\documentclass[12pt]{article} \usepackage{amssymb} \usepackage{graphicx} \usepackage[dcucite]{harvard} \usepackage{amsmath} \usepackage{color}\usepackage{setspace} \usepackage{booktabs} \usepackage{hyperref} \usepackage[T1]{fontenc} \usepackage{threeparttable} \usepackage{array} \usepackage{longtable} \usepackage{subcaption} \usepackage{caption} \usepackage{natbib} \setcounter{MaxMatrixCols}{10} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \setlength{\topmargin}{0.1in} \setlength{\headheight}{0in} \setlength{\headsep}{0in} \setlength{\topskip}{0in} \setlength{\textheight}{8.5in} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \setlength{\textwidth}{6.5in} \makeatletter \setlength{\@fptop}{0pt} \makeatother \begin{document} \title{Modeling Multiple Country-Specific Tariff Rate Quotas} \author{David Riker} \date{\vspace{1.5in}% \today} \thispagestyle{empty} { % set font to helvetica (arial) to make it 508-compliant \fontfamily{phv}\selectfont \begin{center} {\Large \textbf{MODELING MULTIPLE COUNTRY-SPECIFIC}} \\ \vspace{0.5in} {\Large \textbf{TARIFF RATE QUOTAS}} \\ \vspace{1.5in} {\Large David Riker} \\ \vspace{0.75in} {\large ECONOMICS WORKING PAPER SERIES}\\ Working Paper 2026--02--A \\ \vspace{0.5in} U.S. INTERNATIONAL TRADE COMMISSION \\ 500 E Street SW \\ Washington, DC 20436 \\ \vspace{0.5in} February 2026 \end{center} \vfill \noindent The views expressed solely represent the opinions and professional research of the author. The content of the working paper is not meant to represent the views of the U.S. International Trade Commission, any of its individual Commissioners, or the United States government. \newpage \thispagestyle{empty} % remove headers, footers, and page numbers from cover page \begin{flushleft} Modeling Multiple Country-Specific Tariff Rate Quotas \\ David Riker\\ Office of Economics Working Paper 2026--02--A \\ February 2026 \\~\\ \end{flushleft} \vfill \begin{abstract} \noindent I develop an industry-specific simulation model of trade policy with multiple country-specific tariff rate quotas. I use the model to simulate a hypothetical expansion of TRQs on South Korea's imports of rice. This example shows that there are diminishing returns to expanding a TRQ : the increase in its imports is less than the increase in the quota level for expansions over a certain threshold quantity. This threshold reflects the supply and demand elasticities in the industry as well as any coinciding changes in the TRQs on imports from the rest of the world. \end{abstract} \vfill \begin{flushleft} David Riker, Research Division, Office of Economics\\ \href{mailto:david.riker@usitc.gov}{david.riker@usitc.gov}\\ \vspace{0.5in} Suggested Citation: \\ \vspace{0.25in} Riker, D. (2026): "Modeling Multiple Country-Specific Tariff Rate Quotas." U.S. International Trade Commission. Economics Working Paper 2026--02--A. \end{flushleft} } % end of helvetica (arial) font \clearpage \newpage \doublespacing \setcounter{page}{1} \section{Introduction \label{sec: section1}} A tariff rate quota (TRQ) is a two-tiered tariff schedule. TRQs are prevalent in agricultural trade and are also common in safeguard remedies and settlements of other bilateral trade disputes of the United States There is a large economic literature that examines the economic effects of TRQs. Most of the studies model trade in specific agricultural products, including \citeasnoun{Grant2009} for specialty cheeses; \citeasnoun{CCM2011} for rice; \citeasnoun{PouliotLarue2012} for poultry; \citeasnoun{Junker2012} for beef; and \citeasnoun{BeckmanArita2016} for a variety of products with strict sanitary and phytosanitary measures. There are also interesting theoretical models of TRQs, including \citeasnoun{Hranaiova2005}, \citeasnoun{Scoppola2010}, \citeasnoun{HallrenRiker2017}, and \citeasnoun{Riker2024}. Figure 1 illustrates the economics of a TRQ on imports from a single country. The tariff rate is a step function of the quantity of imports from country $a$. This TRQ has three policy parameters. The TRQ's quota level $\bar Q_a$ is the quantity of imports at the vertical part of this step function. The in-quota tariff rate $\tau_a^{in}$ applies when import quantities fall below $\bar Q_a$. The out-of-quota tariff rate $\tau_a^{out}$ applies when import quantities exceed $\bar Q_a$. \begin{figure}[h!] \centering \caption{One-Step Tariff Rate Quota} \includegraphics[width=1.0\textwidth]{Figure 1.png} \\ %\floatfoot{Notes: To be added.