%Paper: ewp-it/9405001
%From: Jason Christian <jason@gordy.ucdavis.edu>
%Date: Tue, 10 May 1994 09:22:57 -0700 (PDT)

%Use Robert Solovay's checksum utility:
%checksum = "13604 1646 7950 67691"
\documentstyle[titlepage,harvard,curves,tabls]{article}
         \author{Jason E. Christian
         \thanks{The author is a Postdoctoral Scholar in the Department
	 of Agricultural Economics at U.C. Davis.}\\{\small\tt
	 jason@gordy.ucdavis.edu}}
         \title{Endogenous Process Innovation under Piracy and
	 Multinational Enterprise}

\newcommand{\Kb}{K_b}
\newcommand{\Lb}{L_b}
\newcommand{\Ka}[1]{K_a^{#1}}
\newcommand{\La}[1]{L_a^{#1}}
\newcommand{\Bm}[1]{\boldmath$#1$\unboldmath}

\newtheorem{proposition}{Proposition}
\begin{document}

\maketitle
 
\begin{abstract}
	 A $2\times 2\times 2$ model with endogenous process innovation
	 describes two regimes for international technology transfer:
	 multinational enterprise, in which the innovating firm receives
	 all rents from foreign and domestic use of the innovation, and
	 piracy, in which some are all of the rents are kept in the
	 technology-receiving country.  Piracy increases the unit
	 requirements for the factor which is scarce in the recipient
	 country, and decreases use of its abundant factor, reducing
	 income in the technology recipient.  Any piracy regime can be
	 dominated by a combination of no-piracy and transfers from the
	 innovator to the recipient, with increases in both welfare in
	 the recipient country and innovator profits.
\end{abstract}

\tableofcontents

%You may go ahead and copy the code used to build the figures in this
%paper, so long as you include a comment in the latex source document
%indicating that you copied them from me.
%
% Before moving on to the text, I put in various commands necessary to
% typeset the figures in the article.  the output from these commands is
% eventually stored in various \savebox bins, which are then used when the
% figure-floats are declared.  To skip over these commands, you may search
% for ```Introduction.''

\setlength{\unitlength}{.75mm}

% I start by defining several components of the economist's graphical
% toolbox. I typeset these into save-boxes, which can then be used
% repeatedly.  All of these were designed using a unit-length of .75 mm;
% to make things larger, one could simply set it to a longer length, such
% as by using \setlength{\unitlength}{1.5mm}, or
% \setlength{\unitlength}{0.1in}.  In addition, it is assumed that the
% figure box will be 105 units square, with the origin in and up five
% units (millimeters by default).  The final
% version is produced by
%   \begin{figure}
%   \centerline{\begin{picture}(105,115)
%   \put(0,10){\usebox{\Figbin}}
%   \end{picture}}
%   \end{picture}
% where the ``final version'' was the result of
%   \begin{picture}(105,105)(5,5)
%   ...
%   \end{picture}
% and was stored in the savebox \Figbin.
% 
% I begin by typesetting an x-y axis and a set of dots which can be used
% to set up a picture. To use them, just enter \put(0,0){\usebox{\Axes}}
% or \put(0,0){\usebox{\Graphdots}}

\newsavebox{\Axes}
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\begin{picture}(105,105)
  \thicklines
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\end{picture}
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% Without something that looks like a demand curve, we wouldn't be
% economists.  So we'll put one in a box for later use with a command that
% will retrieve the box and put letters on the ends.  I do the same for
% two differently-sloped things that could be supply curves.  The
% commands, and their usage, are
%     \Dem{topx}{topy}{toplabel}{bottomlabel}
%     \Supa{bottomx}{bottomy}{toplabel}{bottomlabel}
%     \Supb{bottomx}{bottomy}{toplabel}{bottomlabel}
% where toplabel and bottom labels are the bold-math labels at the top and
% bottom ends of the lines and topx, topy, bottomx, and bottomy are the x
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% I will want a curved line that goes through a specified point, then,
% eventually, becomes steep.  This line will be shifted, so will be set
% once and put in a savebox.  The command to actually use the curve has
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% Of course, when we make a set of axes, we usually like to label them.
% So I define an environment XYgraph, which as two arguments for the x
% and y axis labels, which are set as math. 

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% Typically we designate some points on our graphs with dots, which are
% then labelled.  I assume that these labels, like the labels used later
% on graphed lines and curves, are boldmath versions of the normal-sized
% text font.  I set the labels  2 millimeters northwest of the dot. 
% The arguments of the command are the (x,y) coordinates of the
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\put(#1,#2){
\begin{picture}(100,100)
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}

% A more general form of the \Dot command places the label in a box which is
% centered {x-offset}{y-offset} from the center of the dot.  Usage:
%  \Dotgen{x}{y}{x-offset}{y-offset}{label} 
% 

\newcommand{\Dotgen}[5]{
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}

% Another way of labelling things is to point arrows at them.  Here, I
% assume the labels are math.  To get regular characters, enclose in 
% an \mbox{}.  There are two commands, one where the label is to the right
% of the spot to be pointed at, and one where the label is to the left of
% the thing to be pointed at.  There are 6 (ugh) arguments to these
% commands, which have usage:
%   \RLabel{label-x}{label-y}{dx}{dy}{veclength}{labelmath}
% and
%   \LLabel{label-x}{label-y}{dx}{dy}{veclength}{labelmath}
% label-x and label-y designate the point where the label ends and the
% pointer begins. dx and dy describe the slope of the pointer, and *must*
% be integers between -4 and +4.  veclength is the projection of the
% pointer-vector onto the x-axis; if the pointer is vertical, which will
% look bad, it is the projection onto the y-axis.  veclength must be
% positive. For \RLabel, dx must be negative, for \LLabel dx must be
% positive.
% The pointer comes off the left edge of the
% middle of the label in \RLabel, which is to the right of the target, and
% off the right end of the label in \LLabel, which is to the left of the
% target.
% 
\newcommand{\RLabel}[6]{
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 \thinlines
 \put(1,0){\makebox(0,0)[l]{\footnotesize$\displaystyle #6$}}
 \put(0,0){\vector(#3,#4){#5}}
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\put(#1,#2){\begin{picture}(100,100)
 \thinlines
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 \put(0,0){\vector(#3,#4){#5}}
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% I'm going to typeset the figures into Saveboxes at the top of the file,
% then ``use'' the boxes as appropriate in the document.  That way one can
% read through the text in the finished file more easily, and in addition
% it is easier to build the graphs all at once.
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 }
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\newsavebox{\Thirdfig}
\sbox{\Thirdfig}{
\begin{XYgraph}{K_a^I}{p}
%\put(0,0){\usebox{\Graphdots}}
\Dem{15}{85}{M}{M}
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\newsavebox{\Fourthfig}
\sbox{\Fourthfig}{
\begin{XYgraph}{K_a^I}{p}
%\put(0,0){\usebox{\Graphdots}}
\Dem{15}{85}{M}{M}
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% When I built these graphs, I did them all at once, and printed them out
% for final revision and tweaking, using the following commands:
% 
%   \centerline{\usebox{\Firstfig}}
%   \centerline{\usebox{\Secondfig}}
%   \centerline{\usebox{\Thirdfig}}
%   \centerline{\usebox{\Fourthfig}}

\section{Introduction}
The development and international transfer of new products and processes
is and has been an ubiquitous feature of the modern economy.  In
addition to its major contribution to economic growth in the highly
industrialized countries \cite{ar:JEP-Griliches1988}, the exploitation
of innovations has motivated or facilitated a substantial portion of the
international activities---trade in products, investment, and
licensing---of businesses in these countries \cite{bk:NSB1988}.  Since
both R\&D expenditures and the stock of scientists and engineers
continue to grow,  in the United States and in other industrialized
countries, technological change will likely maintain its important role
in the evolution of the modern economy for the foreseeable future.

Technological change does not affect all agents in an economy equally.
\citeasnoun{ar:OEP-Findlay1959},\citeasnoun{ic:Jones1970} and
\citeasnoun{ic:Chipman1970} showed how differential rates of growth in
the productivity of different factors could change the terms of trade,
factor prices, and the distribution of purchasing power across
individuals and countries.  Technological change might be immiserizing
for some countries.  \citeasnoun{ar:WorldDev-Helleiner1975} proposed in
particular that one regime for the international transfer of
technology, multinational enterprise, encourages a labor-saving bias in
technological change which tends to reduce growth in the national
products of labor-rich less-developed countries.

The ideas contained in the Helleiner proposition had a major impact on
policy concerning the international transfer of technology towards less
industrialized countries.  Codes of Conduct were proposed for the
transfer of technology by multinational enterprises, joint ventures with
local partners were required, remittances of profits were taxed or
restricted, and other policies were developed which tended to discourage
multinational enterprise as a vehicle for the international transfer of
technology.

The analysis of regimes in economics is an exercise in comparisons.  In
order to have normative content, the Helleiner proposition must compare
the regime which is being criticized---multinational enterprise---to some
other arrangement.  The object of the present paper is to make such a
comparison explicit.  A model is developed containing the principal
features of the Helleiner proposition: technology developed in an
industrialized country for transfer within an enterprise to a less
developed country.  The alternative regime is described as ``piracy'':
the transfer of technology without compensation from the innovator to
the less developed country.  It is shown that the move away from a pure
multinational-enterprise regime tends to increase the unit requirements
for the factor which is scarce in $S$, in favor of greater savings of
$L$.  This tends to reduce gross national product in $L$-rich $S$, in
direct contradiction to the Helleiner proposition, an effect which is
reinforced by adverse movement in the terms of trade.  While piracy can
increase utility in $S$ even at the less labor-intensive technology, due
to the appropriation of rents in $S$, it is established that any piracy
regime can be improved upon by a combination of perfect appropriability
of technology---an absence of piracy--- and transfers from the innovator
to the host.  The latter conclusion refines the results of
\citeasnoun{ar:JPE-Feenstra1982}, which asserted the superiority of the
technology-transfer tariff over other taxes on international commerce.

Following a brief discussion of the relation of the model's principal
features to the existing literature,  the assumptions used are listed
explicitly, and the framework for endogenous technological change is
discussed.  The model is then laid out in the case of perfect
appropriability, or pure multinational enterprise.  Modification of the
model follows, to permit uncompensated international transfer of
technology, or piracy.  The comparative-static analysis of an increase
in piracy is then carried out, with evaluation from the standpoint of
welfare in the technology recipient.  Finally, possible extensions are
noted.

The model's conclusions rest upon endogenous variations in the direction
of process innovation.  The innovator has a fixed stock of research
resources, which can produce various combinations of unit input
requirements.  Once the choice is made, two fixed-coefficient techniques
are available:  a pre-existing ``public'' technology, freely available
to all, and the innovative technology.  The ``putty-clay'' formulation
follows \citeasnoun{ar:EJ-Binswanger1974}; the use of an isoquant to
represent innovation possibilities was proposed by
\citeasnoun{ar:EJ-Ahmad1966}.

Attention to issues of unemployed or underemployed resources---the
object of the Helleiner proposition and much of development
economics---requires use of a process-innovation framework.  Recent
advances in the international economics of technological change
\cite
{ar:JPE-Krugman1979,ar:JPE-Helpman1984,ar:RES-Jensen1987,ar:JPE-Grossman1989}
have built upon a product-innovation approach
within the framework of monopolistic competition.  While this framework
provides an equilibrium in the presence of fixed R\&D costs, it cannot
be applied to the question of factor-saving technological change.
However, abandoning this approach requires an alternative specification
of the industrial organization of technological change, which can be
found in the rent-extracting and rent-diffusing approach commonly used
in the industrial-organization literature
\cite{ar:EJ-Dasgupta1980,ar:QJE-Kamien1986,ar:QJE-Katz1986}.
It is useful to think of
independent researchers organizing themselves into a team to develop an
innovation, which is then sold to the highest bidder, using license fees
and royalties to extract the innovator's rent.

With fixed-coefficient technologies used in both countries, failure of
factor-price equalization requires specialization in at least one
country, which implies that one factor is unemployed.  This requirement
is acceptable, indeed desirable, in the present context, which considers
commerce in technology from the standpoint of a country with a surplus
factor in the sense that increases in the employment of that factor do
not involve an increase in its price.  Within a general-equilibrium
framework this requires that the slack  factor is either unemployed and
free or is available at a fixed price, such as a subsistence wage, from
a secondary market, as is described by the dual-market theory of
development economics \cite{bk:Meier1989}.  Alternatively, one might
suppose that the owners of the unemployed resource are paid a transfer
out of gross national product (the earnings of the fully employed
factor), or are part-owners of this resource.

In the pure-multinational-enterprise form of the model, total control of
the innovative technology is the basis for business operations across
countries.  This technology is the headquarters asset of
\citeasnoun{ar:JPE-Helpman1984} or the firm-specific advantage of
\citeasnoun{ar:JIntE-Markusen1984}.  The requirement that technology be
perfectly appropriable is not excessively strong, as
high-appropriability regimes do exist, allowing the innovator reasonable
control over the use of and rents from the new technology
\cite{ar:AER-Levin1988}.  Indeed, the present model analyzes explicitly
variations in appropriability.

\section{Assumptions and Definitions}
\begin{enumerate}
\item The world is made up of two countries, $N$ and $S$, two products, $a$
and $b$, and two factors, $K$ and $L$.
\item The technology for the production of good $b$ is given by the
fixed unit-input requirements $\left(\Kb,\Lb\right)$.  \item Production of good
$a$ can take place either with a freely-available public technology,
given by the fixed unit-input requirements
$\left(\Ka{P},\La{P}\right)$, or with the innovative technology given
by $\left(\Ka{I},\La{I}\right)$.

\item The innovative technology is picked from a set of techniques which
can be achieved with given research resources, found only in Country $N$
and organized within one enterprise, the {\em innovator}.
\item The innovative technology is non-drastic \cite{ic:Arrow1971}, so
that the (single) innovator producing commodity $a$ charges a price
equal to the unit cost under the public technology.
\item The innovator is a price taker in factor markets.
\item All consumers in both countries have identical preferences,
obeying the Cobb-Douglas utility function
\[
U_i=c_{ia}^{\alpha}c_{ib}^{1-\alpha},
\]
where $c_{ij}$ is the consumption by individual $i$ of commodity $j$.

\item Under either the public technology or any achievable innovative
technology the production of commodity $a$ is $L$-intensive relative to
commodity $b$, but not sufficiently so to fully employ factors in
Country $S$.  There is full employment of factors in $N$.  Therefore,
\[
\frac{\bar{K}^S}{\bar{L}^S}<
\frac{K_a^j}{L_a^j}<
\frac{\bar{K}^N}{\bar{L}^N}<
\frac{K_b}{L_b}, \;\;j=I,P,
\]
where $\left(\bar{K}^i,\bar{L}^i\right)$ are the endowments of factors
in country $i$.

\item Under free trade in commodities, let $S$ specialize in the
production of commodity $a$, under either the public or the innovative
technology.

\end{enumerate}

\section{The Induced-Innovation Framework}

To simplify the exposition, let the innovative technology be chosen from
a set of technologies available for given research expenditures.  This
is a restrictive approach, in the spirit in particular of
\citeasnoun{ar:EJ-Ahmad1966}. 

In a more general specification \cite{mi:Christian1993}, there is a
mapping $A(m)$ from the vector $m$ of expenditures in different research
projects into the input-output matrix $A$.  It can be shown that
decreasing returns to expenditures in the various projects are
sufficient to uniquely determine the distribution of research resources
across commodities and research projects, where the research resources
in each enterprise share the innovative rents.  Furthermore, dual to the
$A(m)$ mapping is a minimum-research-effort function, written in the
two-factor case for good $i$ as $h_i\!\left(K_i,L_i\right)$, which gives the
research
resources required to attain unit input requirements $\left(K_i,L_i\right)$.
In
the present context, if there are $\bar{R}$ units of the research
resource, used only in industry $a$, the following minimum  research
efforts hold:
\begin{eqnarray*}
h_a\!\left(K_a^I,L_a^I\right)&=&\bar{R}\\
h_a\!\left(K_a^P,L_a^P\right)&=&0
\end{eqnarray*}

The level set of $h_a$, for fixed expenditures $\bar{R}$, looks just
like the isoquant of elementary production theory. and is often called
an {\em Ahmad isoquant}, following \citeasnoun{ar:EJ-Ahmad1966}.
Unlike the neoclassical isoquant, there is no substitution {\em
ex-post}, and the requirements to attain this set may be substantial.

\section{The Perfect-Appropriability Equilibrium}

Let the innovator in $N$ completely control access to the innovative
technology in industry $a$, and be free to operate in either country.
The operation of a single firm in both countries is a representation of
multinational enterprise, operating here under conditions of perfectly
appropriable technology.  Suppose furthermore that there are no barriers
to trade in commodities.  Under such conditions, a world general
equilibrium can be described.

Factor-market equilibrium in $N$ determines production of commodities
$a$ and $b$ in $N$:
\begin{equation}\label{eqn:K-equil-N}
\bar{K}^N=K_a^N x_a^N + K_bX_b^N
\end{equation}
and
\begin{equation}\label{eqn:L-equil-N}
\bar{L}^N=L_a^N x_a^N + L_bX_b^N,
\end{equation}
where $x_i^N$ is the production of commodity $i$ in country $N$
($i=a,b$).  With non-drastic innovation, the price of commodity $a$,
$p_a$, is the unit cost under the public technology, while the price of
commodity $b$, $p_b$, is its unit cost:
\begin{equation}\label{eqn:pacalc}
p_a=K_a^P w_K^N + L_a^P w_L^N
\end{equation}
and
\begin{equation}\label{eqn:pbcalc}
p_b=K_b^P w_K^N + L_b^P w_L^N,
\end{equation}
where $w_j^N$ is the price in $N$ of factor $j$ ($j=K,L$).  With free
trade, commodity prices in $N$ yield the terms of trade,
\begin{equation}\label{eqn:termsoftrade}
p=\frac{p_a}{p_b}.
\end{equation}
Provided that $p>K_a^I / K_b$, country $S$ will specialize in commodity
$a$, with production limited by the $K$ constraint:
\begin{equation}\label{eqn:S-Kequil}
x_a^S=\frac{\bar{K}^S}{K_a^I}.
\end{equation}
With Cobb-Douglas utility, commodity-market equilibrium requires
\begin{equation}
\label{eqn:goodsequil}
p=\frac{\alpha}{1-\alpha} \frac{X_b^N}{x_a^N + x_a^S},
\end{equation}
where $\alpha$ is the Cobb-Douglas parameter.

For any given innovative technology $\left(K_a^I,L_a^I\right)$, equations
(\ref{eqn:K-equil-N})--(\ref{eqn:goodsequil}) determine the real
variables $x_a^N$, $x_b^N$, $x_a^S$, and $p$, and the ratios
$w_K^N/W_L^N$, $w_K^N/p$ and $w_K^N/p$.  Nominal variables $p_a$, $p_b$,
$w_K^N$ and $w_L^N$ can be found following selection of a numeraire.

To determine the actual technology under perfect appropriability and
free trade, consider the rent-maximization problem for the innovator.
In $N$, the innovator pays market-determined prices to the factors of
production.  In $S$, there remains unemployed $L$, so $L$ is free.
Since under either the public or the innovative technology $S$
specializes in the production of commodity $a$, the alternative cost of
the $K$ employed by the innovator in $S$ is the production foregone
under the public technology.  With no obstacles to use of the public
technology in $S$, this must be the cost actually incurred by the
innovator, so the world-wide profits, in nominal terms, are
\begin{eqnarray}
\nonumber \pi_a^I&=&
  x_a^N\left(p_a - w_K^N K_a^I - W_L^N L_a^I \right)
     + p_a \left(x_a^S - \frac{\bar{K}^S}{K_a^P}\right)\\
  &=& x_a^N\left(p_a - w_K^N K_a^I - W_L^N L_a^I \right)
     + p_a \left( \frac{\bar{K}^S}{K_a^I}- \frac{\bar{K}^S}{K_a^P}\right).
     \label{eqn:profits}
\end{eqnarray}

As the innovating firm by hypothesis recognizes no influence on factor
prices, nor does it recognize influence over product prices or on the
quantity sold.  Rather, the limit price (the public-technology
unit-cost) is accepted as given, and the innovating firm supplies the
entire market.  The firm's choice variables $\left(K_a^I,L_a^I\right)$
enter the maximization problem only as shown in (\ref{eqn:profits}), and
recursive effects on factor prices, commodity prices and quantities are
ignored.

The innovator's rent maximization problem is constrained by the
possibilities given by the Ahmad isoquant, 
\begin{equation}\label{eqn:ahmad}
h\!\left(K_a^I,L_a^I\right)=\bar{R}_a.
\end{equation}
If (\ref{eqn:profits}) is maximized subject to (\ref{eqn:ahmad}), the
the first-order conditions are
\begin{equation}\label{eqn:foncK}
w_K^N x_a^N + p_a\frac{\bar{K}^S}{{\left(K_a^I\right)}^2} + \lambda h_K =
0
\end{equation}
and
\begin{equation}\label{eqn:foncL}
w_L^N x_a^N +  \lambda h_L = 0,
\end{equation}
where $\lambda$ is the multiplier associated with constraint
(\ref{eqn:ahmad}), and $h_j$ is the research requirement for a further
one-unit reduction in the use of factor $j$.  Combining (\ref{eqn:foncK})
and (\ref{eqn:foncL}), profit maximization implies
\begin{equation}
\label{eqn:pi-max}
\frac{h_K}{h_L} = \frac{w_K^N}{w_L^N} + \frac{x_a^S}{x_a^N}
\frac{p_a}{w_L^N K_a^I},
\end{equation}
a condition which is illustrated in Figure \ref{fig:pi-max} at point
\boldmath$A$\unboldmath,
	\begin{figure} 
	\centerline{
	\begin{picture}(105,115)
        \put(0,10){\usebox{\Firstfig}}
	\end{picture}
	}
	\caption{\protect{\boldmath} Public technology ($A_0$) and
	optimal innovation under perfect appropriability ($A$) and
	under pure piracy ($A^P$).
	\protect{\unboldmath}\label{fig:pi-max}}
	\end{figure}
and which completes the model for an economy with free trade in
commodities and perfectly appropriability of innovations.  Note that for
$x_a^S$ close to zero, $h_K/h_L$ is close to $w_K^N/w_L^N$, which is
Kennedy's induced-innovation condition when factor prices in the home
country completely determine the direction of technological change
\cite{ar:EJ-Kennedy1964}.


\section{Equilibrium with Piracy}

Suppose that the authorities in country $S$ permit partial or full
copying of the innovation.  In particular, let producers in $S$ have
access to unit $K$ requirement
\begin{equation}
\label{eqn:K-pir}
K_a^S=\sigma K_a^I + (1-\sigma)K_a^P.
\end{equation}
The innovator must then pay the $K$-resources used in $S$ the value of
their alternative use, or $p_a\left(\bar{K}^S/K_a^S\right)$, so that the
innovator's rent-maximization problem becomes
\begin{equation}
\label{max:pir}
\max_{K_a^I,L_a^I} \left\{
  x_a^N \left(
     p_a - w_K^N K_a^I - W_L^N L_a^N
  \right) + p_a \left(
     \frac{\bar{K}^S}{K_a^I}  - \frac{\bar{K}^S}{K_a^S} 
  \right)
\right\}
\end{equation}
subject to
\[
h\!\left(K_a^I,L_a^I\right)=\bar{R}_a,
\]
with first-order conditions
\begin{equation}
\label{eqn:fonc-PirK}
  w_K^N x_a^N + p_a\left(
    \frac{\bar{K}^S}{{\left(K_a^I\right)}^2} 
    - \frac{\bar{K}^S}{{\left(K_a^S\right)}^2} \frac{dK_a^S}{dK_a^I}
  \right) + \lambda h_K = 0
\end{equation}
and
\begin{equation}\label{eqn:fonc-PirL}
w_L^N x_a^N +  \lambda h_L = 0.
\end{equation}
Substituting from (\ref{eqn:K-pir}), and noting that $dK_a^S/dK_a^I=\sigma$,
equation (\ref{eqn:fonc-PirK}) can be rewritten as
\begin{equation}
\label{eqn:fonc-PirKb}
  w_K^N x_a^N + p_a\left(
    \frac{\bar{K}^S}{{\left(K_a^I\right)}^2} 
    - \frac{\sigma\bar{K}^S}{{\left(\sigma K_a^I +(1-\sigma)K_a^P\right)}^2}
  \right) + \lambda h_K = 0.
\end{equation}
Note that for $\sigma=0$, corresponding to perfect appropriability,
(\ref{eqn:fonc-PirKb}) is identical to (\ref{eqn:fonc-PirK}), while for
$\sigma=1$, or pure piracy, (\ref{eqn:fonc-PirKb}) and (\ref{eqn:fonc-PirL})
yield the Kennedy condition $h_k/h_L=w_K^N/w_L^N$, as at point \Bm{A^P}
in Figure \ref{fig:pi-max}.  For an intermediate value of $\sigma$, the
innovator picks a technology such as that at \Bm{A^\sigma}, which is
more $K$-intensive than the perfect-appropriability point \Bm{A}.

