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Capital–labor substitution and long-run growth in a model with physical and human capital
Mathematical Social Sciences 78 (2015) 106–113
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Capital–labor substitution and long-run growth in a model with physical and human capital Manuel A. Gómez ∗ Departamento de Economía Aplicada II, Facultade de Economía e Empresa, Universidade da Coruña, Campus da Coruña, 15071 A Coruña, Spain
highlights • • • •
We examine the link between factor substitution and endogenous growth. Previous work relied on one-sector models with exogenous long-run growth. We use a one-sector endogenous growth model with physical and human capital. The long-run growth rate is increasing in the elasticity of substitution.
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Article history: Received 12 April 2015 Received in revised form 31 July 2015 Accepted 25 October 2015 Available online 31 October 2015
abstract Most of the literature analyzing the growth-substitutability nexus considers models in which long-run growth is exogenous. We study this link in an endogenous growth model with physical and human capital. We show that for two economies differing uniquely in factor substitutability, the one with the higher elasticity of substitution will have higher long-run growth. This is due to the efficiency effect of a higher factor substitution. Furthermore, if the initial ratio of physical to human capital is below (above) its steadystate value, the economy with the higher elasticity of substitution will have a higher (lower) steady-state physical capital income share. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Is the elasticity of substitution an engine of growth? From a theoretical viewpoint the answer is affirmative in the Solow (1956) model, as it was uncovered in the seminal contributions of de La Grandville (1989) and Klump and de La Grandville (2000). Miyagiwa and Papageorgiou (2003) show that such positive relationship does not necessarily hold in the Diamond (1965) overlapping-generations model. Irmen and Klump (2009) reconcile these findings by introducing possible asymmetries of savings out of factor income. They find that factor substitution has a positive efficiency effect and an ambiguous distribution effect on economic growth. If the savings rate out of capital income is sufficiently high, the efficiency effect dominates and the overall effect is positive, so ‘‘higher factor substitution [. . . ] works as a major engine of growth’’ (Irmen and Klump, 2009, p. 464). Xue and Yip (2012) present a comprehensive characterization of the link between the elasticity of substitution and the steady-state capital and output
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http://dx.doi.org/10.1016/j.mathsocsci.2015.10.004 0165-4896/© 2015 Elsevier B.V. All rights reserved.
per capita in the Solow, Ramsey–Cass–Koopmans, and Diamond models. If initial capital per capita is below its steady-state value, a higher elasticity of substitution generates a higher steady-state income per capita in the Solow and the Ramsey–Cass–Koopmans models, whereas the effect is ambiguous in the Diamond model. These works have two limitations in common. First, all of them consider one-sector models in which long-run growth is exogenously given. As a consequence, factor substitution can affect transitional but not long-run growth. Therefore, the question posed on the link between capital–labor substitution and long-run growth cannot be addressed. Second, they consider that the factors of production are (physical) capital and (raw) labor. However, considering effective labor – i.e., labor adjusted for human capital – rather than raw labor as a factor of production has become relatively standard (e.g., Ben-Porath, 1967, Heckman, 1976, Lucas, 1988). This is justified by an extensive empirical literature that confirms the productive role of human capital (see, e.g., the reviews by Hanushek and Woessmann, 2008, and Glewwe et al., 2014). Our purpose is to overcome these two limitations by introducing effective labor as an input in a model of endogenous growth. This paper studies the relationship between long-run growth and the elasticity of substitution between capital and effective labor. To this end, we consider the one-sector endogenous growth
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
model with physical and human capital (e.g., Barro and Sala-iMartin, 2004, and Gómez, 2003). In this way we achieve two goals. First, the long-run growth rate is endogenized and thus we can examine the effect of factor substitution on the long-run growth rate. Second, we introduce effective labor as a productive input and, therefore, we are still analyzing the effect of the elasticity of substitution between (physical) capital and (effective) labor. The model is admittedly simple in that physical and human capital enter in a symmetric fashion into the model. However, this assumption has also been made, for instance, in the influential works by Mankiw et al. (1992) and Barro et al. (1995) and, as Mankiw et al. (1992, p. 416) argue, ‘‘We believe that, at least for an initial examination, it is natural to assume that the two types of production functions are similar’’. Furthermore, this framework has been recently used, e.g., to study the growth effects of epidemics (Boucekkine et al., 2008), and to compare the development processes in China and India (Kan and Wang, 2013).1 The simplicity of the model will allow us to get clear-cut analytical results. We prove that the long-run growth rate is increasing with the elasticity of substitution between physical and human capital. Hence, for two economies differing uniquely in factor substitutability, the economy with the higher elasticity of substitution will feature a higher long-run growth rate. We show that this is due to the efficiency effect of Klump and de La Grandville (2000); i.e., a higher elasticity of factor substitution increases the productivity of inputs. Furthermore, an increase in the elasticity of substitution has a positive or negative effect on the steady-state physical capital income share depending on whether the baseline ratio of physical to human capital is lower or greater than its steady-state value, respectively. We show that this is consequence of the distribution effect (Irmen and Klump, 2009). Finally, we also study the effect of the elasticity of substitution on the growth rate along the transition.2 Klump and de La Grandville (2000) show that there exists a positive link between factor substitution and transitional growth in the Solow model.3 Therefore, an interesting question is whether this result carries over to the one-sector endogenous growth model with physical and human capital. As an analytical answer is probably intractable, we resort to numerical simulations. Unlike in the Solow model, our results show that a higher elasticity of substitution does not necessarily lead to a higher growth rate of income and physical capital (or human capital) along the transition in the present model. Related research has been recently made. Duffy and Papageorgiou (2000) estimate a CES production function with physical capital and human capital inputs, but they do not analyze the relationship between factor substitution and growth. Irmen (2011) first established a positive relation between the elasticity of substitution and the endogenously determined long-run growth rate, but he considers instead a multi-sector neoclassical model with endogenous capital- and labor-augmenting technical change. Thus, our results reinforce the findings in Irmen (2011). Kan and Wang (2013) analyze the accumulation of physical and human capital in a one-sector endogenous growth model with physical and human capital similar to that considered in this paper. However, they do not study the link between the elasticity of substitution, long-run growth and factor income distribution, as we do. Gómez (in press) shows that factor substitution between public and private inputs is positively related to the optimal (first-best) long-run growth
1 This model is also covered, e.g., in the textbooks by Barro and Sala-i-Martin (2004) and Acemoglu (2009). 2 I thank a referee for suggesting this study. 3 The elasticity of substitution has obviously no effect on long-run growth, which is exogenously given.
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rate and welfare in the Barro (1990) endogenous growth model. In contrast, this paper studies the effect of capital–labor substitution – which is the effect typically analyzed in the literature – in a decentralized economy. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes the effect of factor substitutability on growth and factor income distribution. Section 4 concludes. 2. The model We consider the one-sector endogenous growth model with physical and human capital (Barro and Sala-i-Martin, 2004; Gómez, 2003) with CES technology. The economy is populated by a large number of identical and infinitely lived agents. For simplicity, and without loss of generality, we assume that population is constant and normalized to unity. Thus, individual and aggregate quantities coincide. 2.1. Firms Output Y is produced using physical capital K and human capital (effective labor) H by means of the CES technology Y = F (K , H ) = A[α K ψ + (1 − α)H ψ ]1/ψ , A > 0, 0 < α < 1, ψ < 1, where A is the productivity parameter, α is the distribution parameter and σ = 1/(1 − ψ) is the elasticity of substitution. Denoting y = Y /H and k = K /H, the production function in intensive form can be written as y = f (k) = F (k, 1) = A[α kψ + (1 − α)]1/ψ . Let r and w denote the rental prices of physical and human capital, respectively. Profit maximization entails that the rental prices of physical and human capital are their respective marginal products:
∂F (K , H ) = f ′ (k) = α Aψ [f (k)/k]1−ψ , ∂K ∂F w= (K , H ) = f (k) − kf ′ (k) = (1 − α)Aψ f (k)1−ψ . ∂H r =
(1) (2)
2.2. Agents The representative agent derives utility from consumption C in accordance with the intertemporal utility function ∞
U = 0
C 1−θ − 1 −ρ t e dt , 1−θ
θ > 0, ρ > 0,
(3)
where ρ is the rate of time preference and 1/θ is the elasticity of intertemporal substitution. The agent receives rents from physical and human capital and spends her income on accumulation of physical and human capital and consumption. The agent’s budget constraint is then rK + w H ≥ IK + IH + C ,
(4)
where IK and IH are gross investments in physical and human capital, respectively, which must each be nonnegative. The stocks of physical and human capital evolve according to K˙ = IK − δK K ,
(5)
˙ = IH − δ H H , H
(6)
where δK and δH are the depreciation rates of physical and human capital, respectively.
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M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
The agent maximizes her utility (3) subject to the budget constraint (4), the constraints on physical capital and human capital accumulation (5) and (6), respectively, and the constraints on non-negative investment, given K (0) and H (0). The current-value Hamiltonian of the agent’s problem is C 1−θ − 1
the transversality condition is satisfied and long-run growth is positive. Assumption 1. Parameter values are such that (θ − 1)γ¯ + ρ > 0 and γ¯ > 0.