} \label{fig:1} \end{figure} The figure depicts the equilibrium outcomes for three different levels of demand. In all three cases, the equilibrium consumer price is determined by the intersection of the demand curve for imports from country $a$ (depicted in orange) and the TRQ step function (depicted in green). With relatively low demand $D1$, the quantity of imports is below the quota level at point 1, and the TRQ does not fill. With higher demand $D2$, the quantity of imports equals the quota level -- and the quota fills -- at point 2. With even higher demand $D3$, the quantity of imports exceeds the quota level.\footnote{Imports can exceed the quota level because a TRQ is not an absolute quantitative restraint on imports.} If the intersection occurs on the flat parts of the TRQ function, then the consumer price of the imports is equal to landed duty-paid cost, and this price is not sensitive to small fluctuations in demand. If the intersection is on the vertical part of the TRQ function, then the consumer price is more sensitive to fluctuations in demand. An increase in the quota level $\bar Q_a$ might increase imports $q_a$ by less than the increase in the quota level due to the complicated interactions between the supply and demand for the competing products in the market, which includes imports from other countries and domestic products. Expanding the quota level shifts the TRQ step function to the right, but any increase in the quantity imported from country $a$ also shifts the demand for imports from other countries and domestic products to the left in their respective supply and demand graphs (not depicted in Figure 1). This reduces the consumer prices of some or all of these competing products, and these price reductions feed back into the demand for imports from country $a$, which shifts to the left in Figure 1. For this reason, it is complicated to estimate the increase in $q_a$ in response to an increase in $\bar Q_a$: it requires a formal economic model of all of these supply and demand interactions. In this paper, I develop an industry-specific simulation model with multiple country-specific TRQs for this purpose. I present the equations and data requirements of the model in Section \ref{sec: section2}, and I graph the economic effects of expanding TRQs in Section \ref{sec: section3}. Then I use the model to simulate a hypothetical expansion of one or more of the TRQs on South Korea's imports of rice in Section \ref{sec: section4}. Section \ref{sec: section5} discusses potential extensions of the model, and Section \ref{sec: section6} concludes. \section{Equations and Data Requirements \label{sec: section2}} The model focuses on sales in a single national market that is supplied by three different countries: foreign country $a$, foreign country $b$, and domestic producers $dom$.\footnote{It is straightforward to extend the model to include additional countries that supply the domestic market.} Consumers have constant elasticity of substitution preferences with unit price elasticity of demand for the industry as a whole and an elasticity of substitution for products from different countries equal to $\sigma$. Equation (\ref{eq:1}) is the demand for products from $j \in \{a,b,dom\}$. \begin{equation}\label{eq:1} q_j = \gamma \ \frac{\beta_j \ {p_j}^{-\sigma}}{(p_{dom})^{1-\sigma} \ + \ \beta_a \ (p_a)^{1-\sigma} \ + \ \beta_b \ (p_b)^{1-\sigma}} \end{equation} \noindent $\gamma$ is aggregate expenditure in the market, and $\beta_a$ and $\beta_b$ are preference asymmetry parameters for imports from countries $a$ and $b$.\footnote{The model does not include demand uncertainty, though this can be an important issue when modeling the economic effects of TRQs. Uncertainty is addressed in the TRQ model in \citeasnoun{Riker2024}.} There is perfect competition among all suppliers of the market. The supply curve of domestic producers is upward-sloping. It has a constant elasticity $\epsilon$ and a supply shift parameter $\alpha$. \begin{equation}\label{eq:2} q_{dom} = \alpha \ (p_{dom})^{\epsilon} \end{equation} \noindent If imports from country $j$ are not at country $j$'s TRQ quota level and demand is intersecting one of the flat parts of their TRQ step function, then the supply of imports from country $j$ is perfectly price-elastic, reflecting a "small country" assumption that the cost of supplying imports to this domestic market from each foreign country is constant.\footnote{This simplifying assumption could be relaxed by modifying the equations of the model to include less price-elastic import supply.} \begin{equation}\label{eq:3} p_j = m_j \ (1 \ + \ \tau_j) \ (1 \ + \ f_j) \end{equation} \noindent ${m}_j$ is the marginal cost of imports from country $j$. $\tau_j$ is equal to $\tau_j^{in}$ if $q_j \leq \bar Q_a$ and is equal to $\tau_j^{out}$ if $q_j > \bar Q_a$. $f_j$ is an ad valorem international freight charge on imports from country $j$. In contrast, when the quantity of imports is at the TRQ quota level, the supply of imports is perfectly price-inelastic. The TRQ generates quota rents if imports are at or above the TRQ quota level.