externalities in goods production (w = 0). The third term, which depends on the product of the relative size of both externalities, is nonzero only if both external effects coexist. As Go´mez (2003), let us suppose now that the government subsidizes the stock of human capital at a rate sˆ h . The agent’s budget constraint is then k˙ ¼ ð1 sk Þrk þ ð1 sˆ h Þwuh c þ sˆ h h:
ð22Þ
Handling the first-order conditions as in Section 2, it can be observed that the relationship sˆ h = s h w carries over all the calculations, and so, the dynamics of the market economy are also summarized by Eqs. (12a,b,c), after s h being substituted by sˆ h /w. Thus, from (19), (20a) and sˆ h = s h w, we can derive the following result. Proposition 3. The optimal equilibrium can be decentralized if capital income is not taxed and the stock of human capital is subsidized at a rate sˆ h ¼
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ ð 1 h bÞ y :
The subsidy can be financed by taxing labor income at a constant rate sˆ h ¼
w x wx : þ ð 1 b þ wÞ ð v þ xÞ 1bþw vþx
Lump-sum taxes are not required to balance the government budget. The government’s size, measured as the subsidy (or taxes) share of output, is constant and equals /¼
sˆ h h ¼ y
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ ð 1 bÞ :
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M.A. Go´mez / Economics Letters 88 (2005) 251–259
5. Conclusions We devise a fiscal policy capable of decentralizing the optimal equilibrium in a Lucas-type model with externalities in goods production and in human capital accumulation. The optimal growth path can be decentralized by keeping capital income untaxed and instituting a subsidy to investment in education or to the stock of human capital, which can be financed by a labor income tax. Lump-sum taxation is not needed to balance the government budget either in the steady state or in the transitory phase.
Acknowledgements I thank an anonymous referee for valuable comments. Financial support from Spanish Ministry of Education and Science and FEDER is gratefully acknowledged.
Appendix A
Proof of Proposition 1. Linearizing Eqs.(14a,b,c) around the steady state (x*,q*,u*) yields 0 1 0 1 0 x˙ x x4 a11 @ q˙ A ¼ A@ q q4 A ¼ @ a21 a31 u˙ u u4
a12 a22 a32
10 1 x x4 a13 a23 A@ q q4 A; u u4 a33
where a11 ¼ ð1 bÞAx4b1 u41b ; a12 ¼ x4;
a13 ¼ ð1 bÞAx4b1 u4b þ Bð1 b þ wÞð1 u4Þvþx1 ðv þ xÞ=ð1 bÞ x4; a21 ¼ ð1 bÞðr bÞAx4b2 u41b q4=r; a22 ¼ q4; a23 ¼ ð1 bÞðr bÞAx4b1 u4b q4=r; a31 ¼ 0; a32 ¼ bð1 u4Þu4=ðbð1 u4Þ þ u4ð1 v xÞÞ; a33 ¼ Bð1 b þ wÞð1 u4Þvþx1 u4ðv þ xÞ=ð1 bÞ:
M.A. Go´mez / Economics Letters 88 (2005) 251–259
259
The determinant of A is negative: detð AÞ ¼ Bð1 b þ wÞð1 u4Þvþx1 u4bAx4b1 u41b q4ðv þ xÞ
rð1 u4Þ þ u4ð1 v xÞ b0: rðbð1 u4Þ þ u4ð1 v xÞÞ
Using (15a) and (15b), we obtain a11 þ a22 ¼ q þ
Bð1 b þ wÞð1 u4Þvþx ðr 1Þ Bð1 b þ wÞð1 u4Þvþx1 u4ðv þ xÞ ¼ N0; 1b 1b
where the last equality follows from (15c). The trace of A is then positive: trð AÞ ¼ a11 þ a22 þ a33 ¼ 2Bð1 b þ wÞð1 u4Þvþx1 u4ðv þ xÞ=ð1 bÞN0: Hence, A must have one stable and two unstable eigenvalues.
5
References Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth. Journal of Economic Theory 63, 113 – 142. Chamley, C.P., 1993. Externalities and dynamics in models of dlearning or doingT. International Economic Review 34, 583 – 610. Garcı´a-Castrillo, P., Sanso, M., 2000. Human capital and optimal policy in a Lucas-type model. Review of Economic Dynamics 3, 757 – 770. Go´mez, M.A., 2003. Optimal fiscal policy in the Uzawa–Lucas model with externalities. Economic Theory 22, 917 – 925. Lucas Jr., R.E., 1988. On the mechanics of economic development. Journal of Monetary Economics 22, 3 – 42. Rosen, S., 1976. A theory of life earnings. Journal of Political Economy 84, S45 – S67.
Externalities and fiscal policy in a Lucas-type model Manuel A. Go´mezT Department of Applied Economics, Facultad de Ciencias Econo´micas, Campus de Elvin˜a, 15071, University of A Corun˜a, Spain Received 30 April 2004; received in revised form 22 February 2005; accepted 22 March 2005 Available online 23 May 2005
Abstract This paper devises a fiscal policy capable of decentralizing the optimal growth path in a Lucas-type model when average human capital has an external effect in the goods sector and average learning time has an external effect in human capital accumulation. D 2005 Elsevier B.V. All rights reserved. Keywords: Human capital; Optimal policy; Externalities JEL classification: O41
1. Introduction Several authors have considered externalities in endogenous growth models with human capital accumulation (e.g., Lucas, 1988; Chamley, 1993; Benhabib and Perli, 1994). In this case, optimal growth paths and competitive equilibrium paths may not coincide, although an adequate government intervention can correct this market failure. Actually, Garcı´a-Castrillo and Sanso (2000) and Go´mez (2003) devise fiscal policies that are capable of decentralizing the optimal equilibrium in Lucas’ (1988) model when average human capital has an external effect in goods production. This paper devises a fiscal policy by means of which the optimal equilibrium can be decentralized in a Lucas-type model. We extend the analyses of Garcı´a-Castrillo and Sanso (2000) and Go´mez (2003) to T Tel.: +34 981 167000; fax: +34 981 167070. E-mail address: [email protected]. 0165-1765/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2005.03.001
252
M.A. Go´mez / Economics Letters 88 (2005) 251–259
allow also for the average learning time to have an external effect on human capital accumulation. This issue is interesting since these external effects are also plausible (e.g., Chamley, 1993; Benhabib and Perli, 1994) and one may conjecture that the optimal policy will depend on the class of externalities considered. We assume diminishing point-in-time returns in human capital accumulation, because they have been shown to be essential for capturing the characteristic life-cycle phases of human capital accumulation (e.g., Rosen, 1976). The optimal policy proposed by Garcı´a-Castrillo and Sanso (2000) must rely on lump-sum taxation at least in the transitional phase. More realistically, we assume that the government cannot resort to lump-sum taxes. Interestingly enough, we find that a similar policy to that devised by Go´mez (2003), conveniently modified to correct for an externality of average learning time as well, can decentralize the optimal equilibrium. Section 2 describes the market economy and Section 3 the centralized economy. Section 4 devises the optimal policy. Section 5 concludes.
2. The decentralized economy Consider an economy inhabited by a constant population, normalized to one, of identical infinitelylived agents who derive utility from consumption, c, according to Z l eqt c1r 1 =ð1 rÞdt qN0; rN0: ð1Þ 0
Time endowment is normalized to unity, which can be allocated to work, u, or learning, 1 u. Human capital, h, is accumulated according to h˙ ¼ Bð1 uÞv ð1 uÞx ah
BN0; vN0; xz0; v þ xV1;
ð2Þ
where (1 u)a expresses externalities associated to average learning time. The government taxes capital income at a rate s k , labor income at a rate s h and subsidizes investment in education at a rate s h . The sole cost of education is foregone earnings, w(1u)h, a fraction s h of which is therefore financed by the government. The agent’s budget constraint is k˙ ¼ ð1 sk Þrk þ ð1 sh Þwuh c þ sh wð1 uÞh;
ð3Þ
where r is the interest rate, and w, the wage rate. We shall express (3) equivalently as k˙ ¼ ð1 sk Þrk þ ð1 sˆ h Þwuh c þ sh wh;
ð4Þ
where sˆ h ¼ sh þ sh :
ð5Þ
Output, y, is produced with Cobb–Douglas technology: y ¼ Ak b ðuhÞ1b hwa
AN0; 0bbb1; wz0;
where h a expresses externalities associated to average human capital. Profit maximization by competitive firms implies that r = by/k and w = (1 b)y / (uh).
M.A. Go´mez / Economics Letters 88 (2005) 251–259
253
The government runs a balanced-budget and cannot resort to lump-sum taxation: sk rk þ sh wuh ¼ sh wð1 uÞh; or using (5), sk rk þ sˆ h wuh ¼ sh wh:
ð6Þ
Let J be the current-value Hamiltonian of the agent’s problem, J ¼ c1r 1 =ð1 rÞ þ kðð1 sk Þrk þ ð1 sˆ h Þwuh c þ sh whÞ þ lBð1 uÞv ð1 uÞx a h: The first-order conditions are cr ¼ k;
ð7aÞ
kð1 sˆ h Þwh ¼ lvBð1 uÞv1 ð1 uÞx a h;
ð7bÞ
k˙ =k ¼ q ð1 sk Þr;
ð7cÞ
ˆ h Þwu þ sh wÞ; l˙ =l ¼ q Bð1 uÞv ð1 uÞx a ðk=lÞðð1 s
ð7dÞ
lim kkeqt ¼ lim lheqt ¼ 0:
ð7eÞ
tYl
tYl
Hereafter, let c z = ˙z / z denote the growth rate of the variable z. In what follows, the equilibrium conditions h = h a and 1 u = (1 u)a will be imposed. From (7a) and (7c), we get cc ¼ ð ð1 sk Þby=k qÞ=r:
ð8Þ
Using (6), (4) can be expressed as ck ¼ y=k c=k:
ð9Þ
Log-differentiating (7b) yields ck s˙ˆ h =ð1 sˆ h Þ þ cy cu ¼ cl þ ð1 v xÞcu u=ð1 uÞ þ ch :
ð10Þ
From (7b) and (7d), we get cl ¼ q Bð1 uÞvþx Bð1 uÞvþx1 ðvu þ vsh =ð1 sˆ h ÞÞ:
ð11Þ
The following system characterizes the dynamics of the economy in terms of the variables u, x = kh (1b+w)/(b1) and q = c / k, that are constant in the steady state: cq ¼ ðð1 sk Þb rÞAxb1 u1b =r þ q q=r;
ð12aÞ
cx ¼ Axb1 u1b ð1 b þ wÞBð1 uÞvþx =ð1 bÞ q;
ð12bÞ
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M.A. Go´mez / Economics Letters 88 (2005) 251–259
cu ð1 uÞ=ðbð1 uÞ þ ð1 v xÞuÞ vsh s˙ˆ h vþx1 b1 1b b sk Ax u q þ Bð1 uÞ ð1 b þ wÞð1 uÞ þ vu þ : 1 sˆ h 1 sˆ h ð12cÞ Eq. (12a) is obtained from (8) and (9), and (12b) from (2) and (9). Substituting c l from (11), c k from (7c) and c y from c y = bc k + (1 b)c u + (1 b + w)c h in (10), and then replacing c k from (9) and c h = B(1 u)v+x in the ensuing expression, we get (12c).
3. The centrally planned economy The central planner takes all the relevant information into account. He maximizes (1) subject to k˙ = Ak b u 1b h 1b+x c and h˙ = B(1 u)v+x h. Let J¯ be the current-value Hamiltonian of the planner’s problem, J¯ ¼ c1r 1 =ð1 rÞ þ k¯ Ak b u1b h1bþw c þ l¯ Bð1 uÞvþx h: The first-order conditions are cr ¼ k¯ ;
ð13aÞ
k¯ ð1 bÞAk b ub h1bþw ¼ l¯ ðv þ xÞBð1 uÞvþx1 h;
ð13bÞ
k˙¯ =k¯ ¼ q bAk b1 u1b h1bþw ;
ð13cÞ
l˙¯ =l¯ ¼ q Bð1 uÞvþx k¯ =l¯ ð1 b þ wÞAk b u1b hbþw ;
ð13dÞ
lim k¯ keqt ¼ lim l¯ heqt ¼ 0:
ð13eÞ
t Yl
t Yl
Proceeding as in the case of the market economy, we arrive at cq ¼ ðb rÞAxb1 u1b =r þ q q=r;
ð14aÞ
cx ¼ Axb1 u1b Bð1 b þ wÞð1 uÞvþx =ð1 bÞ q;
ð14bÞ
cu ¼
ð1 b þ wÞBð1 uÞvþx ðð1 bÞð1 uÞ þ ðv þ xÞuÞ ð1 bÞbð1 uÞq : ð1 bÞðbð1 uÞ þ ð1 v xÞuÞ
ð14cÞ
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Equating the growth rates to zero yields the steady state. From (14a) and (14b), we get x4 ¼
Að1 bÞbu41b Bð1 b þ wÞrð1 u4Þvþx þ ð1 bÞq
1=ð1bÞ ð15aÞ
;
q4 ¼ Bð1 b þ wÞðr bÞð1 u4Þvþx =ðð1 bÞbÞ þ q=b:
ð15bÞ
Substituting (15b) into (14c) and simplifying yields Bð1 b þ wÞð1 u4Þvþx1 ðð1 rÞð1 u4Þ þ ðv þ xÞu4Þ ¼ ð1 bÞq:
ð15cÞ
Now, we analyze when (15c) has a solution u* a (0,1). From (15c), we define PðuÞ ¼ Bð1 b þ wÞð1 uÞvþx1 ðð1 rÞð1 uÞ þ ðv þ xÞuÞ ð1 bÞq; which is defined at least in [0,1). P is twice continuously differentiable and monotonically increasing because PVðuÞ ¼ Bð1 uÞvþx2 ð1 b þ wÞðv þ xÞðrð1 uÞ þ uð1 v xÞÞN0;
8uað0; 1Þ:
If v þ x b 1; lim PðuÞ ¼ þl, and so, (15c) has a solution u* a (0,1) if and only if u Y1
Pð0Þ ¼ Bð1 b þ wÞð1 rÞ ð1 bÞqb0; or equivalently, qð1 bÞNBð1 b þ wÞð1 rÞ:
ð16aÞ
If v þ x ¼ 1; lim PðuÞ ¼ Pð1Þ ¼ Bð1 b þ wÞ ð1 bÞq, and so, (15c) has a solution u* a (0,1) if uY1 and only if Pð0Þ ¼ Bð1 b þ wÞð1 rÞ ð1 bÞqb0; Pð1Þ ¼ Bð1 b þ wÞ ð1 bÞqN0; or equivalently, Bð1 b þ wÞNqð1 bÞNBð1 b þ wÞð1 rÞ:
ð16bÞ
As (15b) and (15c) imply q* N 0 and x* N 0 if u* a (0,1), the steady state is feasible. We can also state the following proposition (see the proof in Appendix A). Proposition 1. The steady state of the optimal-growth problem is locally saddle-path stable.
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4. The optimal policy This section devises a fiscal policy capable of decentralizing the optimal equilibrium. First, note that (12b) and (14b) coincide. Comparing (12a) with (14a), they coincide if s k = 0. Equating (12c) and (14c), with s k = 0, yields ðð1 bÞx þ wðv þ xÞÞð1 sˆ h Þu ð1 uÞ1vx sˆ˙ h þ sh ¼ : ð17Þ ð1 bÞv Bv We guess a constant optimal sˆh , i.e., sˆ˙h = 0. Hence, (17) reduces to sh ¼
ð1 b þ wÞx þ wv ð1 sˆ Þu: ð1 bÞv
Solving (6) and (18), with s k = 0, we obtain w x wx þ ; sˆ h ¼ 1 b þ w v þ x ð1 b þ wÞðv þ xÞ which is constant as guessed, and 0 b sˆh b 1. The optimal subsidy is then w x wx þ sh ¼ u; 1 b þ w v þ x ð1 b þ wÞðv þ xÞ which satisfies 0 b s h b 1. Now, (5), (19) and (20a) yield w x wx þ ð1 uÞ; sh ¼ sˆ h sh ¼ 1 b þ w v þ x ð1 b þ wÞðv þ xÞ
ð18Þ
ð19Þ
ð20aÞ
ð20bÞ
which satisfies 0 b s h b 1. These results are consistent with the Pigouvian tax intuition that a subsidy directed at the source of a positive externality will correct the market failure. The following proposition summarizes these findings. Proposition 2. The optimal equilibrium can be decentralized if capital income is not taxed and investment in human capital is subsidized at a rate sh ¼
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ u:
The subsidy can be financed by taxing labor income at a rate sh ¼
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ ð 1 uÞ :
Lump-sum taxation is not required to balance the government budget. Comparing (7a) and (13a), decentralization of the optimal equilibrium requires k = k¯. Hence, c k = ck¯ which, using (7c) and (13c), yields s k = 0. Log-differentiating (7b) and (13b), recalling that a constant sˆh was guessed, decentralization also requires c l = cl¯ ; i.e., using (7d) and (13d), ðk=lÞð1 sˆ h þ sh =uÞð1 bÞAk b u1b hbþw ¼ k¯ =l¯ ð1 b þ wÞAk b u1b hbþw : ð21Þ
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Hence, the optimal policy must be set so that private and social marginal products of human capital in the goods sector, measured in terms of human capital as numeraire, coincide. This can be accomplished by setting s h according to (18), which, thus, corrects for the effects of two divergences: the difference between the private and the social marginal product of human capital in goods production because of the externality associated to average human capital, and the difference ¯ ¯ between the imputed real price of physical capital in the centralized economy, k / l , and that in the market economy, k / l, because of the externality associated to average learning time (see (7b) and (13b)). The government’s budget constraint (6), with s k = 0, provides the additional condition required to determine sˆh and s h . The optimal policy just derived is similar to that devised by Go´mez (2003), conveniently adapted to account for the externality of average learning time as well. Eqs. (20a) and (20b) comprise three terms. The first term, wu / (1 b + w) and w(1 u) / (1 b + w), respectively, depends on the relative size of the externality associated