φ ≥ 0, (rK + w H − IK − IH − C )φ = 0,
(10)
With reversible investments in physical and human capital, the ¯ model features no transitional dynamics. If k0 = K0 /H0 ̸= k, the ratio of physical to human capital jumps immediately to its stationary value k¯ – and the ratio of consumption to physical capital does so – adjusting K and H by discrete amounts so that K + H is kept constant. Thereafter, the economy evolves along its balanced growth path where all variables grow at the common rate γ¯ . However, if investments are irreversible this solution would violate one of the constraints on nonnegative investment. Now, if k0 < k¯ then H is initially abundant relative to K . The desire to lower H relative to K entails that the inequality IH ≥ 0 will be binding in an interval [0, T ], whereas IK = Y − C > 0. Hence, Eq. (8) implies ˙ . Using (11), (5), and that η = 0 and φ = λ, so that −θ C˙ /C = λ/λ ˙ /H = −δH , the dynamics of the economy in terms of k and c can H be described by
λ˙ = ρλ − ∂ H /∂ K = (ρ + δK )λ − φ r ,
(11)
k˙
H =
+ λ(IK − δK K ) + µ(IH − δH H ) 1−θ + φ(rK + w H − IK − IH − C ) + ηIK + ξ IH ,
where λ and µ are the shadow prices of physical and human capital, φ is the multiplier associated with the agent’s budget constraint, and η and ξ are the multipliers associated with the nonnegativity constraints on gross investment. The first-order conditions are
∂ H /∂ C = C −θ − φ = 0, ∂ H /∂ IK = λ − φ + η = 0, IK ≥ 0, η ≥ 0, ηIK = 0, ∂ H /∂ IH = µ − φ + ξ = 0, IH ≥ 0, ξ ≥ 0, ξ IH = 0, rK + w H ≥ IK + IH + C ,
µ ˙ = ρµ − ∂ H /∂ H = (ρ + δH )µ − φw,
(7) (8) (9)
(12)
together with the transversality condition4 lim e−ρ t λK = lim e−ρ t µH = 0.
t →∞
t →∞
c (13)
2.3. Equilibrium Let c ≡ C /H denote the ratio of consumption to human capital. Eqs. (7) and (10) immediately imply that φ > 0 and the agent’s budget constraint (4) is binding, Y = IK + IH + C ,
(14)
where we have used (1) and (2) to substitute for r and w . In the interior solution with IK > 0 and IH > 0, Eqs. (8) and (9) imply that λ = µ = φ . Eqs. (11) and (12) imply that the rental prices on physical and human capital net of depreciation must be equal, r − δK = w − δH ; i.e.,
¯ ′ (k¯ ) − δH . f ′ (k¯ ) − δK = f (k¯ ) − kf
(15)
Log-differentiating (7), using (11) and (1), the growth rate of consumption is constant and equal to
γ¯ =
1 ′ f (k¯ ) − δK − ρ .
θ
k c˙
(16)
It can be readily observed that, in the interior solution, C , K , H, IK , IH and Y must all grow at the same rate γ¯ . Gross investments in physical and human capital are positive since IK /K = γ¯ + δK and IH /H = γ¯ + δH > 0. The resource’s constraint (14) entails that the steady-state ratio of consumption to human capital is c¯ = (1 + k¯ )[(θ − 1)γ¯ + ρ]. It is feasible (i.e., strictly positive) if the transversality condition (13), which is equivalent to (θ − 1)γ¯ + ρ > 0, is satisfied. Henceforth, we assume that parameter values guarantee that
4 These conditions are sufficient because the current-value Hamiltonian is concave in the states and the controls.
= =
K˙ K C˙ C
− −
˙ H H ˙ H H
= =
f (k) k 1
θ
−
c k
− δK + δH ,
[f ′ (k) − δK − ρ] + δH .
(17) (18)
At time T , the net returns on each type of capital are equalized, ¯ and the constraint IH ≥ 0 becomes non-binding. From k(T ) = k, t = T on, the solution is given by k(t ) = k¯ and c (t ) = c¯ , and all quantities grow at the constant rate γ¯ . The solution of the system (17)–(18) with the three boundary conditions k(0) = k0 , k(T ) = k¯ and c (T ) = c¯ , yields the time paths k(t ) and c (t ) in [0, T ] together with the terminal time T . ¯ the desire to lower K relative to H entails that the If k0 > k, inequality IK ≥ 0 will be binding in an interval [0, T ], whereas IH = Y − C > 0. Hence, Eq. (9) implies that ξ = 0 and φ = µ, so that −θ C˙ /C = µ/µ ˙ . Using (12), (6), and K˙ /K = −δK , the dynamics of the economy can be described by k˙ k c˙ c
= =
K˙ K C˙ C
− −
˙ H H ˙ H H
= −f (k) + c + δH − δK , =
(19)
1 f (k) − kf ′ (k) − δH − ρ − f (k) + c + δH .
θ
(20)
At time T , the net returns on each type of capital are equalized, ¯ and the constraint IK ≥ 0 becomes non-binding. From k(T ) = k, t = T on, the solution is given by k(t ) = k¯ and c (t ) = c¯ , and all quantities grow at the constant rate γ¯ . The solution of the system (19)–(20) with the boundary conditions k(0) = k0 , k(T ) = k¯ and c (T ) = c¯ , yields the time paths k(t ) and c (t ) together with the terminal time T . 3. Factor substitution, growth and factor income distribution Since the works of de La Grandville (1989) and Klump and de La Grandville (2000), it is known that to study the relationship between the elasticity of substitution and economic growth, the underlying CES production function must be normalized. Therefore, we shall first briefly describe the normalization procedure proposed by Klump and de La Grandville (2000). Then, we will examine the link between the elasticity of substitution and longrun growth, long-run factor income distribution, and transitional growth.
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
Differentiating (26) with respect to σ we have that
3.1. The normalized CES production function Following Klump and de La Grandville (2000), the normalized CES production function in intensive form is5
1/ψ
y = f (k, σ ) = A(σ ) α(σ )kψ + [1 − α(σ )]
,
(21)
where the productivity and distribution parameters are
A(σ ) =y0
α(σ ) =
1−ψ
+ m0 k0 + m0
k0
1−ψ k0
1/ψ ,
(22)
+ m0
,
(23)
for given baseline values of k0 , y0 = f (k0 , σ ), and the marginal rate of substitution ∂f
m0 =
f (k0 , σ ) − k0 ∂ k (k0 , σ ) . ∂f (k , σ ) ∂k 0
Thus, Eq. (21) makes explicit the dependence of output in intensive form on the elasticity of factor substitution σ = 1/(1 − ψ). Let π denote the physical capital income share,
π = π (k, σ ) =
∂f ¯ 1 ∂f ¯ (k(σ ), σ ) = (k(σ ), σ ) dσ ∂ k [1 + k¯ (σ )] ∂σ 1 + − [f (k¯ (σ ), σ ) − δH + δK ] [1 + k¯ (σ )]2 ¯ 1 ∂f ¯ dk (k(σ ), σ ) (σ ). + dσ [1 + k¯ (σ )] ∂ k d
k
∂f (k, σ ) ∂k
f (k, σ )
=
1−ψ kψ k0 1−ψ kψ k0 m0
+
,
∂f ¯ 1 ∂f ¯ (k(σ ), σ ) = (k(σ ), σ ) ≥ 0, ¯ ∂k [1 + k(σ )] ∂σ
(27)
with equality if and only if k0 = k¯ (σ ), where
∂f ¯ 1 π0 ¯ (k(σ ), σ ) = − f (k(σ ), σ ) π¯ (σ ) ln ∂σ (σ − 1)2 π¯ (σ ) 1 − π0 + (1 − π¯ (σ )) ln , 1 − π¯ (σ )
(28)
and π¯ (σ ) = π (k¯ (σ ), σ ). The long-run growth rate (16) is
(24)
1
θ
∂f ¯ (k(σ ), σ ) − δK − ρ , ∂k
and so, taking into account (27), we immediately get that
(25)
with equality if and only if k = k0 . Actually, this is consequence of the CES production function being a linear transformation of a general mean of order ψ = 1 − 1/σ – which is increasing in the elasticity of substitution – and that the general mean is an increasing function of its order (e.g., de La Grandville, 2009). Thus, production and production per unit of effective labor are increasing functions of the elasticity of substitution. 3.2. Elasticity of substitution and long-run growth First, we examine the effect of the elasticity of substitution on the long-run growth rate. In the steady state, Eq. (15) shows that the rental prices of physical and human capital net of depreciation are equal; i.e.,
∂f ¯ ∂f ¯ (k(σ ), σ ) − δK = f (k¯ (σ ), σ ) − k¯ (σ ) (k(σ ), σ ) − δH , ∂k ∂k
dγ¯ dσ
(σ ) = =
1 d
θ dσ
∂f ¯ (k(σ ), σ ) ∂k
1 1 ∂f ¯ (k(σ ), σ ) ≥ 0, θ [1 + k¯ (σ )] ∂σ
(29)
with equality if and only if k0 = k¯ (σ ). Hence, the positive growth effect of the elasticity of factor substitution is due the efficiency effect of Klump and de La Grandville (2000). The following proposition summarizes this result.6 Proposition 1. Consider two economies that initially differ only with respect to their elasticity of substitution, both satisfying the conditions in Assumption 1. The economy with the higher elasticity of substitution will have a higher long-run growth rate. Of course, another immediate consequence of the efficiency effect is that the agent in the economy with the higher elasticity of substitution will also enjoy a higher level of intertemporal utility (i.e., welfare). The reason is simple: given that ∂ f (k, σ )/∂σ > 0 for k ̸= k0 , the optimal solution of the low-elasticity economy is feasible in the high-elasticity economy. Hence, the optimal solution of the high-elasticity economy provides a higher welfare level than the optimal solution of the low-elasticity economy. 3.3. Elasticity of substitution and long-run factor income distribution Let us now consider the long-run distributional effect of the elasticity of substitution. From (24), as in Klump and de La Grandville (2000, equation 17), we have that
which can be rewritten as
∂f ¯ 1 (k(σ ), σ ) = [f (k¯ (σ ), σ ) − δH + δK ]. ∂k [1 + k¯ (σ )]
dσ
γ¯ (σ ) =
so that π0 = π (k0 , σ ) = k0 /(k0 + m0 ), which does not depend on the elasticity of substitution. Klump and de La Grandville (2000, Theorem 1) show that y is an increasing function of the elasticity of substitution σ ,
∂f 1 ∂y = (k, σ ) = − f (k, σ ) ∂σ ∂σ (σ − 1)2 π 1 − π0 0 × π ln ≥ 0, + (1 − π ) ln π 1−π
Using (26) to substitute for ∂ f /∂ k, we get that d
1−ψ k0
109
(26)
5 de La Grandville (2009, Ch. 3) provides a detailed derivation of the normalized CES production function.
k¯ (σ ) =
(1 − π0 ) π¯ (σ ) π0 (1 − π¯ (σ ))
σ σ−1
k0 .
