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\title{Structural Equations for PE Models in Group 5\vspace{0.5in}%
}
\author{Riker and Schreiber\thanks{U.S. International Trade Commission.\newline Contact emails: pemodeling@usitc.gov}}
\date{\vspace{1.5in}%
\today}

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{\Large STRUCTURAL EQUATIONS FOR PE MODELS \\
\vspace{0.25in}
IN GROUP 5 (MANY COUNTRIES)} \\
\vspace{1.00in}
{\Large David Riker and Samantha Schreiber} \\
\vspace{0.25in}
{U.S. International Trade Commission, Office of Economics} \\
\vspace{0.75in}

{\Large March 2020}
\vspace{1.00in}

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\begin{abstract}
\noindent This paper presents the structural equations for the fifth group of Euler method simulation models of changes in trade policy that are available for download on the USITC's PE Modeling Portal at \url{https://www.usitc.gov/data/pe\_modeling/index.htm.}


\end{abstract}

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\noindent The models described in this paper are the result of ongoing professional research of USITC staff and are solely meant to represent the professional research of individual authors. These papers are not meant to represent in any way the views of the U.S. International Trade Commission or any of its individual Commissioners. Please address correspondence to david.riker@usitc.gov.

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\section{Introduction}

There are a pair of  many country trade models that have been translated into a user-friendly spreadsheet format.

\section{Tariff Model with Two National Markets}

This model includes two source countries that are also destination markets, the domestic economy ($d$) and the foreign economy ($f$). Equations (1) through (6) describe this multi-region model.

\begin{equation}\label{eq:1}
P_d = \left( (p_d)^{1-\sigma} \ + \ b_{fd} \ (p_f \ \tau_{fd})^{1-\sigma} \right) ^\frac{1}{1-\sigma}
\end{equation}

\begin{equation}\label{eq:2}
P_f = \left( b_{df} \ (p_d \ \tau_{df})^{1-\sigma} \ + \  (p_f)^{1-\sigma} \right) ^\frac{1}{1-\sigma}
\end{equation}

\begin{equation}\label{eq:3}
q_d = k_d \ (P_d)^{\eta_d} \ \left(\frac{p_d}{P_d} \right)^{-\sigma} \ + \ k_f \ (P_f)^{\eta_f} \ \left(\frac{p_d \ \tau_{df}} {P_f} \right)^{-\sigma} \ b_{df}
\end{equation}

\begin{equation}\label{eq:4}
q_f = k_d \ (P_d)^{\eta_d} \ \left(\frac{p_f \ \tau_{fd}}{P_d} \right)^{-\sigma} \ b_{fd} \ + \ k_f \ (P_f)^{\eta_f} \ \left(\frac{p_f} {P_f} \right)^{-\sigma}
\end{equation}

\begin{equation}\label{eq:5}
q_d = a_d \ (p_d)^{\epsilon_d}
\end{equation}

\begin{equation}\label{eq:6}
q_f = a_f \ (p_f)^{\epsilon_f}
\end{equation}

\noindent This model simulates the effects of tariff changes of either country, on prices and volumes of imports and domestic production in both of the markets ($d$ and $f$).\footnote{There is also a tariff model with three national markets available on the portal as a Mathematica notebook.}

\section{Many Country Model of Country-Specific Supply Shocks}

This model includes up to five source countries and as many destination markets. It is an endowment economy model with exogenous supplies in each country and the product is not differentiated by country source. $c_j$ is the quantity of the product consumed in country $j$.

\begin{equation}\label{eq:7}
c_j = E_j \ p^{\phi}
\end{equation}

\noindent $E_j$ is aggregate expenditure in country $j$, $p$ is the price of the product, and $\phi<0$ is the price elasticity of demand in each country. $y_j$ is the inelastic supply of the product in each country. Equation (8) is the global market clearing condition.

\begin{equation}\label{eq:8}
\sum_j c_j = \sum_j y_j
\end{equation}

\noindent Equation (9) is the implies global equilibrium price.

\begin{equation}\label{eq:9}
p = \left( \frac{\sum_j E_j}{\sum_j y_j} \right)^{-\frac{1}{\phi}}
\end{equation}

\noindent Finally, Equation (10) is the quantity of net improts of country $j$
\begin{equation}\label{eq:10}
nx_j = c_j \ - \ y_j
\end{equation}

\noindent The model simulates the effects of country-specific supply shocks on global prices and country-specific trade flows and consumption levels.

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