In the analysis of the innovator's maximization problem, the changes in
output in $N$ which are necessary to maintain factor market equilibrium
are ignored:  it is assumed that the innovator takes factor-market
conditions as parametric, and does not calculate the changes in relative
prices that necessarily follows a change in the good $a$ production
technology.  However, in full equilibrium, quantities and prices used in
the optimization problem must also clear factor and commodity markets.


To analyze the effects of a piratical policy upon welfare in $S$, it is
necessary to consider the effects of the policy upon commodity prices.
Consumption possibilities for $S$ depend on the terms of trade for
exports from $S$, which in turn depend upon output in $S$ and $N$ of the
two goods.  As $\sigma$ increases, $K_a^I$ increases and $L_a^I$
decreases, so that production of commodity $a$ in $N$ must increase if
factor-market equilibrium is to be maintained.  Does this increase
outweigh the decrease in production in $S$ following the increase in the
unit $K$ requirement?  If so, does the consequent adverse effect on the
terms of trade outweight the increase in the earnings of $S$ resources
(GNP) following greater piracy?  Answers to these questions are
developed in the comparative-static exercise which follows.

\section{Comparative Statics: Increased Piracy}

The model with piracy can usefully be reduced, which permits a fairly
transparent graphical analysis.  Equations (\ref{eqn:K-equil-N}) and
(\ref{eqn:L-equil-N}) can be solved to give $x_a^N$ in terms of the
parameters and the endogenous commodity-$a$ technology.  Using these
substitutions, as well as (\ref{eqn:S-Kequil}), equation
(\ref{eqn:goodsequil}) can be rewritten as
\begin{equation}
\label{eqn:pricesched}
 p=\frac{\alpha}{1-\alpha} \frac{
  x_b^N \!\left( K_a^I,L_a^I \right)
  }{
  x_a^N \!\left( K_a^I,L_a^I \right) +
  x_a^S \!\left( K_a^I \right)
  }.
\end{equation}
Let commodity $b$ be the numeraire.  Then $p=r_a$, and equations
(\ref{eqn:pacalc}) and (\ref{eqn:pbcalc}) can be solved to give $w_j^N$
in terms of the parameters and $p$.  Making the resulting substitutions,
as well as those for commodity quantities, into (\ref{eqn:fonc-PirKb})
and (\ref{eqn:foncL}), and combining,
\begin{equation}
\label{eqn:innovsched}
\frac{h_K}{h_L}\!\left(K_a^I,L_a^I\left)
= \frac{w_K^N}{w_L^N}\!\right(p\right) 
  +\frac{p\bar{K}^S}
    {w_L^N \!\left( p \left)
      \cdot x_a^N \!\left( K_a^I,L_A^I \right)\right.\right.}
    \left[\frac{1}{\left[K_a^I\right]^2} 
      - \frac{\sigma}{\left[K_a^S\right]^2} \right],
\end{equation}
where the parentheses in (\ref{eqn:pricesched}) and
(\ref{eqn:innovsched}) indicate the endogenous terms in the
implicit-function partially-reduced forms for included endogenous
quantities and ratios.  Equations (\ref{eqn:pricesched}),
(\ref{eqn:innovsched}) and the the innovation constraint
(\ref{eqn:ahmad}) form a three-equation partially reduced system to
determine $p$, $K_a^I$ and $L_a^I$, where  equation
(\ref{eqn:pricesched}) describes the combinations of the three
remaining endogenous variables which are consistent with world
commodity-market equilibrium and with factor-market equilibrium in $N$,
while equations (\ref{eqn:ahmad}) and (\ref{eqn:innovsched}) describe
rent-maximizing innovation at equilibrium factor prices and commodity
prices and quantities.

Points consistent with market equilibrium (\ref{eqn:pricesched}) are
traced in $\left(p,K_a^I\right)$ space of Figure \ref{fig:increase} as the
schedule \Bm{M}\Bm{-}\Bm{M}.  To find the slope of \Bm{M}\Bm{-}\Bm{M},
differentiate
(\ref{eqn:pricesched}) with respect to $K_a^I$, where $L_a^I$ is
interpreted as an implicit function of $K_a^I$ using (\ref{eqn:ahmad}),
so that $dL_a^I=-h_K/h_L$:
\begin{equation}
\label{eqn:MMslope}
\left.\frac{dp}{dK_a^I}\right|_{MM} =
   \frac{\alpha}{(1-\alpha)\left(x_a^N + x_a^S \right)^2}
   \left\{
       \left(x_a^N + x_a^S\right) \frac{dx_b^N}{dK_a^I} - 
	x_b^N\left( \frac{dx_a^N}{dK_a^I} \frac{dx_a^S}{dK_a^I}  \right)
    \right\}.
\end{equation}
In the Appendix it is shown that a sufficient condition for the
right-hand side of (\ref{eqn:MMslope}) to be negative is that $h_K/h_L$ be
non-negative.  As this holds by hypothesis, the \Bm{M}\Bm{-}\Bm{M} schedule is
drawn with negative slope in Figure \ref{fig:increase}.  For points above
and to the right of \Bm{M}\Bm{-}\Bm{M}, demand for commodity $a$ is
insufficient to
clear the market in $N$ for factor $L$, which is equivalent to excess
demand for factor $K$; this imbalance can be corrected either by a lower
price for $L$-intensive commodity $a$, or by reduced demand for factor
$K$, achieved by lovering the unit $K$ requirement of commodity $a$. 

Full equilibrium of the model involves simultaneous satisfaction of the
rent maximization conditions (\ref{eqn:ahmad}) and
(\ref{eqn:innovsched}).  Combinations of $K_a^I$, $p$, $\sigma$, and
(implicitly) $L_a^I$ which are consistent with rent maximization can be
found.  For a given intellectual-property rule in $S$,
such as $\sigma^\ast$, and terms of trade $p^\ast$, one can find the
innovator's best choice of technology, characterized by the unit
$K$-requirement $K^\ast$.  If either $p$ or $\sigma$ changes, then the
optimizing choice of innovative technology also changes. 

{	\begin{figure} 
	\centerline{
	\begin{picture}(105,115)
        \put(0,10){\usebox{\Secondfig}}
	\end{picture}
	}
	\caption{Effects of increasing piracy:  innovation-choice
	schedule moves from $T$$-$$T(\sigma^\ast)$ to $T'$$-$$T'(\sigma')$,
	constant-utility schedule shifts from $U$$-$$U(\sigma^\ast)$ to
	$U'$$-$$U'(\sigma')$; equilibrium point $E'$ represents increased
	utility in $S$.\label{fig:increase}}
	\end{figure}
	}
For given $\sigma$, combinations of $K_a^I$ and $p$ consistent with rent
maximization can be plotted, such as, for $\sigma=\sigma^\ast$, the
schedule \Bm{T}\Bm{-}\Bm{T(\sigma^\ast)} of Figure \ref{fig:increase}.
Changes in $\sigma$ correspond to shifts in the \Bm{T}\Bm{-}\Bm{T}
schedule.  In the Appendix it is shown that if the second-order
conditions hold for rent-maximization(\ref{max:pir}), and holding
constant the quantities of good $a$ that are expected to be produced in
country $N$,, then (1) the
\Bm{T}\Bm{-}\Bm{T} schedule is positively sloped, and (2) an increase
in $\sigma$---increased piracy---shifts \Bm{T}\Bm{-}\Bm{T} to the
right.

For given $\sigma$, such as $\sigma^\ast$, general equilibrium is
represented by the intersection of the \Bm{M$$-$$M} and \Bm{T$$-$$T}
schedules, as at \Bm{E^\ast} in Figure \ref{fig:increase}, with
technology $K^\ast$ and commodity price $p^\ast$.  As the \Bm{M$$-$$M}
schedule remains constant for changes in $\sigma$, the full-equilibrium
effects of increasing piracy in $S$ can be represented by the move from
\Bm{E^\ast} to \Bm{E'}, with lower $p$ and higher $K_a^I$.

To describe the shape and movement of \Bm{T$$-$$T}, apply
(\ref{eqn:ahmad}) to (\ref{eqn:innovsched}), differentiate totally, and
rearrange to get
\begin{eqnarray*}
\lefteqn{\left[ H' + \frac{p}{w_L^N x_a^N} \frac{x_a^S}{K_a^I}
   \left( \tau_K + \frac{2(1-\tau)}{K_a^I} \right) \right] dK_a^I = }\\
   &&\left[W' + \frac{1-\tau}{w_L^N x_a^N}
     \frac{\bar{K}^S}{\left(K_a^I\right)^2}
     \left(1-\frac{p}{w_L^N}\frac{\partial w_L^N}{\partial p} \right)
      \right] dp -
      \frac{p}{w_L^N x_a^N} \frac{\bar{K}^S}{\left(K_a^I\right)^2}
        \tau_\sigma d\sigma,
\end{eqnarray*}
where $H\!\left(K_a^I\right) \equiv h_K/h_L$, $W\!(p)\equiv w_K/w_L$,
and $\tau\!\left(K_a^I,\sigma\right)=\sigma\left(K_a^I/K_a^S\right)^2$,
so that
\begin{equation}
\label{eqn:TTdpdK}
\left.\frac{dp}{dK_a^I}\right|_{TT}=
\frac
{H'+\frac{p}{w_L^N x_a^N} \frac{x_a^S}{K_a^I} \left( \tau_K +
  \frac{2(1-\tau)}{K_a^I}\right)}
{W' + \frac{1-\tau}{w_L^N x_a^N} \frac{\bar{K}^S}{\left(K_a^I\right)^2}
   \left(1-\frac{p}{w_L^N} \frac{\partial w_L^N}{\partial p} \right)}
>0,
\end{equation}
\begin{equation}
\label{eqn:TTdKdsig}
\left.\frac{dK_a^I}{d\sigma}\right|_{TT}=
\frac
{-\frac{p}{w_L^N x_a^N} \frac{\bar{K}^S}{\left(K_a^I\right)^2} \tau_\sigma}
{H'+\frac{p}{w_L^N x_a^N} \frac{x_a^S}{K_a^I} \left( \tau_K +
  \frac{2(1-\tau)}{K_a^I}\right)}
>0,
\end{equation}
and
\begin{equation}
\label{eqn:TTdpdsig}
\left.\frac{dp}{d\sigma}\right|_{TT}=
 \frac
{W' + \frac{1-\tau}{w_L^N x_a^N} \frac{\bar{K}^S}{\left(K_a^I\right)^2}
   \left(1-\frac{p}{w_L^N} \frac{\partial w_L^N}{\partial p} \right)}
{\frac{p}{w_L^N x_a^N} \frac{\bar{K}^S}{\left(K_a^I\right)^2} \tau_\sigma}
<0.
\end{equation}
The derivation of the signs of equations
(\ref{eqn:TTdpdK})--(\ref{eqn:TTdpdsig}) is shown in the Appendix.

An increase in $\sigma$ unambiguously reduces production in $S$, as well
as decreasing production in $N$ of commodity $b$.  But the
terms-of-trade deterioration and reduced output need not reduce welfare
in $S$:  the owners in $S$ of factor $K$ receive greater payments,
measured in physical units of commodity $a$.  To determine whether
welfare in $S$ improves  or deteriorates following increased piracy, it
is necessary to consider effects on welfare of simultaneous changes in
factor income and commodity prices.

It is straightforward to identify combinations of $K_a^I$ and terms of
trade $p$ which are consistent with constant welfare in country $S$,
where ``welfare'' is a function of consumption in $S$ of the two
commodities, and can be measured by the Cobb-Douglas utility function.
GNP for the constrained optimization in $S$ is derived solely from
payments to the owners in $S$ of factor $K$, which are employed
exclusively by the innovator for the production of commodity $a$.  These
payments are made (or at least are measured) in units of the exportable
good.  Since the best alternative use of these resources, at equilibrium
world prices, is the production of the commodity using the public, or
pirated, technology, purchasing power in terms of the numeraire is
\[
Y^S=p\left( \frac{\bar{K}^S}{K_a^S\!\left(K_a^I,\sigma\right)} \right).
\]

To find the combinations of $p$ and $K_a^I$ consistent with given
utility, derive the indirect utility function, which with Cobb-Douglas
preferences is 
\begin{equation}
\label{eqn:indutil}
U\!\left(p,K_a^I,\sigma\right) =
  C \frac{\bar{K}^S}{K_a^S\!\left(K_a^I,\sigma\right)} p^{1-\alpha}
\end{equation}
where $C$ is an irrelevant constant.  For any combination
$\left(p,K_a^I\right)$ such as $\left(p^\ast,K^\ast\right)$ on the
\Bm{M$$-$$M} schedule in Figure \ref{fig:increase}, there is a level set
of $U$ for given policy $\sigma^\ast$,
\[
U\!\left(p,K_a^I\right)=U\!\left(p^\ast,K^\ast,\sigma^\ast\right),
\]
traced as \Bm{U$$-$$U(\sigma^\ast)} in Figure \ref{fig:increase}, with
changes in $\sigma$ represented by shifts in the schedule.
Differentiating (\ref{eqn:indutil}) totally for constant $U$ and
simplifying gives
\[
0= - \frac{p\sigma}{K_a^S} dK_a^I 
  + (1-\alpha)dp - \frac{p}{K_a^S}\left( K_a^I - K_a^P \right) d\sigma,
\]
so that the slope and shifts of \Bm{U$$-$$U(\sigma^\ast)} are:
\begin{equation}
\label{eqn:UUdpdK}
\left.\frac{dp}{dK_a^I}\right|_{UU}= 
   \frac{p\sigma}{(1-\alpha)K_a^S} > 0,
\end{equation}
\begin{equation}
\label{eqn:UUdKdsig}
\left.\frac{dK_a^I}{d\sigma}\right|_{UU}= 
   \frac{K_a^P-K_a^I}{\sigma} > 0,
\end{equation}
and
\begin{equation}
\label{eqn:UUdpdsig}
\left.\frac{dp}{d\sigma}\right|_{UU}=
\frac{\left(K_a^I-K_a^P\right)p}{(1-\alpha)K_a^S}<0.
\end{equation}
Points above and to the left of \Bm{U$$-$$U(\sigma^\ast)} represent
increased utility compared to the situation at $(p^\ast,K^\ast)$.  An
increase in $\sigma$ from $\sigma^\ast$ to $\sigma'$ is represented in
Figure \ref{fig:increase} by a rightward shift in \Bm{U$$-$$U}.

The two schedules \Bm{U$$-$$U(\sigma^\ast)} and \Bm{U'$$-$$U'(\sigma')}
show the combinations $(p,K_a^I)$ consistent with the same utility,
given two intellectual-property regimes $\sigma^\ast$ and $\sigma'$.
With greater appropriation by $S$ of innovation rents, at $\sigma'$, the
original combination $(p^\ast,K^\ast)$ yields increased utility: $E^\ast$
is now above and to the left of the level curve representing the pre-change
utility level.  But $E^\ast$ is no longer an equilibrium point.  The
\Bm{T$$-$$T} schedule has also shifted to the right, so that the
equilibrium point moves down to the right on the \Bm{M$$-$$M} schedule,
to $E'$.  As drawn, $E'$ is above and to the left of
\Bm{U'$$-$$U'(\sigma')}, indicating that the increase in $\sigma$ has
improved welfare in $S$.