(30)
6 Appendix A shows that this result also holds in a model where investments in physical and human capital are fixed fractions of output (e.g., Mankiw et al., 1992).
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M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
Differentiating this expression with respect to σ , we get that dk¯
(σ ) = −
dσ
k¯ (σ ) k(σ ) ln σ (σ − 1) k0
¯
which depends on the relative position of the baseline ratio of physical to human capital with respect to its stationary value. The indirect effect is given by
σ k¯ (σ ) dπ¯ + (σ ). (σ − 1) π¯ (σ )(1 − π¯ (σ )) dσ
(31)
Appendix B shows that the effect of the elasticity of substitution on the long-run physical capital income share can be obtained as dπ¯
(σ ) =
dσ
1
(1 + k¯ (σ ))(σ − 1)
π¯ (σ ) ln
+ k¯ (σ )(1 − π¯ (σ )) ln
π¯ (σ ) π0
1 − π0
1 − π¯ (σ )
,
(32)
(1) = π0 (1 − π0 ) ln
dσ
k(σ )
¯
k0
.
Thus, we immediately have that sgn
dπ¯ dσ
(1) = sgn(k¯ (σ ) − k0 ).
(33)
If σ ̸= 1, Eq. (32) entails that sgn
dπ¯ dσ
(σ ) = sgn(σ − 1) sgn(π¯ (σ ) − π0 ).
(34)
Given that
π(σ ¯ ) − π0 = π¯ (σ )(1 − π0 ) 1 −
k0
σ σ−1
k¯ (σ )
we have that sgn(π¯ (σ )−π0 ) = sgn(σ −1) sgn(k¯ (σ )−k0 ). Plugging this result into (34), and recalling (33), we finally conclude that sgn
dπ¯ dσ
(σ ) = sgn(k¯ (σ ) − k0 ).
Thus, we can state the following proposition. Proposition 2. Consider two economies that initially differ only with respect to their elasticity of substitution, both satisfying the conditions in Assumption 1. If the initial ratio of physical to human capital is below its steady-state value, the economy with the higher elasticity of substitution will have a higher steady-state physical capital income share. If the initial ratio of physical to human capital is above its steady-state value, the economy with the higher elasticity of substitution will have a lower steady-state physical capital income share. To get some insight on this result note that differentiating
π(σ ¯ ) = π (k¯ (σ ), σ ) with respect to σ we have that dπ¯ dσ
(σ ) =
∂π ¯ ∂π ¯ dk¯ (k(σ ), σ ) + (k(σ ), σ ) (σ ). ∂σ ∂k dσ
(35)
Therefore, the effect of the elasticity of substitution on the steadystate physical capital income share can be decomposed as the sum of the (direct) distribution effect, ∂π /∂σ (Irmen and Klump, 2009), and the effect of a change in k as a consequence of the change in the elasticity of substitution, (∂π /∂ k)(dk/dσ ). Klump and de La Grandville (2000, equation 10) find that the distribution effect is given by
The sign of this indirect effect depends on the effect of σ on the steady-state ratio of physical to human capital and on the elasticity of substitution being greater or lower than unity. In any case, the direct distribution effect dominates the indirect effect, so the overall effect depends only on the relative position of the baseline ratio of physical to human capital with respect to its baseline value.7
Finally, we examine the link between factor substitution and the growth rate along the transition. This link is positive in the Solow model, as shown by Klump and de La Grandville (2000), so we are interested on whether this result carries over to the model considered in this paper. Due to the complexity of the dynamic systems involved, this issue is probably analytically intractable in the present model, so we have to resort to numerical simulations. Only for illustrative purposes, let us consider the baseline point y0 = 0.25, k0 = 1 and m0 = 0.01, together with the parameter values ρ = 0.03, ψ = −2, θ = 1.05 and δ = δK = δH = 0.1365— which is set so that the steady-state growth rate is 2%. The ensuing steady-state values are k¯ = 4.6416, c¯ = 0.1749, and g¯ = 0.02. ¯ the transitional dynamics of the economy is Given that k0 < k, described by the system (17)–(18). Fig. 1(a) depicts in solid line the evolution of the growth rates of income, physical capital and consumption.8 To illustrate the effect of an increase in the elasticity of factor substitution, Fig. 1(a) depicts in dashed line the evolution of the growth rates when ψ = −1; so that the elasticity of substitution (ES) increases from σ = 0.33 to σ = 0.5. The corresponding steady-state values are k¯ = 10, c¯ = 0.3521 and g¯ = 0.0402. According with our previous analytical results (see also Appendix C), the steady-state ratio of physical to human capital and the long-run growth rate in the low-elasticity economy are lower than the corresponding ones in the high-elasticity economy. On the dynamic effects, Fig. 1(a) shows that an increase in the elasticity of substitution causes an unambiguous increase in the growth rates of income, physical capital and consumption at any time along the transition. As a result, income and physical capital will also be higher at any time in the economy with the higher elasticity of substitution. Let us now increase the value of θ from 1.05 to 4 and, as in the former case, set δ = δK = δH = 0.0775 so that the steadystate growth rate is 2%. With this parameterization, the steady state of the economy is k¯ = 4.6416, c¯ = 0.5077 and g¯ = 0.02 – ¯ and so, the transitional dynamics when ψ = −2. Again k0 < k, is described by the system (17)–(18). Fig. 1(b) depicts the growth rates of income, physical capital and consumption when ψ = −2 (solid line) and ψ = −1 (dashed line). In this case, an increase in the elasticity of substitution causes an unambiguous decrease in the growth rate of physical capital along the transition. The growth rate of income in the high-elasticity economy starts below and, eventually, overtakes the corresponding one in the low-elasticity
7 Appendix C shows that some of the former expressions can be simplified if we assume identical depreciation rates, δ = δK = δH . Furthermore, it shows that in ¯
∂π ¯ 1 k¯ (σ ) (k(σ ), σ ) = 2 π¯ (σ )[1 − π¯ (σ )] ln , ∂σ σ k0
(37)
3.4. Elasticity of substitution and transitional growth
if σ ̸= 1 and, taking the limit as σ tends to 1, dπ¯
∂π ¯ dk¯ (σ − 1) dk¯ (k(σ ), σ ) (σ ) = π¯ (σ )[1 − π¯ (σ )] (σ ). ∂k dσ dσ σ k¯ (σ )
(36)
this case we also have that sgn ddσk (σ ) = sgn(k¯ (σ ) − k0 ). 8 The growth rate of human is simply −δ until the steady state is reached at time
T , and thereafter it is the long-run growth rate γ¯ .
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
111
(a) Higher ES implies higher transitional growth rate of physical capital (θ = 1.05, δK = δH = 0.1365).
(b) Higher ES implies lower transitional growth rate of physical capital (θ = 4, δK = δH = 0.0775).
(c) Higher EOS implies first lower and then higher transitional growth rate of physical capital (θ = 2,
δK = δH = 0.1175).
Fig. 1. Dynamic effects of increasing the Elasticity of Substitution (ES). Note: Common parameter values are y0 = 0.25, k0 = 1, m0 = 0.01, ρ = 0.03, ψ = −2 (solid line),
ψ = −1 (dashed line).
economy. Finally, Fig. 1(c) considers the case in which θ = 2 and δ = δK = δH = 0.1175. Now, the time paths of the growth rates of income and physical capital in the high-elasticity economy are first below and then above the corresponding ones in the low-elasticity economy.9 4. Conclusions This paper has analyzed the link between the elasticity of substitution, long-run growth and factor income distribution. Most of the previous literature has studied this issue in one-sector models with (physical) capital and raw labor as inputs in which long-run growth is exogenously given. In contrast, we consider a one-sector endogenous growth model in which physical capital and human capital are the factors of production. We have shown that for two economies differing only in factor substitutability, if the initial ratio of physical to human capital is below its steadystate value, the economy with the higher elasticity of substitution will have a higher steady-state physical capital income share. This result should be reversed if the initial ratio of physical to human capital is above its steady-state value. In any case, the economy with the higher elasticity of substitution will have a higher longrun growth rate (and welfare). This result reinforces the finding first established in Irmen (2011) that the elasticity of substitution positively affects the endogenously determined long-run growth
9 The case in which k > k¯ is not illustrated because, given the symmetrical 0 role that physical and human capital play in this model, the implications for the growth rates of income, consumption and human capital along the transition would be similar to the ones discussed above.
rate of the economy. Finally, our numerical results show that a higher elasticity of substitution does not necessarily lead to a higher growth rate of income and physical capital (or human capital) along the transition. This sharply contrasts with what happens in the Solow model, in which there is an unambiguously positive link between factor substitution and the growth rate of income and (physical) capital along the transition (Klump and de La Grandville, 2000). The one-sector model with physical and human capital considered in this paper is admittedly simple. Its simplicity has the advantage that it allows to get explicit analytical results. However, an interesting extension would be to study theoretically the growth-substitutability nexus in more general endogenous growth models. This will be the subject of future research. Acknowledgments The author thanks the valuable comments of two anonymous referees. Financial support from the Spanish Ministry of Economics and Competitiveness through Grant ECO2014-57711-P is gratefully acknowledged. Appendix A. The model with exogenous saving rates Let us assume that gross investments on physical and human capital represent a constant fraction of output; i.e., IK = sK Y and IH = sH Y , with 0 < sK < 1, 0 < sH < 1, and sK + sH ≤ 1. Then, the stocks of physical and human capital evolve according to the dynamic equations K˙ = sK Y − δK K ,
˙ = sH Y − δH H . H
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M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
Along a balanced growth path, K , H and Y grow at the same constant rate
γ¯ (σ ) = sK
f (k¯ (σ ), σ ) k¯ (σ )
− δK = sH f (k¯ (σ ), σ ) − δH .