\footnote{If imports exceed the TRQ quota level, then there are no quota rents on out-of-quota imports, but there are still quota rents on in-quota imports.} The form of import administration -- for example whether the quotas are allocated by an auction -- affects the magnitude of foreign exporters' revenue gains from expanding the TRQ. I assume that the TRQ quotas allocated to each country are auctioned, and the quota rents are fully captured by the auctioning government.\footnote{\citeasnoun{Skully2001} provides a useful introduction to the economics of TRQ administration. Section \ref{sec: section5} discusses an extension of the model that relaxes this assumption about quota rents.} The competitive foreign exporters only receive their ex-factory cost of supplying the market. Under this assumption, the revenue of the exporters in country $j$ is $m_j \ q_j$ for all equilibrium outcomes. The data requirements of this industry-specific, single market model of trade policy are fairly limited. The model requires estimates of the trade elasticity of substitution $\sigma$ and the domestic supply elasticity $\epsilon$; the initial and new TRQ parameters for imports from each of the countries (including the in-quota rate $\tau_j^{in}$, the out-of-quota rate $\tau_j^{out}$, and the quota level $\bar Q_j$), estimates of international freight rates $f_j$, and the initial consumer and ex-factory values of sales in the market from each source country, both domestic and foreign. Given estimates of elasticities $\sigma$ and $\epsilon$ and the initial market equilibrium values of $p_a^0$, $p_b^0$, $p_{dom}^0$, $q_a^0$, $q_b^0$, and $q_{dom}^0$, equations (\ref{eq:4}), (\ref{eq:5}), and (\ref{eq:6}) are the formulas for calibrating parameters $\alpha$, $\beta_j$, and $\gamma$. \begin{equation}\label{eq:4} \beta_j = \frac{p_{j}^0 \ q_{j}^0} {p_{dom}^0 \ q_{dom}^0} \ \left( \frac{p_j^0}{p_{dom}^0} \right)^{\sigma - 1} \end{equation} \begin{equation}\label{eq:5} \gamma = \frac{q_a^0}{\beta_a} \ \left(\frac{(p_a^0)^{1 - \sigma}}{(p_{dom}^0)^{1-\sigma} \ + \ \beta_a \ (p_a^0)^{1-\sigma} \ + \ \beta_b \ (p_b^0)^{1-\sigma}}\right) \end{equation} \begin{equation}\label{eq:6} \alpha = q_{dom}^0 \ (p_{dom}^0)^{-\epsilon} \end{equation} \noindent I calibrate $m_j$ to $\frac{p_j^0}{(1+\tau_j^{in}) \ (1+f_j)}$ if the quantity of imports from country $j$ is below the TRQ quota level in the initial equilibrium and to $\frac{p_j^0}{(1 \ + \ \tau_j^{out}) \ (1+f_j)}$ if the quantity of imports is above this quota level. If the quantity of imports is exactly at the quota level, then $p_j^0 = m_j \ (1+\tau_j^{in}) \ (1+f_j) \ \phi_j^0$, where $\phi_j^0$ represents the initial quota rents included in consumer prices. In this case, I calibrate $m_j$ to $\frac{p_j^0}{(1+\tau_j^{in}) \ (1+f_j) \ \phi_j^0}$. The model predicts the new equilibrium in the market after a change in the country-specific TRQ. In this three-country model with two TRQ countries, there are three alternative new equilibrium outcomes for imports from each TRQ country. In total, there are nine combinations of outcomes to consider.\footnote{In general, with $N$ TRQ countries, there are $3^N$ combinations of outcomes to consider.} For example, one of the combinations has a quantity of imports from country $a$ below their TRQ quota level and a quantity of imports from country $b$ at their TRQ quota level. For each of the nine combinations, the model simulates new consumer prices based on three equations. For all nine combinations, equation (\ref{eq:7}) for the domestic product is one of the three equations. \begin{equation}\label{eq:7} \gamma \ \frac{{p_{dom}}^{-\sigma}}{(p_{dom})^{1-\sigma} \ + \ \beta_a \ (p_a)^{1-\sigma} \ + \ \beta_b \ (p_b)^{1-\sigma}} = \alpha \ (p_{dom})^{\epsilon} \end{equation} \noindent In addition, equation (\ref{eq:8}) applies to imports from each country $j$ if the quantity of these imports is below $\bar Q_j$. \begin{equation}\label{eq:8} p_j = m_j \ (1 \ + \ \tau_j^{in}) \ (1 \ + \ f_j) \end{equation} \noindent Equation (\ref{eq:9}) applies to imports from each country $j$ if the quantity of these imports is at $\bar Q_j$. \begin{equation}\label{eq:9} \gamma \ \frac{\beta_j \ {p_j}^{-\sigma}}{(p_{dom})^{1-\sigma} \ + \ \beta_a \ (p_a)^{1-\sigma} \ + \ \beta_b \ (p_b)^{1-\sigma}} = \bar Q_j \end{equation} \noindent Finally, equation (\ref{eq:11}) applies to imports from each country $j$ if the quantity of these imports is above $\bar Q_j$. \begin{equation}\label{eq:11} p_j = m_j \ (1 \ + \ \tau_j^{out}) \ (1 \ + \ f_j) \end{equation} \noindent After solving for the prices of the three products, the model calculates quantities using equation (\ref{eq:1}). Finally, the model determines which of the nine combinations is actually the new equilibrium. Returning to the example above, the combination with the quantity of imports from country $a$ below the TRQ quota level and the quantity of imports from country $b$ at the TRQ quota level is the new equilibrium if and only if $q_a$ is less than $\bar Q_a$ when the tariff rate is $\tau_a^{in}$, $q_b$ is greater than or equal to $\bar Q_b$ when the tariff rate is $\tau_b^{in}$, and $q_b$ is less than or equal to $\bar Q_b$ when the tariff rate is $\tau_b^{out}$. The Appendix provides Python code that implements this model. \section{The Economic Effects of a TRQ Expansion \label{sec: section3}} The model can be used to estimate the economic effects of expanding one of the TRQs. Figure 2 depicts the qualitative effects of two alternative TRQ expansions. The initial TRQ quota level for imports from country $a$ is labeled $\bar Q_a$, and the demand curve for these imports is labeled $D$. The first policy change is a relatively small expansion of the TRQ on imports from country $a$, from $\bar Q_a$ to $\bar Q_a'$. This reduces the demand for the domestic product and imports from country $b$. $p_b$ declines if and only if the equilibrium for imports from country $b$ are on the vertical part of the TRQ step function for imports from country $b$. $p_{dom}$ always declines, because domestic supply is upward-sloping. The demand curve for imports from country $a$ shifts to the left to $D'$. In the new equilibrium, imports from country $a$ fill their new quota level $\bar Q_a'$, and the increase in the quantity of imports ($\Delta q_a$) \textbf{is equal to} the increase in the TRQ quota level ($\Delta \bar Q_a$). Figure 2 also depicts a second case with a larger expansion of the TRQ, from $\bar Q_a$ to $\bar Q_a''$. For this larger policy change, demand shifts further to $D''$. The new TRQ does not fill in the new equilibrium, and the increase in the quantity of imports \textbf{falls short of} the increase in the TRQ quota level. \begin{figure}[h!] \centering \caption{Two Alternative TRQ Expansions} \includegraphics[width=1.0\textwidth]{Figure 2.png} \\ %\floatfoot{Notes: To be added.} \label{fig:2} \end{figure} \newpage \noindent Whether imports increase one-for-one with the TRQ expansion, or fall short, will depend on the magnitude of the policy change and also on economic conditions in the specific industry. \section{Illustrative Simulation \label{sec: section4}} Next, I apply the model to the restrictive TRQs on South Korean imports of rice. In this specific application of the three-country model, the domestic producer $dom$ is South Korea (KOR), country $a$ with the expanding TRQ quota level is the United States (USA), and country $b$ is an aggregate of imports from the rest of the world (ROW). The sources of data for the TRQ simulation model will depend on the specific application. In this example, I calibrate the model to data for the South Korean rice market from \citeasnoun{USDA2025} and the Global Trade Atlas.\footnote{The Global Trade Atlas data are available at \url{https://marketplace.spglobal.com}.} Table 1 lists the country-specific TRQs quota levels in South Korea in 2024. The in-quota tariff rate was 5\%, and the out-of-quota tariff rate was 513\%. \begin{table}[tbph] \centering \begin{threeparttable} \caption{South Korean Rice TRQs in 2024} \begin{tabular}{p{6cm} r } \toprule Country & Metric Tons \\ \midrule United States & 132,304 \\ China & 157,195 \\ Vietnam & 55,112 \\ Thailand & 28,494 \\ Australia & 15,595 \\ \midrule Most Favored Nation & 20,000 \\ \bottomrule \end{tabular} \end{threeparttable} \end{table} I set the elasticity of substitution to 5 based on the estimate in \citeasnoun{USITC2025}. I set the domestic supply elasticity to 3 based on the description of the oversupply of Korean rice producers in \citeasnoun{USDA2025}. I set the international freight rates to 10\% based on the ratio of cost-in-freight to customs values of US-Korea rice trade from the U.S. International Trade Commission's Dataweb.\footnote{The Dataweb trade data are available at \url{https://dataweb.usitc.gov}.