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The subsidy to human capital accumulation in an endogenous growth model: A comparative dynamics analysis
JAIME ALONSO-CARRERA University of Vigo Vigo, Spain
The SubsMy to Human Capital Accumulation in an Endogenous Growth' Model: A Comparative Dynamics Analysis* This paper analyzes both the growth and the dynamic effects of the subsidy to human capital investment in a two-sector endogenous growth model. We show that the subsidy is growthincreasing° and it determines the dynamic behavior of the physical and human capital variables. Moreover, the economy reacts instantaneously to unanticipated changes in the subsidy rate. We prove that the jolt caused by the marginal introduction of the subsidy depends on whether the inverse of the elasticity ofintertemporal substitution in consumption is larger than the elasticity of marginal productivity of labor with respect to physical capital.
1. Introduction This paper analyzes the effects that a subsidy to human capital investment has on the development patterns of an economy represented by a twosector model of endogenous growth with physical and human capital accumulation, In this model, labor in efficiency units can be reproduced in an unbounded fashion through the accumulation of human capital. In the spirit of Rebelo and Stokey (1995) there is a market sector supplying hmnan capital competitively. Individuals buy this human capital so as to increase the efficiency of their labor. The growth models with endogenous accumulation of human and physical capital are the subjects of an important branch of the recent literature on growth theory. In this type of model, the growth rate of income per capita is endogenously determined by the interaction among the technologies that allow for the accumulation of physical and human capital, the preferences of individuals and government policy variables. Thus, fiscal policy can have not only level effects, but also growth effects. Several theoretical studies in this literature have examined the mechanism by which alternative *I am indebted to Jordi Caball6, Erik Canton, M. J. Freire-Ser6n, Howard Petith and three anonymous referees for their very useful discussions and comments. Of course, all errors that remain are entirely my own. Financial support from the Spanish Ministry of Education through DGICYT grant PB95-0130-C02-01 is gratefully acknowledged.
Journal of Macroeconomics, Summer 2000, Vol. 22, No. 3, pp. 409~31 Copyright © 2000 by Louisiana State University Press 0164-0704/2000/$1.50
409
Jaime Alonso-Carrera fiscal policies affect growth (see, e.g., Lucas 1990; King and Rebelo 1990; Rebelo 1991; Rebelo and Stokey 1995; Milesi-Ferretti and Roubini 1998). There is, however, one aspect of this literature that deserves further consideration. While these studies have basically reduced their analysis to the effects on the balanced-growth path equilibrium, they have not sufficiently focused on the effects on the transitional dynamics of this type of model. Besides the growth effects, fiscal policy can also determine the equilibrium path along which the economy converges to the balanced-growth path equilibrium. Hence, it seems reasonable to investigate how different fiscal policies affect the equilibrium dynamics of this class of models. Accordingly, Bond, Wang and Yip (1996) analyze the effects that factor taxes have on the stability properties of the model. Ortigueira (1998) characterizes the transitional dynamics of a model where human capital accumulation is a non-market activity in the presence of labor and capital income taxes. Nevertheless, these studies still do not analyze how changes in the fiscal policy parameters affect the dynamic behavior of the model. Mino (1996) makes the latter analysis for capital income taxation. However, since he cannot precisely compute the transitional dynamics of his model, his results on the dynamic effects are based on conjectures about these dynamics. Finally, Devereux and Love (1995) numerically study the dynamic effects of government spending shocks in a two-sector endogenous growth model with an endogenous labor supply. The central concern of this paper is to characterize the equilibrium dynamics analytically in the presence of a constant subsidy to human capital investment as well as the dynamic effects of permanent shocks in the subsidy rate. We extend Hall's (1971) comparative dynamics analysis from the neoclassical growth model. More precisely, we explicitly calculate the competitive equilibrium path followed by the economy in the presence of a constant subsidy. After doing so, we analyze the effect that an unanticipated and permanent change in the subsidy rate has on the previous equilibrium path. We perform a positive analysis without normative conclusions. In our model, the laissez faire equilibrium is Pareto optimal since there are neither externalities nor other assumptions that may violate the first welfare theorem. There are other studies that investigate the welfare implications of subsidy policy in calibrated versions of models with externalities (see, e.g., HSfert 1996). They base their numerical analysis on the comparison, in terms of utility, between the entire path of consumption in the laissez faire economy and in the subsidized economy. The presence of a subsidy to human capital investment does not alter the stability property of the model, although ,it does alter the behavior of physical and human capital along the transition path. In the absence of fiscal
410
Subsidy to Huraan Capital Accumulation in an Endogenous Growth Model policy, recent papers have found three parametric scenarios determining the transitional dynamics of this type of models (see, e.g., Caball6 and Santos 1993; Mulligan and Sala-i-Martin 1993; Chamley 1993 and Faig 1995). If the economy is endowed with a relatively scarce stock of human capital (i.e., a high ratio between physical and human capital), then the growth rate of human capital along the adjustment process is higher than, lower than or equal to its stationary value depending on structural parameters. Our analysis shows that the presence of the subsidy expands the region of the parameter space in which the first growth scenario occurs. Regarding the dynamic effects of changes in the subsidy rate, we show that the control variables adjust instantaneously to the new equilibrium path leading the economy to the new balanced-growth path equilibrium. Unanticipated shocks to the subsidy rate generate a substitution and an income effect in the intertemporal allocation of consumption, which are determined by the costs and benefits of investing an additional unit of income in acquiring new human capital. On one hand, the instantaneous reduction in the after-subsidy price of new human capital encourages individuals to increase their investments in human capital. The magnitude of this substitution effect is inversely related to the elasticity of intertemporal substitution since the increase in human capital investments will be more attractive if agents are more willing to intertemporally substitute consumption. On the other hand, since the abundance of new human capital will reduce the return on the total investments in human capital, the subsidy shock induces individuals to maintain their level of human capital investments and to allocate the increase in their disposable income to present consumption. This income effect is determined by the elasticity of marginal productivity of labor with respect to physical capital. The rule determining the overall effect of these two countervailing forces is then quite simple. We prove that the net effect depends on whether the inverse of the elasticity of intertemporal substitution in consumption is larger than the elasticity of marginal productivity of labor with respect to physical capital. The paper is organized as follows. In Section 2 we present the model and define the competitive equilibrium of the economy. Section 3 characterizes the dynamic behavior of all endogenous variables in the presence of the subsidy. The growth and dynamic effects of changes in the subsidy rate are analyzed in Section 4. Section 5 presents the conclusions of the paper.
2. The Model
We consider an infinite horizon, continuous time, endogenous growth model with physical and human capital accumulation. The economy consists
411
Jaime Alonso-Carrera
of competitive firms, a representative household and a government. The competitive equilibrium is achieved in a decentralized manner through perfect competition among firms and optimizing behavior of the household. Production In our economy there are two types of firms which produce physical goods, Y(t), and new human capital, E(t), respectively. The first type uses physical and human capital as inputs, whereas the second only uses human capital. The stock of physical capital K(t) hired from the household is then employed entirely in the physical goods sector. On the other hand, the entire stock of human capital h(t) is distributed competitively between both sectors of production. Thus, if u(t) denotes the fraction of human capital allocated to the physical goods sector, then the efficiency units of labor used in that sector are u(t)h(t). Physical goods are produced according to a constant returns to scale technology Y(t) = F(K(t), u(t)h(t)) = u(t)h(t)f(zl(t)), where zl(t) = K(t)/ (u(t)h(t)) is the ratio of physical capital to labor in efficiency units in the physical goods sector. The output per efficiency units of labor function, f(zt(t)), is assumed to satisfy the standard neoclassical properties f'(zl(t)) > 0 ,
(1)
f"(zl(t)) < O,
and the Inada conditions for each zl(t) > 0. Physical goods may be either used for consumption or for investment in physical capital. In the sector producing new human capital, we postulate a linear production function as in Uzawa (1965) and Lucas (1988). Then E(t) = 7(1 - u(t))h(t), where 7 > 0 is the scale factor. Let us denote the price of physical goods, price of new human capital, rental rate of physical capital and rental rate of human capital by pl(t), p2(t), R(t) and V(t), respectively. Since firms behave competitively, the value of the marginal productivity of each input is equal to its rental rate. Hence, if human capital is perfectly mobile across sectors, the profit maximization conditions are R(t) = pl(t)f'(zl(t)), V(t) = pl(t)(f(zl(t)) -f'(zl(t))zt(t))
= 7 Pz(t) .
(2)
Household The representative household owns the stocks of physical and human capital, which she supplies, inelastically, to firms. She derives utility from consuming C(t) units of physical goods at each moment in time. Preferences are represented by the discounted lifetime utility 412
Subsidy to Human Capital Accumulation in an Endogenous Growth Model (3)
where p, > 0 is the constant subjective rate of time preferences, and o > 0 denotes the. inverse of the constant elasticity of intertemporal substitution. In maximizing U, the household, faces the fl0w budget constraint
R(t)K(t) + V ( t ) h ( t ) -
T(t) = pl(t)C(t) + pt(t)Ii(t)
s)p2(t)I2(t),
+ (1 -
(4)
where Ii(t) is the. investment in physical capital, I2(t) is the investment in new human capital, T(t) is a lump-sum tax, and s is a time-invariant subsidy per unit of income invested in human capital. Income at time t, which is provided: by renting both capital stocks to firms, is divided between consumption and investments, in physical capital' and human capital. Thus, the laws ofmotiort for both capital stocks are then given by
6K(t),
(5)
I2(t) - rlh(t),
(6)
K(t)= Ii(t)-
l~(t) =
where ~i ___ 0 and t 1 -> 0, are the constant rates of depreciation of physical and human capital., respectively. The ratioua~ behavior of the representative household is then guided by the maximization~ of (3) subject to (4), (5), (6), K (0) = K0, h(0) = h0, and: the non-negative constraints on all variables. This problem is a dynamic optimization problem with control variables C(t), tl(t), and Is(t) and state variables K(t) and h(t). Hence, by standard procedure, we find the first-order conditions, and rearrange, the expressions to summarize the necessary conditions for optimality by the following two dynamic equations:
R(t) -
-
pl(t)
+
pa(t)
6
pl(t)
-
v(t) (1 - s)p2(t)
+
p2(t) -
-
pz(t)
c(t> F. c(t)
-
T~Lp~(t)
-
-
~,
(7)
] 6 -
p
.
(s)
Equation (7) is the non-arbitrage condilJon for the household's portfolio selection between physical and human capital investment at each moment in time. This condition states that the after-subsidy net rate of return plus the capital gain from both types of capital should be equal. On the other 413
Jaime Alonso-Carrera hand, Equation (8) establishes the optimal intertemporal allocation of consumption. The first-order conditions are also sufficient if the following transversality conditions hold: lim
e -~
21(t)K(t)
=
0
and
lim
e -~
22(t)h(t)
=
0 ,
(9)
where Ll(t) and ~,2(t) are the costate variables associated with K(t) and h(t), respectively.
The Government The government in this economy faces a balanced budget constraint at each moment in time, which is given by T(t) = s p2(t)[2(t) •
(10)
It is assumed that the subsidy rate s is kept constant over time, and only marginal variations in discrete moments of time can be introduced.
