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\begin{center}
\textbf{Appendix A. Estimation of FDI composition data }
\end{center}

The data on direct investment and multinational enterprises compiled by the
U.S. Bureau of Economic Analysis (BEA) is used to estimate the FDI
composition data for selected East Asian economies. The estimated figures
are reflecting only the patterns of the operating behaviour of
majority-owned nonbank foreign affiliates of the United States. The BEA
dataset is conventionally used in almost all of the literature for
cross-country comparison of FDI behaviour, but it is important to note that
the presented FDI composition data is therefore influenced by factors such
as U.S.'s bilateral trade relationship with a particular host economy, and
that the aggregate direct investment patterns of the U.S. do not necessarily
reflect the direct investment patterns of other major developed economies. \
\ 

On the dataset, BEA surveys of U.S. direct investment abroad are tabulated
based on reported financial and operating data of foreign affiliates. Unlike
most broad level foreign direct investment datasets based solely on Balance
of Payments statistics, this database focuses not only on annual direct
investment position data, but also data on the activities of multinational
enterprises. The BEA maintains an annual time series, though not all the
variables have data recorded for the entire period of 1982-98. Based on
BEA's definition, a foreign affiliate is a foreign business enterprise in
which there is a direct ownership stake, or more specifically, with at least
a ten percent equity stake. A majority-owned affiliate is a foreign business
enterprise in which the U.S. entity would have at least 51 percent equity
holdings. While statistical classification does distinguish between parent
companies, majority-, and minority-owned foreign affiliates, not all
statistics are made available. For instance, there is no research and
development (R\&D) expenditure and value added data available for
minority-owned foreign affiliates, therefore limiting the scope of our
coverage to only the majority-owned affiliates as it is practically
impossible to know how much of the aggregate amount of minority-owned
affiliates' sales figure belong to horizontally or vertically-integrated
enterprises.

Conceptually, to estimate the different MNC composition from an aggregate
figure reported by country, we first distinguish between the shares of
foreign affiliates with horizontally- and vertically-integrated operations.
Most empirical studies in the literature adopt a sales-based classification
approach in the tradition of Horstmann-Markusen-Venables (HMV), where
Horizontal MNC is set up in a host economy to serve solely the markets of
the host economy, while affiliates with Vertical MNC mode engage in exports.
We follow the same tradition and calculate the export-to-value added ratio
[which is a more accurate measure than export-to-sales] for each host
country, and use the ratio calculated for each year to separate the share of
Horizontal FDI from the combined shares of Vertical and non-mandated MNC.

Based on the HMV definition, in any particular location/host economy, the
MNC with a vertical operation is the most skilled intensive, followed by a
MNC with a horizontal operation, and lastly plants of MNCs, which in our
context, refers to the non-mandated MNCs. This, coupled with our relatively
strict definition for Vertical MNCs as consisting of only firms with
innovation at the world frontier, allow us to make a further assumption
consistent with the theoretical framework of HMV and ours to estimate the
shares of Vertical MNCs: Only firms with Vertical MNC mode conduct R\&D in
the host economy, while for other modes of operation, R\&D is conducted at
the headquarters or parents company. Given the significance of production
fragmentation in the Asia and Pacific region [Athukorola, 2005], the ratio
of affiliates' R\&D expenditure over the aggregate R\&D expenditure spent by
parent companies for the entire region is used. Hence, the shares of
Vertical MNC from the earlier residual figures for all vertically integrated
operations is determined by the R\&D expenditure spent in the host economy,
adjusted by the total spent by parent companies for the overall region.

Lastly, consistent with how we define non-mandated FDI as including
technological licensing, the series for royalties and technological
licensing fees received by U.S. parent companies are added to the estimated
non-mandated MNC figures for each host economy. Time series on the
composition of each FDI type from 1999-2008 are then computed for each host
country. These can subsequently be applied to the FDI statistics on U.S.
direct investment position abroad [on a historical cost basis] compiled
directly from Balance of Payments to obtain the estimated time series for
U.S. FDI\ compositions in monetary amount for each destination country.

Conditioned on having to use the HMV approach, the estimated FDI
compositions do capture the actual heterogeneous industrial structure and
FDI-promoting policies of the East Asian economies well. The predominant
shares of Horizontal MNCs in most economies are consistent with the HMV
prediction. Both Singapore and Taiwan have much larger shares of
non-mandated MNCs than Horizontal MNCs despite their developed status due to
their historical role of being component parts' manufacturing and export
platforms. Japan, Singapore, and South Korea have the largest shares of
Vertical FDI, though there does not appear to be a consistent pattern [and
in the case of Japan, Vertical FDI appears to have declined in recent
years]. The two unique characteristics associated with Taiwan and South
Korea commonly documented in the technological capabilities literature---the
industrial success of the former is driven by small and medium enterprises
conducting basic, non-cutting edge activities; the latter does not involve
much in the value chains of regional MNCs' production fragmentation---are
also reflected in the estimated FDI compositions.

\begin{center}
\pagebreak

\textbf{Appendix B. Technical Notes }
\end{center}

As noted in the text, the dynamic system of the model economy consists of
nine equations [four differential equations, five static equations] that
determine the evolutions of $m_{t}^{R}$, $m_{t}^{I}$, $Q_{t}^{R}$, and $%
z_{t}^{C}$.

\textbf{Domestic Sectors}

First, starting from the final goods market equilibrium, the expression for
private investment $I_{t}$, (40) is substituted into (22) to get%
\begin{equation*}
\dot{K}_{t}=[(1-\gamma \eta -\tau )Y_{t}-L_{t}C_{t}^{a}]-\delta K_{t},
\end{equation*}%
or equivalently 
\begin{subequations}
\label{A}
\begin{equation}
\frac{\dot{K}_{t}}{K_{t}}=[(1-\gamma \eta -\tau )(\frac{Y_{t}}{K_{t}}%
)-z_{t}^{C}]-\delta ,  \tag{A.1}  \label{A1}
\end{equation}%
where $z_{t}^{C}=L_{t}C_{t}^{a}/K_{t}$.