to average human capital, and fully corrects for this external effect in the absence of externalities in the educational sector (x = 0). This is the optimal policy derived by Go´mez (2003). The second term, xu / (v + x) and x(1 u) / (v + x), respectively, depends on the relative size of the externality associated to average learning time, and fully corrects for this external effect in absence of externalities in goods production (w = 0). The third term, which depends on the product of the relative size of both externalities, is nonzero only if both external effects coexist. As Go´mez (2003), let us suppose now that the government subsidizes the stock of human capital at a rate sˆ h . The agent’s budget constraint is then k˙ ¼ ð1 sk Þrk þ ð1 sˆ h Þwuh c þ sˆ h h:
ð22Þ
Handling the first-order conditions as in Section 2, it can be observed that the relationship sˆ h = s h w carries over all the calculations, and so, the dynamics of the market economy are also summarized by Eqs. (12a,b,c), after s h being substituted by sˆ h /w. Thus, from (19), (20a) and sˆ h = s h w, we can derive the following result. Proposition 3. The optimal equilibrium can be decentralized if capital income is not taxed and the stock of human capital is subsidized at a rate sˆ h ¼
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ ð 1 h bÞ y :
The subsidy can be financed by taxing labor income at a constant rate sˆ h ¼
w x wx : þ ð 1 b þ wÞ ð v þ xÞ 1bþw vþx
Lump-sum taxes are not required to balance the government budget. The government’s size, measured as the subsidy (or taxes) share of output, is constant and equals /¼
sˆ h h ¼ y
ð 1 bw þ w þ v þx x ð 1 b þwxwÞ ð v þ xÞ Þ ð 1 bÞ :
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5. Conclusions We devise a fiscal policy capable of decentralizing the optimal equilibrium in a Lucas-type model with externalities in goods production and in human capital accumulation. The optimal growth path can be decentralized by keeping capital income untaxed and instituting a subsidy to investment in education or to the stock of human capital, which can be financed by a labor income tax. Lump-sum taxation is not needed to balance the government budget either in the steady state or in the transitory phase.
Acknowledgements I thank an anonymous referee for valuable comments. Financial support from Spanish Ministry of Education and Science and FEDER is gratefully acknowledged.
Appendix A
Proof of Proposition 1. Linearizing Eqs.(14a,b,c) around the steady state (x*,q*,u*) yields 0 1 0 1 0 x˙ x x4 a11 @ q˙ A ¼ A@ q q4 A ¼ @ a21 a31 u˙ u u4
a12 a22 a32
10 1 x x4 a13 a23 A@ q q4 A; u u4 a33
where a11 ¼ ð1 bÞAx4b1 u41b ; a12 ¼ x4;
a13 ¼ ð1 bÞAx4b1 u4b þ Bð1 b þ wÞð1 u4Þvþx1 ðv þ xÞ=ð1 bÞ x4; a21 ¼ ð1 bÞðr bÞAx4b2 u41b q4=r; a22 ¼ q4; a23 ¼ ð1 bÞðr bÞAx4b1 u4b q4=r; a31 ¼ 0; a32 ¼ bð1 u4Þu4=ðbð1 u4Þ þ u4ð1 v xÞÞ; a33 ¼ Bð1 b þ wÞð1 u4Þvþx1 u4ðv þ xÞ=ð1 bÞ:
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The determinant of A is negative: detð AÞ ¼ Bð1 b þ wÞð1 u4Þvþx1 u4bAx4b1 u41b q4ðv þ xÞ
rð1 u4Þ þ u4ð1 v xÞ b0: rðbð1 u4Þ þ u4ð1 v xÞÞ
Using (15a) and (15b), we obtain a11 þ a22 ¼ q þ
Bð1 b þ wÞð1 u4Þvþx ðr 1Þ Bð1 b þ wÞð1 u4Þvþx1 u4ðv þ xÞ ¼ N0; 1b 1b
where the last equality follows from (15c). The trace of A is then positive: trð AÞ ¼ a11 þ a22 þ a33 ¼ 2Bð1 b þ wÞð1 u4Þvþx1 u4ðv þ xÞ=ð1 bÞN0: Hence, A must have one stable and two unstable eigenvalues.
5
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