The Competitive Equilibrium After having studied the supply-side and the demand-side of the economy, we must analyze the market equilibrium conditions. The marketclearing conditions for the physical goods and for the new human capital are, respectively, Ii(t)
=
F(K(t), u(t)h(t)) - C(t),
(11)
I2(t)
--
7(1 - u(t))h(t).
(12)
Therefore, conditions (2), (5), (6), (7), (S), (9), (10), (11) and (12) form a system of equations that can be solved for the competitive equilibrium of the economy given the initial stocks of physical and human capital. Since the actual data show a positive investment in physical and human capital, we shall focus on cases in which the economy exhibits interior balanced-growth path (BGP, henceforth) equilibria, along which u(t) takes a constant value in the open interval (0,1), and the paths {C(t), Ii(t), I2(t), K(t), h(t), T(t), R(t), V(t), pl(t), p2(t)} grow at a constant and strictly positive rate. We can show that the properties of the interior BGP equilibria are invariant to the presence of the subsidy to human capital investment. First, physical capital, human capital and consumption still grow at the same rate, denoted by g*, and the relative price of physical goods in units of new human capital is constant along a BGP equilibrium_ We can also prove that the 414
Subsidy to H u m a n Capital A c c u m u l a t i o n in an Endogenous G r o w t h Model
uniqueness and local saddle stability of the interior BGP equilibria also hold with the presence of the subsidy. 1 For that purpose, we can just write a pseudo-social planning problem as is suggested by Jones and Manuelli (1990). 2 Thus, from their Theorem 2 we know that the solution of this modified problem coincides with our competitive equilibrium. Moreover, this problem is exactly the same as the original problem in Lucas' model (1988) without considering externalities. For the latter model Caball6 and Santos (1993) prove existence, uniqueness and local saddle-path stability of the BGP equilibria. Evidently, our conditions for the existence of an interior BGP equilibrium are slight modifications of theirs because of the presence of the subsidy. In particular, these conditions in our model transform into [7/(1 s)] - q - p > 0 and 7(1 - or) < (1 - s)[p + I1(1 - cr)]. However, the economic meaning of such conditions is identical to the one given by the aforementioned authors. The first of the previous conditions guarantees a strictly positive growth rate g*, whereas the second ensures that the transversality conditions (9) hold. Finally, we must note that the existence of the interior BGP equilibrium is compromised with the introduction of a positive subsidy for a given set of parameters. Effectively, while the first of the previous conditions are clearly more likely to be satisfied in presence of the subsidy, the opposite is true for the second condition. -
3. Equilibrium Dynamics in the Presence of a Subsidy The concern of this section is to show how the presence of a constant subsidy to human capital investment affects the behavior of the endogenous variables during the transition to the interior BGP equilibrium. This subsidy has no effects on the uniqueness and stability properties of the BGP equilibrium. However, it alters the dynamic behavior of the economy outside this BGP equilibrium. In particular, the presence of a constant subsidy affects the dynamic distribution of human capital between both sectors. 1In a model with physical capital as an input in the human capital sector, Bond, Wang and Yip (1996) prove that taxes on capital income in the sector producing physical capital and a subsidy to labor cost in the sector producing new human capital may generate either general instability or indeterminacy. As we assume that physical capital does not participate in producing new human capital, both instability and indeterminacy are not possible results in our model. eThis problem would consist of maximizing U subject to
K(t) = F(K(t), u(t)h(t)) - C(t) - ~n(t), /~(t) = "~(1 - u(t))h(t)
-
~lh(t) + qb(t),
where ¢5(t) is an endowment sequence and ~ = 7/(1 - s).
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Jaime Alonso-Carrera
Hence, the subsidy to human capital investment qualitatively determines the dynamic adjustment of the endogenous variables to the interior BGP equilibrium. Thus, the levels and the growth rate to which these variables will converge also depend on the values of the subsidy rate. To characterize the equilibrium dynamics outside [he interior BGP equilibrium, we first represent these dynamics through a complete dynamic system on variables that are constant alongthe BGP equilibrium. More precisely, we define the following three variables: z(t) = K(t)/h(t), x(t) = C(t)/ h(t) and co(t) = V(t)/R(t). The dynamic equations for z(t) and x(t) are easily derived from expressions defining the competitive equilibrium as
z(t)- zl(t~ ) z(t)~+ z(t)(/~)
)
y - 6 + ~1 - x(t),
(13)
(14) where u(t) was replaced by the :ratio z(t)/zl(t ). To get a dynamic equation for co(t), we make use ofthe procedure devised by Uzawa (1964):to show that :this rental ratio implicitly explains the ~evolutionof zl(t), u(t) and the relative :price of physical goods in units of new :human capital. Thus, from (2) we get o~(t) - f(zl(t)) f'(zl(t)) _
p(t)
px(t)
(15)
zi(t) ,
_
p2(t )
7 f(zl(t))
-
(16)
f'(z~(t))zl(t)
"
By assumptions (1), the variables Zl(t ) and p(t) are :uniquely determined by c0(t) and they will be denoted by Zl(co(:t))and p(co(t)), :respectively. Applying the implicit function theorem to (15) and (16), we :respectively get that z{(co(t)) > 0 and p'(co(t)) < 0. :Finally, from (t6) we get p(t)/p(t) = [p'(co(t))/p(co(t))]Co(t). Manipulating :this relationship with the equations defining the =competitive equilibrium, we get the following law of motion for co(t): Zl(°9(t)) ~(t)
-
#(t)zi(co(t))
[
y (~
-
f'(zl(co~t))) + 5] s)
-
~
(17)
-
where :~(t) is the .elasticity of marginal productivity ,of labor with respect to physical capital; [hat is, 416
Subsidy to H u m a n Capital Accumulation in an Endogenous Growth Model
f"(Zl(co(t)))(Zl(co(t)))2 fl(t) = -f(zt(eo(t))) - f'(zl(co(t)))z!(co(t))
> 0 "
(18)
The ordinary differential Equations (13), (14) and (17) constitute a complete dynamic system with respect to z(t), x(t) and c0(t), the first one being a state,like variable and the remaining two control-like variables. Then, since z(0) is the ratio between the initial stocks of physical and human capital, given these initial stocks the system composed of Equations (13), (14) and (17) form a complete description of the competitive equilibrium paths of the economy. The evolution of z(t), x(t) and co(t) in a local neighborhood of the BGP equilibrium can be approximated by the linearization of the dynamic system composed of Equations (13), (14) and (17). More precisely, since the BGP equilibrium is locally saddle,path stable, the previous approximation is given by the linear, unidimensional, stable manifold corresponding to the unique negative eigenvalue of the dynamic system. From the linearization of this system in Appendix A, we get that this stable manifold is given by z(t) .= z* + exp{a3~t}(z(O) ~ z * ) ,
(19a)
x(t) = x* + a2a(a3.--,a -- a11.).+ alaaZl (z(t) ~ z*) ,
(19b)
~713a33 - a~a co(t) = co* +
tZZl - aa3(all - aaa) . . . . (z(t) - z * ) , a13a33 -- a23
(19c)
where z*, x* and to* denote respectively the stationary values of z(t), x(t) and co(t), and the a~js are the coefficients of the linearized system, Given an initial condition z(0), Equations (19b) and (19c) approximate the unique pair of the initial values of x and co which lead the economy to the BGP equilibrium. ~ At this point, one can see that the subsidy to human capital investment alters the dynamic behavior of the economy outside the BGP equilibrium. In particular, the subsidy is relevant to determine the dynamic behavior of human capital qualitatively, and so plays a crucial role in determining the speed and the direction of convergence to the BGP equilibrium. One can see from the Appendix that the speed of convergence, which is given by 3Note that (19b) and (19c) are nothingbut the linear approximationof policyfunctionsofx and co,respectively.The coefficientsof (z(t) - z*) in these equationsare respectivelythe second and the third normalizedcomponentsof the eigenvectorassociatedto the negativeeigenvalue. A detailedderivationof the linearstablemanifold(19)is availablefromthe authoruponrequest. 417
Jaime Alonso-Carrera -a33, depends on the subsidy rate. However, the sign of this dependence is unknown under a general production function for the physical goods sector. If we instead assume a Cobb-Douglas technology, i.e., Y(t) = AK(t)~(u(t)h(t))l - 8, then we get that the speed of convergence is a strictly increasing and strictly convex function of the subsidy rate. More precisely, in this case the speed of convergence is given by - a ~ 3 = [(1 - 1~)/~][(7/(1 s)) + 5 - q]. One can numerically check that a positive subsidy shock drives speed of convergence up moderately, and only for quite large subsidy rates this speed rises more than proportionately. 4 The following results characterize the dynamic adjustment process, and so the dependence of the direction of convergence on the subsidy rate. The presence of the subsidy only affects the direction of convergence ofu(t). The dynamic behavior of all other variables does not qualitatively change with the presence of the subsidy.
PROPOSITION 1. Consider the economy described by Equations (2), (5), (6), (7), (8), (9), (I0), (11) and (12). (i) If z(O) < z*, then x(O) < x* and on(O) < co*. Hence, consumption and physical capital both grow more rapidly than human capital in the transition to the BGP equilibrium. (ii) If z(O) > z*, then x(O) > x* and co(O) > co*. Hence, consumption and physical capital both grow slower than human capital during the transition to the BGP equilibrium.
Proof Using Appendix A, one can show that the coefficients of (z(t) - z*) in (19b) and (19c) are both positive. From Proposition 1 we can establish the transitional behavior of zl, p and u.5 For that purpose, we introduce the notation or(t) to denote the elasticity of marginal productivity of physical capital with respect to labor in efficiency units at time t, i.e.,
a(t) = -f"(zl(co(t)))(zl(co(t))) f'(z,(co(t)))
> O.
(20)
4By assuming a Cobb-Douglas technology for the production of physical goods, Ortigueira and Santos (1997) find that the Uzawa-Lucas model implies a much higher speed of convergence than the standard neoclassical model. They get values around 0.2 for -a33. 5Bond, Wang and Yip (1996) reduce the dynamic analysis to the space (x,z) making use of the fact that the value function is homogeneous of degree 1 - cy. However, their analysis is not complete. With this method, we could not fully characterize the dynamic behavior of the economy. We could not analyze the instantaneous response of the rental ratio, relative prices or the work effort devoted in each sector to changes in the structural parameters, such as the rate of human capital subsidy.
418
Subsidy to Human Capital Accumulation in an Endogenous Growth Model CortoLLanY 1. Consider the economy described in Proposition 1. If z(O) < z* (z(O) > z*), then zl(O) < z~ (resp. zl(O) > z~) and p(O) > p* (resp.
p(O) < p*). Proof
It follows directly from Proposition 1 and Equations (15) and (16).
PROPOSITION 2. Consider the economy described in Proposition 1. Let fl* and a* be respectively the values at the BGP equilibrium of the elasticities fl(t) and a(t) defined in (18) and (20). Starting from z(O) < z* (z(O) > z*), the following statements hold: (i) u(O) > u* (resp. u(O) < u*) if and only if gJ > O; (ii) u(O) < u* (resp. u(O) > u*) if and only if gJ < O; and (iii) u(O) = u* if and only if ~P = O, where
'e-
o~*(~r~.~- #*) (1 -7 s)
+6-
s7 r/ + s( 1' --~ "
Proof. For the purpose of this proof we will compute the slope of the policy function ofu at the BGP equilibrium. Since u = Z/Zl, the derivative of u with respect to z is given by du
dz
zl(co) -
zz{(co)(dog/dz)
(zl(co)) 2
(21)
Taking dm/dz from (19), and using (20) and Appendix A, we compute the derivative (21) at the BGP equilibrium as follows:
~ZZ(z.x.,u.)