The steady-state ratio of physical to human capital satisfies that10 sK
f (k¯ (σ ), σ ) k¯ (σ )
∂f
where we have used (27) to substitute for ddσ [ ∂ k (k¯ (σ ), σ )]. Substituting (31) into the former expression, and using (30), we have that dπ¯ dσ
(σ ) =
− δK = sH f (k¯ (σ ), σ ) − δH .
Differentiating this expression with respect to σ , after simplification, we get that dk¯ dσ
(σ ) =
sK [k¯ (σ )]2 d − sH f (k¯ (σ ), σ ) . dσ sK f (k¯ (σ ), σ ) k¯ (σ )
On the other hand, we have that
∂f ¯ dk¯ ∂f ¯ f (k¯ (σ ), σ ) = (k(σ ), σ ) (σ ) + (k(σ ), σ ) dσ ∂k dσ ∂σ sH k¯ (σ ) d = π¯ (σ ) 1 − f (k¯ (σ ), σ ) sK dσ ∂f ¯ + (k(σ ), σ ), ∂σ d
and, therefore, sK d f (k¯ (σ ), σ ) = dσ sK (1 − π¯ (σ )) + π¯ (σ )sH k¯ (σ ) ∂f ¯ (k(σ ), σ ) ≥ 0, ×
∂σ
with equality if and only if k0 = k¯ (σ ). Hence, differentiating the long-run growth rate with respect to σ we get that dγ¯ dσ
(σ ) = sH
d f (k¯ (σ ), σ ) ≥ 0,
dσ
¯ Thus, we can state the following with equality if and only if k0 = k. result. Proposition 3. Consider two economies that initially differ only with respect to their elasticity of substitution. Then the economy with the higher elasticity of substitution will have a higher long-run growth rate. Appendix B. Derivation of dd πσ¯ (σ) Differentiating the physical capital income share at the steady state,
π(σ ¯ ) = π (k¯ (σ ), σ ) =
k¯ (σ )
∂f ¯ (k(σ ), σ ),
f (k¯ (σ ), σ ) ∂ k
with respect to σ , we have that dπ¯ dσ
∂f ¯ (k(σ ), σ ) f (k¯ (σ ), σ ) dσ ∂ k ∂f f (k¯ (σ ), σ ) − k¯ (σ ) ∂ k (k¯ (σ ), σ ) dk¯ + (σ ) dσ [f (k¯ (σ ), σ )]2 k¯ (σ ) ∂f ¯ ∂f ¯ − ( k(σ ), σ ) (k(σ ), σ ) ∂k [f (k¯ (σ ), σ )]2 ∂σ π¯ (σ )(1 − π¯ (σ )) dk¯ = (σ ) dσ k¯ (σ ) π¯ (σ ) − k¯ (σ )(1 − π¯ (σ )) ∂ f ¯ − (k(σ ), σ ), ∂σ f (k¯ (σ ), σ )
(σ ) =
k¯ (σ )
d
10 If δ = δ we simply have that k¯ (σ ) = k¯ = s /s . K H K H
1
σ
π¯ (σ )(1 − π¯ (σ )) ln
k0
k¯ (σ ) (σ − 1)[π¯ (σ ) − k¯ (σ )(1 − π¯ (σ ))] ∂ f ¯ + (k(σ ), σ ) ∂σ (1 + k¯ (σ ))f (k¯ (σ ), σ ) π¯ (σ )(1 − π¯ (σ )) (1 − π0 )π¯ (σ ) = ln (σ − 1) π0 (1 − π¯ (σ )) (σ − 1)[π¯ (σ ) − k¯ (σ )(1 − π¯ (σ ))] ∂ f ¯ + (k(σ ), σ ). ∂σ (1 + k¯ (σ ))f (k¯ (σ ), σ )
Finally, substituting (28) into the former expression, after simplification, we get (32). Appendix C. The model with identical depreciation rates, δ =
δK = δH
If δ = δK = δH we have that r = w at the steady state so, using (1) and (2), the steady-state ratio of physical to human capita can be expressed as k¯ (σ ) =
α(σ ) 1 − α(σ )
1/(1−ψ)
= k0 m−σ 0 .
The effect of the elasticity of substitution on this ratio can be readily obtained from
¯ k¯ (σ ) k(σ ) (σ ) = −k¯ (σ ) ln m0 = ln . dσ σ k0 dk¯
If the baseline ratio of physical to human capital is the stationary one, k0 = k¯ (i.e., the marginal rate of substitution is unitary, m0 = 1), then changes in the elasticity of substitution have no effect on the steady-state ratio of physical to human capital. If the baseline ratio of physical to human capital is lower than its stationary value, k0 < k¯ (i.e., m0 < 1), then dk¯ /dσ > 0, so the higher the elasticity of substitution the higher the steady-state ratio of physical to human capital. If k0 > k¯ (i.e., m0 > 1), then dk¯ /dσ < 0, so the higher the elasticity of substitution the lower the steady-state ratio of physical to human capital. In summary, the higher the elasticity of substitution is, the farther from its baseline value is the steadystate ratio of physical to human capital. We can state the following result.11 Proposition 4. Consider two economies that initially differ only with respect to their elasticity of substitution, both satisfying the conditions in Assumption 1. If the initial ratio of physical to human capital is below its steady-state value, the economy with the higher elasticity of substitution will have a higher steady-state ratio of physical to human capital. If the initial ratio of physical to human capital is above its steady-state value, the economy with the higher elasticity of substitution will have a lower steady-state ratio of physical to human capital.
11 This result does not necessarily hold when the depreciation rates of physical and human capital are different. In this case both terms in (31) are of different signs, and the overall effect may be uncertain. Extensive numerical simulations show that a change in the sign of dk¯ /dσ as σ increases is very unlikely but it cannot be discarded. It happens, for instance, for the parameter values y0 = 0.192, k0 = 2, ¯
m0 = 0.5, δK = 0.1 and δH = 0, when we have that ddσk (0.19) = 0.7269 > 0 ¯ with k¯ (0.19) = 0.8208 < k0 , ddσk (0.2) = 0.0782 > 0 with k¯ (0.2) = 0.8245 < k0 , dk¯ dσ
(0.3) = −0.9747 < 0 with k¯ (0.3) = 0.7492 < k0 , and with k¯ (0.6) = 0.5137 < k0 .
dk¯ dσ
(0.6) = −0.5882 < 0
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
In this case, we can also get a simplified expression for the steady-state physical capital income share as ∂f
π(σ ¯ ) = π (k¯ (σ ), σ ) =
k¯ (σ ) ∂ k (k¯ (σ ), σ ) f (k¯ (σ ), σ )
=
k¯ (σ ) 1 + k¯ (σ )
,
and, therefore, dπ¯ dσ
(σ ) =
1
dk¯
[1 + k¯ (σ )]2 dσ
(σ ) =
1
σ
π¯ (σ )[1 − π¯ (σ )] ln
k(σ )
¯
k0
.
In this case the indirect effect on the physical capital income share (37) can be obtained as
¯ ∂π ¯ dk¯ (σ − 1) k(σ ) . (k(σ ), σ ) (σ ) = π¯ (σ )[1 − π¯ (σ )] ln ∂k dσ σ2 k0 The sign of this indirect effect depends on the relative position of the baseline ratio of physical to human capital with respect to its steady-state value and on the elasticity of substitution being greater or lower than unity. In any case it is now evident that the direct distribution effect (36) dominates the indirect effect, so the overall effect (35) depends only on the relative position of the baseline ratio of physical to human capital. References Acemoglu, D., 2009. An Introduction to Modern Economic Growth. Princenton University Press. Barro, R.J., 1990. Government spending in a simple model of endogenous growth. J. Polit. Econ. 98 (5), S103–S125. Barro, R.J., Mankiw, N.G., Sala-i Martin, X., 1995. Capital mobility in neoclassical models of growth. Amer. Econ. Rev. 85 (1), 103–115. Barro, R.J., Sala-i-Martin, X., 2004. Economic Growth, second ed. The MIT Press. Ben-Porath, Y., 1967. The production of human capital and the life cycle of earnings. J. Polit. Econ. 75 (4), 352–365.