Differentiating $z_{t}^{C}$ with respect to time, we have $\dot{z}%
_{t}^{C}/z_{t}^{C}=n+(\dot{C}_{t}^{a}/C_{t}^{a})-\dot{K}_{t}/K_{t}$. This,
combined with (2), (16), and (\ref{A1}), delivers the first-order
differential equation for $z_{t}^{C}$: 
\end{subequations}
\begin{equation}
\frac{\dot{z}_{t}^{C}}{z_{t}^{C}}=n+[\sigma \alpha -(1-\gamma \eta -\tau )](%
\frac{Y_{t}}{K_{t}})+z_{t}^{C}-\sigma (\rho +\delta )+\delta .  \tag{A.2}
\label{A2}
\end{equation}

Next, for the imitation sector, (7) can be rewritten as

\begin{equation*}
\frac{\dot{M}_{t}^{I}}{M_{t}^{I}}=(n_{FH,t})^{\psi _{1}^{I}}[1+\psi
_{2}^{I}n_{FV,t}\frac{M_{t}^{R}}{M_{t}^{I}}]\theta _{U,I,t},
\end{equation*}%
or equivalently,

\begin{equation}
\frac{\dot{M}_{t}^{I}}{M_{t}^{I}}=(n_{FH,t})^{\psi _{1}^{I}}[1+\psi
_{2}^{I}n_{FV,t}\frac{M_{t}^{R}}{M_{t}^{I}}](\theta _{U,t}-\theta _{U,Y,t}).
\tag{A.3}  \label{A3}
\end{equation}

Given that $\dot{m}_{t}^{I}/m_{t}^{I}=\dot{M}_{t}^{I}/M_{t}^{I}-\dot{K}%
_{t}/K_{t}$, the first-order differential equation for $m_{t}^{I}$ is
obtained by combining (\ref{A3}) and (\ref{A1}):%
\begin{equation}
\frac{\dot{m}_{t}^{I}}{m_{t}^{I}}=(n_{FH,t})^{\psi _{1}^{I}}[1+\psi
_{2}^{I}n_{FV,t}\frac{m_{t}^{R}}{m_{t}^{I}}](\theta _{U,t}-\theta
_{U,Y,t})-(1-\gamma \eta -\tau )(\frac{Y_{t}}{K_{t}})+z_{t}^{C}+\delta . 
\tag{A.4}  \label{A4}
\end{equation}

From the first-order condition (8) in the imitation sector, and using (7),%
\begin{equation}
w_{t}^{U}=(\frac{1}{1+\Lambda ^{I}})(\frac{R_{t}^{I}}{L_{t}}%
)(n_{FH,t})^{\psi _{1}^{I}}[1+\psi _{2}^{I}n_{FV,t}\frac{M_{t}^{R}}{M_{t}^{I}%
}]M_{t}^{I}.  \tag{A.5}  \label{A5}
\end{equation}

This is then equated to the first-order condition (17) in the final goods
sector, $w_{t}^{U}=(\frac{\beta ^{U}}{1+\Lambda ^{Y}})Y_{t}/L_{U,Y,t}$ to get%
\begin{equation}
\frac{Y_{t}}{M_{t}^{I}}=\frac{1+\Lambda ^{Y}}{\beta ^{U}(1+\Lambda ^{I})}%
R_{t}^{I}\theta _{U,Y,t}(n_{FH,t})^{\psi _{1}^{I}}[1+\psi _{2}^{I}n_{FV,t}%
\frac{M_{t}^{R}}{M_{t}^{I}}].  \tag{A.6}  \label{A6}
\end{equation}

From (11)-(12), $R_{t}^{I}=(1-\eta )\nu \gamma (Y_{t}/M_{t}^{I})$.
Substituting the expression into (\ref{A6}) and replacing $Y_{t}/M_{t}^{I}$\
out of the equation yields a solution for the proportion of unskilled labour
(of total population) in the production of the final good $\theta _{U,Y,t}$:%
\begin{equation}
\theta _{U,Y,t}=\frac{\beta ^{U}(1+\Lambda ^{I})}{(1+\Lambda ^{Y})(1-\eta
)\nu \gamma }(n_{FH,t})^{-\psi _{1}^{I}}[1+\psi _{2}^{I}n_{FV,t}\frac{%
m_{t}^{R}}{m_{t}^{I}}]^{-1}.  \tag{A.7}  \label{A7}
\end{equation}

Similarly, for the innovation sector, rewrite (9) as

\begin{equation}
\frac{\dot{M}_{t}^{R}}{M_{t}^{R}}=(n_{FV,t})^{\psi _{1}^{R}}[1+\psi _{2}^{R}(%
\frac{m_{t}^{I}}{m_{t}^{R}})]\theta _{S,R,t},  \tag{A.8}  \label{A8}
\end{equation}%
or equivalently,%
\begin{equation}
\frac{\dot{M}_{t}^{R}}{M_{t}^{R}}=(n_{FV,t})^{\psi _{1}^{R}}[1+\psi _{2}^{R}(%
\frac{m_{t}^{I}}{m_{t}^{R}})](\theta _{S,t}-\theta _{S,Y,t}).  \tag{A.9}
\label{A9}
\end{equation}

Given that $\dot{m}_{t}^{R}/m_{t}^{R}=\dot{M}_{t}^{R}/M_{t}^{R}-\dot{K}%
_{t}/K_{t}$, the first-order differential equation for $m_{t}^{R}$ is
obtained by combining (\ref{A9}) and (\ref{A1}):%
\begin{equation}
\frac{\dot{m}_{t}^{R}}{m_{t}^{R}}=(n_{FV,t})^{\psi _{1}^{R}}[1+\psi _{2}^{R}(%
\frac{m_{t}^{I}}{m_{t}^{R}})](\theta _{S,t}-\theta _{S,Y,t})-(1-\gamma \eta
-\tau )(\frac{Y_{t}}{K_{t}})+z_{t}^{C}+\delta .  \tag{A.10}  \label{A10}
\end{equation}

Substituting (\ref{A9}) and (11) into the first-order condition (10), yields%
\begin{equation}
w_{t}^{S}=(\frac{1}{1+\Lambda ^{R}})(\frac{Q_{t}^{R}}{L_{t}}%
)(n_{FV,t})^{\psi _{1}^{R}}[1+\psi _{2}^{R}(\frac{m_{t}^{I}}{m_{t}^{R}}%
)]M_{t}^{R}.  \tag{A.11}  \label{A11}
\end{equation}

