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

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

Dirk Bethmann*

Department of Economics
Humboldt University Berlin
Spandauer Straβe 1
D-10178 Berlin
[email protected]

June 13, 2005

Abstract

This paper extends the class of stochastic AK growth models with a closed-form
solution to the case where there are two capital goods in the model. To be precise,
we consider the Uzawa-Lucas model of endogenous growth with human and physical
capital. The extension holds, even if an external effect in the use of human capital
in goods production occurs. Using the “guess and verify” method, we determine
the value function of the social planner in the centralized economy and the value
function of the representative agent in the decentralized case. We show that the
introduction of income taxes on wages and of a subsidy on physical capital earnings
is able to help the decentralized economy in reaching the social optimum, while
keeping the policy maker’s budget balanced. Then the time series implications of
the model’s solution are derived. In Appendix to the paper the uniqueness of the
value functions is proved by using an alternative method.

Key words: closed-form solution, value function, saddle path stability, endogenous
growth

JEL Classifications: C61, C62

*The support of the Collaborative Research Center 649 ‘Economic Risk’ is gratefully acknowledged. I
would like to thank Michael C. Burda, Harald Uhlig, Mark Weder and seminar participants at Humboldt
University for helpful comments. My thanks go also to Prof. Paulo Brito, Prof. Roger Farmer, and par-
ticipants of the XIII^th Summer School of the European Economic Association ‘Endogenous Fluctuations’
in Lisbon, 9-14 September 2002.

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