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{\LARGE \textbf{Modeling the Distribution of Tariff Revenue to Firms within a Partial Equilibrium Framework}} \\
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{\Large Paul Phillips} \\
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{\large ECONOMICS WORKING PAPER SERIES}\\
Working Paper 2025-08-B \\ \vspace{0.5in}
U.S. INTERNATIONAL TRADE COMMISSION \\
500 E Street SW \\
Washington, DC 20436 \\
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August 2025 \\
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\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.
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Modeling the Distribution of Tariff Revenue to Firms within a Partial Equilibrium Framework \\
Paul Phillips \\
Economics Working Paper 2025-08-B\\
August 2025 \\~\\
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\begin{abstract}
\noindent In this paper, I expand upon the current partial equilibrium modeling literature by discussing how to incorporate the distribution of tariff revenue to firms into these models. Rather than assume all firms are identical, I model firms as having a continuum of heterogeneous productivities, so that model users can compute equilibrium results on the fraction of firms that enter or leave the market in any given counterfactual situation. I then solve the model under various counterfactual scenarios given theoretical parameter choices, and illustrate the ways in which modeling outcomes are sensitive to parameter values. While this paper's simulations do not use any real-world data or correspond to any real-world countries, its calibration and estimation discussion takes into account current data limitations and practical challenges in solving the model.
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Paul Phillips \\
Research Division, Office of Economics\\
U.S. International Trade Commission\\
Paul.Phillips@usitc.gov \\
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Suggested citation: \\
Phillips, Paul. "Modeling the Distribution of Tariff Revenue to Firms within a Partial Equilibrium Framework." U.S. International Trade Commission. Office of Economics Working Paper 2025-08-B.
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\section{Introduction}
\indent The revenue generated by tariffs may be distributed in several different ways, with potential recipients including the government, consumers, or firms. Examples of policies that distributed tariff revenue to firms include the 2019 Market Facilitation Program, which provided direct payments to farmers impacted by trade disruptions, and the Byrd Amendment, which distributed the revenue generated from anti-dumping or countervailing duties among firms that filed the original anti-dumping or subsidy complaints.\footnote{For more information on the Market Facilitation Program, see the following \href{https://www.farmers.gov/sites/default/files/documents/Market_Facilitation_Program-Fact_Sheet-Sept.pdf}{USDA guide}. For more information on the Byrd Amendment, see the following \href{https://www.usitc.gov/glossary/term/byrd-amendment-continued-dumping-and-subsidy-offset-act}{information} provided by the USITC.} Although the distribution of tariff revenue to firms has been implemented as a policy, the international trade literature has given significantly less attention to modeling it. In this paper, I contribute to existing scholarship on partial equilibrium modeling by introducing a method that incorporates tariff transfer payments to firms within a static, single-stage partial equilibrium model. \\
\indent The model introduced in this paper can represent any number of countries or industries, as well as any particular choice of country or industry. Lump-sum transfer payments in the model either arrive unexpectedly after firms have already made decisions about prices and production, or firms know they are receiving some form of payment and take this expected transfer into account when making production decisions. The modeling framework also accommodates different methods for calculating the payments; for example, firms could receive only the tariff revenue collected on imports in their specific industry, or in a multi-industry setup they could receive some proportion of the tariff revenue collected across all imports in the economy. \\
\indent Consumers in the model substitute between domestically produced goods and imports across any number of foreign countries, with tariffs applying to foreign imports. I assume that total consumption from all sources remains constant in any potential counterfactual. Monopolistically competitive firms decide what quantities to send to each destination, and must pay a fixed cost in order to send their production along that route. Unlike the majority of partial equilibrium models in the literature, this model follows Melitz (2003) by featuring heterogeneous firms with varying productivities, and each origin-destination pair $ji$ has a cutoff productivity level below which firms in country $j$ cannot export to $i$. This firm heterogeneity allows model users to analyze how lump-sum transfer payments based on tariff revenue would affect the fraction of firms able to produce, an analysis that would not be possible in a model with perfect competition or monopolistic competition with identical firms. Lump-sum payments effectively lower the fixed costs necessary for operating, reducing the minimum productivity necessary for firms to stay in business. \\
\indent I discuss how to compute equilibria in counterfactual situations where tariffs increase. Due to the difficulty of calibrating certain model parameters, this computational method solves for the percentage changes between baseline and counterfactual equilibria, rather than the absolute values of counterfactual equilibria. I then illustrate the model's capabilities by applying this method under different theoretical scenarios. As part of the discussion, I analyze the difference in equilibria between theoretical simulations in which transfer payments arrive unexpectedly and theoretical simulations in which firms factor these payments into their production decisions. \\
\indent The rest of the paper proceeds as follows. Section 2 introduces a partial equilibrium model with heterogeneous firm productivities, while Section 3 goes through what the lump-sum distribution of tariff revenue looks like within the model. Section 4 explains how to solve the model given hypothetical data and parameters, and Section 5 presents the results of solving the model using hypothetical parameters. Section 6 concludes.
\subsection{Literature Review}
The economics literature, as well as canonical models of international trade presented in graduate courses, most frequently model transfers of tariff revenue as a lump-sum payment to \textit{consumers}, rather than firms. These payments appear on the right-hand side of consumers' budget constraints, increasing the amount of money they have available to spend. Academic papers that use such a feature within their model including Alessandria et al. (2025), Bagwell and Lee (2020), and Lashkaripour and Lugovskyy (2019); with the exception of Alessandria et al. (2025), tariff revenue distribution is one of many features used in these papers' modeling and not the main focus of analysis. Jafarey and Lahiri (2024) and Lahiri and Nasim (2006) do feature payments to firms as the primary focus of their analysis, but limit their discussion to rebates for tariffs on intermediate goods, as a way to compensate firms that may sell to a domestic market but may have to pay tariffs on intermediate inputs that they import from abroad. This paper discusses a general structure for modeling lump-sum tariff payments in a wide variety of situations.\\ 
\indent This paper also contributes to a series of working papers by Commission staff that embed heterogeneous-productivity firms within a standard partial equilibrium model. Barbe, Chambers, Khachaturian, and Riker (2017) and Mueller and Riker (2020) discuss this embedding process more generally, and Khachaturian and Riker (2016) uses a heterogeneous-productivity partial equilibrium model in order to analyze the effect of decreased fixed costs on international trade in services. The modeling framework in this paper uses heterogeneous productivity in firms to quantify the effects of tariff transfer payments on firms of varying productivities, providing an estimate for the proportion of firms that would otherwise be forced to exit the market but are able to remain operational due to the lump-sum transfer they receive. \\
\indent More broadly, this paper complements a wider array of academic literature on trade with heterogeneous firms. The seminal paper in this body of scholarship is Melitz (2003), the first paper that used a Pareto firm productivity distribution to model the effects of trade liberalization. Helpman, Melitz, and Yeaple (2004) build on this framework by using a model with Pareto-distributed productivities to explain how firms decide between exporting to another country and increasing foreign direct investment in their home country. Demidova (2008), Demidova and Rodriguez-Clare (2009), and Felbermayr, Jung, and Larch (2013) make further theoretical contributions to general equilibrium modeling with Pareto-distributed productivities, while Spearot (2016) and Melitz and Redding (2015) calibrate heterogeneous-productivity firm models using data inputs. While this paper does not present any data-driven results, it does discuss in theory how its model could be calibrated, as well as what data inputs would play a role in this calibration process.