This is then equated to the first-order condition (17) in the final goods
sector, $w_{t}^{S}=(\frac{\beta ^{S}}{1+\Lambda ^{Y}})Y_{t}/L_{S,Y,t}$.
Rearranging the expression and dividing by $K_{t}$ yields%
\begin{equation}
\frac{Y_{t}}{K_{t}}=\frac{(1+\Lambda ^{Y})}{\beta ^{S}(1+\Lambda ^{R})}%
Q_{t}^{R}\theta _{S,Y,t}(n_{FV,t})^{\psi _{1}^{R}}[1+\psi _{2}^{R}(\frac{%
m_{t}^{I}}{m_{t}^{R}})]m_{t}^{R},  \tag{A.12}  \label{A12}
\end{equation}%
which can be rearranged to obtain an equation for the proportion of skilled
labour (of total population) in the production of the final good, $\theta
_{S,Y,t}$:%
\begin{equation}
\theta _{S,Y,t}=\frac{\beta ^{S}(1+\Lambda ^{R})}{(1+\Lambda ^{Y})}(\frac{%
Y_{t}}{K_{t}})[Q_{t}^{R}(m_{t}^{R})]^{-1}(n_{FV,t})^{-\psi _{1}^{R}}[1+\psi
_{2}^{R}(\frac{m_{t}^{I}}{m_{t}^{R}})]^{-1}.  \tag{A.13}  \label{A13}
\end{equation}

The fourth and last differential equation to derive is for $Q_{t}^{R}$. From
(13), (14), and (16),%
\begin{equation}
\frac{\dot{Q}_{t}^{R}}{Q_{t}^{R}}=[\alpha (\frac{Y_{t}}{K_{t}})-\delta
]-(1-\eta )\gamma (1-\nu )(\frac{Y_{t}}{K_{t}})(\frac{1}{Q_{t}^{R}})(\frac{1%
}{m_{t}^{R}}).  \tag{A.14}  \label{A14}
\end{equation}

The static equation for $Y_{t}/K_{t}$ is given by (48), which is derived
after applying the assumptions of $\beta ^{S}+\beta ^{U}-\varrho \iota =0$
and $(\gamma /\eta )+\alpha +\varrho =1$: 
\begin{equation}
\frac{Y_{t}}{K_{t}}=\frac{\Theta _{2}}{[(\theta _{t}^{S,Y})^{\beta
^{S}}(\theta _{t}^{U,Y})^{\beta ^{U}}]^{-1/(1-\gamma )}}\left\{
(m_{t}^{I})^{\nu (1-\eta )/\eta }(m_{t}^{R})^{(1-\nu )(1-\eta )/\eta
}\right\} ^{\gamma /(1-\gamma )}.  \tag{A.15}  \label{A15}
\end{equation}

The remaining two static equations to derive are for $\theta _{S,t}$ and $%
\theta _{U,t}$. Before that, from the first-order conditions (17), dividing
the expression for $w_{t}^{U}$ by the expression for $w_{t}^{S}$ yields the
relative wage ratio: 
\begin{equation}
\frac{w_{t}^{U}}{w_{t}^{S}}=(\frac{\beta ^{U}}{\beta ^{S}})(\frac{L_{S,Y,t}}{%
L_{U,Y,t}})=(\frac{\beta ^{U}}{\beta ^{S}})(\frac{\theta _{S,Y,t}}{\theta
_{U,Y,t}}).  \tag{A.16}  \label{A16}
\end{equation}

Then, substituting the wage ratio (\ref{A16}) into (4), the threshold
cognitive ability can be expressed as 
\begin{equation}
\hat{a}_{t}=[\frac{\beta ^{U}}{\beta ^{S}(1-\Gamma )}\frac{\theta _{S,Y,t}}{%
\theta _{U,Y,t}}]^{1/\xi }.  \tag{A.17}  \label{A17}
\end{equation}

Substituting this expression for $\hat{a}_{t}$ into both (5) and (6) would
yield

\begin{equation}
\theta _{U,t}=1-a_{m}^{\chi }[\frac{\beta ^{U}}{\beta ^{S}(1-\Gamma )}\frac{%
\theta _{S,Y,t}}{\theta _{U,Y,t}}]^{-\chi /\xi },\text{ }  \tag{A.18}
\label{A18}
\end{equation}%
and%
\begin{equation}
\text{\ }\theta _{S,t}=\frac{\chi a_{m}^{\chi }}{\chi -1}[\frac{\beta ^{U}}{%
\beta ^{S}(1-\Gamma )}\frac{\theta _{S,Y,t}}{\theta _{U,Y,t}}]^{(1-\chi
)/\xi },  \tag{A.19}  \label{A19}
\end{equation}%
respectively for $\theta _{U,t}$ and $\theta _{S,t}$.

\bigskip

\textbf{Foreign Sector}

The derivations of key equations in the foreign sector work as follows.
Given the nested objective function, (22), even though shadow investment
prices are taken as as given by experts, we can similarly write a
theoretical aggregate shadow investment price index, $P_{F}$, as in
Brambilla et al. (2009). This is given by 
\begin{equation}
P_{F}=\{(\int_{0}^{N_{F}}[\int_{0}^{M_{t}^{I}}\gamma _{1,t}^{\sigma
^{F}}P_{s,FH,t}^{1-\sigma ^{F}}ds+\int_{0}^{M_{t}^{R}}\gamma _{2,t}^{\sigma
^{F}}P_{s,FV,t}^{1-\sigma ^{F}}]^{\frac{1}{1-\sigma ^{F}}})^{1-\theta
^{F}}ds]dj\}^{\frac{1}{1-\theta ^{F}}},  \tag{A.20}  \label{A20}
\end{equation}%
where $P_{s,k},k=FV,FH$ are the implicit, theoretical shadow investment
prices across different intermediate varieties.

Next, by assuming that each firm is small within its group, imposing a
standard monopolistic competition pricing condition of constant mark-up
would yield (23), which is rewritten here as%
\begin{equation}
P_{s,t}=\left( \frac{\sigma ^{F}}{\sigma ^{F}-1}\right) \left( \varpi
_{s,t}\right) .  \tag{A.21}  \label{A21}
\end{equation}

Given these theoretical shadow prices, solving the optimisation problem
associated with the nested objective function, (22), would yield a series of
theoretical investment demand functions for variety $s$ and `productivity
requirement'-induced shadow quality $j$ of 
\begin{equation}
q_{j,s,t}=\gamma _{\varkappa ,t}^{\sigma ^{F}}\left( \frac{y_{t}^{F}}{%
P_{t}^{F}}\right) \left( \frac{P_{j,t}}{P_{t}^{F}}\right) ^{-\theta
^{F}}\left( \frac{P_{j,t}}{P_{s,t}}\right) ^{\sigma ^{F}},  \tag{A.22}
\label{A22}
\end{equation}%
where $\varkappa =1$ or $2$, and $P^{F}$, $P_{s}$, $P_{j}$ are the host
economy-specific theoretical shadow price indices for the aggregate, variety 
$s$, and shadow investment quality $j$ respectively.

