A COMPARISON OF PARTIAL EQUILIBRIUM MODELS
OF TARIFF RATE QUOTAS
Ross Hallren
David Riker
ECONOMICS WORKING PAPER SERIES
Working Paper 201709B
U.S. INTERNATIONAL TRADE COMMISSION
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September 2017
Office of Economics
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A Comparison of Partial Equilibrium Models of Tariff Rate Quotas
Ross Hallren and David Riker
Office of Economics Working Paper 201709B
September 2017
ABSTRACT
In this short paper we consider the impact of a TRQ on industryspecific imports and domestic production using two different types of partial equilibrium models, an Armington CES model and a Krugman CES model of trade. In the Armington model with only adjustment on the intensive margin of trade, a TRQ that fills has the same effect on trade as a flat tariff at the outofquota rate. On the other hand, when there is also adjustment on the extensive margin of trade, as in a Krugman or Melitz model of trade, the two policies are not equivalent and the inquota rate has an effect on trade and domestic production even when the TRQ fills.
Ross Hallren
Office of Economics, Research Division
Ross.Hallren@usitc.gov
David Riker
Office of Economics, Research Division
David.Riker@usitc.gov
A tariff rate quota (TRQ) is a tariff schedule with a step: there is an inquota tariff rate on import volumes below the quota volume and a higher outofquota tariff rate on imports above the quota volume. In this paper we model the impact of a TRQ on the volumes of subject imports and domestic production using two alternative partial equilibrium (PE) models of international trade. Our first PE model is an Armington (1969) CES model with perfect competition and infinitely elastic supply. Our second PE model is a Krugman (1980) CES model with product differentiation, fixed costs of trade and production, and monopolistic competition.
We describe the two models and then we run several simulations of changes in trade policy, using illustrative data inputs, and compare the estimates from the alternative models. We focus on a scenario in which the TRQ fills, which means that the volume of subject imports is greater than or equal to the quota volume. In the Armington CES model, whenever the TRQ fills, the outofquota tariff rate affects the volumes of trade and domestic production, but the inquota rate and the TRQ quota amount do not affect trade or domestic production. In this model, a TRQ with a specific ad valorem outofquota rate has the same effect on trade and domestic production as a flat tariff at the same ad valorem rate.
In contrast, in a Krugman CES model with fixed costs of trade and production, all of the parameters of the TRQ (the inquota rate and quota volume, as well as the outofquota rate) affect the volumes of trade and domestic production, even if the TRQ fills and the outofquota rate applies. In this case, the inquota rate affects the incentives for foreign producers to enter the market by transferring inframarginal tariff revenues.
In the first model, we assume that there is perfect competition and the elasticity of supply for each producer is infinitely elastic. There are constant returns to scale and factor prices are set in the broader economy and are not significantly affected by the policy changes (a partial equilibrium assumption). Although this is a restrictive assumption about the elasticity of supply, it makes the Armington model more comparable to the Krugman model discussed below.
The prices of producers in source country $j$ are equal to their exogenous marginal costs ${c}_{j}$.
${p}_{j}={c}_{j}$ (1)
Equation (2) is the industry price index for the market.
$P={\left({\displaystyle \sum}_{j}{b}_{j}{\left({p}_{j}\left(1+{\tau}_{j}\right)\right)}^{1\sigma}\right)}^{\frac{1}{1\sigma}}$ (2)
The ${b}_{j}$ parameters represent potential asymmetries in preferences for the domestic and foreign varieties indexed by $j$. The variable ${\tau}_{j}$ is the applicable tariff rate in the home country on varieties from source country $j$.
Equation (3) is the demand function corresponding to the CES preferences.
${q}_{j}=Y{b}_{j}{P}^{\sigma 1}{\left({p}_{j}\left(1+{\tau}_{j}\right)\right)}^{\sigma}$ (3)
The variable $Y$ is total expenditures on all varieties of the product. We assume that $Y$ is a constant share of aggregate expenditures (an assumption that there are CobbDouglas preferences between the different industries) and that aggregate expenditures are exogenous (another partial equilibrium assumption). The tariff rate on subject imports is equal to the inquota rate ${\tau}_{s}^{in}$ if ${q}_{s}$ is less than the quota volume and is equal to the outofquota rate ${\tau}_{s}^{out}$ if ${q}_{s}$ is greater than the quota volume. (The subscript $s$ indicates imports subject to the TRQ.) We focus on the second scenario, where the TRQ fills and marginal tariff rate is ${\tau}_{s}^{out}$.
Equation (4) calibrates the preference parameters if initial prices are set equal to one.
${b}_{j}={m}_{j}$ (4)
The variable ${m}_{j}$ is the initial market share for the varieties from source country $j$.
We solve for the change in the volume of subject imports and domestic production due to the TRQ, holding $Y$ and ${c}_{j}$ fixed, by substituting equations (1) and (4) into equations (2) and (3). We consider three alternative trade policy scenarios. In all of the scenarios, the baseline tariff rate is 0 percent. In the first alternative, the outofquota tariff rate is 20 percent and the inquota rate is 0 percent. In the second alternative, the outofquota rate is higher (30 percent, rather than 20 percent) but the inquota rate is still 0 percent. In the third alternative, both the inquota rate and the outofquota rate are 20 percent. This third alternative is equivalent to a flat 20 percent tariff rate on all import volumes. The model includes three source countries (or groups of countries) for the products of the industry: domestic production ( $d$ ), subject imports ( $s$ ) and imports from the rest of the world ( $r$ ).
We simulate the effects of the policies on the volumes of subject imports and domestic production using the specific model inputs listed in Table 1. To illustrate the differences among the policy alternatives, we make several assumptions about market shares and elasticities. We assume that domestic producers have a 60 percent market share, while subject imports have a 30 percent share. We assume that the total size of the market is 100 units, while the quota volume is only 5 units. Finally, we assume that the elasticity of substitution among varieties from different sources is 4.
Table 1: Inputs and Estimated Outcomes in the Armington CES PE Model