\section{Model}
Because this model is a partial equilibrium model, each industry has its own separate set of equilibrium conditions, and in the case of multiple industries the model assumes that prices and consumption within the various industries do not influence one another. Since industries for the most part are separate from one another, the equations in this section do not index variables by industry.\footnote{The distribution of tariff revenue discussed in section 3 may have cross-industry effects, depending on how it is performed. In discussing modeling scenarios with cross-industry payments, industry-level indexing would become necessary.}
\subsection{Consumers}
Goods in this model come from any one of $N$ origin countries, indexed by $j$. The model further subdivides consumption from each origin according to varieties $z$, and consumers in country $i$ choose among origins and varieties to maximize the following CES utility function:
\begin{align*}
\big(\sum\limits_{j=1}^N \int\limits_0^{m_{ji}} c_{ji}(z)^{\frac{\sigma_i-1}{\sigma_i}}dz\big)^{\frac{\sigma_i}{\sigma_i-1}}
\end{align*}
Consumers in country $i$ substitute with elasticity $\sigma_i$ among goods of different varieties and from different origin countries. $m_{ji}$ is an equilibrium allocation representing the total measure of goods that are exported from $j$ to $i$; a higher $m_{ji}$ indicates that country $j$ is sending a higher volume of shipments to country $i$. \\
\indent Let $E_i$ denote total consumer expenditure on a given industry in country $i$. As is common in partial equilibrium models,\footnote{See Riker and Hallren (2017).} I assume that $E_i = Y_i P_i^{-\theta}, \ \theta=1$, and that $Y_i$ does not change in any counterfactual situations. The consumer's first-order conditions lead to the following equation:
\begin{align}
E_{ji}(z) = Y_i P_{i}^{\sigma_i-1}P_{ji}(z)^{1-\sigma_i} \ \forall z
\end{align}
where $P_{i}$ is the aggregate price index for country $i$ and $P_{ji}(z)$ is the price of variety $z$ when shipped from country $j$ to country $i$. The aggregate price index $P_i$ is given by
\begin{align*}
P_i^{1-\sigma_i} = \sum\limits_{j=1}^N \int\limits_0^{m_{ji}} P_{ji}(z)^{1-\sigma_i}dz
\end{align*}
\subsection{Firms}
Goods are produced by a continuum of firms, each with productivity $x$. I assume, without loss of generality, that each variety $z$ maps to a productivity $x$, and I accordingly use the notation $x(z)$. A firm in country $j$ exporting variety $z$ to country $i$ maximizes the following profit function:
\begin{align*}
p_{ji}(z)q_{ji}(z) - \frac{w_j\tau_{ji}q_{ji}(z)}{x(z)} - w_jf_{ji}
\end{align*}
In other words, the firm earns revenue from selling output $q_{ji}(z)$ at price $p_{ji}(z)$, while paying variable costs $\frac{w_j\tau_{ji}q_{ji}(z)}{x(z)}$ and fixed costs $w_jf_{ji}$. A higher productivity $x(z)$ lowers the cost of producing $z$, while a higher tariff $\tau_{ji}$ raises the cost of sending variety $z$ from $j$ to $i$ with $\tau_{jj} = 1$. $f_{ji}$, the fixed cost of production, varies based on the origin country and destination country. Finally, $w_j$ is the unit cost of producing the good; if labor is the only input in production, then $w_j$ would simply equal the wage. I take these production inputs to be the numeraire good, and so $w_j$ is normalized to 1; since $w_j$ always equals 1, it will not appear in any future equations. \\
\indent Market clearing indicates that $q_{ji}(z) = c_{ji}(z) \ \forall z$. Using (1) to replace $c_{ji}(z)$ in the firm's profit function and taking first-order conditions,\footnote{The appendix goes into all mathematical derivations in greater detail.} I find that
\begin{align*}
p_{ji}(z) = \frac{\sigma_i}{\sigma_i-1}\frac{\tau_{ji}}{x(z)}
\end{align*}
where $\frac{\sigma_i}{\sigma_i-1}$ is the firm's markup over marginal cost $\frac{\tau_{ji}}{x(z)}$.
\subsection{Productivity Distribution}
Firm productivity $x$ in country $j$ follows the Pareto distribution
\begin{align*}
G_j(x) = 1 - x^{-\gamma_j}
\end{align*}
which has the density function
\begin{align*}
\frac{dG_j(x)}{dx} = \gamma_j x^{-\gamma_j-1}
\end{align*}
I refer to $\gamma_j$ as the `shape parameter', and a higher value of $\gamma_j$ indicates that the distribution of firms in country $j$ will be more clustered toward lower productivities.\footnote{Note that both the elasticity of substitution and shape parameter are allowed to differ by country. If modelers do not have enough data to estimate $\gamma_j$ and $\sigma_i$ separately for every country, then can assume $\gamma_j = \gamma$ and $\sigma_i = \sigma$ for all $j$ and $i$.}  \\ 
\indent The properties of the Pareto distribution both make the model more tractable and ensure the existence of a cutoff productivity, $\bar x_{ji}$, for all destination-origin pairs $ji$ such that all $j$ firms with productivities higher than $\bar x_{ji}$ export to $i$ and all $j$ firms with productivities lower than $\bar x_{ji}$ do not export to $i$ because they cannot make enough operating profit to recoup their fixed costs. $\bar x_{ji}$ depends on fixed costs $f_{ji}$, tariff rates $\tau_{ji}$, and prices, and differs by origin and destination. Given the differing levels of profitability along each route, a firm with a given productivity may choose to sell goods within its own domestic market without exporting, and a firm with a given productivity may choose to export to one destination country but not another. \\
\indent Let $\mu_j$ represent the measure of \textit{potential} firms in country $j$. The measure of firms in $j$ able to export to $i$ is then given by 
\begin{align*}
m_{ji} = \mu_j\int\limits_{\overline x_{ji}}^{\infty}dG_j(x) = \mu_j\bar x_{ji}^{-\gamma_j}
\end{align*} 
where $\bar x_{ji}^{-\gamma_j}$ is the fraction of total firms in $j$ with sufficiently high productivities to export to $i$.
\subsection{Equilibrium Equations}
Country $i$'s expenditure on goods from country $j$, or alternatively the value of imports from $j$ to $i$, is given by
\begin{align}
E_{ji} &= \mu_j \int\limits_{\overline x_{ji}}^{\infty} c_{ji}(x)p_{ji}(x)dG_j(x) \nonumber \\
&= \mu_j E_{i} P_i^{\sigma_i-1}\big(\frac{\tau_{ji}w_j\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\frac{\overline x_{ji}^{\sigma_i-1-\gamma_j}}{\gamma_j-(\sigma_i-1)}
\end{align}
This last step requires an assumption that $\gamma_j \geq \sigma_i-1$ for consumption to be non-negative. Likewise, the aggregate price in country $i$ is given by
\begin{align}
P_i^{1-\sigma_i} &= \sum\limits_{j=1}^N \big[\mu_j \int\limits_{\overline x_{ji}}^{\infty} p_{ji}(x)^{1-\sigma_i}dG_j(x)\big] \nonumber \\
&= \frac{\gamma_j}{\gamma_j-(\sigma_i-1)}\big(\frac{\sigma_i }{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j (\tau_{ji})^{1-\sigma_i} \overline x_{ji}^{\sigma_i-\gamma_j-1}
\end{align}
An equilibrium is characterized by consumption expression (2), price index expression (3), and finally zero-profit conditions that use the definition of $\overline x_{ji}$ as the boundary point where a country $j$ firm is indifferent between selling in country $i$ and not selling in country $i$. In other words:
\begin{align*}
f_{ji} &= \pi(\overline x_{ji}) \\
&= p_{ji}(\overline x_{ji})c_{ji}(\overline x_{ji}) - \frac{\tau_{ji}c_{ji}(\overline x_{ji})}{\overline x_{ji}}
\end{align*}
Substituting for $p_{ji}(\overline x_{ji})$ and $c_{ji}(\overline x_{ji})$ and doing some algebra, I get
\begin{align}
f_{ji} &= Y_i P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1}(\frac{\sigma_i}{\sigma_i-1})^{1-\sigma_i}\frac{1}{\sigma_i}
\end{align}
Equations (2)-(4) are sufficient to solve for an equilibrium of this model without transfer payments, given values for $\gamma_j, \sigma_i, f_{ji}, \tau_{ji}$ and $\mu_j$.
\section{Distribution of Tariff Revenue}
A higher value of $\tau_{ji}$ raises the minimum productivity necessary to export from $j$ to $i$, and some firms would exit that market. The exodus of exporting firms would eliminate some of the competition faced by domestic firms producing and selling goods in $i$, lowering the domestic productivity threshold $\bar x_{ii}$. However, if other countries also maintained high tariffs, domestic producers in $i$ would face a higher export barrier $\bar x_{ij}$. This potential loss in exports could decrease industry profits, while an increase in domestic sales could increase profits. The distribution of tariff revenue as lump-sum transfers introduces an additional benefit of higher tariffs for domestic industries, and can compensate those industries for the negative effects that they may incur from higher foreign tariffs. While Section 2 went through the mechanics of a partial equilibrium model as described in Barbe et al. (2017) and Mueller and Riker (2020), this section describes a novel methodology for introducing tariff transfer payments within this partial equilibrium modeling framework. \\
\indent The precise formula for distributing tariff revenue is both hypothetical and unknown. Firms could be paid out the revenue from tariffs imposed on imports of their own industry, or alternatively paid according to some share of tariff revenue collected across the entire economy. Furthermore, firms may not know they are receiving transfer payments until after they have already committed to a production and pricing decision, or alternatively they receive notice of such transfers in advance and consider their expected payments when determining production. This paper therefore discusses multiple possible ways to model the distribution of tariff revenue, taking into account that transfer payments could be collected and/or distributed differently.
\subsection{Revenue Distribution}
I begin with a note clarifying the distinction between total tariff revenue collected across the entire economy, and the amount of tariff revenue received by each firm. The two values are not equivalent, because the economy consists of more than one firm. I set $R_j$ to be the total tariff revenue collected across the entire economy of country $j$, and $r_j$ to be the transfer payment received by each firm within country $j$. \\
\indent As discussed in Section 2, $\mu_j$ gives the total measure of firms in a country $j$. This parameter $\mu_j$ represents the total amount of firms that \textit{could} produce in country $j$, and does not change in any counterfactual situations. The total measure of firms in country $j$ that actually produce for market $i$ is given by $\mu_j x_{ji}^{-\gamma_j}$
and the total measure of firms in $j$ that produce for any market at all, $m_j$, is this expression summed across all possible destinations, or $\mu_j\sum_{i=1}^N \bar x_{ji}^{-\gamma_j}$. In all future discussions, I will refer to this measure by the more concise name of `firm participation.'\\
\indent Assuming that all firms within a country receive the same amount, $R_j$ and $r_j$ are related by the following identity:
\begin{align*}
r_j &= \frac{R_j}{\mu_j\sum_{i=1}^N \bar x_{ji}^{-\gamma_j}}
\end{align*}
The relationship between the two quantities depends on $\mu_j$ and $\gamma_j$, both model parameters, and $\{\bar x_{ji}\}_{i=1}^N$, a set of equilibrium allocations.
