Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case

Hence, the compensation of work effort is reduced by the ratio of the output elasticities
of human capital in the decentralized and centralized economy, i.e. by the ratios of
private and social marginal returns of human capital in goods production. These tax
revenues are then distributed to the owners of the physical capital stock. This result is
very intuitive and leads to an increased goods production in the decentralized economy.⁴
Note that the planner’s and the representative agent’s value functions are identical if we
apply the above condition on τw and use (6).

In the last two sections, we have studied both the centralized as well as the decen-
tralized version of the Uzawa-Lucas Model of Endogenous Growth. We have found the
two value functions and shown that the implied controls satisfy the Euler equations and
the transversality conditions. In the next section, we show that the solutions are saddle
path stable and determine their time-series implications.

5 Stability properties and time series implications of
the solutions

In this section, the aim is twofold. First, we want to determine the stability properties
of the two solutions. Second, we want to characterize the time-series properties. Lucas
(1988) points out that the growth rate of human capital along the balanced growth
path is given by B(1 - ubgp). Furthermore, he shows that the growth rates of physical
capital, output, and consumption are 1^-α+^^γ times the growth rate of human capital.
Mulligan and Sala-i-Martin (1993) use this property in order to introduce transformed
state-like and control-like variables. These new variables remain constant along the
balanced growth path. This stationarity together with the fact that the number of state
variables is reduced by one makes the analysis of growth models much simpler. Benhabib
and Perli (1994) follow this strategy and define the state-like variable xt and the control-
like variable qt . In principle, we apply the same strategy and argue that the DOP
is homogeneous in the initial conditions h0 = ha,0 and k0 . However, as in Bethmann
(2002) and Bethmann and Reiβ (2003), our consideration leads us to a different definition
of the control-like variable qt⁵ .

Because of the homogeneity in the initial conditions of the central planner’s DOP, we
define the state-like variable xt and the control-like variable qt as follows:

xt :=    '+γ and   qt := i-t+γ .

h_t 1^-α                        h_t 1^-α

Similarly, the representative agent’s DOP is homogeneous in its initial conditions. The
only difference is that we must distinguish between the representative agent’s stock of
human capital h and the economy-wide average stock of human capital ha . Therefore we
redefine the state-like variable xt and the control-like variable qt as:

x_t := —⅛_y-  and  q_t := —⅛₇-.

hth_a1^-α                      hth_a1^-α

a,t                              a,t

⁴ Uhlig and Yanagawa (1996) present an opposite result. They study a two period OLG model with
endogenous growth where lower labor income taxes correspond to higher capital income taxes. Thereby
the young generation is able to generate higher savings which in turn lead to higher growth.

⁵ The first paper studies a discrete time version of the deterministic Uzawa Lucas Model of Endogenous
Growth with full depreciation of human and physical capital while the second refers to continuous time
and no depreciation. In both papers, we apply the same definition of q as we do here. On the other
hand, Benhabib and Perli (1994) use q = c/k.

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