\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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\begin{center}
{\Large STRUCTURAL EQUATIONS FOR PE MODELS \\
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IN GROUP 3 \\
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(FIRM HETEROGENEITY, NEW ENTRY, \\
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AND INTELLECTUAL PROPERTY)} \\
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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 May 2019}
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\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}
\vfill
\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 and firm heterogeneity. The second model allows for the entry of imports from a new source. The third values the monopoly profits created by the protection of intellectual property rights. The fourth examines global incentives to innovate.
\section{Model with Fixed Costs and Firm Heterogeneity \label{sec: section2}}
The first model is a two-country \citeasnoun{Melitz2003} model of trade with firm heterogeneity. The model adopts useful simplifications and distributional assumptions in \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, $M_0$ 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 Entry of a New Source of Imports \label{sec: section3}}
The second model addresses the entry of new sources of imports.\footnote{\citeasnoun{Riker2019} presents an extended version of this model.} There are initially three sources of supply to the market, one domestic source ($x$) and two foreign sources ($y$ and $z$). Consumers have CES preferences, with elasticity of substitution $\sigma$ and a price elasticity of total industry demand equal to $\eta$. There is perfect competition in the market, and the supply from all three sources is perfectly elastic (i.e., there are no capacity constraints on production), so $p_j = c_j$ for $j \in \{x,y,z\}$.
Equation (\ref{eq:8}) is the original CES price index for the market, and (\ref{eq:9}) is the initial equilibrium demand for the product from source $j \in \{x,y,z\}$.
\begin{equation}\label{eq:8}
P_0 = \left( (p_{x0})^{1-\sigma} + b_y \ (p_{y0} \ \tau_{y0})^{1-\sigma} + b_z \ (p_{z0} \ \tau_{z0})^{1-\sigma} \right)^{\frac{1}{1-\sigma}}
\end{equation}
\begin{equation}\label{eq:9}
q_{j0} = k \ (P_0)^{\sigma+\eta} \ (p_{j0} \ \tau_{j0})^{-\sigma} \ b_j
\end{equation}
Equations (\ref{eq:10}) through (\ref{eq:12}) calibrate the three demand parameters to initial expenditures and tariff rates, normalizing initial prices to one without loss of generality.
\begin{equation}\label{eq:10}
b_y = \left( \frac{v_{y0}}{v_{x0}} \right) \left( \tau_{y0} \right)^{\sigma-1}
\end{equation}
\begin{equation}\label{eq:11}
b_z = \left( \frac{v_{z0}}{v_{x0}} \right) \left( \tau_{z0} \right)^{\sigma-1}
\end{equation}
\begin{equation}\label{eq:12}
k = v_{x0} \ \left( 1 + b_y \ (\tau_{y0})^{1-\sigma} + b_z \ (\tau_{z0})^{1-\sigma} \right)^{\frac{-\sigma -\eta}{1-\sigma}}
\end{equation}
Entry leads to a fourth source of supply, entrant $e$. The model assumes that source $z$ is an appropriate reference group, meaning that the production costs and perceived quality of the products of the new entrant ($c_e$ and $b_e$) are the same as those of the reference group ($c_z$ and $b_z$). This is not a model of endogenous entry; it is modeling the economic impact conditional on entry.\footnote{The issue of conditional entry is discussed in detail in \citeasnoun{Riker2019}.}
Equation (\ref{eq:13}) is the new equilibrium CES price index that includes source $e$.
\begin{equation}\label{eq:13}
P = \left( (p_x)^{1-\sigma} + b_y \ (p_y \ \tau_y)^{1-\sigma} + b_z \ (p_z \ \tau_z)^{1-\sigma} + b_e \ (p_e \ \tau_e)^{1-\sigma} \right)^{\frac{1}{1-\sigma}}
\end{equation}
\noindent Since supply from each source is perfectly elastic, $p_j = p_{j0} = c_j$ for $j \in \{x,y,z,e\}$. Equation (\ref{eq:14}) is the new equilibrium quantity from source $j$.
\begin{equation}\label{eq:14}
q_{j} = k \ (P)^{\sigma+\eta} \ (p_{j} \ \tau_{j})^{-\sigma} \ b_j
\end{equation}
\noindent Finally, (\ref{eq:15}) is the new equilibrium market share of source $j$.
\begin{equation}\label{eq:15}
m_{j} = \frac{ b_j \ (p_{j} \ \tau_{j})^{1-\sigma}}{(p_{x})^{1-\sigma} + b_y \ (p_{y} \ \tau_y)^{1-\sigma} + b_z \ (p_{z} \ \tau_z)^{1-\sigma} + b_e \ (p_{e} \ \tau_e)^{1-\sigma}}
\end{equation}
\noindent The model can be used to simulate the entry of a new source of imports, for example due to the reduction or removal of a prohibitive tariff on imports from the new source.
\section{Model of the Value of a Monopoly Created by Protecting Intellectual Property Rights \label{sec: section4}}
In the third model, there is a linear demand curve for the products of the market.\footnote{It can be problematic to assume a constant elasticity demand curve in a monopoly model, since the profit-maximizing price will be infinite for an elasticity of one or below in absolute value. For this reason, monopoly models often assume a linear demand curve.}
\begin{equation}\label{eq:16}
Q = a - b \ P
\end{equation}
\noindent Equation (\ref{eq:17}) is the initial price elasticity of total industry demand.
\begin{equation}\label{eq:17}
\eta_0 = \frac{\partial Q_0}{\partial P0} \ \frac{P_0}{Q_0} \ = -b \ \left( \frac{P_0}{Q_0} \right)
\end{equation}
There is initially perfect competition, because intellectual property rights (IPRs) are not protected, so price is equal to marginal cost. Profits are competed to zero by infringing or imitating firms. Equations (\ref{eq:18}) through (\ref{eq:20}) calibrate marginal costs and the two parameters of the demand curve based on the initial equilibrium price and quantity, $P_0$ and $Q_0$.
\begin{equation}\label{eq:18}
c = P_0
\end{equation}
\begin{equation}\label{eq:19}
b = \eta_0 \left( \frac{Q_0}{P_0} \right)
\end{equation}
\begin{equation}\label{eq:20}
a = Q_0 \ \left(1 + \eta_0 \right)
\end{equation}
\indent The protection of IPRs creates a monopoly in the market, and there is a new market equilibrium. Equation (\ref{eq:21}) is monopoly profits, in terms of the new monopoly price $P_m$ and quantity $Q_m$.
\begin{equation}\label{eq:21}
\pi_m = \left( P_m - c \right) \ Q_m
\end{equation}
\noindent Equation (\ref{eq:22}) is the first order condition for monopoly pricing.
\begin{equation}\label{eq:22}
\frac{\partial\pi_m}{\partial P_m} = a - 2 \ b \ P_m + b \ c = 0
\end{equation}
\noindent This first order condition implies the monopoly price in (\ref{eq:23}), the percent change in price in (\ref{eq:24}), and the percentage change in quantity in (\ref{eq:25}).
\begin{equation}\label{eq:23}
P_m= P_0 \ \left( \frac{1+2 \ \eta_0}{2 \ \eta_0} \right)
\end{equation}
\begin{equation}\label{eq:24}
\frac{P_m - P_0}{P_0} = \frac{1}{2} \left( \frac{1}{\eta_0} \right)
\end{equation}
\begin{equation}\label{eq:25}
\frac{Q_m - Q_0}{Q_0} = - \left( \frac{1}{2} \right)
\end{equation}
\noindent Finally, (\ref{eq:26}) is the value of monopoly profits at the new equilibrium, as a function of total industry revenues in the initial equilibrium, $R_0 = P_0 \ Q_0$.
\begin{equation}\label{eq:26}
\pi_m = \left( \frac{1}{4 \ \eta_0} \right) \ R_0
\end{equation}
\section{Model of Trade and Innovation \label{sec: section4}}
The fourth model addresses how the protection of IPRs affects incentives to innovate. It is based on models of trade, product diversity, and monopolistic competition in \citeasnoun{Krugman1980}. It is a simpler, static, partial equilibrium version of the model with innovation and horizontal differentiation in \citeasnoun{GH1989}. As we note below, the same model can be applied -- with specific modifications to the model inputs -- to address innovation that creates vertical differentiation, by reducing production costs or increasing product quality.
Within each industry, consumers have symmetric CES preferences with elasticity of substitution $\sigma$. There are Cobb-Douglas preferences between industries, which implies that the price elasticity of total industry demand is equal to -1.
There is a fixed cost to invent a new variety, $f$, and constant marginal costs of production $c$. The "blueprint" for each variety is non-rival in its use in different countries, so there are global scale economies to innovation, as long as the returns to innovation are ensured by the protection of IPRs.
There are a number of national markets, indexed by $j$, in which IPRs are protected in the initial equilibrium. In each market, there is a continuum of varieties. Each firm prices at a constant mark-up over marginal cost. Equation (\ref{eq:27}) are initial profits in market $j$.
\begin{equation}\label{eq:27}
\pi_j = \frac{1}{\sigma} \ R_{j}
\end{equation}
\noindent $R_j$ are initial revenues in country $j$. The model assumes that laws that protect IPRs create a monopoly in the variety that would otherwise not exist. Unrestricted imitation and infringement would drive the mark-up to zero and eliminate the incentive to develop the additional variety. Equation (\ref{eq:28}) is the initial number of varieties, $N_0$.
\begin{equation}\label{eq:28}
N_0 = \frac{1}{\sigma f} \sum_j \ R_{j}
\end{equation}
With the additional protection of IPRs in country $k$, the equilibrium number of varieties will increase to $N$.
\begin{equation}\label{eq:29}
N = N_0 + \frac{R_k}{\sigma f} = \frac{1}{\sigma f} \left( \sum_j \ R_{j} + R_k \right)
\end{equation}
\noindent Equation (\ref{eq:30}) is the percent change in the total number of product varieties developed. This measure of innovation increases in proportion to the size of the global sum of the IPR-protected national markets.
\begin{equation}\label{eq:30}
\frac{N - N_0}{N_0}= \frac{R_k}{\sum_j \ R_{j}}
\end{equation}
\noindent Equation (\ref{eq:31}) is the simulated change in the value of innovations from protecting IPRs in the additional markets.
\begin{equation}\label{eq:31}
f \ \Delta N = \frac{1}{\sigma} R_k
\end{equation}
If innovation leads to cost reductions then the IPR-protected mark-up is determined by the cost advantage of the technology leader over non-infringing imitators, rather than the reciprocal of the elasticity of substitution. The model can be applied to this alternative scenario by changing the model inputs. If innovation leads to quality reductions, then the mark-up would be based on the quality step.\footnote{Examples of models with vertical differentiation include \citeasnoun{GH1990}, \citeasnoun{GH1991QJE}, and \citeasnoun{GH1991RES}.}
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