\subsection{Payment Values}
If transfer payments are industry-specific, then domestic firms receive a share of the revenue collected from tariffs in that industry. For example, apple producers would receive transfer payments funded by tariffs on imported apples. One implication of this policy is that industries with higher volumes of foreign imports would receive higher payments, as more tariff revenue would be collected from those industries. Conversely, industries with high export demand but low import demand would not benefit to any great degree from the distribution of tariff revenue. \\
\indent Total transfer payments would be given by the expression
\begin{align*}
R_i = \psi_i\sum\limits_{j\neq i}^N (\tau_{ji}-1)\int\limits_{\bar x_{ji}}^{\infty} E_{ji}(x)dG_j(x)
\end{align*}
as imports from all foreign sources and of all productivities would be subject to the tariff rate $\tau_{ji}-1$.\footnote{The firm's profit function in Section 2.2 defined $\tau_{ji}$ as greater than 1, or 1 plus the imposed tariff rate. To keep notation consistent, the formula in this section contains $\tau_{ji}-1$.} The $\psi_i$ parameter ranges from 0 to 1, allowing modelers to vary the magnitude of the payment rather than limit themselves to situations where firms either receive no transfer payments or receive transfers equal to the entirety of the eollected tariff revenue.\\
\indent Alternatively, each industry could receive some share of total tariff revenue collected across the entire economy. Since a discussion of this formulation entails distinguishing between different industries, in the following equations I will index industries by the letter $n$. \\
\indent Total tariff revenue across the entire economy in country $i$ is given by
\begin{align*}
R_{in} = s_n\sum\limits_{n=1}^K\sum\limits_{j\neq i}^N (\tau_{jin}-1)\int\limits_{\bar x_{jin}}^{\infty} E_{jin}(x)dG_{jn}(x),
\end{align*}
summing the industry-specific revenue in Section 3.1.1 across all industries $n$. $s_n$ refers to industry $n$'s share of the total tariff revenue available in country $i$, and a calculation of $R_{in}$ in this setup requires some assumption about what those shares might be. For example, a country might want to compensate industries more adversely affected by foreign tariffs, in which case $s_n$ could reflect industry $n$'s share of total exports from country $i$.
\begin{align*}
R_{in} = \frac{E_{in}}{\sum\limits_{n=1}^K E_{in}}\psi_i\sum\limits_{n=1}^K\sum\limits_{j\neq i}^N (\tau_{jin}-1)\int\limits_{\bar x_{jin}}^{\infty} E_{jin}(x)dG_{jn}(x),
\end{align*}
Alternatively, a country might want to encourage production for the domestic market, in which case $s_n$ could reflect industry $n$'s share of $i$'s share of domestic production for the domestic market:
\begin{align*}
R_{in} = \frac{E_{iin}}{\sum\limits_{n=1}^K E_{iin}}\psi_i\sum\limits_{n=1}^K\sum\limits_{j\neq i}^N (\tau_{jin}-1)\int\limits_{\bar x_{jin}}^{\infty} E_{jin}(x)dG_{jn}(x),
\end{align*}
\subsection{Unexpected vs. Expected Payments}
If transfer payments are unexpected, firms do not consider them when choosing production quantities, and the payments simply arrive after an equilibrium has already been determined. Since firms have already made production decisions, these unexpected payments does not help additional firms to enter the domestic market, but they do alleviate profit losses that domestic firms may incur due to the imposition of foreign tariffs. \\
\indent Researchers may also want to consider a scenario where firms know they are receiving some sort of tariff revenue transfer, and take this transfer payment into account when deciding how much to produce. I assume that while domestic industries expect a payment, their production decisions do \textit{not} consider how payments depend linearly on values of imports from other countries. Such consideration would introduce a counter-intuitive mechanism whereby domestic industries have an incentive to reduce their production in order to stimulate foreign imports, which would in turn increase the tariff revenue from which domestic industries benefit. I consider such an effect to be unrealistic and unlikely, and instead model `expected' transfer payments as a quantity $r_{jn}$ that enters on the left-hand side of each industry's zero-profit condition. The equilibrium in this situation would be the solution to a fixed-point problem where $R_{in}$ is equal to some function $g$ of equilibrium imports and industry-specific tariff rates $\tau_{jin}-1$. \\
\indent Equations (2) and (3) would be unchanged in the case of expected transfers, while (4) would add the payments to its left-hand side. The resultant equilibrium would be the solution to the equations
\begin{align*}
E_{ji} &= \mu_j E_{i} P_i^{\sigma_i-1}\big(\frac{\tau_{ji}w_j\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\frac{\overline x_{ji}^{\sigma_i-1-\gamma_j}}{\gamma_j-(\sigma_i-1)} \\
P_i^{1-\sigma_i} &= \frac{\gamma_j}{\gamma_j-(\sigma_i-1)}\big(\frac{\sigma_i }{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j (\tau_{ji})^{1-\sigma_i} \overline x_{ji}^{\sigma_i-\gamma_j-1} \\
f_{ji} - r_j &= Y_i P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1}(\frac{\sigma_i}{\sigma_i-1})^{1-\sigma_i}\frac{1}{\sigma_i} \\
R_j &= g(\{\tau_{nj}\}_{n=1}^N, \{E_{nj}\}_{n=1}^N)
\end{align*}
Figure 1 uses a timeline format to review the setup discussed in Sections 2 and 3. In the absence of transfer payments, each firm that could potentially produce in the economy finds out its productivity parameter, then determines how much output to produce for each destination based on how this parameter compares to the minimum productivity needed to generate positive profits along that route. If payments are expected, then firms will adjust production decisions accordingly. If the payments come unexpectedly, then firms only find out about them after deciding how much output to produce for each destination, so they only affect final profit levels. 
\begin{figure}
\centering
\begin{tikzpicture}
\draw (1,0) -- (15,0);
\foreach \x in {2,5,8,11,14}
\draw (\x cm,3pt) -- (\x cm,-3pt);
\draw (2,0) node[below,align = center] {\\ \textbf{Productivity} \\ \textbf{parameter} \\ \textbf{revealed}};
\draw (5,0) node[below, align = center] {\\ \textbf{Expected} \\ \textbf{payment} \\ \textbf{announced}};
\draw (8,0) node[below, align = center] {\\ \textbf{Output and} \\ \textbf{destinations} \\ \textbf{chosen}};
\draw (11,0) node[below, align = center] {\\ \textbf{Unexpected} \\ \textbf{payment} \\ \textbf{announced}};
\draw (14,0) node[below, align = center] {\\ \textbf{Calculate} \\ \textbf{profits}};
\end{tikzpicture}
\caption{Model Timeline}
%ALT TEXT: Timeline explaining how the sequences in which various model mechanisms occur.
