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Introduction

How would the imposition of a carbon tax affect the rate of carbon-reducing
technological progress in the U. S. economy? In the conventional
neoclassical framework, it would not: consumers and firms would substitute
away from carbon-intensive energy sources, while the rate of ''autonomous''
carbon-reducing technological change in the economy would be unaffected. In
the most sophisticated application of neoclassical methodology to the study
of carbon abatement, that of Jorgenson and his colleagues, such a tax might
in fact have a ''detrimental'' effect on technological progress. Hogan and
Jorgenson (1991) show that disaggregation and the assumption of non-neutral
technological change imply that a carbon tax would lower the rate of
technological change, an externality that would entail an additional,
typically unaccounted for, cost of climate protection.

Naively, such conclusions might seem counterintuitive. For, one could
suppose that new economic incentives for carbon abatement would draw
entrepenuerial resources to the creation of new carbon-reducing (or energy
efficient) technologies, thereby increasing the rate at which such
technologies appeared. Since this effect would be in addition both to the
substitution away from carbon-intensive fuels and to any underlying
autonomous trends, it would stand to reason that such an entrepeneurial
response would serve to lower the costs of climate protection relative to
conventional estimates.

This type of economic process - the creation of new technologies by
profit-seeking agents - is a central theme of the so-called ''new growth
theory.'' A variety of models in this literature have examined the
consequences of ''endogenizing''\ technological change in the sense of
making it responsive to economic incentives. It would therefore seem natural
that the methodology of new growth theory could be fruitfully brought to
bear on the carbon-abatement problem, and the type of optimistic story
sketched above subjected to rigorous critical scrutiny.

This theme has been pursued in recent papers by Nordhaus (1997) and by
Goulder and Schneider (1999). These researchers examine the implications of
endogenizing the creation of carbon-saving or energy-efficient technology by
explicitly modeling the research and development process, thereby stepping
beyond the ''autonomous trend'' approach that is standard in the climate
policy and integrated assessment arena. Their findings provide a cautionary
note on the degree to which endogenous technological change might lower the
costs of carbon abatement. They conclude that when the key ingredient in
research - human expertise or ''human capital'' - is scarce (in the sense of
either being in fixed supply or being costly to produce), then diverting it
to the creation of climate-friendly technologies would, while contributing
to carbon abatement, also entail an opportunity cost due to a reduction in
technological progress in other fields or to the diversion of resources to
its additional production. This ''crowding out'' serves to offset the
benefits arising from acceleration of research on carbon reduction.

This paper summarizes preliminary results from a further investigation of
this crowding out problem. The well-known model of endogenous technological
change devised by Romer (1990) is extended in a straightforward way to
include two research sectors that compete for human capital and that produce
different types of new knowledge or technological progress. The conclusion
is that a policy intervention to increase the allocation of human capital to
one of the research sectors, thereby accelerating its creation of knowledge,
need not ''crowd out'' progress in the other research sector, even when the
overall supply of human capital is fixed.

The source of this result is the allocation of human capital in the Romer
model. In the laissez-faire equilibrium in that model as well as in this
extension, the presence of intertemporal spillovers from research results in
under-supply of human capital to both research sectors. Thus, there is
''room'' for a diversion of human capital from other applications, such as
manufacturing, into research. This finding is consistent with Goulder's and
Schneider's additional conclusion that the presence of prior distortions can
either exacerbate or mitigate the crowding out problem. The overall
implication is to highlight the importance of empirical studies to estimate
the magnitudes of prior distortions (if any) in markets for carbon-saving or
energy-efficient technologies.