=
a l 3 ~ - a23)" Zl(CO*)(al3aa3
The result then follows from Appendix A. The three parts of Proposition 2, respectively, define the normal, paradoxical and exogenous growth eases that Caball6 and Santos (1993) point out to describe the evolution of the human capital variable in the ease of a non-market human capital sector. Assume that the initial capital ratio is lower than its stationary value. In the normal (paradoxical) growth ease the rate of human capital accumulation during the transition is lower (higher) than the one corresponding to the balanced growth path. These rates are equal when the economy belongs to 419
Jaime Atonso-Carrera the exogenous growth ease, In absence of the subsidy the conditions on parameter values that define each growth case in Proposition 2 are reduced to the ones stated by Cahall6 and Santos (1993, Theorem 5.!), though in our model the creation of new human capital is a market activity. The following corollary establishes the transitional behavior ofu in absence of the subsidy to human capital investment.6 COROLLARY 2. Consider the economy described in Proposition 1. Assume s = O. Starting from z(O) < z* (z(O) > z*), thefoUowing statements hold: (i) If fl* < a, then u(O) > u* (resp. u(O) < u*). (ii) If fl* > a, then u(O) < u* (resp. u(O) > u*). Oii) If fl* = a, then u(O) = u*.
Proof.
It follows from Proposition 2. by imposing s = 0.
Unlike the model without fiscal policy, in aur model preferences, technologies and depreciation rates, as welt as the subsidy rate, determine the growth case to which the economy belongs. In particular, the presence of human capital subsidy expands the region of parameter space in which the normal growth case occurs to the detriment of the regions containing the other growth cases, v More precisely, unlike the ease when s = 0, the conditions ~* > ~"and ~* = ~ are not su~eient in our model for-the paradoxical and exogenous cases to arise, respectively. Alternatively, the condition 13" < cr is not necessary for the economy to belong to the normal growth case when s > 0. Hence0 the presence of the subsidy makes the paradoxical and the exogenous growth cases empirically tess relevant, To show that, we will reinterpretate the conditions in Proposition 2 and Corollary 2. The elasticities a(t) and [~(t) defined in (20) and (t8) can be rewritten as a(t) = (1 O(t))/e(t) and 13(t). = tg(t)/e(t), where 0(t) and s~(t) are respectively the physical capital income's share on the product of the physical goods sector and the elasticity of substitution between capital and labor in the production of physical goods, i,e., -
O(t)
ff(Zl(O~(t)))(Zl(('O(t)')) f(z~(cO( (i )i)
f' (zl(eg(t) ) )[f(zl(o~(t) ) ) - f' (zt(a~(t)) )zl(co(t) ) ] ~(t) . . . . f(~(o)(t)))f'(zi(io(t)))Z~(o)(t)) .
.
.
.
6Mu/tigan and Salaq.Martin (1993), Faig (1995) and Barro and Sala.i-Martin (1995) have also shown this result in the case of C0bb-Douglas production technok~gy ~ad a non-market human capital sector. 7Ortigueira (1998) obtains the opposite result for physical capital taxation in the case where the creation of new human capital is a non-market activity.
420
Subsidy to Human Capital Accumulation in an Endogenous Growth Model Using the previous two equalities, we get cr - 13" = ¢y(1 - (0"/~*)). Since a plausible value for cr is higher than unity (see, e.g, Hansen and Singleton 1983), the previous expression has a positive value unless the elasticity of substitution in the product_ion of physical goods is sufficiently small. More precisely, the paradoxical growth case would only emerge for values of ~* much smaller than 0', in the presence of the subsidy, the previous restriction is even stronger since the condition ~* --- ~ is now not sufficient for the paradoxical ease. This fact makes it clear that the paradoxical and exogenous growth cases may not happen in reality, s
4, Effects of Changes in the Subsidy to Human Capital Investment The analytical characterization of the equilibrium dynamics permits us to analyze the impact that shifts in any structural parameter of the model have on economic growth. Specifically, in this Section "we describe the response of the economy to an unanticipated, permanent, marginal increase in the subsidy to human capital investment. For that purpose, we assume that the economy is initially on the balanced growth path without subsidy, and suddenly the government introduces a marginal iricrease in the subsidy rate from zero. 9 There are effects on the BGP equilibrium and on the equilibrium path converging to the BGP equilibrium.
Growth Vffeas We analyze the impact of a marginal introduction of the subsidy policy on the BGP equilibrium using comparative static arguments. PROPOSITION 3. Consider the economy described in Proposition I. Then, the steady,state effects of a marginal introduction of the subsidy to haman capital investment are: do~*
de*
dx*
dg*
--ds < 0 , - - ~ - < 0 , - - ~ - < 0 , ~
Proof
dp*
>0, ~
de*
>0, ~
du*
<0, ~
<0.
See Appendix B.
SFor instance, under the CES production function, and with empirically plausible values for parameters, the elasticity of substitution ~* is larger than the income capital share O* unless ~* is almost zero. 9Relaxingthis restriction complicates the explanation but changes nothing of the substance.
421
Jaime Alonso-Carrera A marginal increase in s has the effect of raising the after-subsidy return to investment in human capital, so that households are now willing to rent their stock of human capital at a smaller rental rate. Hence, the subsidy makes the human capital relatively inexpensive compared to physical capital. This reduction in the stationary level of the rental ratio induces the firms in both sectors to increase the use of human capital. To meet this increase in the demand for human capital, the creation of new human capital is stimulated and the accumulation of human capital is accelerated by reallocating labor in efficiency units to the sector producing new human capital. Consequently, the long-run rate of economic growth becomes higher in the new BGP equilibrium than in the o l d one. Moreover, the capital ratio and the ratio of consumption to human capital will be smaller in the new BGP equilibrium as a result of the increase in the production of new human capital.
Dynamic Effects We approximate the equilibrium path followed by an economy that is perturbed by an unanticipated, permanent, marginal introduction of the subsidy to human capital investment at some time. In Section 3 we characterized the equilibrium path of an economy with a constant and positive subsidy. At this point, we analyze the transition from the before-subsidy equilibrium path to the after-subsidy one. When the subsidy policy is introduced the state variable z is at its old stationary value z*. The economy is then outside both the new BGP equilibrium and the new stable manifold. Hence, the control variables co and x must jump instantaneously to place the economy on the new stable manifold. Otherwise, local convergence to the new BGP equilibrium would not be possible, and the transversality conditions (9) would be locally violated. After this jump, the economy follows the transitional adjustment along the new stable manifold, as described in Section 3. Therefore, the equilibrium path appears to be a smooth function of time, except at the moment at which the subsidy is introduced. In this sub-section we compute and analyze the instantaneous jumps of control variables, and so the point of the new stable manifold reached by the economy after such jumps. We will follow several steps. First, we calculate the point of the old stable manifold corresponding to the moment at which the subsidy is introduced, say t = 0. From (19) we get co(o) =
o9* -
azl
-
a33(an
a13a33
--
-
a33)
(z(0)
-
z*),
x(0) = x* -- a23(a33 -- an) + alaa21 (z(0) -- z*). a13a33
422
--
(22a)
a23
a23
(22b)
Subsidy to Human Capital Accumulation in an Endogenous Growth Model Moreover, since the introduction of the subsidy affects the coefficients ay and the stationary values of z, x and co, we must compute the instantaneous effects of the subsidy on the point given by (22). For that purpose, we differentiate (22) with respect to the subsidy rate. Finally, we evaluate these derivatives at z(0) equal to z*, so that the instantaneous jumps of c0 and x at t = 0 are, respectively1° dee(0) ds
do)* ds
a~l - a33(an - a33)dz* a13a33 - a23 ds '
dx(O)
dx*
a23(a33
ds
ds
-
all)
al~a33
-
+ a23
(23a)
alaazl dz* ds
(23b) "
Since dco*/ds and dz*/ds are negative, and the coefficients of dz*/ds on the right-hand side of (23a) and (23b) are both positive according to Appendix A, it follows that do~(O)/ds > do3*/ds and dx(O)/ds > dx*/ds. Hence, after the instantaneous jumps, the control variables co and x fall monotonically to their new stationary values. However, without any other manipulation, we cannot say anything about the sign or direction of those jumps. The following result establishes the direction of these instantaneous jumps. PROPOSITION 4. Consider the economy described in Proposition 1. After a marginal introduction of a subsidy to human capital investment, the following occurs: (i) The rental ratio jumps up instantaneously. (ii)The consumption to human capital ratio jumps up instantaneously when if* > a, whereas if fl* < a then that ratio jumps down instantaneously.
Proof.
See Appendix C_
Therefore, the time paths of co and x after a marginal increase in s from zero are completely defined by Propositions 2 and 5. They are shown in Figure 1. To complete the comparative dynamic analysis, we describe the time path of p, zl and u after a marginal introduction of the subsidy. PROPOSITION 5. Consider the economy described by Proposition 1. After a 1°Notethat the point of the newstable manifoldreachedby the economyafterthe jolt caused by the marginalintroductionof the subsidyis givenby z(0) = z*, x(O) = x* + (dx(O)/ds)and o3(0) = o3* + (do3(O)/ds),where (z*, x*, o3")is the old steady-state. 423
Jaime Alonso-Carrera (o
x* co*
(o)*)!
(X*)'
t
t-'
t
Figure i. Time paths of (o and x perturbed by a marginal introduction of the subsidy.
marginal introduction of the subsidy to human capital investments.. (i) z~ jumps up instantaneously; and (ii) p and u jump down instantaneously.
Proof. It follows directly from Proposition 4 by noting that zi(O) is an increasing function of re(O), p(O) is a decreasing function of re(O), and u(O) = z(O)/zl
(0),
The analysis of the time paths of zl, p and u after a marginal increase in the subsidy rate from zero is trivial from Proposition 5 and Corollaries 1 and 2. With regard to u, we must note that this variable always jumps down simultaneously with the marginal introduction of s. However, this jump is larger (smaller) than the variation in its stationary value when ~* < s (resp. 13" > ~). Figure 2 illustrates these features of the time path of u after a change in s. A marginal rise in s reduces the after-subsidy price of new human capital, which generates both a substitution and an income effect. On one hand, the subsidy induces households to substitute human eapitai investments for consumption and physical capital investments. This expansion in the demand of new human capital reduces the relative prices and encourages the production of new human capital. Since the sector producing human capital only uses effective labor as an input, the ratio (0 then increases instantaneously and labor is reallocated to this sector. On the other hand, a marginal rise in s increases households' disposable income, and so they can also expand their present consumption. Hence, the increase in human capital investments and the effect of the subsidy on consumption depend on which of these effects dominates. The net effect is determined by the relative costs 424
Subsidy to Human Capital Accumulation in an Endogenous Growth Model U
(u*y
~* <(3 t
Figure 2. Time paths of u perturbed by a marginal introduction of the subsidy.
and benefits of investing an additional unit of income in new human capital. The benefit of this marginal investment is given by the induced increase in future consumption. Thus, the benefit depends positively on ~. The costs of a marginal investment in new human capital are twofold. First, this investment reduces the present consumption. Second, the resulting increase in the stock of human capital will reduce the marginal productivity of labor in the physical goods sector, and so the return on total investments in human capital. While the former cost is inversely related to r~, the latter depends positively on 13". Therefore, the income effect dominates when 13" > ry, whereas if 13" < cr then the substitution effect dominates. The instantaneous effect of s on consumption then depends on the relationship between these two elasticities. Alternatively, the positive effect of the subsidy on human capital investment, and so the realloeation of labor to the human capital sector, is inversely related to 13" and directly to ~.