\end{figure}
\section{Calibration and Estimation}
%overview of the section
\subsection{Calibration}
The model as described in Section 2 and Section 3 features five external parameters: $\gamma_j, \sigma_i, \tau_{ji}, f_{ji}$ and $\mu_j$. Of these parameters, $\gamma_j, \sigma_i$, and $\tau_{ji}$ can be directly estimated from data. The estimation of $\tau_{ji}$ is particularly straightforward as the USITC's DataWeb provides tariff rates at the Harmonized Tariff System 8-digit level. In the event that a researcher wants to analyze a broader sector or industry, they can aggregate the tariff rates for that industry's constituent HTS 8-digit subheadings using either a simple average or weighting the tariff rates by HTS8-level imports. \\
\indent Previous papers have discussed methods for calibrating $\sigma_i$, the elasticity of substitution between goods of different origins, and $\gamma_j$, the shape parameter that determines the distribution of firm productivities. The two main methods used by USITC economists for estimating $\sigma_i$ are the markup method, as introduced in Ahmad and Riker (2019), and the trade cost method, as discussed in Riker (2019).\footnote{Feenstra et al. (2018) uses a Melitz-style model with CES preferences to estimate elasticities of substitution, and discusses other approaches used in the academic literature to perform this estimation.} While the markup method may be more appropriate in models or papers where firm markups are a primary object of interest, the trade cost method estimates elasticities at a more granular product level; in addition, the data used as inputs in the trade cost method has been updated more frequently than the data used as inputs in the markup method. \\
\indent The Pareto shape parameter $\gamma$ appears much less frequently in economic modeling than the CES elasticity, and the literature discussing how to estimate this parameter is accordingly more sparse. Ahmad and Akgul (2019) introduce a maximum likelihood method for identifying Pareto shape parameters, and Ahmad and Akgul (2017) describe a visual method for finding $\gamma$ that involves producing a log-log plot of firm productivity and its lower bound. di Giovanni, Levchenko and Ranci\`{e}re (2011) and Chaney (2008) estimate $\gamma$ with regression-based methods using firm-level data. \\
\indent In addition to parameters $\gamma_j, \sigma_i$ and $\tau_{ji}$, whose estimation I have just discussed, fixed costs $f_{ji}$ and firm measures $\mu_j$ are necessary to solve for an equilibrium. However, $f_{ji}$ and $\mu_j$ do not have any clear counterpart in the data, nor are there values for these parameters that I could impose in theoretical simulations of the model such as the ones performed in this paper. I will therefore discuss how to use alternative model outputs that do exist in the data, namely consumption expenditure and the share of firms that export to each destination. \\
\indent Sources such as the USITC's DataWeb or UN Comtrade document industry-specific exports from one country to another. While direct measures are typically not available for consumption expenditure on domestically produced goods, such measures can generally be imputed.\footnote{The ITC's ITPD database is one source containing such consumption estimations.} Data measures can therefore represent model quantities $E_{ji}$ and $Y_i$ in a baseline (pre-counterfactual) equilibrium, and I use the symbol $\beta_{ji}$ to represent an expenditure share, i.e. $\beta_{ji} = \frac{E_{ji}}{Y_i}$. \\
\indent  I use the symbol $\phi_{ji}$ to denote the share of domestically producing firms in $j$ that ship to destination $i$. This measure typically cannot be computed without detailed firm-level data.\footnote{Papers that use detailed firm-level datasets include Aghion et al. (2022); Mayer, Melitz and Ottaviano (2014); Chaney (2008); and Eaton, Kortum, and Kramarz (2004a, 2004b). While papers with firm-level data more frequently use European datasets, Keller and Yeaple (2009) and Harrigan, Ma, and Shlychkov (2015) are two papers that present findings using firm-level data from the United States.} However, even in the absence of such detailed data modelers can input plausible values of the fraction of firms that export to each destination, and in a two-country model $\phi_{ji}, \ j\neq i$ simply represents the share of firms in $j$ that export at all.\footnote{Bernard et al. (2007) find that around 14 percent of all firms in the U.S. export. The International Trade Administration's Exporters Database also provides information on U.S. firms that export.} Using properties of the Pareto distribution, we know
\begin{align*}
\phi_{ji} &= \frac{1 - G_j(\bar x_{ji})}{1 - G_j(\bar x_{jj})} \\
&= \big(\frac{\bar x_{ji}}{\bar x_{jj}}\big)^{-\gamma_j}
\end{align*}
so it is a function of the ratio between the productivity cutoff necessary to produce at all and the productivity cutoff necessary to export to destination $j$.
\subsection{Solving for an Equilibrium}
This subsection discusses how to solve the model for equilibrium changes in a counterfactual with different tariff rates. While the rest of the paper focuses on this counterfactual, the solution methodology could easily be adapted to analyze other counterfactual situations. For example, the modeling framework can be used to examine a situation where fixed costs change and tariff rates do not, or where tariffs and fixed costs both change.\\
\indent Solving for an equilibrium requires rearranging equations (2)-(4) so that they are in terms of changes in allocations rather than the levels of the allocations themselves. Let $\hat y$ represent the counterfactual level of a variable $y$, so that $\hat\tau_{ji}$ represents counterfactual tariff levels and $\tau_{ji}$ represents baseline tariffs. I will start by explaining how to solve for an equilibrium without any distribution of tariff revenue, then explain how the addition of these transfers into the model changes the solution methodology. \\
\indent In an equilibrium without transfer payments, fixed costs do not change in the counterfactual, so $\hat f_{ji} = f_{ji}$. Because of this relationship, I know that the right-hand side of equation (4) in any counterfactual scenario will always be equal to the right-hand side of equation (4) in the baseline equilibrium. Some algebraic rearrangement gives
\begin{align}
\frac{\hat{\bar x}_{ji}}{\bar x_{ji}} &= \big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)\big(\frac{\hat P_{i}}{P_{i}}\big)^{-1}
\end{align}
Likewise, dividing the LHS and RHS of consumption equation (2) in the counterfactual by their counterparts in the baseline gives
\begin{align}
\frac{\hat E_{ji}}{E_{ji}} &= \big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)^{1-\sigma_i}\big(\frac{\hat P_{i}}{P_{i}}\big)^{\sigma_i-1}\big(\frac{\hat x_{ji}}{x_{ji}}\big)^{\sigma_i-1-\gamma_j}
\end{align}
An important, and useful feature of this setup is that $\mu_j$ drops out of the equation since it does not vary by time. \\
\indent Price index equation (3) is more difficult because variables enter into it additively rather than multiplicatively. The rewriting of (3) comes from moving $P_i^{1-\sigma_i}$ over to the right-hand side of the equation, then using (5) and (2) to replace $\mu_j x_{ji}^{\sigma_i-\gamma_j-1}P_i^{\sigma_i-1}$ with $\beta_{ji}$ and equilibrium ratios. Doing so eventually yields
\begin{align}
1 &= \sum\limits_{j=1}^N \beta_{ji}\big(\frac{\hat P_i}{P_i}\big)^{\gamma_j}\big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)^{-\gamma_j}
\end{align}
In an equilibrium without revenue distribution, I am therefore able to solve for counterfactual percent changes using calibrated values of $\sigma_i$ and $\gamma_j$ as well as data on consumer expenditure. Note that I am able to solve for counterfactual \textit{levels} of consumer expenditure, and not just percent changes, because I know $E_{ji}$, so given $E_{ji}$ and the ratio $\frac{\hat E_{ji}}{E_{ji}}$ I can get an absolute level of counterfactual consumption. \\
\indent With firms receiving lump-sum payments, the situation changes slightly. Lump-sum payments effectively alter the fixed costs for firms to enter the market, so (5) is no longer true. Instead I have that
\begin{align}
\frac{\hat{\bar x}_{ji}}{\bar x_{ji}} &= \big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)\big(\frac{\hat P_{i}}{P_{i}}\big)^{-1}\big(\frac{\hat f_{ji}}{f_{ji}}\big)^{\frac{1}{\sigma_i-1}}
\end{align}
where $\hat f_{ji} = f_{ji} - r_j$. Unfortunately, I do not observe levels of fixed costs $f_{ji}$. I outline a method for finding this new equilibrium using $\phi_{ji}$, the fraction of domestically producing firms in $j$ that export to $i$. \\
\indent I rewrite the ratio $\frac{\hat f_{ji}}{f_{ji}}$ as
\begin{align*}
1 - \frac{r_j}{f_{ji}}
\end{align*}
In the event that transfer payments are unexpected, I observe quantity $R_j$ (it is equivalent to total tariff revenue), but not $r_j$, because I don't know the total quantity of firms. Each individual firm would face a change in fixed costs of $1 - \frac{r_j}{f_{ji}}$. I correspondingly define $F_{ji} = f_{ji}m_{ji}$ as the total fixed cost amount paid by all firms using the route from $j$ to $i$. This total fixed cost quantity is 
\begin{align*}
F_{ji} = Y_i\frac{(\gamma_j - (\sigma_i-1))\beta_{ji}}{\gamma_j\sigma_i}
\end{align*}
and the counterfactual change in fixed costs\footnote{For the complete algebraic derivation, see the appendix.} is
\begin{align}
1 - \frac{r_j}{f_{ji}} &= \frac{R_j}{F_{ji}}\big(\sum_k \big(\frac{\phi_{jk}}{\phi_{ji}}\big)\big(\frac{\hat m_{jk}}{m_{jk}}\big)\big)^{-1}
\end{align}
Remember that the ratio $\frac{\hat m_{jk}}{m_{jk}}$ is equal to $\big(\frac{\hat x_{jk}}{x_{jk}}\big)^{-\gamma_j}$. Once that substitution is made, the ratio $\frac{r_j}{F_{ji}}$ can be written in terms of known quantities and counterfactual ratios. \\
\indent In the event that transfers arrive unexpectedly, I solve for an equilibrium with (6)-(8), then use the ratios $\frac{\hat m_{jk}}{m_{jk}}$ to find the change in fixed costs for each origin-destination pair. Applying this change in fixed costs, I am able to find counterfactual changes in profits and the size of the industry. \\
\indent If firms expect payments in advance, then equations (5)-(7) change slightly. (5) becomes (8), and (7) becomes
\begin{align}
1 = \sum\limits_{j=1}^N \beta_{ji}\big(\frac{\hat P_i}{P_i}\big)^{\gamma_j}\big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)^{-\gamma_j}\big(\frac{\hat f_{ji}}{f_{ji}}\big)^{1-\frac{\gamma_j}{\sigma_i-1}}
\end{align}
The equilibrium then consists of (6), (8), and (10). \\
\indent Finally, I present a formula for the change in country $j$'s total firm participation:
\begin{align}
\frac{\hat m_{j}}{m_j} = \frac{\sum\limits_{i=1}^N \phi_{ji}\big(\frac{\hat{\overline x}_{ji}}{\overline x_{ji}}\big)}{\sum\limits_{i=1}^N \phi_{ji}}
\end{align}
The estimation of $\frac{\hat m_{j}}{m_j}$ would not be possible without knowledge of $\{\phi_{ji}\}_{i=1}^N$.
