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Revision48793f55e517a804bc9e8b23cf007fface63a837 (tree)
Zeit2018-12-17 20:20:51
AutorLorenzo Isella <lorenzo.isella@gmai...>
CommiterLorenzo Isella

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

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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
@@ -0,0 +1,172 @@
1+% \documentclass[12pt,a4paper]{article}
2+\documentclass[14pt, a4paper]{extarticle}
3+\usepackage[utf8x]{inputenc}
4+\usepackage[english]{babel}
5+\usepackage{url}
6+\usepackage{graphicx}
7+\usepackage{amsmath}
8+\usepackage{xcolor}
9+\usepackage{caption}
10+\usepackage{hyperref}
11+\usepackage{esvect}
12+% for placeholder text
13+\usepackage{lipsum}
14+\usepackage[margin=0.5in]{geometry}
15+\usepackage{mathtools}% Loads amsmath
16+
17+\title{Some Clarification on Global Trade Definitions}
18+% \author{Lorenzo Isella}
19+\date{}
20+
21+\begin{document}
22+\maketitle
23+
24+% \abstract{
25+% We give the definitions of growth rate and we introduce the main
26+% formulas for the calculations of the composite growth rate along a
27+% multi-period time span. After illustrating the shortcomings inherent to
28+% a straightforward calculation of the
29+% growth rate of trade flows, we suggest a methodology to bypass these
30+% issues which borrows from the theory of measurement of investment returns.
31+% }
32+\section{World Trade, World Imports and World Exports}
33+In trade statistics, we talk about the world total imports and/or
34+exports and of total trade when we sum the two, but I think there is
35+some ambiguity in the definitions (or at least things are not as plain
36+vanilla as one may think).
37+Note that we never talk about intra-EU imports or intra-EU exports,
38+but we talk about world imports, when de facto they coincide with
39+intra-world imports.
40+
41+Let us look at a simple example to fix the ideas
42+
43+\begin{figure}[htb]
44+ \begin{center}
45+\scalebox{.5}{\input{flows.pdf_t}} %the difference is just this part
46+\caption{Example of trade flows between three countries $A$, $B$ and $C$.}
47+\label{world}
48+\end{center}
49+\end{figure}
50+
51+In Figure 1 we have a world consisting only of countries $A$, $B$ and
52+$C$.
53+We assume there are no tariffs or shipping costs, so an export of $20$
54+to $A$ from $C$ is also described as an import of $20$ from $C$ to $A$.
55+
56+We now introduce the notation
57+
58+\begin{equation}
59+\overleftrightarrow{AB}= \overrightarrow{AB} + \overleftarrow{AB}
60+ \end{equation}
61+where we mean that the total trade between $A$ and $B$,
62+$\overleftrightarrow{AB}$, is given by the exports to $B$ from $A$,
63+$\overrightarrow{AB}$, plus the imports from $B$ to $A$,
64+$\overleftarrow{AB}$.
65+Since it does not matter which country we consider as a
66+reporter and which one as a partner, the following properties hold
67+
68+\begin{equation}\label{total}
69+\overleftrightarrow{AB}= \overleftrightarrow{BA}
70+\end{equation}
71+because the total trade between $A$ and $B$ coincides with
72+the trade between $B$ and $A$ and
73+
74+\begin{equation}\label{symmetry}
75+\overrightarrow{AB} = \overleftarrow{BA}
76+\end{equation}
77+
78+i.e. the exports to B from A coincide with the imports from A to B.
79+
80+The total world trade is a well-defined quantity given by three trade
81+flows
82+
83+\begin{equation}
84+W=\overleftrightarrow{AB}+\overleftrightarrow{BC}+\overleftrightarrow{CA}=
85+\overrightarrow{AB} + \overleftarrow{AB} + \overrightarrow{BC} +
86+\overleftarrow{BC}+ \overrightarrow{CA} + \overleftarrow{CA}
87+\end{equation}
88+
89+which can be broken down into three pairs of imports/exports.