5. Conclusion This paper has analyzed the growth and dynamic effects of the subsidy to human capital investment in a two-sector model of endogenous growth with physical and human capital accumulation. The presence of the subsidy does not alter the uniqueness and the stability properties of the BGP equilibrium. However, the incidence analysis has shown that human capital subsidy is growth-increasing. Moreover, the presence of the subsidy qualitatively determines the dynamic behavior of the human capital, and so the speed and direction of convergence to the BGP equilibrium. Regarding the dynamic effects of a marginal introduction of the subsidy, we have found that the economy reacts instantaneously. As a result, we 425
Jaime Alonso-Carrera obtain that consumption, the ratio between the rental rates of both types of capital, relative prices, and the fraction of human capital allocated to each sector all jump instantaneously when the subsidy is introduced. The direction of these jumps depends on the relationship between the elasticity of marginal productivity of labor with respect to physical capital and the inverse of the elasticity of intertemporal substitution in consumption. In any case, the subsidy always generates a strictly positive welfare cost since in our model the first welfare theorem holds. This welfare cost depends on the growth scenario to which the economy belongs. Thus, given the response of the consumption to changes in the subsidy rate, the welfare cost would be larger in the paradoxical growth case than in the normal one. A first extension of the paper could then be to use our analysis to evaluate the welfare cost of subsidizing the accumulation of human capital given an externality from the stock of human capital as in Lucas (1988). In this paper we have maintained the assumption of an exogenous labor supply. We conclude with a brief comment on how allowing for an endogenous labor-leisure choice may alter the effects of the subsidy to human capital investment. Milesi-Ferreti and Roubini (1998) emphasize that the growth effects of the subsidy in this type of models depend on the specification of the leisure activity. If leisure is modeled as raw time, then the growth effects of the subsidy are equal to the ones obtained in this paper, whereas if leisure is defined as quality time, the growth effects of the subsidy are larger. Regarding the initial impacts of changes in the subsidy rate one must expect that they are qualitatively determined by the same parameters as in this paper but also by the elasticity of substitution between consumption and leisure and the elasticity of leisure with respect to human capital, since both of these elasticities drive the instantaneous response of the labor supply. Future research should confirm this conjecture. Received: July 1998 Final version: October 1999
References Barro, Robert j., and Xavier Sala-i-Martin. Economic Growth, New York: McGraw Hill Inc., 1995. Bond, Eric W., Ping Wang, and Chong K. Yip. "A General Two Sector Model of Endogenous Growth with Human Capital and Physical Capital."Jourhal of Economic Theory 68, no. 1 (1996): 149-73. Caballr, Jordi, and Manuel S. Santos. "On Endogenous Growth with Physical
426
Subsidy to Human Capital Accumulation in an Endogenous Growth Model and Human Capital." Journal of Political Economy 101, no. 6 (1993): 1042-68. Chamley, Christophe. "Externalities and Dynamics in Models of 'Learning or Doing'." International Economic Review 34, no. 3 (1993): 583-609. Deveretlx, Michael B., and David R. F. Love. "The Dynamics Effects of Government Spending Policies in a Two-Sector Growth Model." Journal of Money, Credit, and Banking 27, no. 1 (1995): 232-56. Faig, Miguel. "A Simple Economy with Human Capital: Transitional Dynamics, Technology Shocks and Fiscal Policies." Journal of Macroeconomics 17, no. 3 (1995): 421-46. Hall, Robert E. "The Dynamic Effects of Fiscal Policy in an Economy with Foresight.'" Review of Economic Studies 38 (1971): 229-44, Hansen, Lars P., and Kenneth J. Singleton. "Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns." Journal of Political Economy 91, no. 2 (1983): 249-65. Htfert, Andreas. "Subsidizing Human Capital. Is there a Conflict between Growth and Welfare?" Swiss Institute for Business Cycles Research (KOF-ETHZ), 1996. Mimeo. Jones, Larry E., and Rodolfo Manuelli. "'A Convex Model of Equilibrium Growth: Theory and Policy Implications." Journal of Political Economy 98, no. 5 (1990): 1008-38. King, Robert G., and Sergio Rebelo. "Public Policy and Economic Growth: Developing Neoclassical Implications.'" Journal of Political Economy 98, no. 5 (1990): S126-S150. Lucas, Robert E., Jr. "On Mechanics of Economics Development." Journal of Monetary Economics 22, no. 1 (1988): 3-42. • "Supply-Side Economics: An Analytical Review." Oxford Economic Papers 42 (1990): 293-316. Milesi-Ferretti, Gian Maria, and Nouriel Roubini. "On the Taxation of Human and Physical Capital in Models of Endogenous Growth." Journal of Public Economics 70, no. 2 (1998): 237-54. Mino, Kazuo. "Analysis of a Two-Sector Model of Endogenous Growth with Capital Income Taxation." International Economic Review 37, no. 1 (1996): 227-53. Mulligan, Casey B., and Xavier Sala-i-Martin. "Transitional Dynamics in Two-Sector Models of Endogenous Growth." Quarterly Journal of Economics 108, no. 3 (1993): 739-73. Ortigueira, Salvador. "Fiscal Policy in an Endogenous Growth Model with Human Capital Accumulation." Journal of Monetary Economics 42, no. 2 (1998): 323-55. Ortigueira, Salvador, and Manuel Santos. "On Speed of Convergence in
427
Jaime Alonso-Carrera Endogenous Growth Models." American Economic Review 87, no. 3 (1997): 383-99. Rebelo, Sergio. "Long Run Policy Analysis and Long Run Growth." Journal of Political Economy 99, no. 3 (1991): 500-21. Rebelo, Sergio, and Nancy L. Stokey. "Growth Effects of Flat-Rate Taxes." Journal of Political Economy 103, no. 3 (1995): 519-50. Uzawa, Hirofumi. "Optimal Growth in a Two-Sector Model of Capital Accumulation." Review of Economic Studies 31 (1964): 1-24. • "Optimal Technical Changes in an Aggregate Model of Economic Growth." International Economic Review 6 (1965): 18-31.
Appendix Linearization of the Dynamic System For the purpose of this appendix, we will first characterize the BGP equilibrium. From the equations defining the competitive equilibrium, we get f'(zl(c°*)) = [
'
+
6
-
u . = [7[,~(a - s) -
(1 - s) z*
=
X•
--
r/
] ,
(A1)
lJ - , ~ ( 1 a7(1
s)(,~ - J-) + p(a - ~)J -
r/ - p ,
(A3)
z{co*)u*
zl(c~*)
2;1(60"))
(~)
s)
(A4) - - ZI((d) 'g)
(A5)
(.7- 17(1- s) - p ( 1 - s) + a6(1 - s))] a(1 - s)
Linearizing the dynamic system around the steady state (z*, x*, o~*), we have
l x(t)/=[ol 00 o23//x(t) x*, ¢(t)] a,~_l Leo(t) co* -,
l r (t) -
where the elements a 0 of the coefficient matrix are given by
428
Subsidy to Human Capital Accumulation in an Endogenous Growth Model
(a~(t)/*
= 27u* +
(ae(t) I* -
ai3 = \ 0 - ~ /
/e~(~o*))
(? + 8 - q)
f(zl(oo*)) -- f'(zl(cO*))Zl(O9*)
ZI((.O* )
sT > 0 , (1 -- 8)
(A6)
+ - -
zi(co)*(y), ,,3 z;(co*)z* ~ z ) -[- (Z,l((.o,))------------------~ [f'(zl((-D*))zl((.o :~) - f(zl((-D*))]
-~i(~o*)~* Fh,u , + f(z~(o~*)) ~_ rcc]/,a,~o*))z/co*)l < o z~(o~*) J ' 1_
=
~1
2y~*
\oz(t)/
z~7g-*)
(~'3C(t) /*
X*I[;' (z 1( ('0g ) )Z~-<('0* )
(A7)
(A8)
z~(oo*) + (1 - ~-----~¢~* > o, Zl(O)*)'YZ=gl
(Ag)
(a6)(t)l* _ Zl(°J*) f,,(z~(~o,)) = aaa = \&o(t)/ fl*
f(zl(co*)) - f'(zl(co*))zl(oo*) <0. zl(co*)
(A10)
The unique negative eigenvalue of the linearized system is equal to aaa. This negative eigenvalue defines a unidimensional stable manifold of the BGP equilibrium.
Comparative Static Analysis of the BGP Equilibrium This appendix evaluates the impact of a marginal introduction of the subsidy on the BGP equilibrium. First, from Appendix A, we get
dco* = y < 0 ds s=o f"(zl(co*))z~(co*)
ds ~=o
\ ds I,=0/
(B1)
~r~
- ---7- < 0 ,
(B2)
429
Jaime Alonso,Carrera
+ [(~ - ~)(z - 1) + p]
[
(d°9*l ~[ (-7 - r/)(~r1) + P-] d
x z~(o~*) \-'~-'1,=0)
z1(co*)7]<0
-
'
(B3)
Using (A2) and (18), we can rewrite the previous effects in a way that will be more useful in the next appendix. More precisely, we get
ds s=o
dx:t:
\ ds Is=o]
(dz$l
~
(B2')
'
/ r((Zl((°:t:))L - f'(gl(°)*))Zl((-/)*)
ds ~=o = \ ds 1~=o¢
] + 7u*
~1(-~~
~)(o. +
fl)z •
~/~
(B3') Finally, from (A2), (A3), (15) and (16) the following is also true
dg*
i
--<0, ds ~=0
" ~-~=o
7
= ->0 a
<0 'ds2 ~=o
dp*
>0.
' ds~ ~=o
The Instantaneous Impact of the Subsidy on the Control Variables Consider first the instantaneous impact of a marginal introduction of s on 09. By substituting (B1) and (B2') for dco*/ds and dz*/ds, respectively, and using (A6)-(A10) and the definition of [3", we transform (23a) into the following expression: dog(0) ds
a13(7o'z* - 2fl*a33) ~r1~*zt1(o)*)(a13a33
--
a23)'
which, from Appendix A, is always positive. On the other hand, using the previous procedure with (23b), we also get the initial impact of s on x. Thus, substituting (B2') and (B3') for dz*/ ds and dx*/ds, respectively, and using (A6)-(A10) and the definition of 13", we get that the initial impact of s on x is
430
Subsidy to Human Capital Accumulation in an Endogenous Growth Model dx(O) ds
(or - fl*)al3(a3a) 2 zl(co* ) ~(al3aaa - a2a)
Therefore, we conclude from Appendix A that dx(O)/ds2 is positive when 13' > or, whereas if 13' < ¢y then it is negative.