\section{Simulated Results}
In this section, I present results of simulating the model after selecting a series of hypothetical values for the parameter inputs. Because simulation results are theoretical, they are intended to showcase the model's capabilities rather than provide commentary on specific policies or countries. \\
\indent I begin with the simplest possible scenario in which I run simulations with one industry and two countries that are identical in every possible way. Bilateral tariffs are symmetric, and increase from 5 percent in the baseline version of the model to 25 percent in the counterfactual. I then illustrate additional situations where the countries differ slightly from one another, or alternatively are identical but conduct trade with a third country that is unaffected by the increase in tariffs. Finally, I run simulations where the two countries conduct trade in more than one industry.
\subsection{Identical Countries}
Initial simulations feature two identical countries, which both have shape parameter $\gamma_j = 4$ and elasticity of substitution $\sigma_i = 3$. Furthermore, both countries have aggregate expenditure $Y = 100$, and spend 70\% of their total expenditure on domestically produced goods compared with 30\% on imported goods. \\
\indent Finally, I assume that 20 percent of domestically producing Country 1 firms also export to Country 2 ($\phi_{12} = .2$). The results that I present will focus on Country 1, and I do not need to set a value for $\phi_{21}$ since providing $\phi$ values for Country 2 is not necessary to compute counterfactual changes in Country 1. \\
\indent Figure 2 shows the results of simulating an increase in tariffs on trade between Country 1 and Country 2. I display the results of varying $\psi$, which governs the size of the transfer payments firms get; if $\psi = 1$, then Country 1 firms would receive all of the revenue generated by imposing tariffs on Country 2 imports. Figure 2a) displays the percent change in total firm participation, $\frac{\hat m_1}{m_1}-1$, and Figure 2b) displays the percent change in total country 1 profits, $\frac{\hat\pi_1}{\pi_1}-1$.
\begin{figure}[h!]
	\centering
	\begin{subfigure}[b]{0.495\linewidth}
		\includegraphics[width = \linewidth]{symmetry1.png}
		\caption{Change in firm participation}
	\end{subfigure}
	\begin{subfigure}[b]{0.495\linewidth}
		\includegraphics[width = \linewidth]{symmetryprof1.png}
		\caption{Change in total profit}
	\end{subfigure}
	\caption{Counterfactual outcomes as a function of payment sizer}
	%ALT TEXT: Graphs showing how results vary according to payment size.
\end{figure} \\
\indent As shown in Figure 2, firm participation increases in $\psi$ at a roughly linear rate in the scenario where firms expect to receive payments, while profits increase in $\psi$ in a roughly linear fashion under both scenarios.  If $\psi$ is set at .5, meaning that firms receive half of the maximum possible payment value, firm participation would increase by about 23.9\% in the expected case, while profits would increase by 26.4\% in the unexpected case and 14.4\% in the expected case. \\
\indent Table 1 provides the results of simulating an increase in tariffs when I choose $\psi$ so that the transfer given to Country 1 firms equals the total tariff fees those firms must pay when exporting to Country 2. Rows describe Country 1 equilibrium allocations of interest, while columns respectively provide counterfactual changes in a scenario with no transfer payments, a scenario with unexpected transfer payments, and a scenario with expected transfer payments.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Allocation & No payments & Unexpected payments & Expected payments  \\ \hline
Percent change in firm participation & 7.87\% & 7.87\% & 37.1\% \\ \hline
Percent change in domestic production & 17.7\% & 17.7\% & 21.8\% \\ \hline
Absolute change in domestic production & 12.4 & 12.4 & 15.3 \\ \hline
Percent change in imports from Country 2 & -42.4\% & -42.4\% & -50.5\%\\ \hline
Absolute change in imports from Country 2 & -12.4 & -12.4 & -15.3 \\ \hline
Percent change in total profits & -0.01\% & 52.7\% & 26.4\%\\ \hline
Absolute change in total profits & $-1.87\times 10^{-3}$ & 8.79 & 4.40 \\ \hline
\end{tabular}
\caption{Model simulation results for identical countries}
%ALT TEXT: Counterfactual results under no transfers, unexpected transfers, and expected transfers. In this case, the two countries are exactly the same.
\label{table:1}
\end{table} \\
When firms expect payments to arrive, they know that their fixed costs will be lower. This decrease in barriers to entry lowers the productivity cutoff $\overline x$ and allows a greater number of firms to produce in Country 1. If the payments are unexpected, then they do not enable additional firms to produce in Country 1, but they do contribute to profits of Country 1 firms that are already operating. \\
\indent Without transfer payments, an increase in the tariff rate between Countries 1 and 2 causes total firm participation to increase by 7.79\%, as an influx of firms that produce for the domestic market outweighs an exodus of firms no longer able to export to Country 2. Profits are essentially unchanged because, since the countries are identical in every way, the increase in revenue from domestic sales is equal to the decrease in revenue from exporting. The introduction of unexpected transfers substantially benefits firms, with a tariff change leading to a 52.7\% increase in profits. In the simulation with expected transfers, the increase in firm profits is lower than in the simulation with unexpected transfers, but the increase in domestic production is higher. While a greater number of operating firms increases production within Country 1, the larger number of less productive firms present in the market undercuts market power and lowers profits compared to a situation where those additional firms do not enter.
\subsection{Non-Identical Countries}
I now consider two alternate scenarios where the countries are not identical. In the first scenario, the countries no longer have identical initial expenditure shares; in particular, Country 1 spends only 20\% of its total expenditure on imports from Country 2, while Country 2 continues to spend 30\% of its expenditure on imports from Country 1. In the second scenario, I alter the relative sizes of each country's economy by increasing total expenditure in Country 1 to 200, while total expenditure in Country 2 remains at 100. \\
\indent Table 2 displays results from these scenarios, with all rows showing counterfactual changes or counterfactual percent changes. In these simulations and in all future simulations, I continue to set $\psi$ so that Country 1 firms do not receive more than the value of the tariffs that they pay.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{4em}{Allocation} & \multicolumn{3}{|c|}{$\beta_1 = (.8,.2)$} & \multicolumn{3}{|c|}{$Y_1 = 200$} \\ 
& None & Unexpected & Expected & None & Unexpected & Expected \\ \hline
Firm participation & 2.41\% & 2.41\% & 18.9\% & 7.87\% & 114\% & 39.1\% \\ \hline
Domestic production & 11.1\% & 11.1\% & 12.8\% & 17.7\% & 17.7\% & 21.8\% \\ \hline
Domestic production (absolute) & 8.94 & 8.94 & 10.2 & 28.4 & 28.4 & 34.9\\ \hline
Country 2 imports & -44.7\% & -44.7\% & -51.1\% & -42.4\% & -42.4\% & -50.9\% \\ \hline
Country 2 imports (absolute) & -8.93 & -8.93 & -10.2 & -16.6 & -16.6 & -20.3 \\ \hline
Total profits & -3.16\% & 27.0\% & 12.3\% & 7.29\% & 69.3\% & 38.4\% \\ \hline
Profits (absolute) & -0.58 & 4.95 & 2.26 & 0.909 & 12.0 & 6.60 \\ \hline
\end{tabular}
\caption{Model simulation results with non-identical countries}
%ALT TEXT: Counterfactual results under no payments, unexpected payments, and expected payments when the countries aren't symmetric.
\label{table:1}
\end{table} \\
\indent As shown in Table 2, if domestic production is given a higher weight in initial expenditure (columns 2-4), counterfactual increases in firm participation and total profits are lower compared to their values in Table 1. Profits in the absence of transfer payments now decrease by 3.16\%, while unexpected transfers cause profits to increase by 27\% and expected transfers cause profits to increase by 12.3\%. The boost to domestic production provided by tariffs is less valuable if Country 1 already had a higher weight placed on domestic production, and transfers are likewise less beneficial for Country 1 firms because Country 1 has a lower import value from which to generate tariff revenue. Meanwhile, a decrease in international trade more adversely affects Country 1 firms than it does in Table 1, because Country 2 is now more export-dependent. \\
\indent An increase in the size of Country 1's economy relative to Country 2's economy (columns 5-7) results in greater increases in firm participation and total profits compared to the case where both countries are identical. Even though percent changes in domestic production and imports are the same as they were in Table 1, tariffs are now re-orienting Country 1 firms toward a domestic market that is larger and therefore offers a higher revenue base. When transfer payments are expected, firms further internalize the greater size of Country 1's economy, resulting in greater increases in domestic production compared to Table 1.
\subsection{Simulation with Three Countries}
I next simulate counterfactual results in a setup where Country 1 and Country 2 are once again identical but also trade with a third country, Country 3. Instead of allocating 30\% of initial consumer expenditure to Country 2 imports, Country 1 now allocates 20\% to imports from Country 2 and 10\% to imports from Country 3, while Country 2 allocates 20\% to imports from Country 1 and 10\% to imports from Country 3. Country 3, meanwhile, is more import-focused, with 25\% of its expenditure going to imports from Country 1 and 25\% also going to imports from Country 2. Country 3 otherwise has all the same parameters as Countries 1 and 2, and $\phi_{13} = .2$ so that 20\% of firms producing in Country 1 export to Country 3, just as 20\% of firms producing in Country 1 export to Country 2. \\
\indent Crucially, while Country 1 and Country 2 increase tariffs on each other, tariffs on imports from Country 3 remain at 5\%. Table 3 shows the results of this counterfactual, with $\psi$ set so that firms receive transfers according to the total value of foreign tariffs that they pay.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Allocation & No transfers & Unexpected transfers & Expected transfers  \\ \hline
Percent change in firm participation & 1.60\% & 1.60\% & 22.5\% \\ \hline
Percent change in domestic production & 11.2\% & 11.2\% & 14.9\% \\ \hline
Absolute change in domestic production & 7.82 & 7.82 & 10.4 \\ \hline
Percent change in imports from Country 2 & -44.7\% & -44.7\% & -52.2\%\\ \hline
Absolute change in imports from Country 2 & -8.93 & -8.93 & -10.2 \\ \hline
Percent change in total profits & -0.966\% & 51.5\% & 34.3\%\\ \hline
Absolute change in total profits & -0.185 & 9.02 & 6.57 \\ \hline
\end{tabular}
\caption{Results for simulations featuring a third country}
%ALT TEXT: Counterfactual results under no transfers, unexpected transfers, and expected transfers for a scenario where two identical countries trade with a third country.
\label{table:1}
\end{table} \\
When the model features a third country whose tariff rates are unchanged, Country 1 industries do not benefit as much from the tariff increase on Country 2. The percent changes in firm participation and domestic production in Table 3 are smaller for all three simulations than the corresponding numbers in Table 1. In addition, the adverse impact of tariffs on imports from Country 2 is stronger (more negative) in Table 3 than in Table 1 for a similar reason; as tariffs make Country 2 imports less desirable, Country 1 consumers have another location to which they can shift their consumption. The addition of Country 3 does not make an enormous difference to counterfactual changes, however, as most results differ from Table 1 by 1-2 percentage points.\\
\indent Percent changes in profits compared to Table 1 are more negative in the case of no transfer payments and less positive in the case of unexpected transfer payments, as Country 3 absorbs some of the benefits that would go to Country 1 producers. However, in the case where transfer payments are expected, profit increases are actually \textit{higher} than in Table 1. The increase in tariffs on Country 2 results in higher exports from Country 1 to Country 3, and when firms are aware of a decrease in fixed costs they can take advantage of this shift in export destinations more effectively, overcoming the loss in revenue from exports to Country 2.\footnote{If analysis is restricted to profits from the sale of goods in Country 1 and Country 2, the percent change in Country 1 profits with expected payments is smaller with the addition of a third country. The higher profit increase observed in Table 3 is therefore entirely caused by an increase in profit from exports to Country 3.}
\subsection{Tariff Transfer Payment Simulation with Industry Weights}
Finally, I solve this model for counterfactual changes in a scenario where tariff revenue is allocated among different industries, rather than each industry receiving the revenue generated from imports in that industry. I consider two new counterfactuals: one in which tariff revenue is distributed based on a given industry's share of domestic revenue (to incentivize domestic production) and one in which tariff revenue is distributed based on a given industry's share of country 1's exports (to compensate industries that are more adversely affected by foreign tariffs). The two industries differ only in their initial consumer expenditure shares, with Industry 1 identical to the industry in the previous section and Industry 2 having the following matrix:
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|}
\hline
 & Country 1 & Country 2  \\ \hline
Country 1 & 0.6 & 0.5 \\ \hline
Country 2 & 0.4 & 0.5\\ \hline
\end{tabular}
\caption{Expenditure shares for Industry 2 in multi-industry scenario}
%ALT TEXT: Parameters representing initial consumer expenditure shares for a second industry.
\label{table:1}
\end{table} \\
Compared to Industry 1, Industry 2 is more import-dependent in terms of consumption for country 1, as well as more export-facing for firms in country 1. Table 4 displays the results of comparing industry-weighted results for Industry 1 with the unexpected and expected results for Industry 1 displayed in Table 1.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Allocation & Unexpected, & Unexpected, & Unexpected, & Expected, & Expected, & Expected,  \\ 
& industry-only & domestic & export & industry-only & domestic & export \\ \hline
Firm participation & 52.7\% & 52.7\% & 52.7\% & 37.1\% & 44.9\% & 33.7\% \\ \hline
Domestic production & 17.7\% & 17.7\% & 17.7\% & 21.8\% & 22.7\% & 21.4\% \\ \hline
Country 2 Imports & -42.4\% & -42.4\% & -42.4\% & -50.5\% & -53.0\% & -50.0\%\\ \hline
Total profits & 52.7\% & 68.6\% & 47.8\% & 26.4\% & 33.2\% & 23.3\% \\ \hline
\end{tabular}
\caption{Counterfactual percent changes for different payment formulas}
%ALT TEXT: Counterfactual results under unexpected and expected transfer payments for a variety of possible transfer formulas.
\label{table:1}
\end{table} \\
\indent Since Industry 1 consumption is more domestically-oriented than Industry 2 consumption, the weighting of unexpected transfer payments based on domestic production benefits Industry 1, and profit increases are substantially higher than in Table 2 when Industry 1's transfers come exclusively from Industry 1 tariff revenue. When unexpected tariff transfers are weighted based on exports, however, Industry 1 becomes slightly worse off than in Table 2 because consumers in Country 2 import more from Country 1 in Industry 2 than they do in Industry 1, and so the weighting scheme penalizes Industry 1 relative to an industry-only transfer scheme. 
\section{Conclusion}
In this paper, I contribute to the partial equilibrium literature by introducing a method for modeling the distribution of tariff revenue as lump-sum transfers to firms. The methodology can be applied to a wide variety of countries or industries, and models both payments that firms do not expect in advance as well as expected payments that firms incorporate into their production and pricing decisions. Firms in the model have heterogenous productivities represented by a Pareto distribution, and this setup allows me to compute equilibrium changes in the fraction of firms that choose to operate in the economy. \\
\indent I next perform a series of counterfactual simulations in which I estimate the effect of a change in tariffs given hypothetical parameter choices. While the paper does not apply observed data or model any real-life countries, it includes a discussion of what types of data would generally be available and how a researcher might use this data to compute equilibrium solutions to the model. Due to data limitations, the paper presents most results in terms of counterfactual percentage changes rather than counterfactual quantities. \\
\indent The model is quite versatile and allows its users to perform a multitude of counterfactuals beyond the one presented in Section 5. More broadly, future research in this area can build on Jafarey and Lahiri (2024) and Lahiri and Nasim (2006) by building a partial equilibrium model in which firms receive rebates to compensate them for the tariffs they face on imported intermediate inputs. Such a project would require a multi-stage model, which adds computational complexity to the one-stage model described here. In addition, future researchers could develop a dynamic partial equilibrium model in order to represent industries or situations where the effect of transfer payments is not instantaneous, or alternatively where firms receive transfers at different points in time. 
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\bibitem{} Mayer, Thierry, Marc Melitz, and Gianmarco Ottaviano. ``Market Size, Competition, and the Product Mix of Exporters." \textit{American Economic Review} 104, no. 2 (2014): 495-536.
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\end{thebibliography} 
\section{Appendix: Equilibrium Equations}
This section goes in detail through the mathematical processes for using the model to generate an equilibrium. \\
I begin by deriving expressions for aggregate consumption and prices. Recall that the total value of consumption originating in $j$ and ending in $i$ is
\begin{align*}
E_{ji} &= \mu_j \int\limits_{\overline x_{ji}}^{\infty} c_{ji}(x)p_{ji}(x)dG_j(x) \\
&= \mu_j \int\limits_{\overline x_{ji}}^{\infty} E_{i}(x)P_i^{\sigma_i-1}p_{ji}(x)^{1-\sigma_i}dG_j(x)
\end{align*} 
where the second line comes from replacing $c_{ji}(x)$ with its equivalent given by (1). \\
\indent Replacing $p_{ji}$ with its equivalent expression from the firms' FOCs and $dG_j(x)$ with the derivative of $G_j(x)$:
\begin{align*}
E_{ji} &= \mu_j E_{i}(x)P_i^{\sigma_i-1}\int\limits_{\overline x_{ji}}^{\infty} \big(\frac{\tau_{ji}\sigma_i}{(\sigma_i-1)x}\big)^{1-\sigma_i} \gamma_j x^{-\gamma_j-1}dx \\
&= \mu_j E_{i}(x)P_i^{\sigma_i-1}\big(\frac{\tau_{ji}\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\gamma_j\int\limits_{\overline x_{ji}}^{\infty} x^{\sigma_i-\gamma_j-2}dx \\
&= \mu_j E_{i}(x)P_i^{\sigma_i-1}\big(\frac{\tau_{ji}\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\frac{\gamma_j}{\sigma_i-\gamma_j-1}(\lim\limits_{x\to\infty}x^{\sigma_i-1-\gamma_j}-\overline x_{ji}^{\sigma_i-1-\gamma_j}) \\
&= \mu_j E_{i}(x)P_i^{\sigma_i-1}\big(\frac{\tau_{ji}\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\frac{\gamma_j\overline x_{ji}^{\sigma_i-1-\gamma_j}}{\gamma_j-(\sigma_i-1)}
\end{align*}
The last step requires the assumption that $\gamma_j \geq \sigma_i-1$. \\
\indent If I observe consumer expenditure share $\beta_{ji} = \frac{E_{ji}}{E_{i}}$, then I end up with
\begin{align*}
\beta_{ji} &= \gamma_j\mu_j P_i^{\sigma_i-1}\big(\frac{\tau_{ji}\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\frac{\overline x_{ji}^{\sigma_i-1-\gamma_j}}{\gamma_j-(\sigma_i-1)}
\end{align*}
The evaluation of an aggregate price expression follows a similar process. Recall that the aggregate price in country $i$ is given by
\begin{align*}
P_i^{1-\sigma_i} = \sum\limits_{j=1}^N \big[\mu_j \int\limits_{\overline x_{ji}}^{\infty} p_{ji}(x)^{1-\sigma_i}dG_j(x)\big]
\end{align*}
Evaluation of this expression involves the same substitutions for $p_{ji}(x)$ and $dG_j(x)$ that were done in evaluating $E_{ji}$ above. After some algebra and integral calculus we find
\begin{align*}
P_i^{1-\sigma_i} = \big(\frac{\sigma_i }{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j \frac{\gamma_j}{\gamma_j-(\sigma_i-1)}(\tau_{ji})^{1-\sigma_i} \overline x_{ji}^{\sigma_i-\gamma_j-1}
\end{align*}
An equilibrium is characterized by consumption expressions, a price index expression, and finally zero-profit conditions that use the definition of $\overline x_{ji}$ as the boundary point where a country $j$ firm is indifferent between selling in country $i$ and shutting down. In other words:
\begin{align*}
f_{ji} &= \pi(\overline x_{ji}) \\
&= Y_i P_i^{\sigma_i-1}p_{ji}(\overline x_{ji})^{1-\sigma_i} - \frac{\tau_{ji}Y_i P_i^{\sigma_i-1}p_{ji}(\overline x_{ji})^{-\sigma_i}}{\overline x_{ji}} \\
&= Y_i P_i^{\sigma_i-1}\big((\frac{\sigma_i \tau_{ji}}{(\sigma_i-1)\overline x_{ji}})^{1-\sigma_i}-\frac{\tau_{ji}}{\overline x_{ji}}(\frac{\sigma_i \tau_{ji}}{(\sigma_i-1)\overline x_{ji}})^{-\sigma_i}\big) \\
&= Y_i P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1}\big((\frac{\sigma_i}{\sigma_i-1})^{1-\sigma_i} - (\frac{\sigma_i}{\sigma_i-1})^{-\sigma_i}\big) \\
&= Y_i P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1}(\frac{\sigma_i}{\sigma_i-1})^{1-\sigma_i}\frac{1}{\sigma_i}
\end{align*}
The zero-profit condition, consumption equation, and aggregate price equation fully characterize the equilibrium.
\subsection{Counterfactual Equilibria}
A chief purpose of this model is to investigate counterfactual situations where the tariff rate changes. The standard procedure for this exercise involves using data for an initial equilibrium to calibrate parameters, then solve for a new equilibrium with these parameters where $\tau_{ji}$ is different. However, given an inability to observe or calibrate $f_{ji}$ and $\mu_j$, I solve the model in terms of \textit{percent} changes in variables rather than in terms of the variables themselves.\\
\indent Let's start with fixed costs. In a world where tariffs change, fixed costs are assumed to stay constant. I therefore know that
\begin{align*}
Y_i P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1}(\frac{\sigma_i}{\sigma_i-1})^{1-\sigma_i}\frac{1}{\sigma_i} &= \hat Y_i \hat P_i^{\sigma_i-1}\hat \tau_{ji}^{1-\sigma_i} \hat {\bar x}_{ji}^{\sigma_i-1}(\frac{\sigma_i}{\sigma_i-1})^{1-\sigma_i}\frac{1}{\sigma_i} \\
\Rightarrow P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1} &= \hat P_i^{\sigma_i-1}\hat \tau_{ji}^{1-\sigma_i} \hat {\bar x}_{ji}^{\sigma_i-1} \\
\Rightarrow \frac{P_i\overline x_{ji}}{\tau_{ji}} &= \frac{\hat P_i\hat{\bar x}_{ji}}{\hat \tau_{ji}}
\end{align*}
Rearranging the price equation yields
\begin{align*}
1 &= \big(\frac{\sigma_i}{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j \frac{\gamma_j}{\gamma_j-(\sigma_i-1)}(\tau_{ji})^{1-\sigma_i} \overline x_{ji}^{\sigma_i-\gamma_j-1}P_i^{\sigma_i-1} \\
1 &= \big(\frac{\sigma_i}{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j \frac{\gamma_j}{\gamma_j-(\sigma_i-1)}(\hat \tau_{ji})^{1-\sigma_i} \hat{\bar x}_{ji}^{\sigma_i-\gamma_j-1}\hat P_i^{\sigma_i-1} \ \text{in counterfactual}
\end{align*}
The goal is to use my rearranged zero-profit condition to replace $\overline x_{ji}$ and $P_i$, since I do not know either of these two variables. Using algebra I can do some rewriting.
\begin{align*}
\hat{\bar x}_{ji}^{\sigma_i-1-\gamma_j}\hat P_i^{\sigma_i-1}\hat \tau_{ji}^{1-\sigma_i} &= \big(\frac{\hat{\bar x}_{ji} \hat P_i}{\hat \tau_{ji}}\big)^{\sigma_i-1-\gamma_j} \hat P_i^{\gamma_j}\hat\tau_{ji}^{-\gamma_j} \\
&= \big(\frac{\bar x_{ji} P_i}{ \tau_{ji}}\big)^{\sigma_i-1-\gamma_j} \hat P_i^{\gamma_j}\hat \tau_{ji}^{-\gamma_j}
\end{align*}
Next I turn to the consumption equation. The goal is, again, to isolate $x_{ji}$ and $P_i$ so that I may eventually replace them. Rearranging the items in the consumption equation gives
\begin{align*}
\overline x_{ji}^{\sigma_i-1-\gamma_j} P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i} &= \frac{\beta_{ji}(\gamma_j-(\sigma_i-1))}{\gamma_j\mu_j}(\frac{\sigma_i}{\sigma_i-1})^{\sigma_i-1} \\
\Rightarrow \big(\frac{\bar x_{ji} P_i}{ \tau_{ji}}\big)^{\sigma_i-1-\gamma_j}P_i^{\gamma_j}\tau_{ji}^{-\gamma_j} &= \frac{\beta_{ji}(\gamma_j-(\sigma_i-1))}{\gamma_j\mu_j}(\frac{\sigma_i}{\sigma_i-1})^{\sigma_i-1}
\end{align*}
Going back to the price index equation,
\begin{align*}
1 &= \big(\frac{\sigma_i}{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j\frac{\gamma_j}{\gamma_j-(\sigma_i-1)}\big(\frac{\bar x_{ji} P_i}{ \tau_{ji}}\big)^{\sigma_i-1-\gamma_j} \hat P_i^{\gamma_j}\hat \tau_{ji}^{-\gamma_j} \\
&= \big(\frac{\sigma_i}{\sigma_i-1}\big)^{1-\sigma_i}\sum\limits_{j=1}^N \mu_j \frac{\gamma_j}{\gamma_j-(\sigma_i-1)}\frac{\beta_{ji}(\gamma_j-(\sigma_i-1))}{\gamma_j\mu_j}(\frac{\sigma_i}{\sigma_i-1})^{\sigma_i-1}P_i^{-\gamma_j}\tau_{ji}^{\gamma_j}\hat P_i^{\gamma_j}\hat \tau_{ji}^{-\gamma_j} \\
&= \sum\limits_{j=1}^N \beta_{ji}\big(\frac{\hat P_i}{P_i}\big)^{\gamma_j}\big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)^{-\gamma_j}
\end{align*}
I can therefore use algebraic substitutions to rewrite the price index equation in terms of ratios between old and new prices and tariffs, or alternatively the percentage changes in those items. The only unknown object is the ratio $\frac{\hat P_i}{P_i}$, and I can solve for this ratio algebraically. \\
\indent Ratios of equilibrium objects similarly characterize changes in consumption and changes in the cutoff productivity:
\begin{align*}
\frac{\hat E_{ji}}{E_{ji}} &= \big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)^{1-\sigma_i}\big(\frac{\hat P_{i}}{P_{i}}\big)^{\sigma_i-1}\big(\frac{\hat P_{i}}{P_{i}}\big)^{\sigma_i-1-\gamma_j} \\
\frac{\hat{\bar x}_{ji}}{\bar x_{ji}} &= \big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)\big(\frac{\hat P_{i}}{P_{i}}\big)^{-1}
\end{align*}
\subsection{Solving for an Equilibrium with Tariff Transfer Payments}
When transfer payments enter the picture, the situation in subsection 7.1 above changes slightly. The payments effectively alter the fixed costs for firms to enter the market, so (5) is no longer true. Instead we have that
\begin{align*}
\frac{\hat{\bar x}_{ji}}{\bar x_{ji}} &= \big(\frac{\hat \tau_{ji}}{\tau_{ji}}\big)\big(\frac{\hat P_{i}}{P_{i}}\big)^{-1}\big(\frac{\hat f_{ji}}{f_{ji}}\big)^{\frac{1}{\sigma_i-1}} \\
\Rightarrow \frac{\hat{\bar x}_{ji} \hat P_{ji}}{\hat\tau_{ji}} &= \frac{\bar x_{ji} P_{ji}}{\tau_{ji}}\big(\frac{\hat f_{ji}}{f_{ji}}\big)^{\frac{1}{\sigma_i-1}}
\end{align*}
As before, we want to replace $x_{ji}$ and $P_i$, but the process is slightly different now that it involves fixed costs. 