These results are preliminary for several reasons in addition to the fact
that only the most basic properties of the model are analyzed. First, the
Romer model itself, while providing a fundamental theoretical benchmark in
the new growth literature, exhibits a ''scale effect'' that limits it
empirical applicability. (This point is discussed in more detail in the
paper.) Second, the usual restriction to steady state or balanced growth
analysis, which is adhered to here, in this case results in knife-edge
sensitivity of the model to underlying parameters, so that the results are
not robust. Finally, this

The paper is organized as follows. I begin with a qualitative overview of
the Romer model. I then summarize its key details along with a description
of its extension for the present analysis. The following two sections
present the decentralized and optimal solutions of the model, respectively,
followed by discussion of how human capital allocations can be corrected. I
then present and discuss certain caveats to the analysis, including the
subject of scale effects in new growth models in general and the Romer model
in particular. I end with comments on the implications of the analysis for
further research.

Description of the Model

Romer's 1990 analysis builds on the following fundamental observation:

There are two symmetric (but not identical) ''industries,'' each comprising
three sectors as in the Romer model. In the first industry, a competitive
output sector employs human capital $H_{Y}$ and non-durable intermediate
goods $y$ to produce final output $Y$ that can be used either directly in
consumption or in the production of new intermediate goods. The intermediate
goods sector, in turn, combines this output with ''designs'' for new
products (in a manner to be described below) that it purchases from the
third, research sector. The research sector employs only human capital $%
H_{A} $ to produce these designs, which are indexed as $A.$ Given an
existing number $A$ of designs, aggregate production for the final output
sector is given by

\[
Y=H_{Y}^{1-\alpha }\dint_{0}^{A}y(i)^{\alpha }di, 
\]

where $0<\alpha <1.$ This representation incorporates technological change
as increasing variety.

Production of new designs by the research sector is given by

\[
\dot{A}=\delta _{A}H_{A}A, 
\]

where $\delta _{A}$ is a fixed productivity parameter.

The second industry is structured symmetrically, with final output denoted
as $Z$, the human capital employed in the final output sector as $H_{Z}$,
the intermediate good as $z$, human capital in research as $H_{B}$, and the
number of designs at a given time as $B.$ Thus, aggregate production for the
final output sector is given by

\[
Z=H_{Z}^{1-\beta }\dint_{0}^{B}z(j)^{\beta }dj, 
\]

where $0<\beta <1$ and it is assumed that $\alpha \neq \beta $, and the
production of new designs by the research sector is given by

\[
\dot{B}=\delta _{B}H_{B}B, 
\]

where $\delta _{B}$ is again a productivity parameter and it is assumed that 
$\delta _{A}\neq \delta _{B}$.

A representative household consumes quantities $C_{1},C_{2}$of the two final
goods, respectively, according to the utility function

\[
\dint_{0}^{\infty }\ln \left( C_{1}^{\eta }C_{2}^{1-\eta }\right) e^{-\rho
t}dt,
\]

where $\rho $ is the rate of time preference and $0<\eta <1.$ Utility is
maximized subject to the budget constraint

\[
\dint_{0}^{\infty }\left( q_{1}C_{1}+q_{2}C_{2}\right) e^{-r^{^{\prime }}t}dt%
\leq W_{0},
\]

where $r^{^{\prime }}$ is the nominal rate of interest and $W_{0}$ is the
present value of wealth at time zero.

Finally, the aggregate supply of human capital $H$ is assumed fixed, and the
human capital supply constraint is

\[
H_{Y}+H_{A}+H_{Z}+H_{B}=H. 
\]

Solution of the Model for a Balanced Growth Path

I now summarize the derivation of a balanced growth path for the model, that
is, an equilibrium in which the key variables grow at the same, constant
rate. Again, these calculations closely follow those in Romer (1990) and
Barro and Sala-i-Martin (1995).

First, letting $q_{1}$ denote the price of $Y,$ profit maximization by the
representative firm gives the wage rate $w$as

\[
w=q_{1}(1-\alpha )\frac{Y}{H_{Y}}, 
\]

and yields an inverse demand function

\[
p_{Y}(i)=\alpha q_{1}H_{Y}^{1-\alpha }y(i)^{\alpha -1}, 
\]

where $y(i)$ is the $i^{th}$intermediate good and $p_{Y}(i)$ its price.

