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



stock of human capital is equal to zero. The schooling technology implies that the
potential marginal and average product of human capital coincide and are equal to
B ,
whereas the realized marginal and average products are equal to
B (1 - ut). Note that
the depreciation rate of human capital is 100 percent per period.

2.3 The goods sector

We assume an infinitely large number of profit-maximizing firms producing a single good.
They are using a Cobb-Douglas technology in physical capital
kt and effective work htut .
Furthermore, the average skill of workers
ha,t has a positive influence on total factor
productivity. Hence, output
yt is determined by:

yt = Atktα (utht)1 hγa,t.                                   (5)

The parameter α is the output elasticity of physical capital and we assume α (0, 1).
The parameter
γ is non-negative and measures the degree of the external effect of human
capital. If we set
ut equal to one, we get the potential output in the goods sector. The
homogeneity of the agents implies that:

ha,t =ht,     tN0.                                  (6)

The state variable At denotes total factor productivity. Throughout this chapter, we
assume that ln
At follows a first-order autoregressive process, i.e.:

ln At+1 = ρln At + εt,     t N0    and ε ~ N (0,σ2) .         (7)

This assumption is a generalization of Bethmann (2002), where A was taken as fixed.
The firm has to rent physical and human capital on perfectly competitive factor markets.
In the decentralized economy, the representative firm’s profit Π in period
t is given by:

Π (kt, ht ; At, ha,t) = Atktα (utht)1 hγa,t - rtkt - wtutht,

where the semicolon indicates that the whole paths of ha,t and At are treated as exogenous
by the representative firm. The first-order necessary conditions for the profit-maximizing
factor demands are:

_ ∂yt _ αyt   sιnd   ,,,x = dyt   (1 (1-α)yt                        (8)

rt ∂kt = kt   and  wt (utht) = utht .                    (8)

These market-clearing factor prices ensure that the zero-profit condition holds. Inserting
the prices into the agent’s budget constraint (3) yields:

τrαyt + τw(1 - α)yt = ct + kt+1,     t N0.                   (9)

2.4 The state sector in the decentralized economy

In each period t, we require the state’s budget to be balanced. Therefore:

(τr - 1) rtkt = (1 - τw) wtutht                            (10)

must hold for all t N0 . This means that if we consider a tax on physical capital returns,
we are subsidizing work effort at the same time and vice versa. This remark ends the
presentation of the model. In Section 3, we solve the centralized version of this model.



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