431
The SubsMy to Human Capital Accumulation in an Endogenous Growth' Model: A Comparative Dynamics Analysis* This paper analyzes both the growth and the dynamic effects of the subsidy to human capital investment in a two-sector endogenous growth model. We show that the subsidy is growthincreasing° and it determines the dynamic behavior of the physical and human capital variables. Moreover, the economy reacts instantaneously to unanticipated changes in the subsidy rate. We prove that the jolt caused by the marginal introduction of the subsidy depends on whether the inverse of the elasticity ofintertemporal substitution in consumption is larger than the elasticity of marginal productivity of labor with respect to physical capital.
1. Introduction This paper analyzes the effects that a subsidy to human capital investment has on the development patterns of an economy represented by a twosector model of endogenous growth with physical and human capital accumulation, In this model, labor in efficiency units can be reproduced in an unbounded fashion through the accumulation of human capital. In the spirit of Rebelo and Stokey (1995) there is a market sector supplying hmnan capital competitively. Individuals buy this human capital so as to increase the efficiency of their labor. The growth models with endogenous accumulation of human and physical capital are the subjects of an important branch of the recent literature on growth theory. In this type of model, the growth rate of income per capita is endogenously determined by the interaction among the technologies that allow for the accumulation of physical and human capital, the preferences of individuals and government policy variables. Thus, fiscal policy can have not only level effects, but also growth effects. Several theoretical studies in this literature have examined the mechanism by which alternative *I am indebted to Jordi Caball6, Erik Canton, M. J. Freire-Ser6n, Howard Petith and three anonymous referees for their very useful discussions and comments. Of course, all errors that remain are entirely my own. Financial support from the Spanish Ministry of Education through DGICYT grant PB95-0130-C02-01 is gratefully acknowledged.
Journal of Macroeconomics, Summer 2000, Vol. 22, No. 3, pp. 409~31 Copyright © 2000 by Louisiana State University Press 0164-0704/2000/$1.50
409
Jaime Alonso-Carrera fiscal policies affect growth (see, e.g., Lucas 1990; King and Rebelo 1990; Rebelo 1991; Rebelo and Stokey 1995; Milesi-Ferretti and Roubini 1998). There is, however, one aspect of this literature that deserves further consideration. While these studies have basically reduced their analysis to the effects on the balanced-growth path equilibrium, they have not sufficiently focused on the effects on the transitional dynamics of this type of model. Besides the growth effects, fiscal policy can also determine the equilibrium path along which the economy converges to the balanced-growth path equilibrium. Hence, it seems reasonable to investigate how different fiscal policies affect the equilibrium dynamics of this class of models. Accordingly, Bond, Wang and Yip (1996) analyze the effects that factor taxes have on the stability properties of the model. Ortigueira (1998) characterizes the transitional dynamics of a model where human capital accumulation is a non-market activity in the presence of labor and capital income taxes. Nevertheless, these studies still do not analyze how changes in the fiscal policy parameters affect the dynamic behavior of the model. Mino (1996) makes the latter analysis for capital income taxation. However, since he cannot precisely compute the transitional dynamics of his model, his results on the dynamic effects are based on conjectures about these dynamics. Finally, Devereux and Love (1995) numerically study the dynamic effects of government spending shocks in a two-sector endogenous growth model with an endogenous labor supply. The central concern of this paper is to characterize the equilibrium dynamics analytically in the presence of a constant subsidy to human capital investment as well as the dynamic effects of permanent shocks in the subsidy rate. We extend Hall's (1971) comparative dynamics analysis from the neoclassical growth model. More precisely, we explicitly calculate the competitive equilibrium path followed by the economy in the presence of a constant subsidy. After doing so, we analyze the effect that an unanticipated and permanent change in the subsidy rate has on the previous equilibrium path. We perform a positive analysis without normative conclusions. In our model, the laissez faire equilibrium is Pareto optimal since there are neither externalities nor other assumptions that may violate the first welfare theorem. There are other studies that investigate the welfare implications of subsidy policy in calibrated versions of models with externalities (see, e.g., HSfert 1996). They base their numerical analysis on the comparison, in terms of utility, between the entire path of consumption in the laissez faire economy and in the subsidized economy. The presence of a subsidy to human capital investment does not alter the stability property of the model, although ,it does alter the behavior of physical and human capital along the transition path. In the absence of fiscal
410
Subsidy to Huraan Capital Accumulation in an Endogenous Growth Model policy, recent papers have found three parametric scenarios determining the transitional dynamics of this type of models (see, e.g., Caball6 and Santos 1993; Mulligan and Sala-i-Martin 1993; Chamley 1993 and Faig 1995). If the economy is endowed with a relatively scarce stock of human capital (i.e., a high ratio between physical and human capital), then the growth rate of human capital along the adjustment process is higher than, lower than or equal to its stationary value depending on structural parameters. Our analysis shows that the presence of the subsidy expands the region of the parameter space in which the first growth scenario occurs. Regarding the dynamic effects of changes in the subsidy rate, we show that the control variables adjust instantaneously to the new equilibrium path leading the economy to the new balanced-growth path equilibrium. Unanticipated shocks to the subsidy rate generate a substitution and an income effect in the intertemporal allocation of consumption, which are determined by the costs and benefits of investing an additional unit of income in acquiring new human capital. On one hand, the instantaneous reduction in the after-subsidy price of new human capital encourages individuals to increase their investments in human capital. The magnitude of this substitution effect is inversely related to the elasticity of intertemporal substitution since the increase in human capital investments will be more attractive if agents are more willing to intertemporally substitute consumption. On the other hand, since the abundance of new human capital will reduce the return on the total investments in human capital, the subsidy shock induces individuals to maintain their level of human capital investments and to allocate the increase in their disposable income to present consumption. This income effect is determined by the elasticity of marginal productivity of labor with respect to physical capital. The rule determining the overall effect of these two countervailing forces is then quite simple. We prove that the net effect depends on whether the inverse of the elasticity of intertemporal substitution in consumption is larger than the elasticity of marginal productivity of labor with respect to physical capital. The paper is organized as follows. In Section 2 we present the model and define the competitive equilibrium of the economy. Section 3 characterizes the dynamic behavior of all endogenous variables in the presence of the subsidy. The growth and dynamic effects of changes in the subsidy rate are analyzed in Section 4. Section 5 presents the conclusions of the paper.
2. The Model
We consider an infinite horizon, continuous time, endogenous growth model with physical and human capital accumulation. The economy consists
411
Jaime Alonso-Carrera
of competitive firms, a representative household and a government. The competitive equilibrium is achieved in a decentralized manner through perfect competition among firms and optimizing behavior of the household. Production In our economy there are two types of firms which produce physical goods, Y(t), and new human capital, E(t), respectively. The first type uses physical and human capital as inputs, whereas the second only uses human capital. The stock of physical capital K(t) hired from the household is then employed entirely in the physical goods sector. On the other hand, the entire stock of human capital h(t) is distributed competitively between both sectors of production. Thus, if u(t) denotes the fraction of human capital allocated to the physical goods sector, then the efficiency units of labor used in that sector are u(t)h(t). Physical goods are produced according to a constant returns to scale technology Y(t) = F(K(t), u(t)h(t)) = u(t)h(t)f(zl(t)), where zl(t) = K(t)/ (u(t)h(t)) is the ratio of physical capital to labor in efficiency units in the physical goods sector. The output per efficiency units of labor function, f(zt(t)), is assumed to satisfy the standard neoclassical properties f'(zl(t)) > 0 ,
(1)
f"(zl(t)) < O,
and the Inada conditions for each zl(t) > 0. Physical goods may be either used for consumption or for investment in physical capital. In the sector producing new human capital, we postulate a linear production function as in Uzawa (1965) and Lucas (1988). Then E(t) = 7(1 - u(t))h(t), where 7 > 0 is the scale factor. Let us denote the price of physical goods, price of new human capital, rental rate of physical capital and rental rate of human capital by pl(t), p2(t), R(t) and V(t), respectively. Since firms behave competitively, the value of the marginal productivity of each input is equal to its rental rate. Hence, if human capital is perfectly mobile across sectors, the profit maximization conditions are R(t) = pl(t)f'(zl(t)), V(t) = pl(t)(f(zl(t)) -f'(zl(t))zt(t))
= 7 Pz(t) .
(2)
Household The representative household owns the stocks of physical and human capital, which she supplies, inelastically, to firms. She derives utility from consuming C(t) units of physical goods at each moment in time. Preferences are represented by the discounted lifetime utility 412
Subsidy to Human Capital Accumulation in an Endogenous Growth Model (3)
where p, > 0 is the constant subjective rate of time preferences, and o > 0 denotes the. inverse of the constant elasticity of intertemporal substitution. In maximizing U, the household, faces the fl0w budget constraint
R(t)K(t) + V ( t ) h ( t ) -
T(t) = pl(t)C(t) + pt(t)Ii(t)
s)p2(t)I2(t),
+ (1 -
(4)
where Ii(t) is the. investment in physical capital, I2(t) is the investment in new human capital, T(t) is a lump-sum tax, and s is a time-invariant subsidy per unit of income invested in human capital. Income at time t, which is provided: by renting both capital stocks to firms, is divided between consumption and investments, in physical capital' and human capital. Thus, the laws ofmotiort for both capital stocks are then given by
6K(t),
(5)
I2(t) - rlh(t),
(6)
K(t)= Ii(t)-
l~(t) =
where ~i ___ 0 and t 1 -> 0, are the constant rates of depreciation of physical and human capital., respectively. The ratioua~ behavior of the representative household is then guided by the maximization~ of (3) subject to (4), (5), (6), K (0) = K0, h(0) = h0, and: the non-negative constraints on all variables. This problem is a dynamic optimization problem with control variables C(t), tl(t), and Is(t) and state variables K(t) and h(t). Hence, by standard procedure, we find the first-order conditions, and rearrange, the expressions to summarize the necessary conditions for optimality by the following two dynamic equations:
R(t) -
-
pl(t)
+
pa(t)
6
pl(t)
-
v(t) (1 - s)p2(t)
+
p2(t) -
-
pz(t)
c(t> F.
-
T~Lp~(t)
-
-
~,
(7)
] 6 -
p
.
(s)
Equation (7) is the non-arbitrage condilJon for the household's portfolio selection between physical and human capital investment at each moment in time. This condition states that the after-subsidy net rate of return plus the capital gain from both types of capital should be equal. On the other 413
Jaime Alonso-Carrera hand, Equation (8) establishes the optimal intertemporal allocation of consumption. The first-order conditions are also sufficient if the following transversality conditions hold: lim
e -~
21(t)K(t)
=
0
and
lim
e -~
22(t)h(t)
=
0 ,
(9)
where Ll(t) and ~,2(t) are the costate variables associated with K(t) and h(t), respectively.
The Government The government in this economy faces a balanced budget constraint at each moment in time, which is given by T(t) = s p2(t)[2(t) •
(10)
It is assumed that the subsidy rate s is kept constant over time, and only marginal variations in discrete moments of time can be introduced.