In the intermediate goods sector, there is an infinite number of firms
(indexed by the interval $0$ to $A$),each of which purchases a design from
the research sector and combines it with output from the final goods sector
to produce its variety of intermediate good. Each of these firms monopolizes
production of its particular variety. Having incurred the cost of a design,
each solves the following problem at every point in time:

\[
\max p_{Y}(y)y-q_{1}y. 
\]

The solution to this problem gives the price charged and quantity produced
by each intermediate firm as

\[
p_{Y}=\frac{q_{1}}{\alpha },\;y=\alpha ^{\frac{2}{1-\alpha }}H_{Y}, 
\]

and the resulting monopoly profit as

\[
\pi _{Y}=q_{1}\frac{1-\alpha }{\alpha }y. 
\]

The market for new designs is competitive, so that the price of a design at
every date is equal to the present value of the monopoly profit earned from
its use in intermediate production.Thus, in steady state, this price $P_{A}$%
is given by

\[
P_{A}=\frac{1}{r}q_{1}\frac{1-\alpha }{\alpha }y. 
\]

The condition for equality of wages between the manufacturing and the
research sector in the $Y$ industry is

\[
q_{1}(1-\alpha )\frac{Y}{H_{Y}}=\delta _{A}P_{A}A, 
\]

which upon substitution and simplification yields the condition

\[
H_{Y}=\frac{r}{\delta _{A}\alpha }. 
\]

By the symmetry imposed, the parallel derivation for the $Z$industry yields
the condition

\[
H_{Z}=\frac{r}{\delta _{B}\beta }. 
\]

Next are equations relating outcomes in the two industries. First, equality
of wages between the two research sectors is equivalent to

\[
\delta _{A}P_{A}A=\delta _{B}P_{B}A, 
\]
which upon subsitution and simplification yields

\[
\frac{A}{B}=\frac{q_{2}}{q_{1}}\frac{(1-\beta )}{(1-\alpha )}\frac{\beta ^{%
\frac{2\beta }{1-\beta }}}{\alpha ^{\frac{2\alpha }{1-\alpha }}}. 
\]
Now let $\gamma $ be the common growth rate of the key variables in the
balanced growth path. The growth rate of new designs in the two research
sectors will be the same, that is,

\[
\gamma =\frac{\dot{A}}{A}=\delta _{A}H_{A}=\frac{\dot{B}}{B}=\delta
_{B}H_{B}. 
\]

Finally, with the form of utility specified above, the consumer's
optimization can be treated as a two-stage problem, in which the intra- and
inter-temporal components of the maximization are solved in turn. Then the
consumer's problem can be written as

\begin{gather*}
\max \dint_{0}^{\infty }\ln (C)e^{-\rho t}dt \\
\text{subject to}\dint_{0}^{\infty }qCe^{-r^{^{\prime }}t}dt\leq W_{0,}
\end{gather*}

where $C\equiv C_{1}^{\eta }C_{2}^{1-\eta }$ is composite consumption and $%
q\equiv \left( \frac{q_{1}}{\eta }\right) ^{\eta }\left( \frac{q_{2}}{1-\eta 
}\right) ^{1-\eta }$ is the ideal price index. Thus, the Euler equation for
the consumer's optimization problem is

\[
\frac{\dot{C}}{C}=r-\rho 
\]

where $r=r^{^{\prime }}-\frac{\dot{q}}{q}$ is the real interest rate, so
that along the balanced growth path,

\[
\frac{\dot{C}}{C}=\gamma . 
\]