Given (22), (\ref{A20}), substitute (\ref{A21}) into (\ref{A22}) yields the
theoretical investment demand of expert with shadow quality (perceived
productivity) preference $j$ matched to product variety $s$, as in 
\begin{equation}
q_{j,s,t}=\left( \frac{\sigma ^{F}}{\sigma ^{F}-1}\right) ^{-\sigma
^{F}}\gamma _{\varkappa ,t}^{\sigma ^{F}}\varpi _{j,s,t}^{-\sigma
^{F}}P_{F,t}^{\theta ^{F}-1}P_{j,t}^{\sigma ^{F}-\theta ^{F}}y_{t}^{F}. 
\tag{A.23}  \label{A23}
\end{equation}

If the `market' for foreign experts' productivity preference is in
equilibrium, symmetry implies that the average quality preference for the
collective pool of foreign experts of a same variety $s$ would equalize, $%
q_{j,s,t}=$ $q_{s,t}$.\footnote{%
Given that the shadow quality difference is driven by perceived
heterogeneity among productivity of domestic workers and therefore implicit
in nature, the symmetric equilibrium result imposed for the collective
experts of a same variety pool therefore holds on average for variety $s$.}

Accounting for the asymmetry of productivity requirement-induced cost [where 
$\varpi _{s,t}=\varpi _{j,s,t}$ holds for non-mandated and Horizontal MNCs,
and $\varpi _{s,t}^{\phi }=\varpi _{j,s,t}^{\phi }$ holds for Vertical
MNCs], (\ref{A21}) and (\ref{A23}) allow us to express an evaluative
`individual value' function for a typical foreign expert $j$ in variety $s$
of either non-mandated, Horizontal, or Vertical investment mode as 
\begin{equation}
\pi _{FP,j,t}(\varpi _{j,s,t})=\left( \frac{(\sigma ^{F}-1)^{\sigma
^{F}-1}y_{t}^{F}}{\left( \sigma ^{F}\right) ^{\sigma ^{F}-1}}\right) (\varpi
_{FP,j,s,t})^{1-\sigma ^{F}}P_{F,t}^{\theta ^{F}-1}P_{0}^{\sigma ^{F}-\theta
^{F}}-F_{0},  \tag{A.24}  \label{A24}
\end{equation}%
\begin{equation}
\pi _{FH,j,t}(\varpi _{j,s,t})=\left( \frac{(\sigma ^{F}-1)^{\sigma
^{F}-1}y_{t}^{F}}{\left( \sigma ^{F}\right) ^{\sigma ^{F}-1}}\right) \gamma
_{1,t}^{\sigma ^{F}}(\varpi _{FH,j,s,t})^{1-\sigma ^{F}}P_{F,t}^{\theta
^{F}-1}P_{j,t}^{\sigma ^{F}-\theta ^{F}}-(F_{0}+F_{1})P_{0},  \tag{A.25}
\label{A25}
\end{equation}%
\begin{equation}
\pi _{FV,j,t}(\varpi _{j,s,t})=\left( \frac{(\sigma ^{F}-1)^{\sigma
^{F}-1}y_{t}^{F}}{\left( \sigma ^{F}\right) ^{\sigma ^{F}-1}}\right) \gamma
_{2,t}^{\sigma ^{F}}(\varpi _{FV,j,s,t}^{\phi })^{1-\sigma
^{F}}P_{F,t}^{\theta ^{F}-1}P_{j,t}^{\sigma ^{F}-\theta
^{F}}-(F_{0}+F_{2})P_{0},  \tag{A.26}  \label{A26}
\end{equation}%
respectively.

As the effect of non-mandated FDI is not modelled in this study, the default
baseline investment price, $P_{0}$ is normalised to one. Further, with $%
q_{j,s,t}=$ $q_{s,t}$ derived earlier, in equilibrium, the standard symmetry
assumption also implies that the shadow price index for quality (perceived
productivity) $j$ within the same variety $s$ would be the same for the
collective pool of workers matched to variety $s$, hence $P_{j}=P_{s}$ is
imposed. Drawing on the idea of Allanson and Montagna (2005), we can then
use a simplification approach for these indices where a time-invariant
parameter generalising the degree of pricing competition in host economy,
Lerner Index, $LI$ is introduced as a structural parameter in place of $%
P_{j}=P_{s}=\frac{\theta ^{F}}{\theta ^{F}-1}\bar{\varpi}$, where $\theta
^{F}$ is simply written in place of $\sigma ^{F}$ in the (\ref{A21})
expression for some average value of $\bar{\varpi}$. Substituting in the
expression for the average value, $\bar{\varpi}$, we can further express $LI$
as 
\begin{equation}
LI=P_{j}=P_{s}=\frac{\theta ^{F}}{\theta ^{F}-1}\frac{\chi }{\chi -1}\frac{%
a_{m}}{a_{FP.\tilde{a}_{\min }}},  \tag{A.27}  \label{A27}
\end{equation}%
where the definition $\bar{\varpi}=\bar{a}/\tilde{a}$, the mean ability
expression of a Pareto distribution [$\bar{z}=$ $\chi z_{\min }/(\chi -1)$
for some $F(z)$], and the assumption $a_{FP}=\tilde{a}/\tilde{a}_{\min
}\forall t$ are applied [the second part of the expression in right-hand
side is used for the parameterisation of $a_{FP}$].