The Competitive Equilibrium After having studied the supply-side and the demand-side of the economy, we must analyze the market equilibrium conditions. The marketclearing conditions for the physical goods and for the new human capital are, respectively, Ii(t)
=
F(K(t), u(t)h(t)) - C(t),
(11)
I2(t)
--
7(1 - u(t))h(t).
(12)
Therefore, conditions (2), (5), (6), (7), (S), (9), (10), (11) and (12) form a system of equations that can be solved for the competitive equilibrium of the economy given the initial stocks of physical and human capital. Since the actual data show a positive investment in physical and human capital, we shall focus on cases in which the economy exhibits interior balanced-growth path (BGP, henceforth) equilibria, along which u(t) takes a constant value in the open interval (0,1), and the paths {C(t), Ii(t), I2(t), K(t), h(t), T(t), R(t), V(t), pl(t), p2(t)} grow at a constant and strictly positive rate. We can show that the properties of the interior BGP equilibria are invariant to the presence of the subsidy to human capital investment. First, physical capital, human capital and consumption still grow at the same rate, denoted by g*, and the relative price of physical goods in units of new human capital is constant along a BGP equilibrium_ We can also prove that the 414
Subsidy to H u m a n Capital A c c u m u l a t i o n in an Endogenous G r o w t h Model
uniqueness and local saddle stability of the interior BGP equilibria also hold with the presence of the subsidy. 1 For that purpose, we can just write a pseudo-social planning problem as is suggested by Jones and Manuelli (1990). 2 Thus, from their Theorem 2 we know that the solution of this modified problem coincides with our competitive equilibrium. Moreover, this problem is exactly the same as the original problem in Lucas' model (1988) without considering externalities. For the latter model Caball6 and Santos (1993) prove existence, uniqueness and local saddle-path stability of the BGP equilibria. Evidently, our conditions for the existence of an interior BGP equilibrium are slight modifications of theirs because of the presence of the subsidy. In particular, these conditions in our model transform into [7/(1 s)] - q - p > 0 and 7(1 - or) < (1 - s)[p + I1(1 - cr)]. However, the economic meaning of such conditions is identical to the one given by the aforementioned authors. The first of the previous conditions guarantees a strictly positive growth rate g*, whereas the second ensures that the transversality conditions (9) hold. Finally, we must note that the existence of the interior BGP equilibrium is compromised with the introduction of a positive subsidy for a given set of parameters. Effectively, while the first of the previous conditions are clearly more likely to be satisfied in presence of the subsidy, the opposite is true for the second condition. -
3. Equilibrium Dynamics in the Presence of a Subsidy The concern of this section is to show how the presence of a constant subsidy to human capital investment affects the behavior of the endogenous variables during the transition to the interior BGP equilibrium. This subsidy has no effects on the uniqueness and stability properties of the BGP equilibrium. However, it alters the dynamic behavior of the economy outside this BGP equilibrium. In particular, the presence of a constant subsidy affects the dynamic distribution of human capital between both sectors. 1In a model with physical capital as an input in the human capital sector, Bond, Wang and Yip (1996) prove that taxes on capital income in the sector producing physical capital and a subsidy to labor cost in the sector producing new human capital may generate either general instability or indeterminacy. As we assume that physical capital does not participate in producing new human capital, both instability and indeterminacy are not possible results in our model. eThis problem would consist of maximizing U subject to
K(t) = F(K(t), u(t)h(t)) - C(t) - ~n(t), /~(t) = "~(1 - u(t))h(t)
-
~lh(t) + qb(t),
where ¢5(t) is an endowment sequence and ~ = 7/(1 - s).
415
Jaime Alonso-Carrera
Hence, the subsidy to human capital investment qualitatively determines the dynamic adjustment of the endogenous variables to the interior BGP equilibrium. Thus, the levels and the growth rate to which these variables will converge also depend on the values of the subsidy rate. To characterize the equilibrium dynamics outside [he interior BGP equilibrium, we first represent these dynamics through a complete dynamic system on variables that are constant alongthe BGP equilibrium. More precisely, we define the following three variables: z(t) = K(t)/h(t), x(t) = C(t)/ h(t) and co(t) = V(t)/R(t). The dynamic equations for z(t) and x(t) are easily derived from expressions defining the competitive equilibrium as
z(t)- zl(t~ ) z(t)~+ z(t)(/~)
)
y - 6 + ~1 - x(t),
(13)
(14) where u(t) was replaced by the :ratio z(t)/zl(t ). To get a dynamic equation for co(t), we make use ofthe procedure devised by Uzawa (1964):to show that :this rental ratio implicitly explains the ~evolutionof zl(t), u(t) and the relative :price of physical goods in units of new :human capital. Thus, from (2) we get o~(t) - f(zl(t)) f'(zl(t)) _
p(t)
px(t)
(15)
zi(t) ,
_
p2(t )
7 f(zl(t))
-
(16)
f'(z~(t))zl(t)
"
By assumptions (1), the variables Zl(t ) and p(t) are :uniquely determined by c0(t) and they will be denoted by Zl(co(:t))and p(co(t)), :respectively. Applying the implicit function theorem to (15) and (16), we :respectively get that z{(co(t)) > 0 and p'(co(t)) < 0. :Finally, from (t6) we get p(t)/p(t) = [p'(co(t))/p(co(t))]Co(t). Manipulating :this relationship with the equations defining the =competitive equilibrium, we get the following law of motion for co(t): Zl(°9(t)) ~(t)
-
#(t)zi(co(t))
[
y (~
-
f'(zl(co~t))) + 5] s)
-
~
(17)
-
where :~(t) is the .elasticity of marginal productivity ,of labor with respect to physical capital; [hat is, 416
Subsidy to H u m a n Capital Accumulation in an Endogenous Growth Model
f"(Zl(co(t)))(Zl(co(t)))2 fl(t) = -f(zt(eo(t))) - f'(zl(co(t)))z!(co(t))
> 0 "
(18)
The ordinary differential Equations (13), (14) and (17) constitute a complete dynamic system with respect to z(t), x(t) and c0(t), the first one being a state,like variable and the remaining two control-like variables. Then, since z(0) is the ratio between the initial stocks of physical and human capital, given these initial stocks the system composed of Equations (13), (14) and (17) form a complete description of the competitive equilibrium paths of the economy. The evolution of z(t), x(t) and co(t) in a local neighborhood of the BGP equilibrium can be approximated by the linearization of the dynamic system composed of Equations (13), (14) and (17). More precisely, since the BGP equilibrium is locally saddle,path stable, the previous approximation is given by the linear, unidimensional, stable manifold corresponding to the unique negative eigenvalue of the dynamic system. From the linearization of this system in Appendix A, we get that this stable manifold is given by z(t) .= z* + exp{a3~t}(z(O) ~ z * ) ,
(19a)
x(t) = x* + a2a(a3.--,a -- a11.).+ alaaZl (z(t) ~ z*) ,
(19b)
~713a33 - a~a co(t) = co* +
tZZl - aa3(all - aaa) . . . . (z(t) - z * ) , a13a33 -- a23
(19c)
where z*, x* and to* denote respectively the stationary values of z(t), x(t) and co(t), and the a~js are the coefficients of the linearized system, Given an initial condition z(0), Equations (19b) and (19c) approximate the unique pair of the initial values of x and co which lead the economy to the BGP equilibrium. ~ At this point, one can see that the subsidy to human capital investment alters the dynamic behavior of the economy outside the BGP equilibrium. In particular, the subsidy is relevant to determine the dynamic behavior of human capital qualitatively, and so plays a crucial role in determining the speed and the direction of convergence to the BGP equilibrium. One can see from the Appendix that the speed of convergence, which is given by 3Note that (19b) and (19c) are nothingbut the linear approximationof policyfunctionsofx and co,respectively.The coefficientsof (z(t) - z*) in these equationsare respectivelythe second and the third normalizedcomponentsof the eigenvectorassociatedto the negativeeigenvalue. A detailedderivationof the linearstablemanifold(19)is availablefromthe authoruponrequest. 417
Jaime Alonso-Carrera -a33, depends on the subsidy rate. However, the sign of this dependence is unknown under a general production function for the physical goods sector. If we instead assume a Cobb-Douglas technology, i.e., Y(t) = AK(t)~(u(t)h(t))l - 8, then we get that the speed of convergence is a strictly increasing and strictly convex function of the subsidy rate. More precisely, in this case the speed of convergence is given by - a ~ 3 = [(1 - 1~)/~][(7/(1 s)) + 5 - q]. One can numerically check that a positive subsidy shock drives speed of convergence up moderately, and only for quite large subsidy rates this speed rises more than proportionately. 4 The following results characterize the dynamic adjustment process, and so the dependence of the direction of convergence on the subsidy rate. The presence of the subsidy only affects the direction of convergence ofu(t). The dynamic behavior of all other variables does not qualitatively change with the presence of the subsidy.
PROPOSITION 1. Consider the economy described by Equations (2), (5), (6), (7), (8), (9), (I0), (11) and (12). (i) If z(O) < z*, then x(O) < x* and on(O) < co*. Hence, consumption and physical capital both grow more rapidly than human capital in the transition to the BGP equilibrium. (ii) If z(O) > z*, then x(O) > x* and co(O) > co*. Hence, consumption and physical capital both grow slower than human capital during the transition to the BGP equilibrium.
Proof Using Appendix A, one can show that the coefficients of (z(t) - z*) in (19b) and (19c) are both positive. From Proposition 1 we can establish the transitional behavior of zl, p and u.5 For that purpose, we introduce the notation or(t) to denote the elasticity of marginal productivity of physical capital with respect to labor in efficiency units at time t, i.e.,
a(t) = -f"(zl(co(t)))(zl(co(t))) f'(z,(co(t)))
> O.
(20)
4By assuming a Cobb-Douglas technology for the production of physical goods, Ortigueira and Santos (1997) find that the Uzawa-Lucas model implies a much higher speed of convergence than the standard neoclassical model. They get values around 0.2 for -a33. 5Bond, Wang and Yip (1996) reduce the dynamic analysis to the space (x,z) making use of the fact that the value function is homogeneous of degree 1 - cy. However, their analysis is not complete. With this method, we could not fully characterize the dynamic behavior of the economy. We could not analyze the instantaneous response of the rental ratio, relative prices or the work effort devoted in each sector to changes in the structural parameters, such as the rate of human capital subsidy.
418
Subsidy to Human Capital Accumulation in an Endogenous Growth Model CortoLLanY 1. Consider the economy described in Proposition 1. If z(O) < z* (z(O) > z*), then zl(O) < z~ (resp. zl(O) > z~) and p(O) > p* (resp.
p(O) < p*). Proof
It follows directly from Proposition 1 and Equations (15) and (16).
PROPOSITION 2. Consider the economy described in Proposition 1. Let fl* and a* be respectively the values at the BGP equilibrium of the elasticities fl(t) and a(t) defined in (18) and (20). Starting from z(O) < z* (z(O) > z*), the following statements hold: (i) u(O) > u* (resp. u(O) < u*) if and only if gJ > O; (ii) u(O) < u* (resp. u(O) > u*) if and only if gJ < O; and (iii) u(O) = u* if and only if ~P = O, where
'e-
o~*(~r~.~- #*) (1 -7 s)
+6-
s7 r/ + s( 1' --~ "
Proof. For the purpose of this proof we will compute the slope of the policy function ofu at the BGP equilibrium. Since u = Z/Zl, the derivative of u with respect to z is given by du
dz
zl(co) -
zz{(co)(dog/dz)
(zl(co)) 2
(21)
Taking dm/dz from (19), and using (20) and Appendix A, we compute the derivative (21) at the BGP equilibrium as follows:
~ZZ(z.x.,u.)
