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\title{Tariffs and Trade Diversion with Multiple Export Markets}
\author{Ward M. Reesman and David Riker}
\date{\vspace{1.5in}%
\today}

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\begin{center}
{\Large \textbf{TARIFFS AND TRADE DIVERSION}} \\ 
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{\Large \textbf{WITH MULTIPLE EXPORT MARKETS}} \\
\vspace{0.75in}
{\Large Ward M. Reesman} \\
\vspace{0.25in}
{\Large David Riker} \\

\vspace{0.75in}

{\large ECONOMICS WORKING PAPER SERIES}\\ 
Working Paper 2025--01--B \\ \vspace{0.5in}
U.S. INTERNATIONAL TRADE COMMISSION \\
500 E Street SW \\
Washington, DC 20436 \\
\vspace{0.5in}
January 2025
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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. Please address correspondence to ward.reesman@usitc.gov or david.riker@usitc.gov.


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Tariffs and Trade Diversion with Multiple Export Markets \\
Ward M. Reesman and David Riker\\
Office of Economics Working Paper 2025--01--B\\ 
January 2025 \\~\\
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\begin{abstract}
\noindent We develop a partial equilibrium model that can estimate the impact of a change in access to an export market on the pattern of international trade and the prices paid to exporting agricultural producers in a country. The model includes barriers to shifting exports to other countries, limited flexibility in production levels, and stock management that can smooth price fluctuations over time. 
\end{abstract}

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Ward M. Reesman, Research Division, Office of Economics\\
\href{mailto:ward.reesman@usitc.gov}{ward.reesman@usitc.gov}\\
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David Riker, Research Division, Office of Economics\\
\href{mailto:david.riker@usitc.gov}{david.riker@usitc.gov}\\
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\section{Introduction \label{sec: section1}}

Foreign market access restrictions on agricultural exports can put downward pressure on prices paid to agricultural producers engaged in export markets. This price pressure is mitigated if exporters can divert their shipments to their domestic market or other export destinations, reduce their production in the short run, and manage stocks to smooth price fluctuations over time.

In this research note, we develop a partial equilibrium (PE) model that can estimate the impact of a change in access to an export market on the pattern of international trade and the prices paid to exporting agricultural producers.\footnote{Our model is similar to the PE model of tariff changes in \citeasnoun{RS2020-11-B} and the multi-country PE models of the economic effects of tariffs and supply shocks in \citeasnoun{RikerSchreiber2020}.} The model reflects important features of trade in agriculture, including barriers to shifting exports to other countries, limited flexibility in production levels, and storage, and it can be easily estimated using information on market shares.

The remainder of this research note is organized into four sections. Section \ref{sec: section2} presents the basic PE model and derives a reduced-form expression for the price effects of tariff changes. Section \ref{sec: section3} adds several types of impediments to trade diversion that modify this reduced-form expression. Section \ref{sec: section4} adds a price elasticity of supply. Section \ref{sec: section5} adds storage and inter-temporal price arbitrage.

\section{Basic PE Model \label{sec: section2}}

A single homogeneous commodity product $j$ is produced by one exporting country and sold across $N$ markets, including a domestic market. Output can be redirected if market access is limited by an increase in tariffs in an export destination. Production is price-inelastic, reflecting inflexibility within crop cycles. Consumers have nested constant elasticity of substitution (CES) demands. The elasticity between different agricultural products is equal to one, and the elasticity between varieties of product $j$ is equal to $\sigma_j > 1$. Under this setup, the quantity of product $j$ exported to country $k$ ($q_{jk}$) is given by
%
\begin{equation}\label{eq:1}
q_{jk} = \gamma_{jk} \ E_k \ \left( P_{jk} \right)^{\sigma_j \ - \ 1} \ (p_{jx} \ ( 1 \ + \ \tau_{jk} ))^{-\sigma_j} \ \beta_{jk}.
\end{equation}

\noindent $E_k$ is aggregate expenditure in country $k$, and $\gamma_{jk}$ is the Cobb-Douglas expenditure share on product $j$ in this country. $p_{jx}$ is the producer price of the exporting country, which is common to all export markets $k$ and is the focus of our model. $\tau_{jk}$ is the tariff rate on exports to country $k$. $\beta_{jk}$ is an exogenous preference parameter on imports into country $k$.\footnote{This parameter absorbs other forms of bilateral trade costs not subject to the policy change.} $P_{jk}$ is the CES price index for all product $j$ consumed in country $k$, given by 
%
\begin{equation}\label{eq:2}
P_{jk} = \left(  (p_{jk})^{1 \ - \ \sigma_j} \ + \beta_{jk} \ (p_{jx} \ ( 1 \ + \ \tau_{jk} ) )^{1 \ - \ \sigma_j}\right)^{\frac{1}{1 \ - \ \sigma_j}}.
\end{equation}

\noindent Finally, equation (\ref{eq:3}) is the adding-up constraint on the exporter's total supply of product $j$ ($\Bar Q_j$),
%
\begin{equation}\label{eq:3}
\sum_k \ q_{jk} = \Bar Q_j
\end{equation}
%
where the summation over market $k$ demand includes the domestic market in addition to all export destinations. In other words, the quantity of $j$ demanded across all $N$ markets is equal to the total supply ($\bar Q_j$), and markets clear.

We log-linearize (\ref{eq:1}), (\ref{eq:2}), and (\ref{eq:3}) to derive a first-order, reduced-form expression for the percent change in the producer price of the exporting country ($\hat p_{jx}$) in terms of the changes in the tariff rate ($\Delta \ \tau_{jk}$), holding all else equal.\footnote{We use the notation $\hat x$ to indicate the log-derivative of variable $x$, which is approximately equal to the percent change ($\hat x = \frac{x'-x}{x} = \frac{\Delta x}{x}$). This traditional definition of $\hat x$ is different from the alternative "hat algebra" notation in \citeasnoun{dek2007}, in which $\hat x = \frac{x'}{x}$.} $E_k$, $\Bar Q_j$, $p_{jk}$, and $\tau_{jk}$ are exogenous variables in the model, while $p_{jx}$ and $q_{jk}$ are endogenous market equilibrium outcomes.\footnote{The assumption that $E_k$ and $p_{jk}$ are exogenous variables is usually called the small country assumption in models of trade policy.} Equation (\ref{eq:1}) therefore becomes
%
\begin{equation}\label{eq:4}
\hat q_{jk} = \big( \left(\sigma_j \ - \ 1 \right) \ \mu_{jk} \ - \ \sigma_j \big) \ \left( \frac{\Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}} \ + \ \hat p_{jx} \right)
\end{equation}
%
where $\mu_{jk}$ is defined as the exports' share of country $k$'s total expenditure on product $j$, i.e. $\mu_{jk} = \frac{ p_{jx} \ (1 + \tau_{jk}) \ q_{jk}} {\gamma_{jk} \ E_k}$. This can be thought of as a trade share, and is a common object throughout workhorse models of international trade.

Similarly, we can rewrite (\ref{eq:3}) as
%
\begin{equation}\label{eq:5}
\sum_k \ \theta_{jk} \ \hat q_{jk} = 0
\end{equation}
%
with $\theta_{jk}$ defined as the share of the total shipments of product $j$ produced in the exporting country that are sent to country $k$, i.e., $\theta_{jk} = \frac{q_{jk}}{\Bar Q_j}$.\footnote{While this shipment share is denoted in terms of quantity, given uniform pricing it is equivalent to a share denoted in terms of trade value.}

Combining (\ref{eq:4}) and (\ref{eq:5}) yields
%
\begin{equation}\label{eq:6}
\sum_k \theta_{jk} \ \big( \left( \sigma_j \ - \ 1 \right) \ \mu_{jk} \ - \ \sigma_j \big) \ \left( \frac{\Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}} \ + \ \hat p_{jx} \right) = 0
\end{equation}
%
which can be inverted to obtain a reduced-form expression for the percent change in the price of the exporter ($\hat p_{jx}$) in response to a change in the tariff rate $\frac{\Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}}$:
%
\begin{equation}\label{eq:7}
\hat p_{jx} = \sum_k \left( \frac{\theta_{jk} \ \left( \left( \sigma_j \ - \ 1 \right) \ \mu_{jk} \ - \ \sigma_j \right) } { \sum_{k'} \theta_{jk'} \ \left( \left(\ \sigma_j \ - \ 1 \right) \ \mu_{jk'} \ - \sigma_j \right) } \right) \ \left( \frac{- \Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}} \right).
\end{equation}

\noindent Equation (\ref{eq:7}) clearly identifies the data requirements of the reduced-form estimate: the price effect depends on the magnitude of the tariff change ($\frac{\Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}}$), the elasticity of substitution ($\sigma_j$), the shares of the total shipments of the exporter that are sent to each country ($\theta_{jk}$ and $\theta_{jk'}$), and the trade share of the exporter within each country ($\mu_{jk'}$ and $\mu_{jk}$). The absolute value of the price elasticity of demand that the exporter faces in country $k$ is decreasing in $\mu_{jk}$, and the shipment shares $\theta_{jk}$ indicate the importance of country $k$ as an alternative destination for the exports. Neither of these shares are exogenous fundamentals in the model; they are both equilibrium outcomes prior to the change in the tariff rate. The shares implicitly incorporate exogenous model fundamentals like $\beta_{jk}$, $\gamma_{jk}$, $\tau_{jk}$, $E_k$, $p_{jk}$, and $\Bar Q_j$.

\section{Adding Impediments to Trade Diversion \label{sec: section3}}

Equation (\ref{eq:3}) assumes linear transformation as shipments of the agricultural product are diverted to other countries, subject to ad valorem tariffs and possibly other trade costs that vary by country. The tariff rates are represented in the model by $\tau_{jk}$, and other ad valorem trade costs that remain fixed are absorbed in $\beta_{jk}$. Both are implicit in equilibrium $\mu_{jk}$ and $\theta_{jk}$. There are several tractable alternatives for including greater impediments to trade diversion in the model while still maintaining a reduced-form expression for price effects that is similar to (\ref{eq:7}).

\subsection{Non-linear Transformation  \label{subsec: subsection3.1}} 

One alternative is non-linear transformation (or diversion) as shipments shift between countries, with constant elasticity. Equation (\ref{eq:8}) is an alternative to the adding-up constraint in (\ref{eq:3}) that has non-linear diversion. $\lambda_j$ is the constant elasticity of transformation for product $j$. Imperfect transformation on the supply side might reflect differences in the production process for goods exported to different national markets, for example due to differences in national sanitary and phytosanitary measures.\footnote{These measures as discussed at length in \citeasnoun{SPS2021}.} It also might reflect limitations in distribution and shipping, for example due to capacity constraints in switching between national marketing, shipping, and distribution channels.

When there is non-linear transformation in trade, the system of (\ref{eq:3}), (\ref{eq:5}), and (\ref{eq:7}) is replaced by 
%
\begin{equation}\label{eq:8}
\sum_k \ (q_{jk})^{\lambda_j} = (\Bar Q_j)^{\lambda_j},
\end{equation}
%
\begin{equation}\label{eq:9}
\sum_k \ \lambda_j \ (\theta_{jk})^{\lambda_j} \ \hat q_{jk} = 0,
\end{equation}
%
\begin{equation}\label{eq:10}
\hat p_{jx} = \sum_k \left( \frac{(\theta_{jk})^{\lambda_j} \ \left( \left( \sigma_j \ - \ 1 \right) \ \mu_{jk} \ - \ \sigma_j \right) } { \sum_{k'} (\theta_{jk'})^{\lambda_j} \ \left( \left(\ \sigma_j \ - \ 1 \right) \ \mu_{jk'} \ - \sigma_j \right) } \right) \ \left( \frac{-\Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}} \right).
\end{equation}

\noindent Equation (\ref{eq:10}) is very similar to (\ref{eq:7}): it just substitutes $\theta_{jk}$ for $(\theta_{jk})^{\lambda_j}$. The two expressions are identical if $\lambda_j =1$.

\subsection{Strict Geographic Segmentation \label{subsec: subsection3.2}}

A second alternative for including impediments to trade diversion is to assume strict geographic segmentation between groups of national markets. For example, supply to export markets could be completely separate from supply to the domestic market.\footnote{Another possibility is that there could be segmentation of export destinations by geographic region.} This segmentation redefines the share inputs $\theta_{jk}$ and $\mu_{jk}$ but otherwise does not change the reduced-form expression for price effects in (\ref{eq:7}). For example, if domestic shipments have a completely separate supply than exports, then $\theta_{jk}$ and $\theta_{jk'}$ is the share of countries $k$ and $k'$ among all exports (but not domestic shipments), and likewise the domestic market is not included in either of the summations in the reduced-form expression.

This alternative is a more extreme and stark version of the non-linear transformation alternative, and it would only apply if there is a dedicated supply line to each destination, and no evidence of shifting between destinations. As with the nonlinear transformation alternative, it could reflect international differences in product requirements, like national differences in sanitary and phytosanitary measures, or capacity constraints on marketing, shipping, and distribution in different national markets. Strict segmentation has the practical advantage that it only requires grouping national markets into segments without further quantification, while the non-linear transformation alternative requires a specific estimate of $\lambda_j$, which might be difficult to quantify.

\section{Adding Price Elasticity of Supply \label{sec: section4}}

It is also straightforward to extend the basic PE model to allow for a constant price elasticity of supply from the exporting country, $\epsilon_j >0$, rather than assuming a fixed production level in the exporting country. For example, crop cycles might be shorter than a year, while economic effects might be calculated on an annual basis. In this case, even when production is price-inelastic within a crop cycle, it will have some price elasticity in analysis of annual effects. Returning to the assumption of linear transformation, now (\ref{eq:11}) replaces (\ref{eq:6}), and (\ref{eq:12}) replaces (\ref{eq:7}).\footnote{The two expressions for price effects are identical if $\epsilon_j =0$.}

\begin{equation}\label{eq:11}
\sum_k \theta_{jk} \ \big( \left( \sigma_j \ - \ 1 \right) \ \mu_{jk} \ - \ \sigma_j \big) \ \left( \frac{\Delta \tau_{jk}}{ 1 \ + \ \tau_{jk}} \ + \ \hat p_{jx} \right) = \epsilon_j \ \hat p_{jx}
\end{equation}

\begin{equation}\label{eq:12}
\hat p_{jx} = \sum_k \left( \frac{\theta_{jk} \ \left( \left( \sigma_j \ - \ 1 \right) \ \mu_{jk} \ - \ \sigma_j \right) } { \sum_{k'} \theta_{jk'} \ \left( \left(\ \sigma_j \ - \ 1 \right) \ \mu_{jk'} \ - \sigma_j \right) \ - \ \epsilon_j} \right) \ \left( \frac{-\Delta \ \tau_{jk}}{1 \ + \ \tau_{jk}} \right)
\end{equation}

The data requirements for implementing the above are identical to the base model, with the added requirement that an estimate of $\epsilon_j$ is now needed. Finally, another reason a price elasticity of supply may be appropriate is the possibility of storing the product for future periods, though this is better represented by an explicitly dynamic model, which we discuss in the next section.

\section{Adding Storage \label{sec: section5}}

In general, adding storage and inter-temporal price arbitrage can significantly complicate a model; however, these features can be added to our analysis in a simple way that still maintains the reduced-form expression for price effects with only slight modification. In this extension, there are two time periods, storage, and inter-temporal price arbitrage such that
%
\begin{equation}\label{eq:13}
p_{jx1} \ (1 \ + \ r) \ (1 \ - \ s) = p_{jx2}
\end{equation}
%
\noindent where $p_{jx1}$ and $p_{jx2}$ are the prices of the exporter in periods 1 and 2, $r$ is the interest rate, and $s$ is an ad valorem storage cost. Provided $r, s$ are constant, then $\hat p_{jx1} = \hat p_{jx2} \equiv  \hat p_{jx}$. The reduced-form expression for this time-invariant price effect can then be written as
%
\begin{equation}\label{eq:14}
\hat p_{jx} = \sum_{k,t} \left( \frac{\theta_{jkt} \ \left( \left( \sigma_j \ - \ 1 \right) \ \mu_{jkt} \ - \ \sigma_j \right) } { \sum_{k',t} \theta_{jk't} \ \left( \left( \sigma_j \ - \ 1 \right) \ \mu_{jk't} \ - \sigma_j \right) } \right) \ \left( \frac{-\Delta \ \tau_{jkt}}{1 \ + \ \tau_{jkt}} \right).
\end{equation}

\noindent Equation (\ref{eq:14}) is similar to (\ref{eq:7}); the difference is that trade shares and summation are calculated across time periods $t$, in addition to across countries $k$.

If the change in the tariff rate is permanent ($\Delta \ \tau_{jk1} = \Delta \ \tau_{jk2} > 0$), then the price effect is the same in both periods, and the possibility of storage and inter-temporal price arbitrage does not matter. However, if the change in the tariff rate is temporary and occurs only in the first of the two periods ($\Delta \ \tau_{jk1} > \Delta \ \tau_{jk2} = 0$), then the price effect in the first period is diluted and extended by storage; supply is smoothed over time by adjusting stocks.

Storage costs might be determined by weight or quantity rather than unit dollar value.\footnote{Storage costs are usually proportional to value if related to product obsolescence, depreciation, or insurance value. On the other hand, they are usually determined by weight or quantity if related to physical warehousing costs.} However, the assumption of an ad valorem storage cost $s$ is standard in economic models of inventories and international trade, like \citeasnoun{akm2010} and \citeasnoun{bcdh2019}. It is convenient in our model, because it incorporates inventory considerations in a tractable way, with only a slight modification and re-interpretation of the reduced-form expression for the price effects of the tariff changes.

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