Next, define $\varpi _{FP}=g(a_{FP})$, $\varpi _{FH}=g(a_{FH})$, $\varpi
_{FV}=g(a_{FV})$, where $g(a)=a/\tilde{a}$, as the respective cut-off values
for each FDI type. Setting $\pi _{FP}(\varpi _{FP})=0$, $\pi _{FH}(\varpi
_{FH})=\pi _{FP}(\varpi _{FH})$, and $\pi _{FH}(\varpi _{FV}^{\phi })=\pi
_{FV}(\varpi _{FV}^{\phi })$ using (\ref{A24})-(\ref{A26}), the three
cut-off productivity values for MNCs' internalisation decision (also
expressed in ability value) in any period $t$ are given by 
\begin{equation}
\varpi _{FP,t}=\frac{a_{FP,t}}{\tilde{a}}=\left[ \frac{F_{0}}{\left( (\sigma
^{F}-1)^{\sigma ^{F}-1}y_{t}^{F}/(\sigma ^{F})^{\sigma ^{F}-1}\right)
P_{F,t}^{\theta ^{F}-1}(1)^{\sigma ^{F}-\theta ^{F}}}\right] ^{1/(1-\sigma
^{F})}\text{,}  \tag{A.28}  \label{A28}
\end{equation}%
\begin{equation}
\varpi _{FH,t}=\frac{a_{FH,t}}{\tilde{a}}=\left[ \frac{F_{1}}{\left( (\sigma
^{F}-1)^{\sigma ^{F}-1}y_{t}^{F}/(\sigma ^{F})^{\sigma ^{F}-1}\right)
P_{F,t}^{\theta ^{F}-1}[\gamma _{1,t}^{\sigma ^{F}}(LI)^{\sigma ^{F}-\theta
^{F}}-(1)^{\sigma ^{F}-\theta ^{F}}]}\right] ^{1/(1-\sigma ^{F})}\text{,} 
\tag{A.29}  \label{A29}
\end{equation}%
\begin{equation}
\varpi _{FV,t}=\frac{a_{FV,t}}{\tilde{a}}=\left[ \frac{(F_{2}-F_{0})}{\left(
(\sigma ^{F}-1)^{\sigma ^{F}-1}y_{t}^{F}/(\sigma ^{F})^{\sigma
^{F}-1}\right) P_{F,t}^{\theta ^{F}-1}(LI)^{\sigma ^{F}-\theta ^{F}}[\gamma
_{2,t}^{\sigma ^{F}}-\gamma _{1,t}^{\sigma ^{F}}]}\right] ^{1/[\phi
(1-\sigma ^{F})]}\text{.}  \tag{A.30}  \label{A30}
\end{equation}%
where (\ref{A27}) is substituted in.

To calculate the number of foreign firms by FDI type (or share of FDI
category), recall that the sorting of foreign firms follow that of $1/\varpi 
$. We know that the cumulative distribution function of a typical Pareto
distribution $z$, takes the form of $F(z)=1-(z_{\min }/z)^{\chi }$ for some
minimum of $z$, $z_{\min }$. Let $F(1/\varpi )=F(\tilde{a}/a)$. Further, by
assuming that there is no exit option for MNCs, we can set $a_{FP}=\tilde{a}/%
\tilde{a}_{\min }\forall t$, where $\tilde{a}/\tilde{a}_{\min }$ denotes
some minimum threshold value of entry by foreign firms [a large value along
the ability distribution of host economy]. At any time $t$, the proportion
of the three types of foreign firms can be computed as\ 
\begin{eqnarray}
n_{FP,t} &=&\frac{N_{FP,t}}{N_{F,t}}=\left[ F(1/\varpi _{FH,t})-F(1/\varpi
_{FP,t})\right]  \TCItag{A.31}  \label{A31} \\
&=&\left[ F(\tilde{a}/a_{FH,t})-F(\tilde{a}/a_{FP,t})\right]  \notag \\
&=&1-(\frac{a_{FH,t}}{a_{FP}})^{\chi }-1+(\frac{a_{FP}}{a_{FP}})^{\chi }%
\text{, where }a_{FP}=\tilde{a}/\tilde{a}_{\min }\text{$\forall t$}  \notag
\\
&=&[1-(\frac{a_{FH,t}}{a_{FP}})^{\chi }]\text{ ,}  \notag
\end{eqnarray}%
\begin{eqnarray}
n_{FH,t} &=&\frac{N_{FH,t}}{N_{F,t}}=\left[ F(1/\varpi _{FV,t})-F(1/\varpi
_{FH,t})\right]  \TCItag{A.32}  \label{A32} \\
&=&\left[ F(\tilde{a}/a_{FV,t})-F(\tilde{a}/a_{FH,t})\right]  \notag \\
&=&[1-(\frac{\tilde{a}_{\min }a_{FV,t}}{\tilde{a}})^{\chi }-(1-(\frac{\tilde{%
a}_{\min }a_{FH,t}}{\tilde{a}})^{\chi })]  \notag \\
&=&[(\frac{a_{FH,t}}{a_{FP}})^{\chi }-(\frac{a_{FV,t}}{a_{FP}})^{\chi }]%
\text{,}  \notag
\end{eqnarray}%
\begin{eqnarray}
n_{FV,t} &=&\frac{N_{FV,t}}{N_{F,t}}=\left[ 1-F(1/\varpi _{FV,t})\right] 
\TCItag{A.33}  \label{A33} \\
&=&\left[ 1-F(\tilde{a}/a_{FV,t})\right]  \notag \\
&=&(\frac{a_{FV,t}}{a_{FP}})^{\chi }\text{,}  \notag
\end{eqnarray}%
where $a_{FP}$, $a_{FH}$, $a_{FV}$ give the host economy-specific threshold
values of entry for non-mandated, Horizontal, and Vertical FDI.

Using (\ref{A28})-(\ref{A30}), the variables $y_{t}^{F}$ and $P_{F,t}$ can
easily be substituted out. Specifically, dividing (\ref{A29}) by (\ref{A28}%
), and (\ref{A30}) by (\ref{A28}), the following two threshold conditions
can be derived after some straightforward algebraic manipulations: 
\begin{equation}
a_{FH,t}=\left[ \frac{F_{0}}{F_{1}}((LI)^{\sigma ^{F}-\theta ^{F}}(\gamma
_{1,t})^{\sigma ^{F}}-1)\right] ^{-1/(1-\sigma ^{F})}a_{FP},  \tag{A.34}
\label{A34}
\end{equation}%
and 
\begin{equation}
a_{FV,t}=\left[ \frac{F_{2}-F_{1}}{F_{0}}\frac{1}{(LI)^{\sigma ^{F}-\theta
^{F}}[\gamma _{2,t}^{\sigma ^{F}}-\gamma _{1,t}^{\sigma ^{F}}]}\right]
^{1/[\phi (1-\sigma ^{F})]}a_{FP}^{1/\phi }\tilde{a}^{(\phi -1)/\phi }\text{.%
}  \tag{A.35}  \label{A35}
\end{equation}

For the feedback channel of the state of industrial development in a host
economy to the determinant of FDI composition, the evolution of the two
foreign preferences, $\gamma _{1}$and $\gamma _{2}$ are modelled using a
Weibull distribution, governed by a hazard function of 
\begin{eqnarray}
\gamma _{1} &=&[1-h(\gamma _{2};\omega _{k},\omega _{\lambda })]\gamma _{2} 
\TCItag{A.36}  \label{A36} \\
&=&[1-(\frac{\omega _{k}}{\omega _{\lambda }}(\frac{\gamma _{2}}{\omega
_{\lambda }})^{\omega _{k}-1})]\gamma _{2}\text{,}  \notag
\end{eqnarray}%
where $h(\gamma _{2};\omega _{k},\omega _{\lambda })$ denotes the
`out-of-taste' rate of $\gamma _{2}$, and $\omega _{k}$ and $\omega
_{\lambda }$ are the shape and scale parameter respectively. As $\gamma _{1}$
is given by the expected value of $E(\gamma _{2})$, we can rewrite (\ref{A34}%
) and (\ref{A35}) as 
\begin{equation}
a_{FH,t}=\left[ \frac{F_{0}}{F_{1}}((LI)^{\sigma ^{F}-\theta
^{F}}(Q_{t}^{F}-\Theta _{1}(Q_{t}^{F})^{\omega _{k}})^{\sigma ^{F}}-1)\right]
^{-1/(1-\sigma ^{F})}a_{FP}\text{,}  \tag{A.37}  \label{A37}
\end{equation}%
and 
\begin{equation}
a_{FV,t}=\left[ \frac{F_{2}-F_{1}}{F_{0}}\frac{1}{(LI)^{\sigma ^{F}-\theta
^{F}}[(Q_{t}^{F})^{\sigma ^{F}}-\left( Q_{t}^{F}-\Theta
_{1}(Q_{t}^{F})^{\omega _{k}}\right) ^{\sigma ^{F}}]}\right] ^{1/[\phi
(1-\sigma ^{F})]}a_{FP}^{1/\phi }\tilde{a}^{(\phi -1)/\phi }\text{,} 
\tag{A.38}  \label{A38}
\end{equation}%
where $\Theta _{1}=(\omega _{k}/\omega _{\lambda })(1/\omega _{\lambda
})^{\omega _{k}-1}$. $Q^{F}$, a measure of the state of industrial
development of the host economy, is now written in place of the preference
parameters. As not all developing economies host an innovation sector, the
foreign source economy therefore evaluates all host economies for offshore
investment by setting $\dot{Q}^{F}=\dot{m}_{t}^{I}$ in each period. This
yields $Q_{t}^{F}=w_{m}m_{t}^{I}$, where $w_{m}$ is a multiplicative
constant.

Lastly, substituting (\ref{A33}) into (\ref{A32}), and use the two
expressions of (\ref{A37}) and (\ref{A38}), the equation for $n_{FH,t}$ can
be derived as 
\begin{eqnarray}
n_{FH,t} &=&a_{FP}^{-\chi }(a_{FH,t}{}^{\chi }-n_{FV,t}.a_{FP}^{\chi }) 
\TCItag{A.39}  \label{A39} \\
&=&\left( \frac{a_{FH,t}}{a_{FP}}\right) ^{\chi }-n_{FV,t}  \notag \\
&=&\left[ \frac{F_{0}}{F_{1}}((LI)^{\sigma -\theta }(w_{m}m_{t}^{I}-\Theta
_{1}(w_{m}m_{t}^{I})^{\omega _{k}})^{\sigma ^{F}}-1)\right] ^{-\chi
/(1-\sigma ^{F})}-n_{FV,t}\text{,}  \notag
\end{eqnarray}%
with $n_{FV,t}$ given by 
\begin{eqnarray}
n_{FV,t} &=&\left( a_{FP}^{1/\phi }\tilde{a}^{(\phi -1)/\phi }\right) ^{\chi
}\times  \TCItag{A.40}  \label{A40} \\
&&\left[ \frac{F_{2}-F_{1}}{F_{0}}\frac{1}{(LI)^{\sigma ^{F}-\theta
^{F}}[(w_{m}m_{t}^{I})^{\sigma ^{F}}-\left( w_{m}m_{t}^{I}-\Theta
_{1}(w_{m}m_{t}^{I})^{\omega _{k}}\right) ^{\sigma ^{F}}]}\right] ^{\chi
/[\phi (1-\sigma ^{F})]}.  \notag
\end{eqnarray}

\bigskip \bigskip

The four differential equations (\ref{A2}), (\ref{A4}), (\ref{A10}), and (%
\ref{A14}), and the seven static equations (\ref{A7}), (\ref{A13}), (\ref%
{A15}), (\ref{A17}), (\ref{A18}), (\ref{A39}), and (\ref{A40}) form the
dynamic system of the model economy.

Recall from the final good sector that, to generate endogenous growth, we
impose the knife-edge conditions: \textbf{Assumptions}: $\beta ^{S}+\beta
^{U}-\varrho \iota =0$, $(\gamma /\eta )+\alpha +\varrho =1$.

Specifically, define $m_{t}^{I}=M_{t}^{I}/K_{t}$, $m_{t}^{R}=M_{t}^{R}/K_{t}$%
, and using (\ref{e22}), (\ref{e26}), (\ref{e23}) can be written as

\begin{equation}
Y_{t}=(\theta _{t}^{S,Y})^{\beta ^{S}}(\theta _{t}^{U,Y})^{\beta
^{U}}L_{t}^{\beta ^{S}+\beta ^{U}-\varrho \iota }  \tag{A.41}  \label{A41}
\end{equation}%
\begin{equation*}
\times \left\{ (\gamma \eta \nu ^{\nu }(1-\nu )^{1-\nu })(m_{t}^{I})^{\nu
(1-\eta )/\eta }(m_{t}^{R})^{(1-\nu )(1-\eta )/\eta }(\frac{Y_{t}}{K_{t}}%
)\right\} ^{\gamma }(K_{t})^{(\gamma /\eta )+\alpha +\varrho },
\end{equation*}%
where $(L_{t})^{0}=1$ if and only if $\beta ^{S}+\beta ^{U}-\varrho \iota =0$%
. The level of output, $Y_{t}$, is linear to the private capital stock, $%
K_{t}$, if and only if $(\gamma /\eta )+\alpha +\varrho =1$.

