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\multicolumn{2}{c} {\bfseries Problem Set 11} \\
\multicolumn{2}{c} {Due Tuesday, December 8}
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{\bfseries Human Capital Accumulation and Economic Growth}
This problem set asks you to use dynamic programming to characterize the solution
to a variant of the model of human capital accumulation and economic growth studied
by Robert Lucas, ``On the Mechanics of Economic Development,'' \emph{Journal of
Monetary Economics}, July 1988, pp.3-42 and, long before that, by Hirofumi Uzawa,
``Optimum Technical Change in an Aggregative Model of Economic Growth,''
\emph{International Economic Review}, January 1965, pp.18-31. The specific version
of that model considered here comes from Dirk Bethmann, ``A Closed-Form Solution of
the Uzawa-Lucas Model of Endogenous Growth,'' \emph{Journal of Economics}, January
2007, pp.87-107.
In this model, a representative consumer divides up his or her time into an amount
$u_{t}$ devoted to education, technical training, and other activities that add to
the stock of human capital and an amount $1-u_{t}$ devoted to producing goods and
services for consumption and investment in physical capital. Let $h_{t}$ and $k_{t}
$ denote the stocks of human capital and physical capital at the beginning of each
period $t=0,1,2,.
..$, let $c_{t}$ denote the amount of output consumed during each
period $t=0,1,2,.
..$, and assume that the two stocks evolve according to
\begin{equation}
\gamma u_{t} h_{t} \geq h_{t+1} \tag{1}
\end{equation}
and
\begin{equation}
k_{t}^{\alpha} [(1-u_{t})h_{t}]^{1-\alpha} \geq c_{t} + k_{t+1} \tag{2}