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\title{Some Clarification on Global Trade Definitions}
% \author{Lorenzo Isella}
\date{}

\begin{document}
\maketitle
  
% \abstract{
% We give the definitions of growth rate and we introduce the main
% formulas for the calculations of the composite growth rate along a
% multi-period time span. After illustrating the shortcomings inherent to
% a straightforward calculation of the
% growth rate of trade flows, we suggest a methodology to bypass these
% issues which borrows from the theory of measurement of investment returns. 
% }
\section{World Trade, World Imports and World Exports}
In trade statistics, we talk about the world total imports and/or
exports and of total trade when we sum the two, but I think there is
some ambiguity in the definitions (or at least things are not as plain
vanilla as one may think).
Note that we never talk about intra-EU imports or intra-EU exports,
but we talk about world imports, when de facto they coincide with
intra-world imports.

Let us look at a simple example to fix the ideas

\begin{figure}[htb]       
  \begin{center}  
\scalebox{.5}{\input{flows.pdf_t}} %the difference is just this part
\caption{Example of trade flows between three countries $A$, $B$ and $C$.}
\label{world} 
\end{center}  
\end{figure}

In Figure 1 we have a world consisting only of countries $A$, $B$ and
$C$.
We assume there are no tariffs or shipping costs, so an export of $20$ 
to $A$ from $C$ is also described as an import of $20$ from $C$ to $A$.

We now introduce the notation

\begin{equation}
\overleftrightarrow{AB}= \overrightarrow{AB} + \overleftarrow{AB}
  \end{equation}
where we mean that the total trade between $A$ and $B$,
$\overleftrightarrow{AB}$, is given by the exports  to $B$ from $A$,
$\overrightarrow{AB}$, plus the imports from $B$ to $A$,
$\overleftarrow{AB}$.
Since it does not matter which country we consider as a
reporter and which one as a partner, the following properties hold

\begin{equation}\label{total}
\overleftrightarrow{AB}= \overleftrightarrow{BA} 
\end{equation}
because  the total trade between $A$ and $B$ coincides with
the trade between $B$ and $A$ and

\begin{equation}\label{symmetry}
\overrightarrow{AB} = \overleftarrow{BA}
\end{equation}

i.e. the exports to B from A coincide with the imports from A to B.

The total world trade is a well-defined quantity given by three trade
flows

\begin{equation}
W=\overleftrightarrow{AB}+\overleftrightarrow{BC}+\overleftrightarrow{CA}=
\overrightarrow{AB} + \overleftarrow{AB} + \overrightarrow{BC} +
\overleftarrow{BC}+ \overrightarrow{CA} + \overleftarrow{CA}
\end{equation}  

which can be broken down into three pairs of imports/exports.

A natural definition of world imports is the sum what every country
imports from the rest of the world. In our case this amounts to

\begin{equation}
  \begin{split}
\boxed{W_{imp}}=\overbrace{\overleftarrow{AB}+\overleftarrow{AC}}^\text{A's
  total imports} + \overbrace{\overleftarrow{BA}+
\overleftarrow{BC}}^\text{B's
  total imports} + \overbrace{ \overleftarrow{CA}+ \overleftarrow{CB}}^\text{C's
  total imports} \\  \overset{\rm reorder\;the\;terms }{=}   
  \overleftarrow{BA}+   \overleftarrow{AB}+ \overleftarrow{BC} +  \overleftarrow{CB}+  \overleftarrow{AC} +
  \overleftarrow{CA} 
  \\  \overset{{\rm use\; Equation\;}\eqref{symmetry} }{=} \overrightarrow{AB} + \overleftarrow{AB} + \overrightarrow{BC} +
\overleftarrow{BC}+ \overrightarrow{CA} + \overleftarrow{CA}=\boxed{W}.
\end{split}
  \end{equation}

Since the world is by definition a closed system which has only
internal trade, aggregating the imports of all its countries simply
amounts to estimating the world total trade. The same
result holds if we aggregate the world total exports.

Within this framework, it is straightforward to calculate the share of world trade
represented by the trade exchanges (imports and exports) between $A$
and $B$; from Figure \ref{world} this is given by $(20+15)/100=35\%$.

The calculation of the ratio of  $A$' imports to the world total
imports (which are just the world total trade)
is
also unambiguous. In  figure
\ref{world}, the world total trade amounts to $100$ and  $A$
imports $20$ from $B$ and $20$ from $C$, so $A$'s imports are $40\%$
of the world's total imports. This expression is commonly used, but it
really means that $A$'s imports are $40\%$ of the world's total
trade.


As a matter of fact, the import value never coincides with the export
value, so we can at most say that $A$ is responsible with its imports of
$40\%$ of world's trade as estimated from the import statistics.



The calculation of the ratio of $A$'s total trade (imports plus
export) to the world trade is problematic because it implies
indirectly the double counting of the trade flows.
Let us see what happens if we naively sum the total trade of each
country in the world in Figure \ref{world}


\begin{equation}
  \begin{split}
\overbrace{\overleftrightarrow{AB}+\overleftrightarrow{AC}}^\text{A's
  total trade} + \overbrace{\overleftrightarrow{BA}+
\overleftrightarrow{BC}}^\text{B's
  total trade} + \overbrace{ \overleftrightarrow{CA}+ \overleftrightarrow{CB}}^\text{C's
  total trade} \\  \overset{\rm reorder\;the\;terms }{=}
\overleftrightarrow{AB}+  \overleftrightarrow{BA}+
\overleftrightarrow{BC}+
\overleftrightarrow{CB}+\overleftrightarrow{AC}+
\overleftrightarrow{CA}
\\  \overset{{\rm use\; Equation\;}\eqref{total} }{=} 2\left(\overleftrightarrow{AB}+\overleftrightarrow{BC}+\overleftrightarrow{CA}  \right)=2W.
\end{split}
\end{equation}


As a consequence, a consistent way to define the share of world
trade absorbed by $A$'s total trade (imports plus exports) is to
divide $A$'s imports plus exports by \emph{twice} the world total trade. Based on Figure
\ref{world} this amounts to $(20+30+20+15)/200=42.5\%$.
I think this is what we implicitly due in our statistics when we
divide the sum of imports plus exports e.g. for China by the sum of
world imports and exports. However, the sum of world imports and
exports is de facto \emph{twice} the world trade, measured by import
and export statistics, respectively.

\end{document}
              
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