90+
91+A natural definition of world imports is the sum what every country
92+imports from the rest of the world. In our case this amounts to
93+
94+\begin{equation}
95+ \begin{split}
96+\boxed{W_{imp}}=\overbrace{\overleftarrow{AB}+\overleftarrow{AC}}^\text{A's
97+ total imports} + \overbrace{\overleftarrow{BA}+
98+\overleftarrow{BC}}^\text{B's
99+ total imports} + \overbrace{ \overleftarrow{CA}+ \overleftarrow{CB}}^\text{C's
100+ total imports} \\ \overset{\rm reorder\;the\;terms }{=}
101+ \overleftarrow{BA}+ \overleftarrow{AB}+ \overleftarrow{BC} + \overleftarrow{CB}+ \overleftarrow{AC} +
102+ \overleftarrow{CA}
103+ \\ \overset{{\rm use\; Equation\;}\eqref{symmetry} }{=} \overrightarrow{AB} + \overleftarrow{AB} + \overrightarrow{BC} +
104+\overleftarrow{BC}+ \overrightarrow{CA} + \overleftarrow{CA}=\boxed{W}.
105+\end{split}
106+ \end{equation}
107+
108+Since the world is by definition a closed system which has only
109+internal trade, aggregating the imports of all its countries simply
110+amounts to estimating the world total trade. The same
111+result holds if we aggregate the world total exports.
112+
113+Within this framework, it is straightforward to calculate the share of world trade
114+represented by the trade exchanges (imports and exports) between $A$
115+and $B$; from Figure \ref{world} this is given by $(20+15)/100=35\%$.
116+
117+The calculation of the ratio of $A$' imports to the world total
118+imports (which are just the world total trade)
119+is
120+also unambiguous. In figure
121+\ref{world}, the world total trade amounts to $100$ and $A$
122+imports $20$ from $B$ and $20$ from $C$, so $A$'s imports are $40\%$
123+of the world's total imports. This expression is commonly used, but it
124+really means that $A$'s imports are $40\%$ of the world's total
125+trade.
126+
127+
128+As a matter of fact, the import value never coincides with the export
129+value, so we can at most say that $A$ is responsible with its imports of
130+$40\%$ of world's trade as estimated from the import statistics.
131+
132+
133+
134+The calculation of the ratio of $A$'s total trade (imports plus
135+export) to the world trade is problematic because it implies
136+indirectly the double counting of the trade flows.
137+Let us see what happens if we naively sum the total trade of each
138+country in the world in Figure \ref{world}
139+
140+
141+\begin{equation}
142+ \begin{split}
143+\overbrace{\overleftrightarrow{AB}+\overleftrightarrow{AC}}^\text{A's
144+ total trade} + \overbrace{\overleftrightarrow{BA}+
145+\overleftrightarrow{BC}}^\text{B's
146+ total trade} + \overbrace{ \overleftrightarrow{CA}+ \overleftrightarrow{CB}}^\text{C's
147+ total trade} \\ \overset{\rm reorder\;the\;terms }{=}
148+\overleftrightarrow{AB}+ \overleftrightarrow{BA}+
149+\overleftrightarrow{BC}+
150+\overleftrightarrow{CB}+\overleftrightarrow{AC}+
151+\overleftrightarrow{CA}
152+\\ \overset{{\rm use\; Equation\;}\eqref{total} }{=} 2\left(\overleftrightarrow{AB}+\overleftrightarrow{BC}+\overleftrightarrow{CA} \right)=2W.
153+\end{split}
154+\end{equation}
155+
156+
157+As a consequence, a consistent way to define the share of world
158+trade absorbed by $A$'s total trade (imports plus exports) is to
159+divide $A$'s imports plus exports by \emph{twice} the world total trade. Based on Figure
160+\ref{world} this amounts to $(20+30+20+15)/200=42.5\%$.
161+I think this is what we implicitly due in our statistics when we
162+divide the sum of imports plus exports e.g. for China by the sum of
163+world imports and exports. However, the sum of world imports and
164+exports is de facto \emph{twice} the world trade, measured by import
165+and export statistics, respectively.
166+
167+\end{document}
168+
169+%%% Local Variables:
170+%%% mode: latex
171+%%% TeX-master: t
172+%%% End: