\begin{document}
\title{Structural Equations for PE Models in Group 3\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 3 \\
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(FIRM HETEROGENEITY)} \\
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{\Large David Riker and Samantha Schreiber} \\
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{U.S. International Trade Commission, Office of Economics} \\
\vspace{0.75in}
{\Large August 2019}
\vspace{1.00in}
\end{center}
\begin{abstract}
\noindent This paper presents the structural equations for the third group of industry-specific 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 \label{sec: section1}}
One of the spreadsheet models incorporates fixed costs, firm heterogeneity, and cross-border trade. The second model extends the first to include foreign affiliate sales.
\section{Model with Fixed Costs and Firm Heterogeneity \label{sec: section2}}
The first model is a two-country \citeasnoun{Melitz2003} model of international trade with firm heterogeneity. The industry-specific model adopts useful simplifications and distributional assumptions from \citeasnoun{HMY2004} and \citeasnoun{Chaney2008}.\footnote{\citeasnoun{Khachaturian2016} provides a step-by-step derivation of the model, for an extended version that includes FDI.} Within the industry, there is a continuum of firms supplying differentiated products, with constant elasticity of substitution $\sigma$. The firms vary in their unit labor requirements. Firm-specific productivity has a Pareto distribution with shape parameter $\gamma$.\footnote{The model adopts the standard assumption in the literature that $\gamma > \sigma - 1$. \citeasnoun{DiGiovanni2011} is a source for industry-specific econometric estimates of these parameter values.} There is a fixed cost of production $f_D$ and an incremental fixed cost of exporting $f_X$. $D$ is the aggregate value of domestic shipments in the market, integrated over the mass $n_D$ of domestic suppliers.\footnote{Following \citeasnoun{Chaney2008}, the model assumes that the number of firms that participate in the market is endogenously determined but the numbers of potential market participants, $n_d$ and $n_f$, are exogenous.}
\begin{equation}\label{eq:1}
D = k \ \left( \frac{n_d}{n_d + n_f \ (\tau)^{-\gamma} \left( \frac{f_X}{f_D} \right)^{\frac{-\gamma}{\sigma-1}+1}} \right)
\end{equation}
\noindent $M$ is the aggregate value of imports into the market, integrated over the mass $n_F$ of foreign suppliers. $k$ is total industry expenditures in the market, and $\tau$ is a variable trade cost on imports.
\begin{equation}\label{eq:2}
M = k \ \left( \frac{n_f \ (\tau)^{-\gamma} \left( \frac{f_X}{f_D} \right)^{\frac{-\gamma}{\sigma-1}+1}}{n_d + n_f \ (\tau)^{-\gamma} \left( \frac{f_X}{f_D} \right)^{\frac{-\gamma}{\sigma-1}+1}} \right)
\end{equation}
\noindent Equation (\ref{eq:3}) is the ratio of the value of imports to the value of domestic shipments.
\begin{equation}\label{eq:3}
\frac{M}{D} = (\tau)^{-\gamma} \left( \frac{n_f}{n_d} \right) \left( \frac{f_X}{f_D} \right)^{\frac{-\gamma}{\sigma-1}+1}
\end{equation}
Next, we define $Z_0 = \left( \frac{n_f}{n_d} \right) \left( \frac{f_X}{f_D} \right)^{\frac{-\gamma}{\sigma-1}+1}$. The model calibrates $Z_0$ based on the ratio of the value of imports to the value of domestic shipments in the initial equilibrium and initial trade costs.
\begin{equation}\label{eq:4}
Z_0 = \left( \frac{M_0}{D_0} \right) \ (\tau_0)^\gamma
\end{equation}
\noindent $\tau_0$ is the initial trade cost factor, $M_0$ is the initial value of imports, and $D_0$ is the initial value of domestic shipments. An increase in the fixed cost of exporting $f_X$ decreases $Z$ and relative expenditure on imports, and an increase in the fixed cost of domestic production $f_D$ increases $Z$ and relative expenditure on imports.
\begin{equation}\label{eq:5}
Z = Z_0 \ \left( 1 + \left(\frac{-\gamma}{\sigma-1}+1 \right) \left( \left( \frac{f_X - f_{X0}}{f_{X0}} \right) - \left( \frac{f_D - f_{D0}}{f_{D0}} \right) \right) \right)
\end{equation}
The model simulates the effects of changes in the fixed costs ($f_X$ and $f_D$) and the variable trade cost ($\tau$) on the value of imports ($M$) and the value of domestic shipments ($D$), based on (\ref{eq:6}) and (\ref{eq:7}).
\begin{equation}\label{eq:6}
D = D_0 \ \left( \frac{1+Z_0 \ (\tau_0)^{-\gamma}}{1+Z \ (\tau)^{-\gamma}} \right)
\end{equation}
\begin{equation}\label{eq:7}
M = M_0 \ \left( \frac{1+Z_0 \ (\tau_0)^{-\gamma}}{1+Z \ (\tau)^{-\gamma}} \right) \ \left( \frac{Z (\tau)^{-\gamma}}{Z_0 (\tau_0)^{-\gamma}} \right)
\end{equation}
\section{Model with Foreign Affiliate Sales \label{sec: section3}}
The second model is based on \citeasnoun{HMY2004} as modified in \citeasnoun{Khachaturian2016}. Like the first model, $Z_{P0}$ and $Z_{X0}$ are calibrated to the initial equilibrium in the market.
\begin{equation}\label{eq:8}
Z_{P0} = \left(\frac{n_f}{n_d}\right) \left(\frac{f_P}{f_D}\right)^{\frac{-\gamma}{\sigma-1}+1} = \left(\frac{A_0}{D_0} \right) \left(1-{C_0}^{1-\sigma} \right)^{-\frac{\gamma}{\sigma-1}+1}
\end{equation}
\begin{equation}\label{eq:9}
Z_{X0} = \left(\frac{n_f}{n_d}\right) \left(\frac{f_X}{f_D}\right)^{\frac{-\gamma}{\sigma-1}+1} = \left(\frac{M_0}{D_0} \right) {C_0}^\gamma + {C_0}^{1-\sigma+\gamma} \left(1-{C_0}^{1-\sigma} \right)^{\frac{\gamma}{\sigma-1}-1} Z_{P0}
\end{equation}
\noindent The variables $A_0$, $M_0$, and $D_0$ represent the initial values of foreign affiliate sales, cross-border imports, and domestic sales. $C_0$ is the initial relative variable cost of delivering foreign services supplied to the domestic market, including variable international trade costs. $f_P$ is the incremental fixed cost of foreign affiliate supply, $f_X$ is the fixed cost of cross-border trade, and $f_D$ is the fixed cost of provision by domestic suppliers. $n_d$ and $n_f$ are the number of domestic and foreign firms that can potentially supply the domestic market.\footnote{$n_d$ and $n_f$ are treated as exogenous variables in the partial equilibrium model.} Again, $\sigma$ is the elasticity of substitution, and $\gamma$ is the shape parameter of the Pareto distribution of firm-specific productivity levels.
Changes in the fixed costs of trade affect $Z_P$ and $Z_X$. Equations (\ref{eq:10}) and (\ref{eq:11}) are the updating equations.
\begin{equation}\label{eq:10}
\frac{Z_{P}-Z_{P0}}{Z_{P0}} = \left(1 + \left(\frac{-\gamma}{\sigma-1}+1 \right) \left( \left(\frac{f_{P}-f_{P0}}{f_{P0}} \right) - \left(\frac{f_{D}-f_{D0}}{f_{D0}} \right) \right) \right)
\end{equation}
\begin{equation}\label{eq:11}
\frac{Z_{X}-Z_{X0}}{Z_{X0}} = \left(1 + \left(\frac{-\gamma}{\sigma-1}+1 \right) \left( \left(\frac{f_{X}-f_{X0}}{f_{X0}} \right) - \left(\frac{f_{D}-f_{D0}}{f_{D0}} \right) \right) \right)
\end{equation}
\noindent The equilibrium values of foreign affiliate sales ($A$), cross-border imports ($M$), and domestic sales ($D$) are defined by (\ref{eq:12}), (\ref{eq:13}), and (\ref{eq:14}).
\begin{equation}\label{eq:12}
A = \frac{ E \ Z_P \ \left(1-C^{1-\sigma} \right)^{\frac{\gamma}{\sigma-1}-1}}{Z_P \ \left(1-C^{1-\sigma} \right)^{\frac{\gamma}{\sigma-1}} \ + \ Z_X \ C^{-\gamma} \ + 1}
\end{equation}
\begin{equation}\label{eq:13}
M = \frac{E \ C^{1-\sigma} \left(Z_X \ C^{-\gamma+\sigma-1} - Z_P \left(1-C^{1-\sigma} \right)^{\frac{\gamma}{\sigma-1}-1} \right)}{Z_P \ \left(1-C^{1-\sigma} \right)^{\frac{\gamma}{\sigma-1}} \ + \ Z_X \ C^{-\gamma} \ + 1}
\end{equation}
\begin{equation}\label{eq:14}
D = \frac{E}{Z_P \ \left(1-C^{1-\sigma} \right)^{\frac{\gamma}{\sigma-1}} \ + \ Z_X \ C^{-\gamma} \ + 1}
\end{equation}
\noindent These three values sum to total expenditure on the industry in the market, $E$, which is held constant in model simulations.
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