Commit: 48793f55e517a804bc9e8b23cf007fface63a837 - myprojects-hg-reborn (hg)

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• Revisión: 48793f55e517a804bc9e8b23cf007fface63a837 (tree)
• Tiempo: 2018-12-17 20:20:51
• Autor: Lorenzo Isella <lorenzo.isella@gmai...>
• Commiter: Lorenzo Isella

Log Message

A nice example showing several ways to comment some equations, group terms etc..

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• add: latex-documents/world-trade.tex

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diff -r ae190f689af1 -r 48793f55e517 latex-documents/world-trade.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/latex-documents/world-trade.tex	Mon Dec 17 12:20:51 2018 +0100
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+% \documentclass[12pt,a4paper]{article}
+\documentclass[14pt, a4paper]{extarticle}
+\usepackage[utf8x]{inputenc}
+\usepackage[english]{babel}
+\usepackage{url}
+\usepackage{graphicx}
+\usepackage{amsmath}  
+\usepackage{xcolor}
+\usepackage{caption}  
+\usepackage{hyperref}
+\usepackage{esvect}
+% for placeholder text
+\usepackage{lipsum}
+\usepackage[margin=0.5in]{geometry}
+\usepackage{mathtools}% Loads amsmath
+     
+\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}
+              
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End: