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\title{Translog Partial Equilibrium Model of Trade Policy Changes\vspace{0.5in}%
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\author{Samantha Schreiber\thanks{U.S. International Trade Commission.\newline Contact emails: samantha.schreiber@usitc.gov}}
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{\Large Translog Partial Equilibrium Model of Trade Policy Changes} \\
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{\Large Samantha Schreiber} \\
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{\large ECONOMICS WORKING PAPER SERIES}\\
Working Paper 2019--07--A \\ \vspace{0.5in}
U.S. INTERNATIONAL TRADE COMMISSION \\
500 E Street SW \\
Washington, DC 20436 \\
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July 2019
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\noindent Office of Economics working papers are the result of ongoing professional research of USITC Staff and are solely meant to represent the opinions and 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. Working papers are circulated to promote the active exchange of ideas between USITC Staff and recognized experts outside the USITC and to promote professional development of Office Staff by encouraging outside professional critique of staff research. The author thanks David Riker for helpful comments and related research collaboration. Please address correspondence to samantha.schreiber@usitc.gov.
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Translog Partial Equilibrium Model of Trade Policy Changes\\
Samantha Schreiber\\
Office of Economics Working Paper 2019--07--A\\
July 2019\\~\\
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\begin{abstract}
\noindent We present a sector-specific partial equilibrium model that quantifies the economic impacts of a tariff change on prices and quantities. The model uses transcendental logarithmic (translog) preferences, originally proposed by Christensen, Jorgenson and Lau (1975) and made popular in recent times by Feenstra, instead of the commonly-used constant elasticity of subtitution (CES) preferences.
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Samantha Schreiber, Office of Economics\\
\href{mailto:samantha.schreiber@usitc.gov}{samantha.schreiber@usitc.gov}\\
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\section{Introduction}
Constant elasticity of substitution (CES) is a common demand functional form assumption used in trade policy models with one constant elasticity parameter that characterizes substitutability across all sources of supply\footnote{Or two elasticity parameters for a nested-CES specification, one for domestic goods and an import aggregate, and one for all imported goods.}. This assumption has a number of attractive properties such as limited data requirements and algebraic ease. However, one substitution elasticity may be too restrictive if there are large substitutability differences across goods. In this paper we describe an alternative to CES, the "transcendental logarithmic" or translog model, that allows for greater variety of substitution patterns across pairs of goods.
The translog model was first proposed by Christensen, Jorgenson and Lau \cite{Christensen} but made popular in recent times by Feenstra \cite{Feenstra3,Feenstra,Feenstra4}. The model begins with a quadratic, logarithmic indirect utility function using expenditure-normalized prices\footnote{The use of expenditure-normalized prices $(p_j/M)$ imposes homogeneity.}. By Roy's Identity, we derive a system of budget share equations that represent consumer demand for each good. There are own- and cross-price demand elasticities for each pair of goods, partially restricted in the model to align with theoretical underpinnings.
First we will present the theory behind the translog model in section 2. In section 3, we describe popular restrictions placed on the model to arrive at a solution. Then we show illustrative simulation results with different model inputs in section 4. Section 5 offers areas of further research and concludes.