=
a l 3 ~ - a23)" Zl(CO*)(al3aa3
The result then follows from Appendix A. The three parts of Proposition 2, respectively, define the normal, paradoxical and exogenous growth eases that Caball6 and Santos (1993) point out to describe the evolution of the human capital variable in the ease of a non-market human capital sector. Assume that the initial capital ratio is lower than its stationary value. In the normal (paradoxical) growth ease the rate of human capital accumulation during the transition is lower (higher) than the one corresponding to the balanced growth path. These rates are equal when the economy belongs to 419
Jaime Atonso-Carrera the exogenous growth ease, In absence of the subsidy the conditions on parameter values that define each growth case in Proposition 2 are reduced to the ones stated by Cahall6 and Santos (1993, Theorem 5.!), though in our model the creation of new human capital is a market activity. The following corollary establishes the transitional behavior ofu in absence of the subsidy to human capital investment.6 COROLLARY 2. Consider the economy described in Proposition 1. Assume s = O. Starting from z(O) < z* (z(O) > z*), thefoUowing statements hold: (i) If fl* < a, then u(O) > u* (resp. u(O) < u*). (ii) If fl* > a, then u(O) < u* (resp. u(O) > u*). Oii) If fl* = a, then u(O) = u*.
Proof.
It follows from Proposition 2. by imposing s = 0.
Unlike the model without fiscal policy, in aur model preferences, technologies and depreciation rates, as welt as the subsidy rate, determine the growth case to which the economy belongs. In particular, the presence of human capital subsidy expands the region of parameter space in which the normal growth case occurs to the detriment of the regions containing the other growth cases, v More precisely, unlike the ease when s = 0, the conditions ~* > ~"and ~* = ~ are not su~eient in our model for-the paradoxical and exogenous cases to arise, respectively. Alternatively, the condition 13" < cr is not necessary for the economy to belong to the normal growth case when s > 0. Hence0 the presence of the subsidy makes the paradoxical and the exogenous growth cases empirically tess relevant, To show that, we will reinterpretate the conditions in Proposition 2 and Corollary 2. The elasticities a(t) and [~(t) defined in (20) and (t8) can be rewritten as a(t) = (1 O(t))/e(t) and 13(t). = tg(t)/e(t), where 0(t) and s~(t) are respectively the physical capital income's share on the product of the physical goods sector and the elasticity of substitution between capital and labor in the production of physical goods, i,e., -
O(t)
ff(Zl(O~(t)))(Zl(('O(t)')) f(z~(cO( (i )i)
f' (zl(eg(t) ) )[f(zl(o~(t) ) ) - f' (zt(a~(t)) )zl(co(t) ) ] ~(t) . . . . f(~(o)(t)))f'(zi(io(t)))Z~(o)(t)) .
.
.
.
6Mu/tigan and Salaq.Martin (1993), Faig (1995) and Barro and Sala.i-Martin (1995) have also shown this result in the case of C0bb-Douglas production technok~gy ~ad a non-market human capital sector. 7Ortigueira (1998) obtains the opposite result for physical capital taxation in the case where the creation of new human capital is a non-market activity.
420
Subsidy to Human Capital Accumulation in an Endogenous Growth Model Using the previous two equalities, we get cr - 13" = ¢y(1 - (0"/~*)). Since a plausible value for cr is higher than unity (see, e.g, Hansen and Singleton 1983), the previous expression has a positive value unless the elasticity of substitution in the product_ion of physical goods is sufficiently small. More precisely, the paradoxical growth case would only emerge for values of ~* much smaller than 0', in the presence of the subsidy, the previous restriction is even stronger since the condition ~* --- ~ is now not sufficient for the paradoxical ease. This fact makes it clear that the paradoxical and exogenous growth cases may not happen in reality, s
4, Effects of Changes in the Subsidy to Human Capital Investment The analytical characterization of the equilibrium dynamics permits us to analyze the impact that shifts in any structural parameter of the model have on economic growth. Specifically, in this Section "we describe the response of the economy to an unanticipated, permanent, marginal increase in the subsidy to human capital investment. For that purpose, we assume that the economy is initially on the balanced growth path without subsidy, and suddenly the government introduces a marginal iricrease in the subsidy rate from zero. 9 There are effects on the BGP equilibrium and on the equilibrium path converging to the BGP equilibrium.
Growth Vffeas We analyze the impact of a marginal introduction of the subsidy policy on the BGP equilibrium using comparative static arguments. PROPOSITION 3. Consider the economy described in Proposition I. Then, the steady,state effects of a marginal introduction of the subsidy to haman capital investment are: do~*
de*
dx*
dg*
--ds < 0 , - - ~ - < 0 , - - ~ - < 0 , ~
Proof
dp*
>0, ~
de*
>0, ~
du*
<0, ~
<0.
See Appendix B.
SFor instance, under the CES production function, and with empirically plausible values for parameters, the elasticity of substitution ~* is larger than the income capital share O* unless ~* is almost zero. 9Relaxingthis restriction complicates the explanation but changes nothing of the substance.
421
Jaime Alonso-Carrera A marginal increase in s has the effect of raising the after-subsidy return to investment in human capital, so that households are now willing to rent their stock of human capital at a smaller rental rate. Hence, the subsidy makes the human capital relatively inexpensive compared to physical capital. This reduction in the stationary level of the rental ratio induces the firms in both sectors to increase the use of human capital. To meet this increase in the demand for human capital, the creation of new human capital is stimulated and the accumulation of human capital is accelerated by reallocating labor in efficiency units to the sector producing new human capital. Consequently, the long-run rate of economic growth becomes higher in the new BGP equilibrium than in the o l d one. Moreover, the capital ratio and the ratio of consumption to human capital will be smaller in the new BGP equilibrium as a result of the increase in the production of new human capital.
Dynamic Effects We approximate the equilibrium path followed by an economy that is perturbed by an unanticipated, permanent, marginal introduction of the subsidy to human capital investment at some time. In Section 3 we characterized the equilibrium path of an economy with a constant and positive subsidy. At this point, we analyze the transition from the before-subsidy equilibrium path to the after-subsidy one. When the subsidy policy is introduced the state variable z is at its old stationary value z*. The economy is then outside both the new BGP equilibrium and the new stable manifold. Hence, the control variables co and x must jump instantaneously to place the economy on the new stable manifold. Otherwise, local convergence to the new BGP equilibrium would not be possible, and the transversality conditions (9) would be locally violated. After this jump, the economy follows the transitional adjustment along the new stable manifold, as described in Section 3. Therefore, the equilibrium path appears to be a smooth function of time, except at the moment at which the subsidy is introduced. In this sub-section we compute and analyze the instantaneous jumps of control variables, and so the point of the new stable manifold reached by the economy after such jumps. We will follow several steps. First, we calculate the point of the old stable manifold corresponding to the moment at which the subsidy is introduced, say t = 0. From (19) we get co(o) =
o9* -
azl
-
a33(an
a13a33
--
-
a33)
(z(0)
-
z*),
x(0) = x* -- a23(a33 -- an) + alaa21 (z(0) -- z*). a13a33
422
--
(22a)
a23
a23
(22b)
Subsidy to Human Capital Accumulation in an Endogenous Growth Model Moreover, since the introduction of the subsidy affects the coefficients ay and the stationary values of z, x and co, we must compute the instantaneous effects of the subsidy on the point given by (22). For that purpose, we differentiate (22) with respect to the subsidy rate. Finally, we evaluate these derivatives at z(0) equal to z*, so that the instantaneous jumps of c0 and x at t = 0 are, respectively1° dee(0) ds
do)* ds
a~l - a33(an - a33)dz* a13a33 - a23 ds '
dx(O)
dx*
a23(a33
ds
ds
-
all)
al~a33
-
+ a23
(23a)
alaazl dz* ds
(23b) "
Since dco*/ds and dz*/ds are negative, and the coefficients of dz*/ds on the right-hand side of (23a) and (23b) are both positive according to Appendix A, it follows that do~(O)/ds > do3*/ds and dx(O)/ds > dx*/ds. Hence, after the instantaneous jumps, the control variables co and x fall monotonically to their new stationary values. However, without any other manipulation, we cannot say anything about the sign or direction of those jumps. The following result establishes the direction of these instantaneous jumps. PROPOSITION 4. Consider the economy described in Proposition 1. After a marginal introduction of a subsidy to human capital investment, the following occurs: (i) The rental ratio jumps up instantaneously. (ii)The consumption to human capital ratio jumps up instantaneously when if* > a, whereas if fl* < a then that ratio jumps down instantaneously.
Proof.
See Appendix C_
Therefore, the time paths of co and x after a marginal increase in s from zero are completely defined by Propositions 2 and 5. They are shown in Figure 1. To complete the comparative dynamic analysis, we describe the time path of p, zl and u after a marginal introduction of the subsidy. PROPOSITION 5. Consider the economy described by Proposition 1. After a 1°Notethat the point of the newstable manifoldreachedby the economyafterthe jolt caused by the marginalintroductionof the subsidyis givenby z(0) = z*, x(O) = x* + (dx(O)/ds)and o3(0) = o3* + (do3(O)/ds),where (z*, x*, o3")is the old steady-state. 423
Jaime Alonso-Carrera (o
x* co*
(o)*)!
(X*)'
t
t-'
t
Figure i. Time paths of (o and x perturbed by a marginal introduction of the subsidy.
marginal introduction of the subsidy to human capital investments.. (i) z~ jumps up instantaneously; and (ii) p and u jump down instantaneously.
Proof. It follows directly from Proposition 4 by noting that zi(O) is an increasing function of re(O), p(O) is a decreasing function of re(O), and u(O) = z(O)/zl
(0),
The analysis of the time paths of zl, p and u after a marginal increase in the subsidy rate from zero is trivial from Proposition 5 and Corollaries 1 and 2. With regard to u, we must note that this variable always jumps down simultaneously with the marginal introduction of s. However, this jump is larger (smaller) than the variation in its stationary value when ~* < s (resp. 13" > ~). Figure 2 illustrates these features of the time path of u after a change in s. A marginal rise in s reduces the after-subsidy price of new human capital, which generates both a substitution and an income effect. On one hand, the subsidy induces households to substitute human eapitai investments for consumption and physical capital investments. This expansion in the demand of new human capital reduces the relative prices and encourages the production of new human capital. Since the sector producing human capital only uses effective labor as an input, the ratio (0 then increases instantaneously and labor is reallocated to this sector. On the other hand, a marginal rise in s increases households' disposable income, and so they can also expand their present consumption. Hence, the increase in human capital investments and the effect of the subsidy on consumption depend on which of these effects dominates. The net effect is determined by the relative costs 424
Subsidy to Human Capital Accumulation in an Endogenous Growth Model U
(u*y
~* <(3 t
Figure 2. Time paths of u perturbed by a marginal introduction of the subsidy.
and benefits of investing an additional unit of income in new human capital. The benefit of this marginal investment is given by the induced increase in future consumption. Thus, the benefit depends positively on ~. The costs of a marginal investment in new human capital are twofold. First, this investment reduces the present consumption. Second, the resulting increase in the stock of human capital will reduce the marginal productivity of labor in the physical goods sector, and so the return on total investments in human capital. While the former cost is inversely related to r~, the latter depends positively on 13". Therefore, the income effect dominates when 13" > ry, whereas if 13" < cr then the substitution effect dominates. The instantaneous effect of s on consumption then depends on the relationship between these two elasticities. Alternatively, the positive effect of the subsidy on human capital investment, and so the realloeation of labor to the human capital sector, is inversely related to 13" and directly to ~.