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Boucekkine, R., Diene, B., Azomahou, T., 2008. Growth economics of epidemics: A review of the theory. Math. Social Sci. 15 (1), 1–26. de La Grandville, O., 1989. In quest of the Slutsky diamond. Amer. Econ. Rev. 79 (3), 468–481. de La Grandville, O., 2009. Economic Growth—a Unified Approach. Cambridge University Press, Cambridge, New York. Diamond, P.A., 1965. National debt in a neoclassical growth model. Amer. Econ. Rev. 55 (5), 1126–1150. Duffy, J., Papageorgiou, C., 2000. A cross-country empirical investigation of the aggregate production function specification. J. Econ. Growth 5 (1), 87–120. Glewwe, P., Maïga, E., Zheng, H., 2014. The contribution of education to economic growth: A review of the evidence, with special attention and an application to Sub-Saharan Africa. World Dev. 59, 379–393. Gómez, M.A., 2003. Equilibrium dynamics in the one-sector endogenous growth model with physical and human capital. J. Econom. Dynam. Control 28 (2), 367–375. Gómez, M.A., 2015. Factor substitution is an engine of growth in an endogenous growth model with productive public expenditure. J. Econ. (in press). Hanushek, E.A., Woessmann, L., 2008. The role of cognitive skills in economic development. J. Econ. Lit. 46 (3), 607–668. Heckman, J.J., 1976. A life-cycle model of earnings, learning, and consumption. J. Polit. Econ. 84 (4), S11–S44. Irmen, A., 2011. Steady-state growth and the elasticity of substitution. J. Econom. Dynam. Control 35 (8), 1215–1228. Irmen, A., Klump, R., 2009. Factor substitution, income distribution and growth in a generalized neoclassical model. Ger. Econ. Rev. 10 (4), 464–479. Kan, K., Wang, Y., 2013. Comparing China and India: A factor accumulation perspective. J. Comp. Econ. 41 (3), 879–894. Klump, R., de La Grandville, O., 2000. Economic growth and the elasticity of substitution: Two theorems and some suggestions. Amer. Econ. Rev. 90 (1), 282–291. Lucas, R.J., 1988. On the mechanics of economic development. J. Monetary Econ. 22 (1), 3–42. Mankiw, N.G., Romer, D., Weil, D.N., 1992. A contribution to the empirics of economic growth. Quart. J. Econ. 107 (2), 407–437. Miyagiwa, K., Papageorgiou, C., 2003. Elasticity of substitution and growth: normalized CES in the Diamond model. Econom. Theory 21 (1), 155–165. Solow, R.M., 1956. A contribution to the theory of economic growth. Quart. J. Econ. 70 (1), 65–94. Xue, J., Yip, C.K., 2012. Factor substitution and economic growth: A unified approach. Macroecon. Dyn. 16 (4), 625–656.
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Capital–labor substitution and long-run growth in a model with physical and human capital Manuel A. Gómez ∗ Departamento de Economía Aplicada II, Facultade de Economía e Empresa, Universidade da Coruña, Campus da Coruña, 15071 A Coruña, Spain
highlights • • • •
We examine the link between factor substitution and endogenous growth. Previous work relied on one-sector models with exogenous long-run growth. We use a one-sector endogenous growth model with physical and human capital. The long-run growth rate is increasing in the elasticity of substitution.
article
info
Article history: Received 12 April 2015 Received in revised form 31 July 2015 Accepted 25 October 2015 Available online 31 October 2015
abstract Most of the literature analyzing the growth-substitutability nexus considers models in which long-run growth is exogenous. We study this link in an endogenous growth model with physical and human capital. We show that for two economies differing uniquely in factor substitutability, the one with the higher elasticity of substitution will have higher long-run growth. This is due to the efficiency effect of a higher factor substitution. Furthermore, if the initial ratio of physical to human capital is below (above) its steadystate value, the economy with the higher elasticity of substitution will have a higher (lower) steady-state physical capital income share. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Is the elasticity of substitution an engine of growth? From a theoretical viewpoint the answer is affirmative in the Solow (1956) model, as it was uncovered in the seminal contributions of de La Grandville (1989) and Klump and de La Grandville (2000). Miyagiwa and Papageorgiou (2003) show that such positive relationship does not necessarily hold in the Diamond (1965) overlapping-generations model. Irmen and Klump (2009) reconcile these findings by introducing possible asymmetries of savings out of factor income. They find that factor substitution has a positive efficiency effect and an ambiguous distribution effect on economic growth. If the savings rate out of capital income is sufficiently high, the efficiency effect dominates and the overall effect is positive, so ‘‘higher factor substitution [. . . ] works as a major engine of growth’’ (Irmen and Klump, 2009, p. 464). Xue and Yip (2012) present a comprehensive characterization of the link between the elasticity of substitution and the steady-state capital and output
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http://dx.doi.org/10.1016/j.mathsocsci.2015.10.004 0165-4896/© 2015 Elsevier B.V. All rights reserved.
per capita in the Solow, Ramsey–Cass–Koopmans, and Diamond models. If initial capital per capita is below its steady-state value, a higher elasticity of substitution generates a higher steady-state income per capita in the Solow and the Ramsey–Cass–Koopmans models, whereas the effect is ambiguous in the Diamond model. These works have two limitations in common. First, all of them consider one-sector models in which long-run growth is exogenously given. As a consequence, factor substitution can affect transitional but not long-run growth. Therefore, the question posed on the link between capital–labor substitution and long-run growth cannot be addressed. Second, they consider that the factors of production are (physical) capital and (raw) labor. However, considering effective labor – i.e., labor adjusted for human capital – rather than raw labor as a factor of production has become relatively standard (e.g., Ben-Porath, 1967, Heckman, 1976, Lucas, 1988). This is justified by an extensive empirical literature that confirms the productive role of human capital (see, e.g., the reviews by Hanushek and Woessmann, 2008, and Glewwe et al., 2014). Our purpose is to overcome these two limitations by introducing effective labor as an input in a model of endogenous growth. This paper studies the relationship between long-run growth and the elasticity of substitution between capital and effective labor. To this end, we consider the one-sector endogenous growth
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
model with physical and human capital (e.g., Barro and Sala-iMartin, 2004, and Gómez, 2003). In this way we achieve two goals. First, the long-run growth rate is endogenized and thus we can examine the effect of factor substitution on the long-run growth rate. Second, we introduce effective labor as a productive input and, therefore, we are still analyzing the effect of the elasticity of substitution between (physical) capital and (effective) labor. The model is admittedly simple in that physical and human capital enter in a symmetric fashion into the model. However, this assumption has also been made, for instance, in the influential works by Mankiw et al. (1992) and Barro et al. (1995) and, as Mankiw et al. (1992, p. 416) argue, ‘‘We believe that, at least for an initial examination, it is natural to assume that the two types of production functions are similar’’. Furthermore, this framework has been recently used, e.g., to study the growth effects of epidemics (Boucekkine et al., 2008), and to compare the development processes in China and India (Kan and Wang, 2013).1 The simplicity of the model will allow us to get clear-cut analytical results. We prove that the long-run growth rate is increasing with the elasticity of substitution between physical and human capital. Hence, for two economies differing uniquely in factor substitutability, the economy with the higher elasticity of substitution will feature a higher long-run growth rate. We show that this is due to the efficiency effect of Klump and de La Grandville (2000); i.e., a higher elasticity of factor substitution increases the productivity of inputs. Furthermore, an increase in the elasticity of substitution has a positive or negative effect on the steady-state physical capital income share depending on whether the baseline ratio of physical to human capital is lower or greater than its steady-state value, respectively. We show that this is consequence of the distribution effect (Irmen and Klump, 2009). Finally, we also study the effect of the elasticity of substitution on the growth rate along the transition.2 Klump and de La Grandville (2000) show that there exists a positive link between factor substitution and transitional growth in the Solow model.3 Therefore, an interesting question is whether this result carries over to the one-sector endogenous growth model with physical and human capital. As an analytical answer is probably intractable, we resort to numerical simulations. Unlike in the Solow model, our results show that a higher elasticity of substitution does not necessarily lead to a higher growth rate of income and physical capital (or human capital) along the transition in the present model. Related research has been recently made. Duffy and Papageorgiou (2000) estimate a CES production function with physical capital and human capital inputs, but they do not analyze the relationship between factor substitution and growth. Irmen (2011) first established a positive relation between the elasticity of substitution and the endogenously determined long-run growth rate, but he considers instead a multi-sector neoclassical model with endogenous capital- and labor-augmenting technical change. Thus, our results reinforce the findings in Irmen (2011). Kan and Wang (2013) analyze the accumulation of physical and human capital in a one-sector endogenous growth model with physical and human capital similar to that considered in this paper. However, they do not study the link between the elasticity of substitution, long-run growth and factor income distribution, as we do. Gómez (in press) shows that factor substitution between public and private inputs is positively related to the optimal (first-best) long-run growth
1 This model is also covered, e.g., in the textbooks by Barro and Sala-i-Martin (2004) and Acemoglu (2009). 2 I thank a referee for suggesting this study. 3 The elasticity of substitution has obviously no effect on long-run growth, which is exogenously given.
107
rate and welfare in the Barro (1990) endogenous growth model. In contrast, this paper studies the effect of capital–labor substitution – which is the effect typically analyzed in the literature – in a decentralized economy. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes the effect of factor substitutability on growth and factor income distribution. Section 4 concludes. 2. The model We consider the one-sector endogenous growth model with physical and human capital (Barro and Sala-i-Martin, 2004; Gómez, 2003) with CES technology. The economy is populated by a large number of identical and infinitely lived agents. For simplicity, and without loss of generality, we assume that population is constant and normalized to unity. Thus, individual and aggregate quantities coincide. 2.1. Firms Output Y is produced using physical capital K and human capital (effective labor) H by means of the CES technology Y = F (K , H ) = A[α K ψ + (1 − α)H ψ ]1/ψ , A > 0, 0 < α < 1, ψ < 1, where A is the productivity parameter, α is the distribution parameter and σ = 1/(1 − ψ) is the elasticity of substitution. Denoting y = Y /H and k = K /H, the production function in intensive form can be written as y = f (k) = F (k, 1) = A[α kψ + (1 − α)]1/ψ . Let r and w denote the rental prices of physical and human capital, respectively. Profit maximization entails that the rental prices of physical and human capital are their respective marginal products:
∂F (K , H ) = f ′ (k) = α Aψ [f (k)/k]1−ψ , ∂K ∂F w= (K , H ) = f (k) − kf ′ (k) = (1 − α)Aψ f (k)1−ψ . ∂H r =
(1) (2)
2.2. Agents The representative agent derives utility from consumption C in accordance with the intertemporal utility function ∞
U = 0
C 1−θ − 1 −ρ t e dt , 1−θ
θ > 0, ρ > 0,
(3)
where ρ is the rate of time preference and 1/θ is the elasticity of intertemporal substitution. The agent receives rents from physical and human capital and spends her income on accumulation of physical and human capital and consumption. The agent’s budget constraint is then rK + w H ≥ IK + IH + C ,
(4)
where IK and IH are gross investments in physical and human capital, respectively, which must each be nonnegative. The stocks of physical and human capital evolve according to K˙ = IK − δK K ,
(5)
˙ = IH − δ H H , H
(6)
where δK and δH are the depreciation rates of physical and human capital, respectively.