\section{Model}
Denote the price of each good $i\in \{1,2,..n\}$ as $p_i$. There are $n$ total sources of supply, including both domestic and imported varieties. Define $M$ as total expenditures, $\alpha_i$ as a demand shift parameter, $t_i$ as the tariff rate, and $q_i$ as the quantity of each good $i$. Define $\gamma_{ij}$ to be a utility parameter representing the coefficient on the interaction of log prices for $i,j\in \{1,2,...n\}$. The translog indirect utility function $log \ V$ takes the form:
\begin{equation}\label{eq:1}
log V = \alpha_0 + \sum_i \alpha_i \ log(\frac{p_i}{M}) + \frac{1}{2} \sum_i \sum_j \gamma_{ij} \ log(\frac{p_i}{M}) \ log(\frac{p_j}{M})
\end{equation}
\noindent By Roy's identity, the budget share for the $ith$ good is:
\begin{equation}\label{eq:2}
\frac{p_i \ q_i}{M} = - \ \frac{\partial log V}{\partial log p_i} \bigg/ \frac{\partial log V}{\partial log M}
\end{equation}
\noindent Partially differentiating this form of the indirect utility function:
\begin{equation}\label{eq:3}
\frac{\partial log V}{\partial log p_i} = \alpha_i +\sum_j \gamma_{ij} \ log \frac{p_j}{M}
\end{equation}
\begin{equation}\label{eq:4}
- \frac{\partial log V}{\partial log M} = \sum_k \big( \alpha_k + \sum_j \gamma_{kj} \ log \frac{p_j}{M}\big)
\end{equation}
\noindent Define $\alpha_M = \sum_k \alpha_k$ and $\gamma_{Mj} = \sum_i \gamma_{ij}$ for ease of notation. This gives us budget share equations for each $i$:
\begin{equation}\label{eq:5}
\frac{p_i q_i}{M} = \frac{\alpha_i + \sum_j \gamma_{ij} log(\frac{p_j}{M}) }{\alpha_M + \sum_j \gamma_{Mj}log(\frac{p_j}{M})}
\end{equation}
\noindent We use the $\gamma$ parameters to calculate the uncompensated price elasticities in this model as:
\begin{equation}\label{eq:6}
\eta_{ij} = -\delta_{ij} + \frac{\frac{\gamma_{ij}}{w_i} - \sum_j \gamma_{ij}}{-1 + \sum_k \gamma_{Mk} log(\frac{p_k}{M})}
\end{equation}
\noindent where $\delta_{ij}$ is the kronecker delta, equal to one if $i=j$ and zero otherwise, and $w_i$ is the budget share of good $i$. On the supply side, let $\epsilon_i$ be the constant price elasticity of supply, and suppose $b_i$ is a supply parameter. Then the supply equations for each source $i$ are:
\begin{equation}\label{eq:7}
q_i = b_i \ p_i^{\epsilon_i}
\end{equation}
\noindent This gives us $i$ budget share equations and $i$ price unknowns. The user solves the model by supplying initial expenditure data for each good $i$, specifying demand parameters $\gamma_{ij}$, supply elasticity parameters $\epsilon_i$ and the current and new tariff rate policy shock. The rest of the parameters are calibrated to the initial data inputs.
\section{Restrictions}
The first restriction for the translog model, required by theory, is to impose symmetry in the $\gamma$ demand parameters $(\gamma_{ij}=\gamma_{ji})$. For $n$ goods in the model, this limits the number of $\gamma$ inputs to $\frac{n(n+1)}{2}$. An additional restriction popular in the literature \cite{Feenstra} is to impose an additivity constraint so that $\sum_j \gamma_{ij} = 0$. This further limits the number of demand parameter inputs to $\frac{n(n-1)}{2}$. Third, also common in the literature \cite{Christensen,Feenstra,MPRA} is to normalize the $\alpha_i$ demand parameters so that $\sum_i \alpha_i = -1$. With this restriction, you can use $(n-1)$ budget share equations to calibrate the $\alpha_i$'s to initial expenditure data inputs\footnote{As pointed out by Christensen, Jorgenson and Lau, there are more parameters than necessary to qualify as a second-order locally flexible functional form.}.
For three sources of supply (domestic production, subject imports, and non-subject imports), there are nine $\gamma$ demand parameters and three $\alpha$ demand parameters. Imposing the first restriction, $\gamma_{ij}=\gamma_{ji}$ eliminates three of the nine parameters, and imposing the second restriction brings the number of free $\gamma$ parameters to three. Imposing the third restriction, $\sum_i \alpha_i = -1$, and using $n-1$ initial budget share conditions, the modeler can calibrate all three $\alpha$ parameters with initial data inputs.
\section{Illustrative Simulations with Three Goods}
In this section, we assume there are three sources of supply: domestic production $d$, subject imports $s$, and non-subject imports $n$. Using the equations from section 2 and restrictions from section 3, the budget share equations and supply equations are:
\begin{equation}\label{eq:8}
\frac{p_d \ q_d}{M} = \frac{\alpha_d + \gamma_{ds} \ log \frac{p_s \ t_s}{M}+ \gamma_{dn} \ log \frac{p_n}{M} + \gamma_{dd} \ log \frac{p_d}{M}}{
(\gamma_{ds} + \gamma_{ns} + \gamma_{ss}) \ log \frac{p_s \ t_s}{M} + (\gamma_{dn} + \gamma_{nn} + \gamma_{sn}) \ log \frac{p_n}{M} + (\gamma_{dd} + \gamma_{nd} + \gamma_{sd}) \ log \frac{p_d}{M} -1}
\end{equation}
\begin{equation}\label{eq:9}
\frac{p_s \ t_s \ q_s}{M} = \frac{\alpha_s + \gamma_{ss} \ log \frac{p_s \ t_s}{M}+ \gamma_{sn} \ log \frac{p_n}{M} + \gamma_{sd} \ log \frac{p_d}{M}}{
(\gamma_{ds} + \gamma_{ns} + \gamma_{ss}) \ log \frac{p_s \ t_s}{M} + (\gamma_{dn} + \gamma_{nn} + \gamma_{sn}) \ log \frac{p_n}{M} + (\gamma_{dd} + \gamma_{nd} + \gamma_{sd}) \ log \frac{p_d}{M}-1}
\end{equation}
\begin{equation}\label{eq:10}
\frac{p_n \ q_n}{M} = \frac{\alpha_n + \gamma_{ns} \ log \frac{p_s \ t_s}{M}+ \gamma_{nn} \ log \frac{p_n}{M} + \gamma_{nd} \ log \frac{p_d}{M}}{
(\gamma_{ds} + \gamma_{ns} + \gamma_{ss}) \ log \frac{p_s \ t_s}{M} + (\gamma_{dn} + \gamma_{nn} + \gamma_{sn}) \ log \frac{p_n}{M} + (\gamma_{dd} + \gamma_{nd} + \gamma_{sd}) \ log \frac{p_d}{M}-1}
\end{equation}
\begin{equation}\label{eq:11}
q_d = b_d p_d^{\epsilon_d}
\end{equation}
\begin{equation}\label{eq:12}
q_s = b_s p_s^{\epsilon_s}
\end{equation}
\begin{equation}\label{eq:13}
q_n = b_n p_n^{\epsilon_n}
\end{equation}
\noindent We impose all three restrictions described in the third section. Table 1 reports model results under two different sets of elasticity values. The first section of the table shows model input values. The second section reports the calculated uncompensated price elasticity values from the initial model inputs. The third section of the table presents the model results.
\begin{table}
\caption{Illustrative Simulations of Translog Model}
\centering
\begin{tabular}{|l|c|c|}\hline
& Simulation 1 & Simulation 2 \\ \hline
Initial Market Share of Domestic Goods & 70\% & 70\%\\
Initial Market Share of Subject Imports & 20\% & 20\% \\
Initial Market Share of Non-Subject Imports & 10\% & 10\% \\
New Tariff Rate & 10\% & 10\% \\
Price elasticity of supply, domestic good & 2 & 2 \\
Price elasticity of supply, imports & 10 & 10 \\
Gamma, domestic and subject import interaction coefficient & -0.25 & -0.4\\
Gamma, domestic and non-subject import interaction coefficient & -0.1 & -0.3 \\
Gamma, subject and non-subject import interaction coefficient & -0.003 & 0 \\
\hline
Uncompensated price elasticity, domestic good & -1.5 & -2\\
Uncompensated price elasticity, subject import good & - 2.25 & -3 \\
Uncompensated price elasticity, non-subject import good & -2.05 & -4\\
Cross-price elasticity, domestic response to $\Delta$ in $p_s$ & 0.35 & 0.6 \\
Cross-price elasticity, domestic response to $\Delta$ in $p_n$ & 0.15 & 0.4\\
Cross-price elasticity, subject-import response to $\Delta$ in $p_d$ & 1.24 & 2 \\
Cross-price elasticity, non-subject import response to $\Delta$ in $p_d$ & 1.03 & 3\\
\hline
\% change in domestic producer price & 0.79\% & 1.08\% \\
\% change in subject import producer price & -1.69\% & -2.08\% \\
\% change in subject import consumer price & 8.14\% & 7.7\%\\
\% change in non-subject import producer price & 0.08\% & 0.23\%\\
\% change in quantity of domestic shipments & 1.58\% & 2.17\% \\
\% change in quantity of subject imports & -15.67\% & -18.96\% \\
\% change in quantity of non-subject imports & 0.83\% & 2.3\%\\ \hline
\end{tabular}
\label{tab:1}
\end{table}
\section{Conclusion}
The translog model, made popular recently in the literature, is a departure from the traditional constant elasticity of substitution (CES) assumption typically used in partial equilibrium trade models. The translog model allows for greater variety in substitutability across sources of supply, but also requires that the modeler supply additional parameter values which may be difficult to obtain. This model should be combined with an econometric estimation of parameter inputs to generate theory-consistent, data-driven results.
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