\begin{align*}
\hat{\bar x}_{ji}^{\sigma_i-1-\gamma_j}\hat P_i^{\sigma_i-1}\hat \tau_{ji}^{1-\sigma_i} &= \big(\frac{\hat{\bar x}_{ji} \hat P_i}{\hat \tau_{ji}}\big)^{\sigma_i-1-\gamma_j} \hat P_i^{\gamma_j}\hat\tau_{ji}^{-\gamma_j} \\
&= \big(\frac{\bar x_{ji} P_i}{ \tau_{ji}}(\frac{\hat f_{ji}}{f_{ji}})^{\frac{1}{\sigma_i-1}}\big)^{\sigma_i-1-\gamma_j} \hat P_i^{\gamma_j}\hat \tau_{ji}^{-\gamma_j} \\
\Rightarrow \big(\frac{\bar x_{ji} P_i}{ \tau_{ji}}\big)^{\sigma_i-1-\gamma_j}(\frac{\hat f_{ji}}{f_{ji}})^{1-\frac{\gamma_j}{\sigma_i-1}} \hat P_i^{\gamma_j}\hat\tau_{ji}^{-\gamma_j} &= \frac{\beta_{ji}(\gamma_j-(\sigma_i-1))}{\gamma_j\mu_j}(\frac{\sigma_i}{\sigma_i-1})^{\sigma_i-1}
\end{align*}
So the only difference in the price index equation is the presence of the fixed cost term. Adding the fixed cost term to the price index equation produces (10). \\
\indent Since the change in fixed costs comes from the addition of lump-sum transfers, $\frac{\hat f_{ji}}{f_{ji}}$ is simply $\frac{f_{ji}-r_{j}}{f_{ji}}$ or $1-\frac{r_j}{f_{ji}}$. I now outline a series of algebraic steps allowing me to rewrite the ratio $\frac{r_j}{f_{ji}}$ given $\phi_{ji}$, $\beta_{ji}$ and $Y_i$. \\
\indent I first need to calculate a value for $F_{ji}$, the total fixed cost payment across all firms shipping from $j$ to $i$ before transfer payments apply. The per-firm fixed cost amount $f_{ji}$ is difficult to calculate, but $F_{ji}$ comes out of equations (2) and (4). Recalling that $m_{ji} = \mu_j\bar x_{ji}^{-\gamma_j}$ and $\beta_{ji} = \frac{E_{ji}}{E_j}$, (2) is equivalent to
\begin{align*}
\beta_{ji} = m_{ji} P_i^{\sigma_i-1}\big(\frac{\tau_{ji}\sigma_i}{(\sigma_i-1)}\big)^{1-\sigma_i}\frac{\overline x_{ji}^{\sigma_i-1}}{\gamma_j-(\sigma_i-1)}
\end{align*}
Furthermore, multiplying both sides of (4) by $m_{ji}$ changes it to
\begin{align*}
m_{ji}f_{ji} = F_{ji} = Y_i P_i^{\sigma_i-1}\tau_{ji}^{1-\sigma_i}\overline x_{ji}^{\sigma_i-1}\big(\frac{\sigma_i}{\sigma_i-1}\big)^{1-\sigma_i}
\end{align*}
This equation is almost exactly the same as the rewritten version of (2). Putting the two equations together I get\footnote{Note that operating profit, or profit before fixed costs, along route $ji$ is given by $\frac{1}{\sigma_i}\beta_{ji}Y_i$. Since $\gamma_j - (\sigma_i - 1) < \gamma_j$, operating profit along route $ji$ will always be higher than $F_{ji}$.}
\begin{align*}
F_{ji} = \beta_{ji}Y_i\frac{\gamma_j-(\sigma_i-1)}{\gamma_j\sigma_i}
\end{align*}
Now that I have derived a value for $F_{ji}$ in terms of observed quantities, let's use it. Multiplying both numerator and denominator by $m_{ji}$, the baseline firm participation level, and replacing $r_j$ with $R_j$ gives
\begin{align*}
\frac{R_j\frac{m_{ji}}{\sum_k \hat m_{jk}}}{f_{ji}m_{ji}} &= \frac{R_j}{F_{ji}}\big(\frac{\sum_k \hat m_{jk}}{m_{ji}}\big)^{-1} \\
&= \frac{R_j}{F_{ji}}\big(\frac{\sum_k m_{jk}\frac{\hat m_{jk}}{m_{jk}}}{m_{ji}}\big)^{-1}
\end{align*}
Given information on the fraction of firms that export to each destination, I know $\phi_{jk} = \frac{m_{jk}}{m_{jj}}$ for all $k$ and hence $\frac{\phi_{jk}}{\phi_{ji}} = \frac{m_{jk}}{m_{ji}}$ for all $k$ and $i$. Putting this into the above equation gives
\begin{align*}
\frac{R_j}{F_{ji}}\big(\frac{\sum_k \big(m_{ji}\frac{\phi_{jk}}{\phi_{ji}}\big)\big(\frac{\hat m_{jk}}{m_{jk}}\big)}{m_{ji}}\big)^{-1} &= \frac{R_j}{F_{ji}}\big(\sum_k \big(\frac{\phi_{jk}}{\phi_{ji}}\big)\big(\frac{\hat m_{jk}}{m_{jk}}\big)\big)^{-1}
\end{align*}
Remember that the ratio $\frac{\hat m_{jk}}{m_{jk}}$ is equal to $\big(\frac{\hat x_{jk}}{x_{jk}}\big)^{-\gamma_j}$. Once that substitution is made, the ratio $\frac{r_j}{F_{ji}}$ can be written in terms of known quantities and counterfactual ratios. \\
\indent Finally, let's derive the formula (11) that governs total change in firm participation. Remember that $\phi_{ji}$, the fraction of domestically producing firms in $j$ that also ship to $i$, is equivalent to $(\frac{\overline x_{ji}}{\overline x_{ii}})^{-\gamma_j}$. For any two destination countries $i$ and $n$, we therefore know that
\begin{align}
\frac{\phi_{ji}}{\phi_{jn}} = \big(\frac{\overline x_{ji}}{\overline x_{ii}}\big)^{-\gamma_j}\big(\frac{\overline x_{ii}}{\overline x_{jn}}\big)^{-\gamma_j} = \big(\frac{\overline x_{ji}}{\overline x_{jn}}\big)^{-\gamma_j}
\end{align}
I can then rewrite base-period firm participation $m_j$ as follows:
\begin{align*}
m_j &= \mu_j\sum\limits_{i=1}^N \overline x_{ji}^{-\gamma_j} \\
&= \mu_j\sum\limits_{i=1}^N \big(\frac{\phi_{ji}}{\phi_{jn}}\big) \overline x_{jn}^{-\gamma_j} \\
&= \frac{\mu_j}{\phi_{jn}}\sum\limits_{i=1}^N\phi_{ji}\overline x_{jn}^{-\gamma_j}
\end{align*}
I rewrite $\hat m_j$ using a similar process, but with one extra step because the shares $\phi_{ji}$ are only known in the baseline. I first rewrite $\hat m_j$ in terms of counterfactual changes:
\begin{align*}
\hat m_j = \mu_j\sum\limits_{i=1}^N \hat{\overline x}_{ji}^{-\gamma_j} = \mu_j\sum\limits_{i=1}^N \overline x_{ji}^{\gamma_j}\big(\frac{\hat{\overline x}_{ji}}{\overline x_{ji}}\big)^{-\gamma_j}
\end{align*}
and then I can use (12) to substitute for $\overline x_{ji}$.
\begin{align*}
\hat m_j &= \mu_j\sum\limits_{i=1}^N \big(\frac{\phi_{ji}}{\phi_{jn}}\big) \overline x_{jn}^{-\gamma_j}\big(\frac{\hat{\overline x}_{ji}}{\overline x_{ji}}\big)^{-\gamma_j} \\
&= \frac{\mu_j\overline x_{jn}^{-\gamma_j}}{\phi_{jn}}\sum\limits_{i=1}^N\phi_{ji}\big(\frac{\hat{\overline x}_{ji}}{\overline x_{ji}}\big)^{-\gamma_j}
\end{align*}
In comparing $\hat m_j$ and $m_j$, multiple terms cancel out, including $\mu_j$ and any quantity indexed by $n$. Dividing $\hat m_j$ by $m_j$ therefore gives me (11), which is in terms of counterfactual changes and observed inputs.
\end{document}