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M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
The agent maximizes her utility (3) subject to the budget constraint (4), the constraints on physical capital and human capital accumulation (5) and (6), respectively, and the constraints on non-negative investment, given K (0) and H (0). The current-value Hamiltonian of the agent’s problem is C 1−θ − 1
the transversality condition is satisfied and long-run growth is positive. Assumption 1. Parameter values are such that (θ − 1)γ¯ + ρ > 0 and γ¯ > 0.
φ ≥ 0, (rK + w H − IK − IH − C )φ = 0,
(10)
With reversible investments in physical and human capital, the ¯ model features no transitional dynamics. If k0 = K0 /H0 ̸= k, the ratio of physical to human capital jumps immediately to its stationary value k¯ – and the ratio of consumption to physical capital does so – adjusting K and H by discrete amounts so that K + H is kept constant. Thereafter, the economy evolves along its balanced growth path where all variables grow at the common rate γ¯ . However, if investments are irreversible this solution would violate one of the constraints on nonnegative investment. Now, if k0 < k¯ then H is initially abundant relative to K . The desire to lower H relative to K entails that the inequality IH ≥ 0 will be binding in an interval [0, T ], whereas IK = Y − C > 0. Hence, Eq. (8) implies ˙ . Using (11), (5), and that η = 0 and φ = λ, so that −θ C˙ /C = λ/λ ˙ /H = −δH , the dynamics of the economy in terms of k and c can H be described by
λ˙ = ρλ − ∂ H /∂ K = (ρ + δK )λ − φ r ,
(11)
k˙
H =
+ λ(IK − δK K ) + µ(IH − δH H ) 1−θ + φ(rK + w H − IK − IH − C ) + ηIK + ξ IH ,
where λ and µ are the shadow prices of physical and human capital, φ is the multiplier associated with the agent’s budget constraint, and η and ξ are the multipliers associated with the nonnegativity constraints on gross investment. The first-order conditions are
∂ H /∂ C = C −θ − φ = 0, ∂ H /∂ IK = λ − φ + η = 0, IK ≥ 0, η ≥ 0, ηIK = 0, ∂ H /∂ IH = µ − φ + ξ = 0, IH ≥ 0, ξ ≥ 0, ξ IH = 0, rK + w H ≥ IK + IH + C ,
µ ˙ = ρµ − ∂ H /∂ H = (ρ + δH )µ − φw,
(7) (8) (9)
(12)
together with the transversality condition4 lim e−ρ t λK = lim e−ρ t µH = 0.
t →∞
t →∞
c (13)
2.3. Equilibrium Let c ≡ C /H denote the ratio of consumption to human capital. Eqs. (7) and (10) immediately imply that φ > 0 and the agent’s budget constraint (4) is binding, Y = IK + IH + C ,
(14)
where we have used (1) and (2) to substitute for r and w . In the interior solution with IK > 0 and IH > 0, Eqs. (8) and (9) imply that λ = µ = φ . Eqs. (11) and (12) imply that the rental prices on physical and human capital net of depreciation must be equal, r − δK = w − δH ; i.e.,
¯ ′ (k¯ ) − δH . f ′ (k¯ ) − δK = f (k¯ ) − kf
(15)
Log-differentiating (7), using (11) and (1), the growth rate of consumption is constant and equal to
γ¯ =
1 ′ f (k¯ ) − δK − ρ .
θ
k c˙
(16)
It can be readily observed that, in the interior solution, C , K , H, IK , IH and Y must all grow at the same rate γ¯ . Gross investments in physical and human capital are positive since IK /K = γ¯ + δK and IH /H = γ¯ + δH > 0. The resource’s constraint (14) entails that the steady-state ratio of consumption to human capital is c¯ = (1 + k¯ )[(θ − 1)γ¯ + ρ]. It is feasible (i.e., strictly positive) if the transversality condition (13), which is equivalent to (θ − 1)γ¯ + ρ > 0, is satisfied. Henceforth, we assume that parameter values guarantee that
4 These conditions are sufficient because the current-value Hamiltonian is concave in the states and the controls.
= =
K˙ K C˙ C
− −
˙ H H ˙ H H
= =
f (k) k 1
θ
−
c k
− δK + δH ,
[f ′ (k) − δK − ρ] + δH .
(17) (18)
At time T , the net returns on each type of capital are equalized, ¯ and the constraint IH ≥ 0 becomes non-binding. From k(T ) = k, t = T on, the solution is given by k(t ) = k¯ and c (t ) = c¯ , and all quantities grow at the constant rate γ¯ . The solution of the system (17)–(18) with the three boundary conditions k(0) = k0 , k(T ) = k¯ and c (T ) = c¯ , yields the time paths k(t ) and c (t ) in [0, T ] together with the terminal time T . ¯ the desire to lower K relative to H entails that the If k0 > k, inequality IK ≥ 0 will be binding in an interval [0, T ], whereas IH = Y − C > 0. Hence, Eq. (9) implies that ξ = 0 and φ = µ, so that −θ C˙ /C = µ/µ ˙ . Using (12), (6), and K˙ /K = −δK , the dynamics of the economy can be described by k˙ k c˙ c
= =
K˙ K C˙ C
− −
˙ H H ˙ H H
= −f (k) + c + δH − δK , =
(19)
1 f (k) − kf ′ (k) − δH − ρ − f (k) + c + δH .
θ
(20)
At time T , the net returns on each type of capital are equalized, ¯ and the constraint IK ≥ 0 becomes non-binding. From k(T ) = k, t = T on, the solution is given by k(t ) = k¯ and c (t ) = c¯ , and all quantities grow at the constant rate γ¯ . The solution of the system (19)–(20) with the boundary conditions k(0) = k0 , k(T ) = k¯ and c (T ) = c¯ , yields the time paths k(t ) and c (t ) together with the terminal time T . 3. Factor substitution, growth and factor income distribution Since the works of de La Grandville (1989) and Klump and de La Grandville (2000), it is known that to study the relationship between the elasticity of substitution and economic growth, the underlying CES production function must be normalized. Therefore, we shall first briefly describe the normalization procedure proposed by Klump and de La Grandville (2000). Then, we will examine the link between the elasticity of substitution and longrun growth, long-run factor income distribution, and transitional growth.
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
Differentiating (26) with respect to σ we have that
3.1. The normalized CES production function Following Klump and de La Grandville (2000), the normalized CES production function in intensive form is5
1/ψ
y = f (k, σ ) = A(σ ) α(σ )kψ + [1 − α(σ )]
,
(21)
where the productivity and distribution parameters are
A(σ ) =y0
α(σ ) =
1−ψ
+ m0 k0 + m0
k0
1−ψ k0
1/ψ ,
(22)
+ m0
,
(23)
for given baseline values of k0 , y0 = f (k0 , σ ), and the marginal rate of substitution ∂f
m0 =
f (k0 , σ ) − k0 ∂ k (k0 , σ ) . ∂f (k , σ ) ∂k 0
Thus, Eq. (21) makes explicit the dependence of output in intensive form on the elasticity of factor substitution σ = 1/(1 − ψ). Let π denote the physical capital income share,
π = π (k, σ ) =
∂f ¯ 1 ∂f ¯ (k(σ ), σ ) = (k(σ ), σ ) dσ ∂ k [1 + k¯ (σ )] ∂σ 1 + − [f (k¯ (σ ), σ ) − δH + δK ] [1 + k¯ (σ )]2 ¯ 1 ∂f ¯ dk (k(σ ), σ ) (σ ). + dσ [1 + k¯ (σ )] ∂ k d
k
∂f (k, σ ) ∂k
f (k, σ )
=
1−ψ kψ k0 1−ψ kψ k0 m0
+
,
∂f ¯ 1 ∂f ¯ (k(σ ), σ ) = (k(σ ), σ ) ≥ 0, ¯ ∂k [1 + k(σ )] ∂σ
(27)
with equality if and only if k0 = k¯ (σ ), where
∂f ¯ 1 π0 ¯ (k(σ ), σ ) = − f (k(σ ), σ ) π¯ (σ ) ln ∂σ (σ − 1)2 π¯ (σ ) 1 − π0 + (1 − π¯ (σ )) ln , 1 − π¯ (σ )
(28)
and π¯ (σ ) = π (k¯ (σ ), σ ). The long-run growth rate (16) is
(24)
1
θ
∂f ¯ (k(σ ), σ ) − δK − ρ , ∂k
and so, taking into account (27), we immediately get that
(25)
with equality if and only if k = k0 . Actually, this is consequence of the CES production function being a linear transformation of a general mean of order ψ = 1 − 1/σ – which is increasing in the elasticity of substitution – and that the general mean is an increasing function of its order (e.g., de La Grandville, 2009). Thus, production and production per unit of effective labor are increasing functions of the elasticity of substitution. 3.2. Elasticity of substitution and long-run growth First, we examine the effect of the elasticity of substitution on the long-run growth rate. In the steady state, Eq. (15) shows that the rental prices of physical and human capital net of depreciation are equal; i.e.,
∂f ¯ ∂f ¯ (k(σ ), σ ) − δK = f (k¯ (σ ), σ ) − k¯ (σ ) (k(σ ), σ ) − δH , ∂k ∂k
dγ¯ dσ
(σ ) = =
1 d
θ dσ
∂f ¯ (k(σ ), σ ) ∂k
1 1 ∂f ¯ (k(σ ), σ ) ≥ 0, θ [1 + k¯ (σ )] ∂σ
(29)
with equality if and only if k0 = k¯ (σ ). Hence, the positive growth effect of the elasticity of factor substitution is due the efficiency effect of Klump and de La Grandville (2000). The following proposition summarizes this result.6 Proposition 1. Consider two economies that initially differ only with respect to their elasticity of substitution, both satisfying the conditions in Assumption 1. The economy with the higher elasticity of substitution will have a higher long-run growth rate. Of course, another immediate consequence of the efficiency effect is that the agent in the economy with the higher elasticity of substitution will also enjoy a higher level of intertemporal utility (i.e., welfare). The reason is simple: given that ∂ f (k, σ )/∂σ > 0 for k ̸= k0 , the optimal solution of the low-elasticity economy is feasible in the high-elasticity economy. Hence, the optimal solution of the high-elasticity economy provides a higher welfare level than the optimal solution of the low-elasticity economy. 3.3. Elasticity of substitution and long-run factor income distribution Let us now consider the long-run distributional effect of the elasticity of substitution. From (24), as in Klump and de La Grandville (2000, equation 17), we have that
which can be rewritten as
∂f ¯ 1 (k(σ ), σ ) = [f (k¯ (σ ), σ ) − δH + δK ]. ∂k [1 + k¯ (σ )]
dσ
γ¯ (σ ) =
so that π0 = π (k0 , σ ) = k0 /(k0 + m0 ), which does not depend on the elasticity of substitution. Klump and de La Grandville (2000, Theorem 1) show that y is an increasing function of the elasticity of substitution σ ,
∂f 1 ∂y = (k, σ ) = − f (k, σ ) ∂σ ∂σ (σ − 1)2 π 1 − π0 0 × π ln ≥ 0, + (1 − π ) ln π 1−π
Using (26) to substitute for ∂ f /∂ k, we get that d
1−ψ k0
109
(26)
5 de La Grandville (2009, Ch. 3) provides a detailed derivation of the normalized CES production function.