5. Conclusion This paper has analyzed the growth and dynamic effects of the subsidy to human capital investment in a two-sector model of endogenous growth with physical and human capital accumulation. The presence of the subsidy does not alter the uniqueness and the stability properties of the BGP equilibrium. However, the incidence analysis has shown that human capital subsidy is growth-increasing. Moreover, the presence of the subsidy qualitatively determines the dynamic behavior of the human capital, and so the speed and direction of convergence to the BGP equilibrium. Regarding the dynamic effects of a marginal introduction of the subsidy, we have found that the economy reacts instantaneously. As a result, we 425
Jaime Alonso-Carrera obtain that consumption, the ratio between the rental rates of both types of capital, relative prices, and the fraction of human capital allocated to each sector all jump instantaneously when the subsidy is introduced. The direction of these jumps depends on the relationship between the elasticity of marginal productivity of labor with respect to physical capital and the inverse of the elasticity of intertemporal substitution in consumption. In any case, the subsidy always generates a strictly positive welfare cost since in our model the first welfare theorem holds. This welfare cost depends on the growth scenario to which the economy belongs. Thus, given the response of the consumption to changes in the subsidy rate, the welfare cost would be larger in the paradoxical growth case than in the normal one. A first extension of the paper could then be to use our analysis to evaluate the welfare cost of subsidizing the accumulation of human capital given an externality from the stock of human capital as in Lucas (1988). In this paper we have maintained the assumption of an exogenous labor supply. We conclude with a brief comment on how allowing for an endogenous labor-leisure choice may alter the effects of the subsidy to human capital investment. Milesi-Ferreti and Roubini (1998) emphasize that the growth effects of the subsidy in this type of models depend on the specification of the leisure activity. If leisure is modeled as raw time, then the growth effects of the subsidy are equal to the ones obtained in this paper, whereas if leisure is defined as quality time, the growth effects of the subsidy are larger. Regarding the initial impacts of changes in the subsidy rate one must expect that they are qualitatively determined by the same parameters as in this paper but also by the elasticity of substitution between consumption and leisure and the elasticity of leisure with respect to human capital, since both of these elasticities drive the instantaneous response of the labor supply. Future research should confirm this conjecture. Received: July 1998 Final version: October 1999
References Barro, Robert j., and Xavier Sala-i-Martin. Economic Growth, New York: McGraw Hill Inc., 1995. Bond, Eric W., Ping Wang, and Chong K. Yip. "A General Two Sector Model of Endogenous Growth with Human Capital and Physical Capital."Jourhal of Economic Theory 68, no. 1 (1996): 149-73. Caballr, Jordi, and Manuel S. Santos. "On Endogenous Growth with Physical
426
Subsidy to Human Capital Accumulation in an Endogenous Growth Model and Human Capital." Journal of Political Economy 101, no. 6 (1993): 1042-68. Chamley, Christophe. "Externalities and Dynamics in Models of 'Learning or Doing'." International Economic Review 34, no. 3 (1993): 583-609. Deveretlx, Michael B., and David R. F. Love. "The Dynamics Effects of Government Spending Policies in a Two-Sector Growth Model." Journal of Money, Credit, and Banking 27, no. 1 (1995): 232-56. Faig, Miguel. "A Simple Economy with Human Capital: Transitional Dynamics, Technology Shocks and Fiscal Policies." Journal of Macroeconomics 17, no. 3 (1995): 421-46. Hall, Robert E. "The Dynamic Effects of Fiscal Policy in an Economy with Foresight.'" Review of Economic Studies 38 (1971): 229-44, Hansen, Lars P., and Kenneth J. Singleton. "Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns." Journal of Political Economy 91, no. 2 (1983): 249-65. Htfert, Andreas. "Subsidizing Human Capital. Is there a Conflict between Growth and Welfare?" Swiss Institute for Business Cycles Research (KOF-ETHZ), 1996. Mimeo. Jones, Larry E., and Rodolfo Manuelli. "'A Convex Model of Equilibrium Growth: Theory and Policy Implications." Journal of Political Economy 98, no. 5 (1990): 1008-38. King, Robert G., and Sergio Rebelo. "Public Policy and Economic Growth: Developing Neoclassical Implications.'" Journal of Political Economy 98, no. 5 (1990): S126-S150. Lucas, Robert E., Jr. "On Mechanics of Economics Development." Journal of Monetary Economics 22, no. 1 (1988): 3-42. • "Supply-Side Economics: An Analytical Review." Oxford Economic Papers 42 (1990): 293-316. Milesi-Ferretti, Gian Maria, and Nouriel Roubini. "On the Taxation of Human and Physical Capital in Models of Endogenous Growth." Journal of Public Economics 70, no. 2 (1998): 237-54. Mino, Kazuo. "Analysis of a Two-Sector Model of Endogenous Growth with Capital Income Taxation." International Economic Review 37, no. 1 (1996): 227-53. Mulligan, Casey B., and Xavier Sala-i-Martin. "Transitional Dynamics in Two-Sector Models of Endogenous Growth." Quarterly Journal of Economics 108, no. 3 (1993): 739-73. Ortigueira, Salvador. "Fiscal Policy in an Endogenous Growth Model with Human Capital Accumulation." Journal of Monetary Economics 42, no. 2 (1998): 323-55. Ortigueira, Salvador, and Manuel Santos. "On Speed of Convergence in
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Jaime Alonso-Carrera Endogenous Growth Models." American Economic Review 87, no. 3 (1997): 383-99. Rebelo, Sergio. "Long Run Policy Analysis and Long Run Growth." Journal of Political Economy 99, no. 3 (1991): 500-21. Rebelo, Sergio, and Nancy L. Stokey. "Growth Effects of Flat-Rate Taxes." Journal of Political Economy 103, no. 3 (1995): 519-50. Uzawa, Hirofumi. "Optimal Growth in a Two-Sector Model of Capital Accumulation." Review of Economic Studies 31 (1964): 1-24. • "Optimal Technical Changes in an Aggregate Model of Economic Growth." International Economic Review 6 (1965): 18-31.
Appendix Linearization of the Dynamic System For the purpose of this appendix, we will first characterize the BGP equilibrium. From the equations defining the competitive equilibrium, we get f'(zl(c°*)) = [
'
+
6
-
u . = [7[,~(a - s) -
(1 - s) z*
=
X•
--
r/
] ,
(A1)
lJ - , ~ ( 1 a7(1
s)(,~ - J-) + p(a - ~)J -
r/ - p ,
(A3)
z{co*)u*
zl(c~*)
2;1(60"))
(~)
s)
(A4) - - ZI((d) 'g)
(A5)
(.7- 17(1- s) - p ( 1 - s) + a6(1 - s))] a(1 - s)
Linearizing the dynamic system around the steady state (z*, x*, o~*), we have
l x(t)/=[ol 00 o23//x(t) x*, ¢(t)] a,~_l Leo(t) co* -,
l r (t) -
where the elements a 0 of the coefficient matrix are given by
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Subsidy to Human Capital Accumulation in an Endogenous Growth Model
(a~(t)/*
= 27u* +
(ae(t) I* -
ai3 = \ 0 - ~ /
/e~(~o*))
(? + 8 - q)
f(zl(oo*)) -- f'(zl(cO*))Zl(O9*)
ZI((.O* )
sT > 0 , (1 -- 8)
(A6)
+ - -
zi(co)*(y), ,,3 z;(co*)z* ~ z ) -[- (Z,l((.o,))------------------~ [f'(zl((-D*))zl((.o :~) - f(zl((-D*))]
-~i(~o*)~* Fh,u , + f(z~(o~*)) ~_ rcc]/,a,~o*))z/co*)l < o z~(o~*) J ' 1_
=
~1
2y~*
\oz(t)/
z~7g-*)
(~'3C(t) /*
X*I[;' (z 1( ('0g ) )Z~-<('0* )
(A7)
(A8)
z~(oo*) + (1 - ~-----~¢~* > o, Zl(O)*)'YZ=gl
(Ag)
(a6)(t)l* _ Zl(°J*) f,,(z~(~o,)) = aaa = \&o(t)/ fl*
f(zl(co*)) - f'(zl(co*))zl(oo*) <0. zl(co*)
(A10)
The unique negative eigenvalue of the linearized system is equal to aaa. This negative eigenvalue defines a unidimensional stable manifold of the BGP equilibrium.
Comparative Static Analysis of the BGP Equilibrium This appendix evaluates the impact of a marginal introduction of the subsidy on the BGP equilibrium. First, from Appendix A, we get
dco* = y < 0 ds s=o f"(zl(co*))z~(co*)
ds ~=o
\ ds I,=0/
(B1)
~r~
- ---7- < 0 ,
(B2)
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Jaime Alonso,Carrera
+ [(~ - ~)(z - 1) + p]
[
(d°9*l ~[ (-7 - r/)(~r1) + P-] d
x z~(o~*) \-'~-'1,=0)
z1(co*)7]<0
-
'
(B3)
Using (A2) and (18), we can rewrite the previous effects in a way that will be more useful in the next appendix. More precisely, we get
ds s=o
dx:t:
\ ds Is=o]
(dz$l
~
(B2')
'
/ r((Zl((°:t:))L - f'(gl(°)*))Zl((-/)*)
ds ~=o = \ ds 1~=o¢
] + 7u*
~1(-~~
~)(o. +
fl)z •
~/~
(B3') Finally, from (A2), (A3), (15) and (16) the following is also true
dg*
i
--<0, ds ~=0
" ~-~=o
7
= ->0 a
<0 'ds2 ~=o
dp*
>0.
' ds~ ~=o
The Instantaneous Impact of the Subsidy on the Control Variables Consider first the instantaneous impact of a marginal introduction of s on 09. By substituting (B1) and (B2') for dco*/ds and dz*/ds, respectively, and using (A6)-(A10) and the definition of [3", we transform (23a) into the following expression: dog(0) ds
a13(7o'z* - 2fl*a33) ~r1~*zt1(o)*)(a13a33
--
a23)'
which, from Appendix A, is always positive. On the other hand, using the previous procedure with (23b), we also get the initial impact of s on x. Thus, substituting (B2') and (B3') for dz*/ ds and dx*/ds, respectively, and using (A6)-(A10) and the definition of 13", we get that the initial impact of s on x is
430
Subsidy to Human Capital Accumulation in an Endogenous Growth Model dx(O) ds
(or - fl*)al3(a3a) 2 zl(co* ) ~(al3aaa - a2a)
Therefore, we conclude from Appendix A that dx(O)/ds2 is positive when 13' > or, whereas if 13' < ¢y then it is negative.
431