The dynamic system of the economy is characterised by a differential
algebraic system consisting of four first-order differential equations and
seven static equations, as follows: \qquad 
\begin{equation}
\frac{\dot{m}_{t}^{R}}{m_{t}^{R}}=(n_{FV,t})^{\psi _{1}^{R}}[1+\psi _{2}^{R}(%
\frac{m_{t}^{I}}{m_{t}^{R}})](\theta _{S,t}-\theta _{S,Y,t})-(1-\gamma \eta
-\tau )(\frac{Y_{t}}{K_{t}})+z_{t}^{C}+\delta ,  \tag{A.42}  \label{A42}
\end{equation}%
\begin{equation}
\frac{\dot{m}_{t}^{I}}{m_{t}^{I}}=(n_{FH,t})^{\psi _{1}^{I}}[1+\psi
_{2}^{I}n_{FV,t}(\frac{m_{t}^{R}}{m_{t}^{I}})](\theta _{U,t}-\theta
_{U,Y,t})-(1-\gamma \eta -\tau )(\frac{Y_{t}}{K_{t}})+z_{t}^{C}+\delta , 
\tag{A.43}  \label{A43}
\end{equation}%
\begin{equation}
\frac{\dot{z}_{t}^{C}}{z_{t}^{C}}=n+[\sigma \alpha -(1-\gamma \eta -\tau )](%
\frac{Y_{t}}{K_{t}})+z_{t}^{C}-\sigma (\rho +\delta )+\delta ,  \tag{A.44}
\label{A44}
\end{equation}%
\begin{equation}
\frac{\dot{Q}_{t}^{R}}{Q_{t}^{R}}=[\alpha (\frac{Y_{t}}{K_{t}})-\delta
]-(1-\eta )\gamma (1-\nu )(\frac{Y_{t}}{K_{t}})(\frac{1}{Q_{t}^{R}})(\frac{1%
}{m_{t}^{R}}),  \tag{A.45}  \label{A45}
\end{equation}%
of which $m_{t}^{I}$ and $m_{t}^{R}$ are backward-looking state variables,
while $z_{t}^{C}$ and $Q_{t}^{R}$ are forward-looking jump variables.

The seven static equations are%
\begin{equation}
\frac{Y_{t}}{K_{t}}=\frac{\Theta _{2}}{[(\theta _{S,Y,t})^{\beta
^{S}}(\theta _{U,Y,t})^{\beta ^{U}}]^{-1/(1-\gamma )}}\left\{
(m_{t}^{I})^{\nu (1-\eta )/\eta }(m_{t}^{R})^{(1-\nu )(1-\eta )/\eta
}\right\} ^{\gamma /(1-\gamma )},  \tag{A.46}  \label{A46}
\end{equation}

\begin{equation}
\theta _{S,Y,t}=\frac{\beta ^{S}(1+\Lambda ^{R})}{(1+\Lambda ^{Y})}(\frac{%
Y_{t}}{K_{t}})[Q_{t}^{R}(m_{t}^{R})]^{-1}(n_{FV,t})^{-\psi _{1}^{R}}[1+\psi
_{2}^{R}(\frac{m_{t}^{I}}{m_{t}^{R}})]^{-1},  \tag{A.47}  \label{A47}
\end{equation}%
\begin{equation}
\theta _{U,Y,t}=\frac{\beta ^{U}(1+\Lambda ^{I})}{(1+\Lambda ^{Y})(1-\eta
)\nu \gamma }(n_{FH,t})^{-\psi _{1}^{I}}[1+\psi _{2}^{I}n_{FV,t}(\frac{%
m_{t}^{R}}{m_{t}^{I}})]^{-1},  \tag{A.48}  \label{A48}
\end{equation}

\begin{equation}
\theta _{U,t}=1-a_{m}^{\chi }[\frac{\beta ^{U}}{\beta ^{S}(1-\Gamma )}\frac{%
\theta _{S,Y,t}}{\theta _{U,Y,t}}]^{-\chi /\xi },\text{ }  \tag{A.49}
\label{A49}
\end{equation}%
\begin{equation}
\text{\ }\theta _{S,t}=\frac{\chi a_{m}^{\chi }}{\chi -1}[\frac{\beta ^{U}}{%
\beta ^{S}(1-\Gamma )}\frac{\theta _{S,Y,t}}{\theta _{U,Y,t}}]^{(1-\chi
)/\xi },  \tag{A.50}  \label{A50}
\end{equation}%
\begin{equation}
n_{FH,t}=\left[ \frac{F_{0}}{F_{1}}((LI)^{\sigma -\theta
}(w_{m}m_{t}^{I}-\Theta _{1}(w_{m}m_{t}^{I})^{\omega _{k}})^{\sigma ^{F}}-1)%
\right] ^{-\chi /(1-\sigma ^{F})}-n_{FV,t},  \tag{A.51}  \label{A51}
\end{equation}%
\begin{eqnarray}
n_{FV,t} &=&\left( a_{FP}^{1/\phi }\tilde{a}^{(\phi -1)/\phi }\right) ^{\chi
}  \TCItag{A.52}  \label{A52} \\
&&\times \left[ \frac{F_{2}-F_{1}}{F_{0}}\frac{1}{(LI)^{\sigma ^{F}-\theta
^{F}}[(w_{m}m_{t}^{I})^{\sigma ^{F}}-\left( w_{m}m_{t}^{I}-\Theta
_{1}(w_{m}m_{t}^{I})^{\omega _{k}}\right) ^{\sigma ^{F}}]}\right] ^{\chi
/[\phi (1-\sigma ^{F})]},  \notag
\end{eqnarray}%
where $\Theta _{1}=(\omega _{k}/\omega _{\lambda })(1/\omega _{\lambda
})^{\omega _{k}-1}$ and $\Theta _{2}=(\gamma \eta \nu ^{\nu }(1-\nu )^{1-\nu
})^{\gamma /(1-\gamma )}$.