To summarize, the balanced growth path is determined by the following set of
equations:

\begin{eqnarray*}
\gamma &=&r-\rho \\
r &=&\alpha \delta _{A}H_{Y} \\
r &=&\beta \delta _{B}H_{Z} \\
\gamma &=&\delta _{A}H_{A} \\
\gamma &=&\delta _{B}H_{B} \\
\frac{A}{B} &=&\frac{q_{2}}{q_{1}}\frac{(1-\beta )}{(1-\alpha )}\frac{\beta
^{\frac{2\beta }{1-\beta }}}{\alpha ^{\frac{2\alpha }{1-\alpha }}} \\
H &=&H_{Y}+H_{Z}+H_{A}+H_{B}.
\end{eqnarray*}

The equilibrium quantities of human capital allocated to each of its four
potential uses are the particular values of interest for the present
purpose. When these equations are solved, these values turn out to be

\begin{eqnarray*}
H_{Y} &=&\frac{\beta \left( \delta _{A}\delta _{B}H+\rho \left( \delta
_{A}+\delta _{B}\right) \right) }{\delta _{A}\Lambda } \\
H_{Z} &=&\frac{\alpha \left( \delta _{A}\delta _{B}H+\rho \left( \delta
_{A}+\delta _{B}\right) \right) }{\delta _{B}\Lambda } \\
H_{A} &=&\frac{\alpha \beta \delta _{A}\delta _{B}H-\rho \left( \alpha
\delta _{A}+\beta \delta _{B}\right) }{\delta _{A}\Lambda } \\
H_{B} &=&\frac{\alpha \beta \delta _{A}\delta _{B}H-\rho \left( \alpha
\delta _{A}+\beta \delta _{B}\right) }{\delta _{A}\Lambda },
\end{eqnarray*}

where $\Lambda =\alpha \delta _{A}+\beta \delta _{B}+\alpha \beta \left(
\delta _{A}+\delta _{B}\right) .$ 

The Pareto Optimum

The above values for the allocation of human capital in the decentralized
(balanced growth) equilibrium can be compared to those that would be found
by a ''social planner'' who corrected the distortions arising from monopoly
pricing in the intermediate sectors and intertemporal spillovers from
research. The planner maximizes the utility of the representative household
subject to the economy's resource constraints and the equations of motion
for production of new designs, that is, solves the problem

\[
\max \text{imize}\dint_{0}^{\infty }\ln \left( C_{1}^{\eta }C_{2}^{1-\eta
}\right) e^{-\rho t}dt
\]

subject to

\begin{eqnarray*}
Y &=&H_{Y}^{1-\alpha }Ay^{\alpha }=C_{1}+Ay \\
Z &=&H_{Z}^{1-\beta }Bz^{\beta }=C_{2}+Bz \\
\dot{A} &=&\delta _{A}H_{A}A \\
\dot{B} &=&\delta _{B}H_{B}B \\
H &=&H_{Y}+H_{Z}+H_{A}+H_{B}.
\end{eqnarray*}

The current-value Hamiltonian for this problem can be written as

\[
%TCIMACRO{\UNICODE[m]{0x126}}
%BeginExpansion
H\llap{\protect\rule[1.1ex]{.735em}{.1ex}}%
%EndExpansion
=\ln \left[ \left( H_{Y}^{1-\alpha }Ay^{\alpha }-Ay\right) ^{\eta }\left(
H_{Z}^{1-\beta }Bz^{\beta }-Bz\right) ^{1-\eta }\right] +\lambda _{1}\delta
_{A}H_{A}A+\lambda _{2}\delta _{B}B,
\]

where $\lambda _{1},\lambda _{2}$ are co-state variables. This problem can
be solved by first solving the ordinary non-linear programming problem

\begin{eqnarray*}
&&\max %TCIMACRO{\UNICODE[m]{0x126}}
%BeginExpansion
H\llap{\protect\rule[1.1ex]{.735em}{.1ex}}%
%EndExpansion
\\
&&subject\;to \\
H &=&H_{Y}+H_{Z}+H_{A}+H_{B}
\end{eqnarray*}

and then using the result to complete the full dynamic optimization. (See
Seierstad and Sydsaeter 1987, Chapter 4.)

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