Policy 1 
Policy 2 
Policy 3 
Model Inputs 



Elasticity of Substitution ( $\sigma $ ) 
4 
4 
4 
Market Share $\u2013$ Subject Imports ( ${m}_{s}$ ) 
0.30 
0.30 
0.30 
Market Share $\u2013$ Domestic Product ( ${m}_{d}$ ) 
0.60 
0.60 
0.60 
Market Share $\u2013$Imports from the Rest of the World ( ${m}_{r}$ ) 
0.10 
0.10 
0.10 
InQuota Tariff Rate on Subject Imports ( ${\tau}_{s}^{in})$ 
0.0 
0.0 
0.2 
OutofQuota Tariff Rate on Subject Imports ( ${\tau}_{s}^{out})$ 
0.2 
0.3 
0.2 
Quota Volume ( ${q}_{s}^{*}$ ) 
5 
5 
5 
Total Market Size ( $Y$ ) 
100 
100 
100 




Estimated Outcomes 



Volume of subject imports under the TRQ 
16.56 
12.56 
16.56 
Percentage change in the volume of subject imports 
44.80 
58.35 
44.80 
Percentage change in the volume of domestic production 
14.47 
19.54 
14.47 
In all three policy scenarios, the TRQ fills (the volume of subject imports under the TRQ is higher than the quota volume in the TRQ), and the policy reduces the volume of subject imports while increasing domestic production. The inquota rate and the quota volume have no effect on ${q}_{s}$ or ${q}_{d}$, because they do not affect the volume of subject imports. In this model, the TRQ has the same effect on the volume of imports and domestic production as a flat tariff rate set at the TRQ’s outofquota rate. This is illustrated by the equivalence of outcomes in the first and third policy scenarios in Table 1. These two policy scenarios have different implications for the amount of tariff revenue collected, but they have identical implications for the volumes of subject imports or domestic production. In contrast, the higher outofquota rate in the second policy scenario results in a greater reduction in the volume of subject imports and a greater increase in domestic production.
In the second model, there are ${n}_{j}$ different firms from source country $j$. Each firm produces a unique variety. Again,
the model includes three source countries (or groups of countries) that supply
the products of the industry to the domestic market. There are ${n}_{s}$ varieties of subject imports, ${n}_{d}$ varieties of domestic product, and ${n}_{r}$ varieties of imports from the rest of the
world. In this monopolistic competition model, there is entry until profits are
driven to zero, and ${n}_{s}$, ${n}_{d}$ and ${n}_{r}$ are endogenously determined within the model.
The price of producer $j$ is a constant markup over its constant marginal cost ${c}_{j}$.
${p}_{j}=\left(\frac{\sigma}{\sigma 1}\right){c}_{j}$ (5)
The industry price index reflects the number of varieties available in the market.
$P={\left({\displaystyle \sum}_{j}{n}_{j}{b}_{j}{\left({p}_{j}\left(1+{\tau}_{j}\right)\right)}^{1\sigma}\right)}^{\frac{1}{1\sigma}}$ (6)
Equation (3) is the demand function for the sum of all ${n}_{j}$ suppliers from source country $j$.
${q}_{j}={n}_{j}Y{b}_{j}{P}^{\sigma 1}{\left({p}_{j}\left(1+{\tau}_{j}\right)\right)}^{\sigma}$ (7)
Equation (4) is the profits that a firm from source country $j$ earns from selling in the domestic market.
${\pi}_{j}=\frac{1}{\sigma {n}_{j}}Y{b}_{j}{P}^{\sigma 1}{\left({p}_{j}\left(1+{\tau}_{j}\right)\right)}^{1\sigma}{f}_{j}+T{S}_{j}\text{}$ (8)
The variable ${f}_{j}$ is a fixed cost of serving the market from source country $j$, and $T{S}_{j}$ is each supplier’s tariff savings at the outofquota rate in the TRQ, relative to its tariff costs if there were a flat tariff at the outofquota rate in the TRQ. Initially, there is no TRQ, and $T{S}_{s}$ is equal to zero. $T{S}_{d}$ and $T{S}_{r}$ are equal to zero in all of the policy scenarios. Under the TRQ:
$T{S}_{s}=\left({\tau}_{s}^{out}{\tau}_{s}^{in}\right){p}_{s}\frac{{q}_{s}^{*}}{{n}_{s}}$ (9)
Equation (9) assumes that the import quota is allocated equally among the ${n}_{s}$ firms that supply subject imports.
Finally, equation (10) calibrates the preference parameters if the initial prices in the model are set equal to one.
${b}_{j}=\frac{{m}_{j}}{{n}_{j0}}$ (10)
The variable ${n}_{j0}$ is the initial number of suppliers from source country $j$. In order to maximize comparability to the Armington CES model, we assume that ${n}_{d0}={n}_{s0}={n}_{r0}=1$.
We solve for the change in the volume of subject imports and domestic production due to the TRQ, holding $Y$ and ${c}_{j}$ fixed, by solving for the new equilibrium number of firms ${n}_{s}$, ${n}_{d}$ and ${n}_{r}$ based on the zero profit conditions defined by equations (5), (6), (8) and (9). The model inputs in Table 2 are identical to the values in Table 1.
Table 2: Inputs and Estimated Outcomes in the Krugman PE Model

Policy 1 
Policy 2 
Policy 3 
Model Inputs 



Elasticity of Substitution ( $\sigma $ ) 
4 
4 
4 
Market Share $\u2013$ Subject Imports ( ${m}_{s}$ ) 
0.30 
0.30 
0.30 
Market Share $\u2013$ Domestic Product ( ${m}_{d}$ ) 
0.60 
0.60 
0.60 
Market Share $\u2013$Imports from the Rest of the World ( ${m}_{r}$ ) 
0.10 
0.10 
0.10 
InQuota Tariff Rate on Subject Imports ( ${\tau}_{s}^{in})$ 
0.0 
0.0 
0.2 
OutofQuota Tariff Rate on Subject Imports ( ${\tau}_{s}^{out})$ 
0.2 
0.3 
0.2 
Quota Volume ( ${q}_{s}^{*}$ ) 
5 
5 
5 
Total Market Size ( $Y$ ) 
100 
100 
100 




Estimated Outcomes 



Volume of subject imports under the TRQ 
9.70 
6.87 
7.84 
Percentage change in the volume of subject imports 
56.90 
69.47 
65.14 
Percentage change in the volume of domestic production 
20.69 
25.85 
24.93 
Again the TRQ fills, the volume of subject imports declines, and domestic production increases in all three policy scenarios. In this second model, however, there are adjustments in the number of firms participating in the market, often called the extensive margin of trade, and larger absolute changes in the volumes of subject imports and domestic production. Once we incorporate these adjustments on the extensive margin of trade, there are important new distinctions between the policy alternatives. In this monopolistic competition model, the inquota tariff rate and the quota volume affect ${q}_{s}$ or ${q}_{d}$, because they affect the number of market participants. A low inquota rate transfers inframarginal tariff revenues to the foreign producers. The TRQ no longer has the same effect on the volume of imports and domestic production as a flat tariff set at the TRQ’s outofquota rate, so the first and third policy scenarios are no longer equivalent. These two policy scenarios have different implications for the amount of tariff revenue collected, but also for the volume of subject imports. The reduction in the volume of subject imports is much larger for a flat 20 percent tariff than for the TRQ with an inquote rate of 0 percent. The increase in the domestic production is almost as large from raising the inquota rate from 0 percent to 20 percent (switching from the first policy scenario to the third policy scenario) as it is from raising the outofquota rate from 20 percent to 30 percent while keeping the inquota rate at 0 percent (switching from the first policy scenario to the second policy scenario).
We have assessed the effects of a TRQ on imports and domestic production using two different types of partial equilibrium models, an Armington CES model and a Krugman CES model of trade. In the Armington model with only adjustment on the intensive margin of trade, a TRQ that fills has the same effect on trade as a flat tariff at the outofquota rate. In the Krugman model, on the other hand, the two policies are not equivalent and the inquota rate has an effect on trade and domestic production even when the TRQ fills.
A next step would be to consider a Melitz (2003) model of trade with heterogeneity in the firms’ variable costs of supplying the market. In a Melitz model, there is adjustment on the extensive margin of trade, so the impact on subject import volumes and domestic production will depend on all of the parameters of the TRQ, like in the Krugman model, and not only on the outofquota rate. However, now there is an additional complexity: the impact on subject import volumes and domestic production will depend on the allocation of the TRQ. It will depend on whether the dutyfree quota volume is available to any importer, on a firstcomefirstserved basis, allocated to the more productive firms who would serve the foreign markets anyway, or allocated to the marginal firms who would only serve the foreign markets if they can enter dutyfree.
Armington, Paul S. (1969): “A Theory of Demand for Products Differentiated by
Place of Production.” Staff Papers
(International Monetary Fund) 16(1): 15978.
Krugman, P. (1980): “Scale Economies, Product Differentiation, and the Pattern of Trade.” American Economic Review 70(5): 950959.
Melitz, Marc J. (2003): “The Impact of Trade on Aggregate Industry Productivity and IntraIndustry Reallocations.” Econometrica 71: 16951725.