k¯ (σ ) =
(1 − π0 ) π¯ (σ ) π0 (1 − π¯ (σ ))
σ σ−1
k0 .
(30)
6 Appendix A shows that this result also holds in a model where investments in physical and human capital are fixed fractions of output (e.g., Mankiw et al., 1992).
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M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
Differentiating this expression with respect to σ , we get that dk¯
(σ ) = −
dσ
k¯ (σ ) k(σ ) ln σ (σ − 1) k0
¯
which depends on the relative position of the baseline ratio of physical to human capital with respect to its stationary value. The indirect effect is given by
σ k¯ (σ ) dπ¯ + (σ ). (σ − 1) π¯ (σ )(1 − π¯ (σ )) dσ
(31)
Appendix B shows that the effect of the elasticity of substitution on the long-run physical capital income share can be obtained as dπ¯
(σ ) =
dσ
1
(1 + k¯ (σ ))(σ − 1)
π¯ (σ ) ln
+ k¯ (σ )(1 − π¯ (σ )) ln
π¯ (σ ) π0
1 − π0
1 − π¯ (σ )
,
(32)
(1) = π0 (1 − π0 ) ln
dσ
k(σ )
¯
k0
.
Thus, we immediately have that sgn
dπ¯ dσ
(1) = sgn(k¯ (σ ) − k0 ).
(33)
If σ ̸= 1, Eq. (32) entails that sgn
dπ¯ dσ
(σ ) = sgn(σ − 1) sgn(π¯ (σ ) − π0 ).
(34)
Given that
π(σ ¯ ) − π0 = π¯ (σ )(1 − π0 ) 1 −
k0
σ σ−1
k¯ (σ )
we have that sgn(π¯ (σ )−π0 ) = sgn(σ −1) sgn(k¯ (σ )−k0 ). Plugging this result into (34), and recalling (33), we finally conclude that sgn
dπ¯ dσ
(σ ) = sgn(k¯ (σ ) − k0 ).
Thus, we can state the following proposition. Proposition 2. Consider two economies that initially differ only with respect to their elasticity of substitution, both satisfying the conditions in Assumption 1. If the initial ratio of physical to human capital is below its steady-state value, the economy with the higher elasticity of substitution will have a higher steady-state physical capital income share. If the initial ratio of physical to human capital is above its steady-state value, the economy with the higher elasticity of substitution will have a lower steady-state physical capital income share. To get some insight on this result note that differentiating
π(σ ¯ ) = π (k¯ (σ ), σ ) with respect to σ we have that dπ¯ dσ
(σ ) =
∂π ¯ ∂π ¯ dk¯ (k(σ ), σ ) + (k(σ ), σ ) (σ ). ∂σ ∂k dσ
(35)
Therefore, the effect of the elasticity of substitution on the steadystate physical capital income share can be decomposed as the sum of the (direct) distribution effect, ∂π /∂σ (Irmen and Klump, 2009), and the effect of a change in k as a consequence of the change in the elasticity of substitution, (∂π /∂ k)(dk/dσ ). Klump and de La Grandville (2000, equation 10) find that the distribution effect is given by
The sign of this indirect effect depends on the effect of σ on the steady-state ratio of physical to human capital and on the elasticity of substitution being greater or lower than unity. In any case, the direct distribution effect dominates the indirect effect, so the overall effect depends only on the relative position of the baseline ratio of physical to human capital with respect to its baseline value.7
Finally, we examine the link between factor substitution and the growth rate along the transition. This link is positive in the Solow model, as shown by Klump and de La Grandville (2000), so we are interested on whether this result carries over to the model considered in this paper. Due to the complexity of the dynamic systems involved, this issue is probably analytically intractable in the present model, so we have to resort to numerical simulations. Only for illustrative purposes, let us consider the baseline point y0 = 0.25, k0 = 1 and m0 = 0.01, together with the parameter values ρ = 0.03, ψ = −2, θ = 1.05 and δ = δK = δH = 0.1365— which is set so that the steady-state growth rate is 2%. The ensuing steady-state values are k¯ = 4.6416, c¯ = 0.1749, and g¯ = 0.02. ¯ the transitional dynamics of the economy is Given that k0 < k, described by the system (17)–(18). Fig. 1(a) depicts in solid line the evolution of the growth rates of income, physical capital and consumption.8 To illustrate the effect of an increase in the elasticity of factor substitution, Fig. 1(a) depicts in dashed line the evolution of the growth rates when ψ = −1; so that the elasticity of substitution (ES) increases from σ = 0.33 to σ = 0.5. The corresponding steady-state values are k¯ = 10, c¯ = 0.3521 and g¯ = 0.0402. According with our previous analytical results (see also Appendix C), the steady-state ratio of physical to human capital and the long-run growth rate in the low-elasticity economy are lower than the corresponding ones in the high-elasticity economy. On the dynamic effects, Fig. 1(a) shows that an increase in the elasticity of substitution causes an unambiguous increase in the growth rates of income, physical capital and consumption at any time along the transition. As a result, income and physical capital will also be higher at any time in the economy with the higher elasticity of substitution. Let us now increase the value of θ from 1.05 to 4 and, as in the former case, set δ = δK = δH = 0.0775 so that the steadystate growth rate is 2%. With this parameterization, the steady state of the economy is k¯ = 4.6416, c¯ = 0.5077 and g¯ = 0.02 – ¯ and so, the transitional dynamics when ψ = −2. Again k0 < k, is described by the system (17)–(18). Fig. 1(b) depicts the growth rates of income, physical capital and consumption when ψ = −2 (solid line) and ψ = −1 (dashed line). In this case, an increase in the elasticity of substitution causes an unambiguous decrease in the growth rate of physical capital along the transition. The growth rate of income in the high-elasticity economy starts below and, eventually, overtakes the corresponding one in the low-elasticity
7 Appendix C shows that some of the former expressions can be simplified if we assume identical depreciation rates, δ = δK = δH . Furthermore, it shows that in ¯
∂π ¯ 1 k¯ (σ ) (k(σ ), σ ) = 2 π¯ (σ )[1 − π¯ (σ )] ln , ∂σ σ k0
(37)
3.4. Elasticity of substitution and transitional growth
if σ ̸= 1 and, taking the limit as σ tends to 1, dπ¯
∂π ¯ dk¯ (σ − 1) dk¯ (k(σ ), σ ) (σ ) = π¯ (σ )[1 − π¯ (σ )] (σ ). ∂k dσ dσ σ k¯ (σ )
(36)
this case we also have that sgn ddσk (σ ) = sgn(k¯ (σ ) − k0 ). 8 The growth rate of human is simply −δ until the steady state is reached at time
T , and thereafter it is the long-run growth rate γ¯ .
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
111
(a) Higher ES implies higher transitional growth rate of physical capital (θ = 1.05, δK = δH = 0.1365).
(b) Higher ES implies lower transitional growth rate of physical capital (θ = 4, δK = δH = 0.0775).
(c) Higher EOS implies first lower and then higher transitional growth rate of physical capital (θ = 2,
δK = δH = 0.1175).
