II. MÉTODO
2.2. Variables, Operacionalización
The eigenvalues of J , represented by E ϑ , are the solutions of the characteristic equation:
2 (a d) (ad bc) 0
ϑ − + ϑ+ − = . (C.4)
These eigenvalues determine the stability properties of the system. If the Jacobian in (C.3) does not have an eigenvalue with negative real part, the BGP is unstable. If the Jacobian in (C.3) has exactly one eigenvalue with negative real part, the BGP is saddle-path stable. Let-ting the economy start from any initial value of x, c jumps immediately to the corresponding saddle-path value, and over time both x and c converge to their steady-state values given by (22) and (23). In other words, in the case of saddle-path stability the equilibrium-growth path (hereafter EGP) of x and c, on which the variables converge to their balanced-growth values, is unique. If the Jacobian in (C.3) has two eigenvalues with negative real parts, the BGP is fully stable. Irrespective of the initial value of x, both endogenous variables, x and c, will eventually approach their steady-state values. Expressed differently, an infinite number of EGP exist (indeterminacy).
At this point we want to determine the sign of the real parts of the eigenvalues of J . E To be able to do so, we make use of the following theorem:
Theorem 1. The number of roots with negative real parts equal two minus the number of sign changes in the scheme
( ) ( )
1&−Tr JE &Det JE
Proof. This is an application to our special case of the more general Routh-Hurwitz theorem (see Gantmacher, 1960).
Recalling that Tr J
( )
E is equal to the sum and Det J( )
E is equal to the product of the eigenvalues of J (see, e.g., Simon & Blume, 1994, Theorem 23.9), we apply Theorem 1 and E establish that a necessary and sufficient condition for saddle-path stability is Det J( )
E being negative. If Det J( )
E >0, indeterminacy (instability) arises for Tr J( )
E <0(
Tr J( )
E >0)
.The determinant of the Jacobian evaluated at the BGP reads in its general form:
( )
EDet J =ad bc− . After some simplifications, we arrive at:
( ) ( ) { ( ) ( ) ( ) }
We used above our necessary-and-sufficient condition for the existence of a balanced-growth equilibrium, namely G< . Notice that in the following we impose that the parameter values 1 always satisfy the restriction:
(
1 G)
n G(
1) (
1 1)
μρ + + Γ −κ κ ≤ ⎡⎣ + −κ μ + +μ κ μ μ+ Γ − − ⎤⎦ , (C.6) which ensures that γcrit∈[0,1]. We conclude from the preceding discussion that the
balanced-growth equilibrium, given by (22) and (23), is locally saddle-path stable for values of γ be-low γcrit.
In order to investigate the stability properties of our dynamical system for values of γ above γcrit, we consider now the trace of the Jacobian in (C.3). We use (C.5) to substitute for
γ in the equation for the trace. More precisely, we add an (infinitesimal small) number χ∈ℜ to the RHS of (C.5), and plug it into the expression for the trace. By doing so we ++
make sure that we are in the parameter range of a positive Det J
( )
E . Some major cancellation occurs, and we get:Making use of (C.6), one can show after some tedious algebra that Tr J
( )
E is alwaysnega-tive, provided that both eigenvalues of J are of the same sign, i.e., E Det J
( )
E >0. It follows that both eigenvalues of J must be negative, if E γ γ> crit, indicating the existence of multiple EGP that eventually approach the BGP (local indeterminacy).Appendix D: Social optimum
This Appendix considers the social optimum. First, we observe that production of each vintage of consumption goods in industry ω∈[0,1] requires the same technology. Hence it cannot be optimal to produce consumption goods at a lower quality level than state-of-the-art quality. When only goods at the current quality frontier are produced in each industry, subutil-ity (2) becomes:
To see that the social planner will choose the same R&D intensity X in each industry, ob-serve that the quantity of each state-of-the-art product consumed by an individual equals:
( )
to choose the distribution of R&D input across industries that maximizes:( )
( )
( )
( ) ( )
Equation (D.3) implies that at each point in time, the social planner wants to carry out all R&D in industries with the lowest X
( )
ω,t . Executing this policy throughout time will lead to X( )
ω,t =X t( ) and I( )
ω,t =I t( ) for all t (at some point in time all industry-specificWe plug this expression into the discounted utility function (1), while noting that ( )L t =ent
(
L0 ≡1)
, to obtain:As shown in Segerstrom (1998), this maximization problem can be reformulated using ( )
x t instead of ( )φ t as state variable. By solving x t( )=X e0 μφ( )t ent for ( )φ t and plugging the resulting expression into (D.4), the dynamic optimization problem the social planner solves takes the form:30
30 Notice that the time argument of functions is dropped in the remainder of this Appendix for the sake of nota-tional convenience.
( ) ( )
Dropping the constant term ( ) 0
0
∫
, the current-value Hamiltonian for the social planner’s problem is:with ξ denoting the costate variable associated with the state variable x.
For the application of Pontryagin’s maximum principle, the Hamiltonian has to be concave in its control variable I (Chiang & Wainwright 2005, chap. 20), which is indeed the case for our maximization problem. By Pontryagin’s maximum principle, we obtain as first-order condition:
We solve (I) for I and substitute the resulting expression into (II), which yields an equation of motion for the state variable ξ. Again solving (I) for I and plugging the resulting value into the state equation, we obtain a differential equation that governs the evolution of relative R&D difficulty, x.
The resulting two-dimensional system of nonlinear, autonomous differential equations can be written as:
In the balanced-growth equilibrium, (I) implies that x= =ξ 0. The
(
x=0)
isocline is upward sloping and the(
ξ=0)
isocline is downward sloping in( )
x,ξ space. There is a unique inter-section of both curves in the positive orthand of( )
x,ξ space, giving rise to the following bal-anced-growth solution of the social planner’s problem:2
opt n
A κ μρ
ξ ρμ
= + ,
opt A
x n
κμ κ μρ
= + .
Stability inferences of the BGP can be made by setting up the “maximized Hamiltonian“, which is obtained when we substitute the value of I derived from (I) into the Hamiltonian (D.5). According to Proposition 10 in Arrow & Kurz (1970) respectively Theorem 7.14 in Acemoglu (2008), if the maximized Hamiltonian is a strictly concave function of the state variable x for any given ξ, jumping onto the saddle path at time t=0 and staying on the saddle path until the BGP is reached represents an optimal path for the economy. It is easy to show that in our case the second-order derivative of the maximized Hamiltonian is strictly negative. Thus, we can establish that the unique BGP resulting from solving the social plan-ner’s optimal control problem is saddle-path stable.
The socially optimal research intensity is the same as in the decentralized case, which can be seen when setting the state equation equal to its balanced-growth value, x=0. This gives equation (30) in the main text. Further, the socially optimal R&D labor share (equation (31) in the main text) can be calculated by substituting x for opt x in L L nxI = μA.