Whether an increase in $\sigma$ improves or harms welfare in $S$ depends
upon the slopes of the \Bm{T$$-$$T} and  \Bm{U$$-$$U} schedules, and on
the magnitudes of their shifts following a change in $\sigma$. 
Table \ref{tab:resultgrid} shows the relationship between these characteristics
and the welfare results in $S$ of an increase in piracy.
   \begin{table}
   \caption{\label{tab:resultgrid}
     Direction of change in welfare in $S$ following increased piracy of
     innovative technology, for different constant-utility ($U-U$) and
     rent-maximization ($T-T$) schedules.
      }
      \[
      \begin{array}{c||c|c}
      &\rule[-.03125in]{0in}{.375in}\left.\frac{dp}{dK_a^I}\right|_{UU} 
	> \left.\frac{dp}{dK_a^I}\right|_{TT}
      &\left.\frac{dp}{dK_a^I}\right|_{UU} <
\left.\frac{dp}{dK_a^I}\right|_{TT}
      \\\hline\hline
      \rule[-.03125in]{0in}{.375in}\left.\frac{dK_a^I}{d\sigma}\right|_{UU} >
        \left.\frac{dK_a^I}{d\sigma}\right|_{TT}  & + & +/- \\ \hline
      \rule[-.03125in]{0in}{.375in}\left.\frac{dK_a^I}{d\sigma}\right|_{UU} < 
        \left.\frac{dK_a^I}{d\sigma}\right|_{TT}  & +/- & -\\ \hline
      \end{array}
      \]
   \end{table}
Two unambiguous results should be noted.  If \Bm{T$$-$$T} is steeper than
and shifts to the right as least as much as \Bm{U$$-$$U}, then an
increase in $\sigma$ is immiserizing (and a decrease in $\sigma$
would improve welfare in $S$).  If \Bm{T$$-$$T} shifts to the right less
than and is not as steep as \Bm{U$$-$$U}, then an increase in $\sigma$
improves welfare (and a decrease in $\sigma$ is immiserizing).

\section{Two Policy Propositions}

The terms which determine whether or not a particular policy change is
welfare-enhancing depend themselves on the values of $\sigma$.  It is
therefore impossible to draw general policy conclusions about the
behavior of a welfare-maximizing government in $S$.  It is possible,
however, to derive conditions where a no-piracy policy is non-optimal
from the standpoint of $S$; this is the task of Proposition 1.
However, it is also shown, in Proposition 2, that any increase in $\sigma$, for
any initial point, represents deterioration in world welfare, in the
sense that the original point is a potential Pareto improvement.

\begin{proposition}
If {\em(i)} $x_a^S\leq x_a^N$ and if {\em(ii)}
\[
\frac{L_b}{K_b} \geq \frac{L_a^P}{K_a^P+\left(K_a^P-K_a^I\right)} 
   -
\frac{\left(K_a^P-K_a^I\right)w_K^N/w_L^N}{K_a^P+\left(K_a^P-K_a^I\right)},
\]
evaluated at $\sigma=0$, then an increase in the piracy rate $\sigma$
improves welfare in the technology-receiving country $S$.
\end{proposition}

Applying (\ref{eqn:UUdpdK}) to the case where $\sigma=0$, the
\Bm{U$$-$$U} schedule is horizontal at the no-piracy equilibrium,
shown as \Bm{E_0} in Figure \ref{fig:nopirate}.
However, applying (\ref{eqn:UUdKdsig}),
$\lim_{\sigma\rightarrow0}\left.\frac{dK_a^I}{d\sigma}\right|_{UU}=\infty$,
so that the results listed in Table \ref{tab:resultgrid} cannot be used.
Instead, since \Bm{T$$-$$T} has positive slope at $\sigma=0$, a
sufficient condition for welfare-improving piracy is that the downward
shift in \Bm{U$$-$$U} be at least as great as the simultaneous downward
shift in \Bm{T$$-$$T}, or
\[
1\geq\frac{\left(\frac{dp}{d\sigma}\right)_{TT}}
	  {\left(\frac{dp}{d\sigma}\right)_{UU}},
\]
evaluated at $\sigma=0$.  Applying (\ref{eqn:TTdpdsig}) and 
(\ref{eqn:UUdpdsig}), it may be shown that
\begin{equation}
\label{eqn:prop1proof}
\frac{\left(\frac{dp}{d\sigma}\right)_{TT}}
	  {\left(\frac{dp}{d\sigma}\right)_{UU}}
=(1-\alpha)\left(\frac{K_a^I}{K_a^P}\right)^2
  \left(\frac{x_a^S}{x_a^N + K_a^P/K_a^I}\right)
  \left(\frac{pK_b - K_a^P}{K_a^P - K_a^I}\right).
\end{equation}
The first two terms in this expression are individually less than 1; the
third term is strictly less than 1 if condition  {\em(i)} holds; it is
established as a Lemma in the appendix that the last term is less than
or equal to one if and only if condition {\em(ii)} holds.  Thus, the
validity of Proposition 1 is established.
{	\begin{figure} 
	\centerline{
	\begin{picture}(105,115)
        \put(0,10){\usebox{\Thirdfig}}
	\end{picture}
	}
	\caption{Perfect appropriability ($\sigma=0$) is suboptimal for
	$S$ if $\left.\frac{dp}{d\sigma}\right|_{UU} \geq
	 \left.\frac{dp}{d\sigma}\right|_{TT}$ .\label{fig:nopirate}}
	\end{figure}
	}

Note that condition {\em(ii)} is more easily satisfied the greater the
reduction in the unit $K$ requirement:  as $K_a^I$ falls from $K_a^P$ to
0, the last multiplicative terms of (\ref{eqn:prop1proof}) become small.
This is intuitively pleasing:  the greater the unit factor savings from
an innovation, the greater the incentive to steal it.

\begin{proposition}
Let $\pi(\sigma)$ be the innovator's rents if country $S$ practices
intellectual property policy $\sigma$, and let $U(\sigma,T)$ measure
utility in country $S$ with policy $\sigma$, where $S$ receives in
addition a transfer $T$.  Then for any $\sigma>0$, there exists a
transfer $T(\sigma)$ such that {\em(i)} $U(0,T(\sigma))>U(\sigma,0)$
and {\em(ii)} $\pi(o) - T(\sigma) > \pi(\sigma)$. 
\end{proposition}

Proposition 2 states that any piracy, $\sigma>0$, is an inferior
arrangement from the standpoint of both the innovator and country $S$,
since there exist transfers from the innovator to $S$ which, when
accompanied by a no-piracy policy in $S$, provide both higher utility in
$S$ and higher rents to the innovator than are possible under any
positive $\sigma$.  The validity of the proposition is established in
the remainder of this section.

For any $\sigma$, the first-order conditions (\ref{eqn:fonc-PirK}) and
(\ref{eqn:fonc-PirL}), combined with the innovation-constraint
(\ref{eqn:ahmad}), yield the unit $K$ requirement chosen by a
rent maximizing innovator, which may be described as $K_a^I(\sigma)$.
Suppose that an arrangement might be made whereby instead of pirating
technology, resources in $S$ receive  $\bar{K}^S/K_a^S(\sigma)$, the
payment that they would have received with  a policy of $\sigma$-piracy.
At the same time, $S$ sets $\sigma=0$.  That is, resources in $S$ are
paid their alternative costs  $\bar{K}^S/K_a^P$, plus a transfer
$T(\sigma)=\bar{K}^S/K_a^S(\sigma) - \bar{K}^S/K_a^P$.

Applying (\ref{eqn:UUdpdK}), when $\sigma=0$ utility in $S$ is
independent of the coefficient $K_a^I$ that is actually chosen, and the
\Bm{U$$-$$U} schedule becomes a horizontal line passing through the
piracy point \Bm{E'} in Figure \ref{fig:transfer}.  From
	\begin{figure} 
	\centerline{
	\begin{picture}(105,115)
        \put(0,10){\usebox{\Fourthfig}}
	\end{picture}
	}
	\caption{\protect{\boldmath} Transfers to $S$ and no piracy
	($E^\ast$) yields higher utility in $S$ and higher innovation
	rents than does piracy regime $E'$).
	\protect{\unboldmath}\label{fig:transfer}}
	\end{figure}
(\ref{eqn:innovsched}), when $\sigma=0$ the innovator's decision is
independent of the technology actually available in $S$, so the
arrangement can be represented as a simple shift of the \Bm{T$$-$$T}
schedule up and to the left, with general equilibrium at the no-piracy
point \Bm{E^\ast}.  As this point is above the \Bm{U'$$-$$U'} schedule
representing utility at $\sigma>0$, the elimination of the transfer,
accompanied by the transfer, improves utility in $S$, establishing
conclusion ({\em i}) of the proposition. 

To establish conclusion ({\em ii}) of the proposition, note that with
the transfer $T(\sigma)$ resources in $S$ are paid the same as they
would have been with  piracy $\sigma$ and no transfer.  The innovator
could have chosen the new technology represented by \Bm{E'}, and earned
the same rents as would have been received with no transfer.  However,
this is not profit-maximizing; the technology at \Bm{E^\ast} must
therefore yield a higher level of rents.  Thus, the innovator, as well
as $S$, is better off with the transfer and no piracy.


\section{Extensions}

Two extensions to the model deserve mention.  First, it is
straightforward to show that the principal results, as summarized in
Propositions 1 and 2, hold for other $S$ policies $\tau$ which extract a
portion of the total innovative rents generated in $S$.  Rather than
being understood as a function of the proportion of the savings in $K$
which are made public in $S$, $\tau$ can be interpreted as a
proportional tax on repatriated earnings, or as a share of profits which
must be shared with local partners who invested on favorable terms, or
even as a function of the period of patent protection.  In all cases,
the innovator can avoid part of the tax burden by paying less attention
to conditions in $S$ when choosing directions for research.

The second principal extension involves making exogenous not only the
direction but also the degree of effort of research and development.  In
a rent-extracting framework, there are two complementary approaches to
this task.  First, a fixed stock of a research resource might be divided
between industries, with inter-industry equilibrium requiring
equalization of rents per unit of research resource across industries.
Second, research resources might be drawn forth from an elastic supply
by their payment, which again can be specified as a share of rents.  In
either case, piracy in $S$ reduces total rents, which in turn drives
down both the level of research in the industry involved, and reduces
savings of the scarce factor $K$.  The argument presented here is
thereby strengthened. 

One might cite a host of strong assumptions used in the development of
this model as candidates for refinement, such as the requirement that
coefficients of production are constant {\em ex-post}, that the results
of research are known with certainty, or that one factor is free in the
technology-importing country.  Relaxing these assumptions is a
worthwhile program, whether one's goal is to advance the scientific
explanation  and prediction of the rate and direction of technological
change, or simply to demonstrate the impact of optimization in research
upon the political choice of regimes for the international transfer of
technology.  The present article may be interpreted in the latter
context, as a contribution to a debate in political economy, and as a
challenge to adherents of restrictions on multinational enterprise to
show that relaxing the strong assumptions changes significantly the
strong conclusions.

\appendix
\section{Mathematical Appendix}

\subsection{Proof: $h_K/h_L \leq 0 \;\;\Rightarrow\;\; \left.
dp/dK_a^I\right|_{MM} <
0$}

Solving (\ref{eqn:K-equil-N}) and (\ref{eqn:L-equil-N}),
\[
x_a^N=\frac{\bar{K}^N L_b - \bar{L}^N K_b}{K_a^I L_b -K_b L_a^I}
\]
and \[
x_b^N=\frac{\bar{L}^N K_a^I - \bar{K}^N L_a^I}{K_a^I L_b -K_b L_a^I},
\]
while (\ref{eqn:S-Kequil}) gives \[
x_a^S=\frac{\bar{K}^S}{K_a^I}.
\]
Then (\ref{eqn:pricesched}) can be rewritten
\[
 p=\frac{\alpha}{1-\alpha} \frac{
  \bar{L}^N K_a^I - \bar{K}^N L_a^I
  }{
 \bar{K}^N L_b - \bar{L}^N K_b +
  \bar{K}^S L_b - \bar{K}^S K_b \left( L_a^I/ K_a^I \right)
  }
  \]
so that 
\[
\left. \frac{dp}{dK_a^I}\right|_{MM} = \frac{\alpha}{
(1-\alpha)\left( \bar{K}^N L_b - \bar{L}^N K_b + \bar{K}^S L_b -
    \bar{K}^S K_b \left( L_a^I/K_a^I \right) \right) ^2
}
    \left\{M\right\},\]
where, using $dL_a^I/dK_a^I=-h_K/h_L$,
\begin{eqnarray*}
M & = &
\left( \bar{K}^N L_b - \bar{L}^N K_b + \bar{K}^S L_b - \bar{K}^S K_b
  \frac{L_a^I}{K_a^I} \right)
\left( \bar{L}^N + \bar{K}^N \frac{h_K}{h_L} \right)
\\
&&
- \left( \bar{L}^N K_a^I - \bar{K}^N L_a^I \right)
   \left( \bar{K}^S K_b \right)
   \left( \frac{1}{K_a^I}\frac{h_k}{h_L} + \frac{L_a^I}{\left(K_a^I\right)^2}
     \right)
\\
&=& \frac{h_K}{h_L}
\left(\bar{K}^N + \bar{K}^S \right)
   \left( \bar{K}^NL_b - \bar{L}^N K_b \right)
    + \bar{L}^N \left( \bar{K}^N L_b - \bar{L}^N K_b  \right)
\\
&&
 + \frac{\bar{K}^S}{K_a^I}
  \left[
   \bar{L}^N \left( K_a^I L_b - K_b L_a^I  \right)
   - K_b \left( \bar{L}^N K_a^I - \bar{K}^N L_a^I  \right)
\right]
\\
&<&0\\
&&\;\;\;\;\Longleftrightarrow \;\; \frac{dp}{dK_a^I} < 0 .
\end{eqnarray*}
Multiplying through by $\left[ - \left(\bar{K}^N +
\bar{K}^S\right)\left(\bar{K}^N L_b - \bar{L}^N K_b\right)\right]^{-1}<0$
and rearranging, the condition for a downward-sloping \Bm{M$$-$$M}
becomes
\begin{eqnarray*}
\frac{h_K}{h_L}&<&
   \frac{-\bar{L}^N}{\bar{K}^N + \bar{K}^S}
   \left\{ 1 + \frac{\bar{K}^S}{K_a^I}
      \left[
	 \frac{K_a^I L_b - K_b L_a^I}{\bar{K}^N L_b -\bar{L}^N K_b}
	  - \frac{K_b}{\bar{L}^N}
	   \left(
	 \frac{K_a^I \bar{L}^N - \bar{K}^N L_a^I}{\bar{K}^N L_b -\bar{L}^N K_b}
	   \right)
      \right]
   \right\}
   \\
   &&= 
   \frac{-\bar{L}^N}{\bar{K}^N + \bar{K}^S}
   \left\{ 1 + \frac{x_a^S}{x_a^N}
      \left(
      1 - \frac{x_b^N L_b}{\bar{L}^N}
      \right)
   \right\}
   \\
   &&= 
   \frac{-\bar{L}^N}{\bar{K}^N + \bar{K}^S}
   \left\{ 1 + \frac{x_a^S}{x_a^N}
      \left(
      \frac{x_a^N L_a^I}{\bar{L}^N}
      \right)
   \right\} \;<\;0.
\end{eqnarray*}
A sufficient condition is therefore that $h_L$ be negative and $h_K$ be
non-positive, which holds by hypothesis.