In the steady-state, $\dot{m}_{t}^{I}=\dot{m}_{t}^{R}=\dot{z}_{t}^{C}=\dot{Q}%
_{t}^{R}=0$, while $Y_{t}/K_{t}\,$, $\theta _{t}^{S,Y}$, $\theta _{t}^{U,Y}$%
, $\theta _{t}^{U}$, $\theta _{t}^{S}$, $n_{FH,t}$, and $n_{FV,t}$ are
constant. This implies that the growth rate of final output is the same as
the growth rate of the private capital stock in steady-state, which in turn
means the growth rate of aggregate private consumption is the same.

To estimate the dynamics of output growth rate during the transition path, $%
K_{t}$ in (\ref{A15}) is first moved to the RHS to derive an expression for $%
Y_{t}$:

\begin{equation}
Y_{t}=\frac{\Theta _{2}}{[(\theta _{t}^{S,Y})^{\beta ^{S}}(\theta
_{t}^{U,Y})^{\beta ^{U}}]^{-1/(1-\gamma )}}\left\{ (m_{t}^{I})^{\nu (1-\eta
)/\eta }(m_{t}^{R})^{(1-\nu )(1-\eta )/\eta }\right\} ^{\gamma /(1-\gamma
)}K_{t}.  \tag{A.53}  \label{A53}
\end{equation}

Log-differentiating (\ref{A41}) gives

\begin{equation}
\frac{\dot{Y}_{t}}{Y_{t}}=\frac{\dot{K}_{t}^{P}}{K_{t}^{P}}+[\frac{\gamma
\nu (1-\eta )}{(1-\gamma )\eta }]\frac{\dot{m}_{t}^{I}}{m_{t}^{I}}+[\frac{%
\gamma (1-\nu )(1-\eta )}{(1-\gamma )\eta }]\frac{\dot{m}_{t}^{R}}{m_{t}^{R}}%
+(\frac{\beta ^{U}}{1-\gamma })\frac{\dot{\theta}_{t}^{U,Y}}{\theta
_{t}^{U,Y}}+(\frac{\beta ^{S}}{1-\gamma })\frac{\dot{\theta}_{t}^{S,Y}}{%
\theta _{t}^{S,Y}}.  \tag{A.54}  \label{A54}
\end{equation}

Next, substituting time derivative equation of (\ref{A13}) into $\dot{\theta}%
_{t}^{S,Y}/\theta _{t}^{S,Y}$, we get

\begin{eqnarray*}
\frac{\dot{Y}_{t}}{Y_{t}} &=&\frac{\dot{K}_{t}^{P}}{K_{t}^{P}}+[\frac{\gamma
\nu (1-\eta )}{(1-\gamma )\eta }]\frac{\dot{m}_{t}^{I}}{m_{t}^{I}}+[\frac{%
\gamma (1-\nu )(1-\eta )}{(1-\gamma )\eta }]\frac{\dot{m}_{t}^{R}}{m_{t}^{R}}%
+(\frac{\beta ^{U}}{1-\gamma })\frac{\dot{\theta}_{t}^{U,Y}}{\theta
_{t}^{U,Y}}+(\frac{\beta ^{S}}{1-\gamma })\{\frac{\dot{Y}_{t}}{Y_{t}}-\frac{%
\dot{K}_{t}^{P}}{K_{t}^{P}} \\
&&-\frac{\dot{Q}_{t}^{R}}{Q_{t}^{R}}-\frac{\dot{m}_{t}^{R}}{m_{t}^{R}}-\psi
_{1}^{R}\frac{\dot{n}_{FV,t}}{n_{FV,t}}-\frac{\psi _{2}^{R}}{(1+\psi _{2}^{R}%
\frac{m_{t}^{I}}{m_{t}^{R}})}[\frac{m_{t}^{I}}{m_{t}^{R}}(\frac{\dot{m}%
_{t}^{I}}{m_{t}^{I}}-\frac{\dot{m}_{t}^{R}}{m_{t}^{R}})]\}\text{.}
\end{eqnarray*}

Since we can easily derive $\dot{n}_{FV,t}/n_{FV,t}$ as

\begin{equation*}
\frac{\dot{n}_{FV,t}}{n_{FV,t}}=\left( \frac{(-\chi \omega _{k}\sigma ^{F})}{%
\phi (1-\sigma ^{F})}\right) \frac{\dot{m}_{t}^{I}}{m_{t}^{I}},
\end{equation*}%
the expression for final output growth can then be written as

\begin{eqnarray}
\frac{\dot{Y}_{t}}{Y_{t}} &=&\frac{\dot{K}_{t}^{P}}{K_{t}^{P}}+\left[ \frac{%
\gamma \nu (1-\eta )}{(1-\gamma )\eta }(1-\frac{\beta ^{S}}{1-\gamma }-\frac{%
\beta ^{S}(1+\psi _{1}^{R})(-\chi \omega _{k}\sigma ^{F})}{(1-\gamma )[\phi
(1-\sigma ^{F})]})^{-1}\right] \frac{\dot{m}_{t}^{I}}{m_{t}^{I}} 
\TCItag{A.55}  \label{A55} \\
&&+(\frac{\beta ^{U}}{1-\gamma })(1-\frac{\beta ^{S}}{1-\gamma })^{-1}\frac{%
\dot{\theta}_{t}^{U,Y}}{\theta _{t}^{U,Y}}+\{[\frac{\gamma (1-\nu )(1-\eta )%
}{(1-\gamma )\eta }  \notag \\
&&-\frac{\beta ^{S}}{1-\gamma }](1-\frac{\beta ^{S}}{1-\gamma })^{-1}\}\frac{%
\dot{m}_{t}^{R}}{m_{t}^{R}}-(\frac{\beta ^{S}}{1-\gamma })(1-\frac{\beta ^{S}%
}{1-\gamma })^{-1}\frac{\dot{Q}_{t}^{R}}{Q_{t}^{R}}  \notag \\
&&-[(\frac{\beta ^{S}\psi _{2}^{R}}{1-\gamma })(1-\frac{\beta ^{S}}{1-\gamma 
})^{-1}(1+\psi _{2}^{R}\frac{m_{t}^{I}}{m_{t}^{R}})^{-1}][\frac{m_{t}^{I}}{%
m_{t}^{R}}(\frac{\dot{m}_{t}^{I}}{m_{t}^{I}}-\frac{\dot{m}_{t}^{R}}{m_{t}^{R}%
})]\text{.}  \notag
\end{eqnarray}%
\newpage 

\end{document}