Fig. 1. Dynamic effects of increasing the Elasticity of Substitution (ES). Note: Common parameter values are y0 = 0.25, k0 = 1, m0 = 0.01, ρ = 0.03, ψ = −2 (solid line),
ψ = −1 (dashed line).
economy. Finally, Fig. 1(c) considers the case in which θ = 2 and δ = δK = δH = 0.1175. Now, the time paths of the growth rates of income and physical capital in the high-elasticity economy are first below and then above the corresponding ones in the low-elasticity economy.9 4. Conclusions This paper has analyzed the link between the elasticity of substitution, long-run growth and factor income distribution. Most of the previous literature has studied this issue in one-sector models with (physical) capital and raw labor as inputs in which long-run growth is exogenously given. In contrast, we consider a one-sector endogenous growth model in which physical capital and human capital are the factors of production. We have shown that for two economies differing only in factor substitutability, if the initial ratio of physical to human capital is below its steadystate value, the economy with the higher elasticity of substitution will have a higher steady-state physical capital income share. This result should be reversed if the initial ratio of physical to human capital is above its steady-state value. In any case, the economy with the higher elasticity of substitution will have a higher longrun growth rate (and welfare). This result reinforces the finding first established in Irmen (2011) that the elasticity of substitution positively affects the endogenously determined long-run growth
9 The case in which k > k¯ is not illustrated because, given the symmetrical 0 role that physical and human capital play in this model, the implications for the growth rates of income, consumption and human capital along the transition would be similar to the ones discussed above.
rate of the economy. Finally, our numerical results show that a higher elasticity of substitution does not necessarily lead to a higher growth rate of income and physical capital (or human capital) along the transition. This sharply contrasts with what happens in the Solow model, in which there is an unambiguously positive link between factor substitution and the growth rate of income and (physical) capital along the transition (Klump and de La Grandville, 2000). The one-sector model with physical and human capital considered in this paper is admittedly simple. Its simplicity has the advantage that it allows to get explicit analytical results. However, an interesting extension would be to study theoretically the growth-substitutability nexus in more general endogenous growth models. This will be the subject of future research. Acknowledgments The author thanks the valuable comments of two anonymous referees. Financial support from the Spanish Ministry of Economics and Competitiveness through Grant ECO2014-57711-P is gratefully acknowledged. Appendix A. The model with exogenous saving rates Let us assume that gross investments on physical and human capital represent a constant fraction of output; i.e., IK = sK Y and IH = sH Y , with 0 < sK < 1, 0 < sH < 1, and sK + sH ≤ 1. Then, the stocks of physical and human capital evolve according to the dynamic equations K˙ = sK Y − δK K ,
˙ = sH Y − δH H . H
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M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
Along a balanced growth path, K , H and Y grow at the same constant rate
γ¯ (σ ) = sK
f (k¯ (σ ), σ ) k¯ (σ )
− δK = sH f (k¯ (σ ), σ ) − δH .
The steady-state ratio of physical to human capital satisfies that10 sK
f (k¯ (σ ), σ ) k¯ (σ )
∂f
where we have used (27) to substitute for ddσ [ ∂ k (k¯ (σ ), σ )]. Substituting (31) into the former expression, and using (30), we have that dπ¯ dσ
(σ ) =
− δK = sH f (k¯ (σ ), σ ) − δH .
Differentiating this expression with respect to σ , after simplification, we get that dk¯ dσ
(σ ) =
sK [k¯ (σ )]2 d − sH f (k¯ (σ ), σ ) . dσ sK f (k¯ (σ ), σ ) k¯ (σ )
On the other hand, we have that
∂f ¯ dk¯ ∂f ¯ f (k¯ (σ ), σ ) = (k(σ ), σ ) (σ ) + (k(σ ), σ ) dσ ∂k dσ ∂σ sH k¯ (σ ) d = π¯ (σ ) 1 − f (k¯ (σ ), σ ) sK dσ ∂f ¯ + (k(σ ), σ ), ∂σ d
and, therefore, sK d f (k¯ (σ ), σ ) = dσ sK (1 − π¯ (σ )) + π¯ (σ )sH k¯ (σ ) ∂f ¯ (k(σ ), σ ) ≥ 0, ×
∂σ
with equality if and only if k0 = k¯ (σ ). Hence, differentiating the long-run growth rate with respect to σ we get that dγ¯ dσ
(σ ) = sH
d f (k¯ (σ ), σ ) ≥ 0,
dσ
¯ Thus, we can state the following with equality if and only if k0 = k. result. Proposition 3. Consider two economies that initially differ only with respect to their elasticity of substitution. Then the economy with the higher elasticity of substitution will have a higher long-run growth rate. Appendix B. Derivation of dd πσ¯ (σ) Differentiating the physical capital income share at the steady state,
π(σ ¯ ) = π (k¯ (σ ), σ ) =
k¯ (σ )
∂f ¯ (k(σ ), σ ),
f (k¯ (σ ), σ ) ∂ k
with respect to σ , we have that dπ¯ dσ
∂f ¯ (k(σ ), σ ) f (k¯ (σ ), σ ) dσ ∂ k ∂f f (k¯ (σ ), σ ) − k¯ (σ ) ∂ k (k¯ (σ ), σ ) dk¯ + (σ ) dσ [f (k¯ (σ ), σ )]2 k¯ (σ ) ∂f ¯ ∂f ¯ − ( k(σ ), σ ) (k(σ ), σ ) ∂k [f (k¯ (σ ), σ )]2 ∂σ π¯ (σ )(1 − π¯ (σ )) dk¯ = (σ ) dσ k¯ (σ ) π¯ (σ ) − k¯ (σ )(1 − π¯ (σ )) ∂ f ¯ − (k(σ ), σ ), ∂σ f (k¯ (σ ), σ )
(σ ) =
k¯ (σ )
d
10 If δ = δ we simply have that k¯ (σ ) = k¯ = s /s . K H K H
1
σ
π¯ (σ )(1 − π¯ (σ )) ln
k0
k¯ (σ ) (σ − 1)[π¯ (σ ) − k¯ (σ )(1 − π¯ (σ ))] ∂ f ¯ + (k(σ ), σ ) ∂σ (1 + k¯ (σ ))f (k¯ (σ ), σ ) π¯ (σ )(1 − π¯ (σ )) (1 − π0 )π¯ (σ ) = ln (σ − 1) π0 (1 − π¯ (σ )) (σ − 1)[π¯ (σ ) − k¯ (σ )(1 − π¯ (σ ))] ∂ f ¯ + (k(σ ), σ ). ∂σ (1 + k¯ (σ ))f (k¯ (σ ), σ )
Finally, substituting (28) into the former expression, after simplification, we get (32). Appendix C. The model with identical depreciation rates, δ =
δK = δH
If δ = δK = δH we have that r = w at the steady state so, using (1) and (2), the steady-state ratio of physical to human capita can be expressed as k¯ (σ ) =
α(σ ) 1 − α(σ )
1/(1−ψ)
= k0 m−σ 0 .
The effect of the elasticity of substitution on this ratio can be readily obtained from
¯ k¯ (σ ) k(σ ) (σ ) = −k¯ (σ ) ln m0 = ln . dσ σ k0 dk¯
If the baseline ratio of physical to human capital is the stationary one, k0 = k¯ (i.e., the marginal rate of substitution is unitary, m0 = 1), then changes in the elasticity of substitution have no effect on the steady-state ratio of physical to human capital. If the baseline ratio of physical to human capital is lower than its stationary value, k0 < k¯ (i.e., m0 < 1), then dk¯ /dσ > 0, so the higher the elasticity of substitution the higher the steady-state ratio of physical to human capital. If k0 > k¯ (i.e., m0 > 1), then dk¯ /dσ < 0, so the higher the elasticity of substitution the lower the steady-state ratio of physical to human capital. In summary, the higher the elasticity of substitution is, the farther from its baseline value is the steadystate ratio of physical to human capital. We can state the following result.11 Proposition 4. Consider two economies that initially differ only with respect to their elasticity of substitution, both satisfying the conditions in Assumption 1. If the initial ratio of physical to human capital is below its steady-state value, the economy with the higher elasticity of substitution will have a higher steady-state ratio of physical to human capital. If the initial ratio of physical to human capital is above its steady-state value, the economy with the higher elasticity of substitution will have a lower steady-state ratio of physical to human capital.
11 This result does not necessarily hold when the depreciation rates of physical and human capital are different. In this case both terms in (31) are of different signs, and the overall effect may be uncertain. Extensive numerical simulations show that a change in the sign of dk¯ /dσ as σ increases is very unlikely but it cannot be discarded. It happens, for instance, for the parameter values y0 = 0.192, k0 = 2, ¯
m0 = 0.5, δK = 0.1 and δH = 0, when we have that ddσk (0.19) = 0.7269 > 0 ¯ with k¯ (0.19) = 0.8208 < k0 , ddσk (0.2) = 0.0782 > 0 with k¯ (0.2) = 0.8245 < k0 , dk¯ dσ
(0.3) = −0.9747 < 0 with k¯ (0.3) = 0.7492 < k0 , and with k¯ (0.6) = 0.5137 < k0 .
dk¯ dσ
(0.6) = −0.5882 < 0
M.A. Gómez / Mathematical Social Sciences 78 (2015) 106–113
In this case, we can also get a simplified expression for the steady-state physical capital income share as ∂f
π(σ ¯ ) = π (k¯ (σ ), σ ) =
k¯ (σ ) ∂ k (k¯ (σ ), σ ) f (k¯ (σ ), σ )
=
k¯ (σ ) 1 + k¯ (σ )
,
and, therefore, dπ¯ dσ
(σ ) =
1
dk¯
[1 + k¯ (σ )]2 dσ
(σ ) =
1
σ
π¯ (σ )[1 − π¯ (σ )] ln
k(σ )
¯
k0
.
In this case the indirect effect on the physical capital income share (37) can be obtained as
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