\subsection{Slope and $\sigma$-shift of the $T-T$ Schedule}

Let $H\!\left(K_a^I\right)
 =\frac{h_K}{h_L}\!\left(K_a^I,L_a^I\!\left(K_a^I\right)\right)$, 
 $W\!(p)=\left.\left.\frac{w_K^N}{w_L^N}\!\right(p\right)$, and
$\tau\!\left(K_a^I,
 \sigma\right) = \sigma \cdot \left[ \frac{K_a^I} 
			 {K_a^S\!\left(K_a^I,\sigma\right)}  \right]^2$.
Then (\ref{eqn:innovsched}) can be rewritten
\[
H\!\left(K_a^I\right)
= W\!(p) 
  +\frac{p}
    {w_L^N \!\left( p \left)
      \cdot x_a^N \right.\right.}
    \frac{\bar{K}^S}{\left[K_a^I\right]^2} 
      \left[1-\tau\!\left(K_a^I,\sigma\right)\right].
\]
Differentiating totally, substituting $x_a^S=\bar{K}^S/K_a^I$, 
and rearranging,
\begin{eqnarray}
\nonumber\lefteqn{
\left[ 
  H'+\frac{p}{w_L^N K_a^I} \frac{x_a^S}{x_a^N}\left( \tau_K +
  \frac{2(1-\tau)}{K_a^I}\right)
\right] dK_a^I
}\\
\label{eqn:TTdiff}&=&
\left[ 
  W'+\frac{1-\tau}{w_L^N K_a^I} \frac{x_a^S}{x_a^N}\left(
  1-\frac{p}{w_L^N}\frac{\partial w_L^N}{\partial p}\right)
\right] dp 
- \frac{p}{w_L^N K_a^I} \frac{x_a^S}{x_a^N} \tau_\sigma d\sigma.
\end{eqnarray}
The slope along \Bm{T$$-$$T} is therefore
\begin{equation}\label{eqn:TTslope}
\left. \frac{dp}{dK_a^I} \right|_{TT} = \frac{
  H'+\frac{p}{w_L^N K_a^I} \frac{x_a^S}{x_a^N}\left( \tau_K +
  \frac{2(1-\tau)}{K_a^I}\right)
}{
  W'+\frac{1-\tau}{w_L^N K_a^I} \frac{x_a^S}{x_a^N}\left(
  1-\frac{p}{w_L^N}\frac{\partial w_L^N}{\partial p}\right)
}.
\end{equation}
The first term of the denominator of (\ref{eqn:TTslope}) is negative:
\[
W\!(p)=\left.\left.\frac{w_K^N}{w_L^N}\!\right( p \right) =
    \frac{pL_b - L_a^P}{K_a^P - pK_b},
\]
so that 
\begin{eqnarray*}
W'\!(p)&=&\frac{
\left(K_a^P - pK_b\right)L_b - \left(-K_b\right)\left(pL_b-L_a^P\right)
}{
\left(K_a^P - pK_b\right)^2
}\\
&=&
\frac{
K_a^P L_b - K_b L_a^P
}{
\left(K_a^P - pK_b\right)^2
} < 0.
\end{eqnarray*}
To show that the denominator of (\ref{eqn:TTslope}) is negative, it then
suffices to show that the second term is also negative:
\[
w_L^N=\frac{K_a^P-pK_2}{
K_a^P L_b - K_b L_a^P
}
\;\;\Longrightarrow\;\;
\frac{ \partial w_L^N }{ \partial p } = \frac{-K_b}{
K_a^P L_b - K_b L_a^P
}>0
\]
\[
\Longrightarrow\;\;
\frac{p}{w_L^N}
\frac{ \partial w_L^N }{ \partial p } = \frac{-pK_b}{K_a^P-pK_2}>1.
\]

To show that the numerator of (\ref{eqn:TTslope}) is negative, consider
the second-order conditions for the maximization problem (\ref{max:pir}),
which are derived from its first-order conditions, written as
\[
\left.
\begin{array}{rcl}
\displaystyle
-w_K^Nx_a^N - \frac{p \bar{K}^S}{\left(K_a^S\right)^2} (1-\tau) - \lambda
   h_K &=&0,\\
-w_L^Nx_a^N - \lambda h_L &=& 0,\\
h\!\left(K_a^I,L_a^I\right) & = & 0.
\end{array}
\right\}
\]
If the Jacobian matrix of this system is negative definite, then its
solution maximizes (\ref{max:pir}), and
\begin{eqnarray*}
\lefteqn{
  0<\left|
  \begin{array}{ccc}
    2(1-\tau) \frac{p\bar{K}^S}{\left(K_a^I\right)^3} 
     + \frac{p\bar{K}^S}{\left(K_a^I\right)^2}  \tau_K - \lambda h_{KK}
      & -\lambda h_{KL}
      & - h_K\\
    -\lambda h_{LK}
      & -\lambda h_{LL}
      & - h_L\\
    - h_K
      & - h_L
      & 0  \\
  \end{array}
  \right|
  }\\
&=&
-h_L^2\left( 2(1-\tau) \frac{p\bar{K}^S}{\left(K_a^I\right)^3} 
  + \frac{p\bar{K}^S}{\left(K_a^I\right)^2} \tau_K \right) +
  \lambda h_L^2 h_{KK} +
  \lambda h_K^2 h_{LL} - 2\lambda h_K h_L h_{KL},
\end{eqnarray*}
which implies, after substituting $x_a^S=\bar{K}^S/K_a^I$,
\[
\frac{\lambda}{h_L^2}\left( 
   h_L^2 h_{KK} + h_K^2 h_{LL} -2h_K h_L h_{KL}
  \right) > 
    2(1-\tau) \frac{px_a^S}{\left(K_a^I\right)^2} 
     + \frac{px_a^S}{K_a^I} \tau_K.
\]
Since
\[
  \lambda=-\frac{w_L^N x_a^N}{h_L}>0,
\]
the condition becomes
\begin{eqnarray*}
\lefteqn{\frac{1}{h_L^2} \left(
   h_L h_{KK} + \frac{h_K}{h_L} h_K h_{LL} -h_K  h_{KL} -
     \frac{h_K}{h_L} h_L h_{KL}
\right)}\\
&<& - \frac{p}{w_L K_a^I}\frac{x_a^S}{x_a^N} \left(
    \frac{2(1-\tau)}{K_a^I} 
     + \tau_K 
\right).
\end{eqnarray*}
The left-hand side of this expression is the change in the absolute
value of the slope of the Ahmad isoquant as $K_a^I$ increases,
$d\left(h_k/h_L\right)/dK_a^I \equiv H'$.  Since it is less than the
right-hand side, the numerator of (\ref{eqn:TTslope}) is negative, as is
the denominator, and a consequence of the second-order conditions for
rent-maximization is that the \Bm{T$$-$$T} schedule has positive slope.


To show that the \Bm{T$$-$$T} schedule shifts to the right when
$\sigma$ increases, set $dp=0$ in (\ref{eqn:TTdiff}) and solve to find
\begin{equation}
\label{eqn:TTshift}
\frac{dK_a^I}{d\sigma} = \frac{
  -\frac{p}{w_L^N x_A^N}
   \frac{\bar{K}^S}{\left(K_a^I\right)^2}\tau_\sigma
  }{
  H'+\frac{p}{w_L^N K_a^I} \frac{x_a^S}{x_a^N}\left( \tau_K +
  \frac{2(1-\tau)}{K_a^I}\right)
  }.
\end{equation}
It was shown above that the denominator of (\ref{eqn:TTshift}) is
negative if the second-order conditions for rent-maximization are
satisfied.  Then the \Bm{T$$-$$T} schedule shifts to the right with an
increase  in $\sigma$ if and only if
\[
0<t_\sigma=\left(\frac{K_a^I}{K_a^S}\right)^2 \left(1-\frac{\sigma\left(
  K_a^I - K_a^P \right)}{K_a^S} \right),
\]
which holds since $K_a^I<K_a^P$.


\subsection{Proof of Lemma used in Proposition 1}
The lemma used in Proposition 1 is:
\begin{eqnarray}
\nonumber\lefteqn{
\frac{pK_b - K_a^P}{K_a^P - K_a^I} \leq 1 \;\;\;\Leftrightarrow} \\
\label{eqn:lemmaRHS}
&&\frac{L_b}{K_b} \geq \frac {L_a^P}{K_a^P + \left(K_a^P - K_a^I\right)}
  - \frac{\left(K_a^P - K_a^I\right)\frac{w_K^N}{w_L^N}}
	 {K_a^P + \left(K_a^P - K_a^I\right)}
\end{eqnarray}

\begin{description}
\item[Proof]  Multiply (\ref{eqn:lemmaRHS}) through by $w_L/w_K$, add 1 to
both sides, and simplify:
\[
\frac{w_K^N K_b + w_L^N L_b}
     {w_K^N K_B}
\geq
\frac{w_L^N L_a^P + w_K^N K_a^P}
     {\left(2K_a^P - K_a^I\right) w_K}.
\]
The numerator on the right-had side is $p_a$, while the numerator on the
left-hand-side is $p_b$.  Eliminating $w_K$ and cross-multiplying then
gives
\[
2K_a^P -K_a^I \geq \frac{p_a}{p_b} K_b = p K_b
\;\;\Leftrightarrow\;\;
K_a^P -K_a^I \geq p K_b - K_a^P
\;\;\Leftrightarrow\;\;
\frac{p K_b - K_a^P}{K_a^P -K_a^I} \leq 1.
\]
\end{description}

%\bibliographystyle{aer}
%\bibliography{../bib/techcnge}

\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox to\leftmargin{\hrulefill\,\,}}
\fi
\begin{thebibliography}{xx}

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\end{thebibliography}
\end{document}