} Finally, I set $\phi_j$ to 1.17 based on the ratio of the consumer price (from \citeasnoun{USDA2025}) and the landed duty paid value of imports from the United States (calculated from Dataweb U.S. export values, the estimates of $f_j$, and the in-quota tariff rate in South Korea). Table 2 reports model simulations for two alternative expansions of the TRQ quota level for imports from the United States, first a smaller 100,000 metric ton expansion and then a larger 200,000 metric ton expansion (above the 2024 quantity of 132,304 metric tons). For the smaller TRQ expansion, the quantities of imports from the United States and rest of the world both fill the new quota levels. For the larger expansion, the quantity of imports from the rest of the world fills their new quota level, but the quantity of imports from the United States is below the new quota level. \begin{table}[tbph] \centering \begin{threeparttable} \caption{Simulated TRQ Expansions with Base Case Elasticity Values} \begin{tabular}{p{9cm} r r } \toprule $\Delta \bar Q_a$ in MT (USA) & 100,000 & 200,000 \\ \midrule $\Delta q_a$ in MT (USA) & 100,000 & 142,383 \\ $\Delta q_b$ in MT (ROW) & 0 & 0 \\ $\Delta q_{dom}$ in MT (KOR) & -53,757 & -74,860 \\ \midrule $\Delta {m}_a \ q_a$ in \$Mill (USA) & 120 & 171 \\ \bottomrule \end{tabular} \end{threeparttable} \end{table} \newpage \noindent These two simulations demonstrate the declining marginal impact of expanding $\bar Q_a$. The marginal impact of the TRQ expansion can be summarized by the ratio of $\Delta q_a$ to $\Delta \bar Q_a$. There is a quota level above which this ratio becomes zero. In this example, that quantity is 142,383 metric tons. A 200,000 metric ton expansion of $\bar Q_a$ leads to a greater $\Delta q_a$ than a 100,000 metric ton expansion, but a 300,000 metric ton expansion does not lead to a greater $\Delta q_a$ than a 200,000 metric ton expansion. This is depicted in Figure 3. \begin{figure}[h!] \centering \caption{Declining Marginal IMpact of a TRQ Expansion} \includegraphics[width=1.0\textwidth]{Figure 3.png} \\ %\floatfoot{Notes: To be added.} \label{fig:3} \end{figure} \newpage \noindent The threshold for a decline in the marginal impact of the TRQ expansion, 142,383 metric tons, is determined by all of the parameters of the model, including the elasticity of other sources of foreign and domestic supply. Supply inelasticity leads to reductions in the prices of these competing products that in turn reduce the demand for imports from country $a$. This reduces $\Delta q_a$ for a specific $\Delta \bar Q_a$ whenever the newly expanded TRQ does not fill. For example, if domestic supply is more price-inelastic, then demand shifts to the left, and this reduces the ratio of $\Delta q_a$ to $\Delta \bar Q_a$. Likewise, if $\bar Q_b$ is initially filled and stays filled, then $p_b$ falls along with $p_{dom}$, and this further reduces the ratio of $\Delta q_a$ to $\Delta \bar Q_a$. Table 3 repeats the simulation of the 200,000 metric ton expansion with the base case elasticity values and then reports alternative simulations that assume a lower value of the trade elasticity of substitution $\sigma$ or a larger value of the domestic supply elasticity $\epsilon$. When the trade elasticity of substitution is reduced from 5 to 3, the increase in $q_a$ is smaller, only 75,833 metric tons, and the reduction in domestic shipments is also smaller.\footnote{When domestic supply is more price-elastic, 7 rather than 3, the increase in $q_a$ is slightly larger, at 145,741 metric tons.} Figure 4 shows that the TRQ expansion has diminishing returns at a lower quota level when $\sigma$ is lower. \begin{table}[tbph] \centering \begin{threeparttable} \caption{Simulated 200,000 MT Expansion for Alternative Parameter Values} \begin{tabular}{p{8cm} r r r } \toprule $\Delta \bar Q_a$ in MT (USA) & Base Case & $\sigma$ from 5 to 3 & $\epsilon$ from 3 to 7 \\ \midrule $\Delta q_a$ in MT (USA) & 142,383 & 75,833 & 145,741 \\ $\Delta q_b$ in MT (ROW) & 0 & 0 & 0 \\ $\Delta q_{dom}$ in MT (KOR) & -74,860 & -33,008 & -90,293 \\ \midrule $\Delta {m}_a \ q_a$ in \$Mill (USA) & 171 & 91 & 175 \\ \bottomrule \end{tabular} \end{threeparttable} \end{table} \begin{figure}[h!] \centering \caption{Sensitivity to a Lower Trade Elasticity of Substitution} \includegraphics[width=1.0\textwidth]{Figure 4.png} \\ %\floatfoot{Notes: To be added.} \label{fig:4} \end{figure} \newpage As a final comparison, I estimate the simulated effects if the TRQ quota level on imports from the rest of the world were also increased by 200,000 metric tons. The second column of estimates in Table 4 reports this additional simulation. In this case, there would be a smaller increase in South Korean imports from the United States (for the same 200,000 metric ton expansion of the TRQ) and a larger reduction in domestic shipments, because there is now also an increase in the quantity of imports from the rest of the world. This example shows that changes in country-specific TRQs should be modeled in combination and cannot be modeled in isolation. \begin{table}[tbph] \centering \begin{threeparttable} \caption{Simulations of Alternative TRQ Expansions} \begin{tabular}{p{9cm} r r r } \toprule $\Delta \bar Q_a$ in MT (USA) & Expand & \ & Expand \\ \ & U.S. TRQ Only & \ & All TRQs \\ \midrule $\Delta q_a$ in MT (USA) & 142,383 & \ & 123,362\\ $\Delta q_b$ in MT (ROW) & 0 & \ & 200,000 \\ $\Delta q_{dom}$ in MT (KOR) & -74,860 & \ & -171,144 \\ \midrule $\Delta {m}_a \ q_a$ in \$Mill (USA) & 171 & \ & 148 \\ \bottomrule \end{tabular} \end{threeparttable} \end{table} \newpage \section{Potential Extensions of the Model \label{sec: section5}} One potential extension of the model could consider a non-auction allocation of the TRQs, such as a "first come first served" allocation as if it were a lottery. In this alternative version of the model, successful "first come" foreign producers would capture some or all of the TRQ quota rents. The increase in their revenues from the TRQ expansion would be smaller, because the exporters' quota rents per metric ton would decline as the consumer price of imports from country $a$ fell. The equations for the equilibrium prices and quantities would not change, but exporters' revenues would include quota rents and the change in these revenues would be calculated differently. This suggests that it is important to pay attention to the mechanism that actually governs the TRQ allocation in the industry and country in the specific application. The model could also be extended by disaggregating imports from the rest of the world, for example by separating imports from China from the rest of world aggregate and treating them as a separate product in consumer demand. If imports for the rest of the world are already subject to country-specific TRQs that fill, as in the specific simulations in Section \ref{sec: section4}, then disaggregating the rest of world into separate countries will not affect the model's estimates of $\Delta q_a$ and $\Delta q_{dom}$. However, in other applications in which some of the other TRQs are not initially filled or are exceeded, the prices of these other imports will not decline and the ratio of $\Delta q_a$ to $\Delta \bar Q_a$ will be different. To capture this, the model with three country-sources could be converted into a model with four or more country-sources (with three or more foreign countries plus domestic production).\footnote{Ideally the model would disaggregate imports from each of the country-sources with no TRQ or a non-filling TRQ but could aggregate imports from country-sources with filling TRQs, as explained above.} \section{Conclusions \label{sec: section6}} Modeling the economic effects of country-specific TRQs is significantly more complicated than modeling the effects of flat tariff rates, especially when there are many country-specific TRQs already in place. There is no flat tariff rate equivalent to a TRQ, and changes in country-specific TRQs cannot be analyzed in isolation. Despite these complications, the economic effects of the trade policy can be estimated with the practical multi-country model of traded policy developed in this paper. \newpage \bibliographystyle{dcu} \bibliography{biblio} \newpage \fontfamily{phv}\selectfont \noindent \textbf{CODE APPENDIX} \vspace{0.25in} \noindent The model is coded in Python. The code is specific to the application in this working paper, but it could be expanded or adjusted for use in other applications. \vspace{0.05in} \noindent \# -*- coding: utf-8 -*- \vspace{0.05in} \noindent """ \vspace{0.05in} \noindent Created on September 11 2025 \vspace{0.05in} \noindent @author: david.riker \vspace{0.05in} \noindent """ \vspace{0.05in} \noindent from scipy.optimize import root \vspace{0.05in} \noindent \# This file implements the multi-TRQ PE simulation model in basic Python code. "a" and "b" are different sources of foreign supply and "dom" is domestic supply. The one-step TRQs have three possible outcomes: below quota, at quota, or above quota. \vspace{0.25in} \noindent \# MODEL INPUTS \vspace{0.05in} \noindent sigma = 5.0 \#Elasticity of substitution \vspace{0.05in} \noindent epsilon = 3 \#Domestic supply elasticity \vspace{0.05in} \noindent \#Initial quantities \vspace{0.05in} \noindent qa0 = 132304.0 \vspace{0.05in} \noindent qb0 = 408700.0-132304.0 \vspace{0.05in} \noindent qdom0 = 4110000.0-408700.0 \vspace{0.05in} \noindent \#Initial consumer expenditures \vspace{0.05in} \noindent va0 = qa0 \vspace{0.05in} \noindent vb0 = qb0 \vspace{0.05in} \noindent vdom0 = qdom0 \vspace{0.05in} \noindent \#Initial consumer price \vspace{0.05in} \noindent pa0 = va0/qa0 \vspace{0.05in} \noindent pb0 = vb0/qb0 \vspace{0.05in} \noindent pdom0 = vdom0/qdom0 \vspace{0.05in} \noindent \#TRQ parameter - initial quota level \vspace{0.05in} \noindent qbara0 = qa0 \vspace{0.05in} \noindent qbarb0 = qb0 \vspace{0.05in} \noindent \#TRQ parameter - revised quota level \vspace{0.05in} \noindent qbara1 = qa0 + 200000 \vspace{0.05in} \noindent qbarb1 = qb0 \vspace{0.05in} \noindent \#TRQ parameter - initial in-quota rate \vspace{0.05in} \noindent tina0 = 0.05 \vspace{0.05in} \noindent tinb0 = 0.05 \vspace{0.05in} \noindent \#TRQ parameter - revised in-quota rate \vspace{0.05in} \noindent tina1 = 0.05 \vspace{0.05in} \noindent tinb1 = 0.05 \vspace{0.05in} \noindent \#TRQ parameter - initial out-of-quota rate \vspace{0.05in} \noindent touta0 = 5.13 \vspace{0.05in} \noindent toutb0 = 5.13 \vspace{0.05in} \noindent \#TRQ parameter - revised out-of-quota rate \vspace{0.05in} \noindent touta1 = 5.13 \vspace{0.05in} \noindent toutb1 = 5.13 \vspace{0.05in} \noindent \#Indicators for initial TRQ outcome \vspace{0.05in} \noindent if (qa0qbarb0): \vspace{0.05in} CInd13=1.0 \vspace{0.05in} \noindent else: \vspace{0.05in} CInd13=0.0 \vspace{0.05in} \noindent if (qa0==qbara0) and (qb0qbarb0): \vspace{0.05in} CInd23=1.0 \vspace{0.05in} \noindent else: \vspace{0.05in} CInd23=0.0 \vspace{0.05in} \noindent if (qa0>qbara0) and (qb0qbara0) and (qb0==qbarb0): \vspace{0.05in} CInd32=1.0 \vspace{0.05in} \noindent else: \vspace{0.05in} CInd32=0.0 \vspace{0.05in} \noindent if (qa0>qbara0) and (qb0>qbarb0): \vspace{0.05in} CInd33=1.0 \vspace{0.05in} \noindent else: \vspace{0.05in} CInd33=0.0 \vspace{0.05in} \noindent \#International transport cost rate \vspace{0.05in} \noindent fa = 0.1 \vspace{0.05in} \noindent fb = 0.1 \vspace{0.05in} \noindent \#Initial ratio of consumer price to landed duty paid cost \vspace{0.05in} \noindent phia = 1.2 \vspace{0.05in} \noindent phib = 1.2 \vspace{0.05in} \vspace{0.25in} \noindent \# CALIBRATION \noindent \#Marginal cost for imports \vspace{0.05in} \noindent ma = pa0/(1+fa)*((CInd11+CInd12+CInd13)/(1+tina0) \noindent +(CInd21+CInd22+CInd23)/(1+tina0)/phia+(CInd31+CInd32+CInd33)/(1.0+touta0)) \vspace{0.05in} \noindent mb = pb0/(1+fb)*((CInd11+CInd21+CInd31)/(1+tinb0) \noindent +(CInd12+CInd22+CInd32)/(1+tinb0)/phib+(CInd13+CInd23+CInd33)/(1.0+toutb0)) \vspace{0.05in} \noindent \#Calibrated demand asymmetry parameters \vspace{0.05in} \noindent betaa = va0/vdom0*(pa0/pdom0)**(sigma-1.0) \vspace{0.05in} \noindent betab = vb0/vdom0*(pb0/pdom0)**(sigma-1.0) \vspace{0.05in} \noindent\#Initial industry price index \vspace{0.05in} \noindent pind0 = (betaa*(pa0)**(1.0-sigma)+betab*(pb0)**(1.0-sigma) \noindent +(pdom0)**(1.0-sigma))**(1.0/(1.0-sigma)) \vspace{0.05in} \noindent\#Calibrated total expenditure in the market \vspace{0.05in} \noindent gamma = (qa0*pind0**(1.0-sigma)*pa0**(sigma))/betaa \vspace{0.05in} \noindent\#Calibrated supply parameter \vspace{0.05in} \noindent alpha = qdom0*pdom0**epsilon \vspace{0.05in} \vspace{0.25in} \noindent \# SIMULATIONS \vspace{0.05in} \noindent \# Sim 11 \vspace{0.05in} \noindent pa = ma*(1.0+tina1)*(1.0+fa) \vspace{0.05in} \noindent pb = mb*(1.0+tinb1)*(1.0+fb) \vspace{0.05in} \noindent def equation(x): \vspace{0.05in} return gamma*(x)**(-sigma)/(betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma) +(x)**(1.0-sigma))-alpha*x**(epsilon) \vspace{0.05in} \noindent sol = root(equation,pdom0,method='hybr') \vspace{0.05in} \noindent pdom = sol.x \vspace{0.05in} \noindent qa11 = gamma*(pa)**(-sigma)*betaa/ \noindent (betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qb11 = gamma*(pb)**(-sigma)*betab/ \noindent (betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qdom11 = gamma*(pdom)**(-sigma)/ \noindent (betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent if qa11qbarb1: \vspace{0.05in} SInd13 = 1.0 \vspace{0.05in} \noindent else: \vspace{0.05in} SInd13 = 0.0 \vspace{0.05in} \noindent \# Sim 31 \vspace{0.05in} \noindent pa = ma*(1.0+touta1)*(1.0+fa) \vspace{0.05in} \noindent pb = mb*(1.0+tinb1)*(1.0+fb) \vspace{0.05in} \noindent def equation(x): \vspace{0.05in} return gamma*(x)**(-sigma)/(betaa*(pa)**(1.0-sigma) +betab*(pb)**(1.0-sigma)+(x)**(1.0-sigma))-alpha*x**(epsilon) \vspace{0.05in} \noindent sol = root(equation,pdom0,method='hybr') \vspace{0.05in} \noindent pdom = sol.x \vspace{0.05in} \noindent qa31 = gamma*(pa)**(-sigma)*betaa/ \noindent (betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qb31 = gamma*(pb)**(-sigma)*betab/ \noindent (betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qdom31 = gamma*(pdom)**(-sigma)/ \noindent (betaa*(pa)**(1.0-sigma)+betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent if qa31>qbara1 and qb31qbara1 and qb33>qbarb1: \vspace{0.05in} SInd33 = 1.0 \vspace{0.05in} \noindent else: \vspace{0.05in} SInd33 = 0.0 \vspace{0.05in} \noindent \# Sim 12 \vspace{0.05in} \noindent pa = ma*(1.0+tina1)*(1.0+fa) \vspace{0.05in} \noindent def equations(vars): \vspace{0.05in} x,y = vars \vspace{0.05in} return [gamma*(x)**(-sigma)/(betaa*(pa)**(1.0-sigma)+betab*(y)**(1.0-sigma) +(x)**(1.0-sigma))-alpha*x**(epsilon), gamma*betab*(y)**(-sigma)/(betaa*(pa)**(1.0-sigma) +betab*(y)**(1.0-sigma)+(x)**(1.0-sigma))-qbarb1] \vspace{0.05in} \noindent initialguess = [pdom0,pb0] \vspace{0.05in} \noindent sol = root(equations,initialguess) \vspace{0.05in} \noindent pdom = sol.x[0] \vspace{0.05in} \noindent pb = sol.x[1] \vspace{0.05in} \noindent qa12 = gamma*(pa)**(-sigma)*betaa/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qb12 = gamma*(pb)**(-sigma)*betab/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qdom12 = gamma*(pdom)**(-sigma)/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent if qa12qbarb1: \vspace{0.05in} SInd23 = 1.0*(1-SInd13-SInd33) \vspace{0.05in} \noindent else: \vspace{0.05in} SInd23 = 0.0 \vspace{0.05in} \noindent \# Sim 32 \vspace{0.05in} \noindent pa = ma*(1.0+touta1)*(1.0+fa) \vspace{0.05in} \noindent def equations(vars): \vspace{0.05in} x,y = vars \vspace{0.05in} return [gamma*(x)**(-sigma)/(betaa*(pa)**(1.0-sigma)+betab*(y)**(1.0-sigma) +(x)**(1.0-sigma))-alpha*x**(epsilon), gamma*betab*(y)**(-sigma)/(betaa*(pa)**(1.0-sigma) +betab*(y)**(1.0-sigma)+(x)**(1.0-sigma))-qbarb1] \vspace{0.05in} \noindent initialguess = [pdom0,pb0] \vspace{0.05in} \noindent sol = root(equations,initialguess) \vspace{0.05in} \noindent pdom = sol.x[0] \vspace{0.05in} \noindent pb = sol.x[1] \vspace{0.05in} \noindent qa32 = gamma*(pa)**(-sigma)*betaa/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qb32 = gamma*(pb)**(-sigma)*betab/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qdom32 = gamma*(pdom)**(-sigma)/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent if qa32>qbara1: \vspace{0.05in} SInd32 = 1.0*(1-SInd31-SInd33) \vspace{0.05in} \noindent else: \vspace{0.05in} SInd32 = 0.0 \vspace{0.05in} \noindent \# Sim 22 \vspace{0.05in} \noindent def equations(vars): \vspace{0.05in} x,y,z = vars \vspace{0.05in} return [gamma*(x)**(-sigma)/(betaa*(z)**(1.0-sigma)+betab*(y)**(1.0-sigma) +(x)**(1.0-sigma))-alpha*x**(epsilon), gamma*betab*(y)**(-sigma)/ (betaa*(z)**(1.0-sigma)+betab*(y)**(1.0-sigma)+(x)**(1.0-sigma))-qbarb1, gamma*betaa*(z)**(-sigma)/(betaa*(z)**(1.0-sigma)+betab*(y)**(1.0-sigma) +(x)**(1.0-sigma))-qbara1] \vspace{0.05in} \noindent initialguess = [pdom0,pb0,pa0] \vspace{0.05in} \noindent sol = root(equations,initialguess) \vspace{0.05in} \noindent pdom = sol.x[0] \vspace{0.05in} \noindent pb = sol.x[1] \vspace{0.05in} \noindent pa = sol.x[2] \vspace{0.05in} \noindent qa22 = gamma*(pa)**(-sigma)*betaa/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qb22 = gamma*(pb)**(-sigma)*betab/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent qdom22 = gamma*(pdom)**(-sigma)/(betaa*(pa)**(1.0-sigma) \noindent +betab*(pb)**(1.0-sigma)+(pdom)**(1.0-sigma)) \vspace{0.05in} \noindent SInd22 = 1-SInd11-SInd12-SInd13-SInd21-SInd23-SInd31-SInd32-SInd33 \vspace{0.05in} \vspace{0.25in} \noindent \# NEW EQUILIBRIUM OUTCOMES \noindent qa1 = SInd11*qa11+SInd12*qa12+SInd13*qa13+SInd21*qa21+SInd22*qa22 \noindent +SInd23*qa23+SInd31*qa31+SInd32*qa32+SInd33*qa33 \vspace{0.05in} \noindent qb1 = SInd11*qb11+SInd12*qb12+SInd13*qb13+SInd21*qb21+SInd22*qb22 \noindent +SInd23*qb23+SInd31*qb31+SInd32*qb32+SInd33*qb33 \vspace{0.05in} \noindent qdom1 = SInd11*qdom11+SInd12*qdom12+SInd13*qdom13+SInd21*qdom21+SInd22*qdom22 \noindent +SInd23*qdom23+SInd31*qdom31+SInd32*qdom32+SInd33*qdom33 \vspace{0.05in} \vspace{0.25in} \noindent \#ESTIMATED EFFECTS \vspace{0.05in} \noindent print('************') \vspace{0.05in} \noindent print('ESTIMATED EFFECTS') \vspace{0.05in} \noindent print('CInd11',CInd11,'CInd12',CInd12,'CInd13',CInd13,'SInd21',CInd21,'CInd22', CInd22,'CInd23',CInd23,'CInd31',CInd31,'CInd32',SInd32,'CInd33',CInd33) \vspace{0.05in} \noindent print('SInd11',SInd11,'SInd12',SInd12,'SInd13',SInd13,'SInd21',SInd21,'SInd22',SInd22, 'SInd23',SInd23,'SInd31',SInd31,'SInd32',SInd32,'SInd33',SInd33) \vspace{0.05in} \noindent print('************') \vspace{0.05in} \noindent print('delta tina', tina1-tina0, ',','delta tinb =', tinb1-tinb0,) \vspace{0.05in} \noindent print('delta touta', touta1-touta0, ',','delta toutb = ', toutb1-toutb0,) \vspace{0.05in} \noindent print('delta qbara', qbara1-qbara0,',','delta qbarb =', qbarb1-qbarb0,) \vspace{0.05in} \noindent print('delta qa =', qa1-qa0) \vspace{0.05in} \noindent print('delta qb =', qb1-qb0) \vspace{0.05in} \noindent print('delta qdom =', qdom1-qdom0) \vspace{0.05in} \noindent print('delta reva =', 1201*(qa1-qa0)/1000000) \vspace{0.05in} \end{document}