Converting an Armington Model into a Melitz Model: Giving Melitz Sectors to GTAP

Abstract

This chapter describes how to convert an existing Armington CGE model into a Melitz CGE model with minimal changes to the original Armington model. The main task is to add equations to the bottom of the Armington model to form what we call an Armington-to-Melitz or A2M system. With an A2M system, industries can be switched between Armington and Melitz treatments by closure swaps. We use BasicArmington (a simple Armington model) to explain how to create an A2M system. Then we apply the method to a 10-region, 57-commodity version of the frequently applied policy-oriented GTAP model to create a GTAP-A2M system. Using this system, we compare the effects under Armington and Melitz assumptions of a tariff imposed by North America on imports of wearing apparel (Wap). To facilitate the comparison, we decompose the welfare effects for each region into parts attributable to changes in employment, terms of trade and scale-related efficiency. This helps us to understand how each of these factors operates under Armington and Melitz, but it does not give us an intuitive explanation of their net outcome. To explain net outcomes for welfare effects by region we set out an intuitive overarching theory. We check its validity by back-of-the-envelope (BOTE) calculations using GTAP data items and selected simulation results. BOTE calculations enable us to cut through the maze of complications in CGE models to locate, for any specific result, the essential underlying ingredients.

Appendices

Appendix 7.1: GEMPACK Program for an A2M System: Solving BasicArmington and MelitzGE

This appendix contains GEMPACK code for the A2M system that can generate solutions for the BasicArmington model and the MelitzGE model. The code, together with zipped simulations from Sect. 7.4 can be downloaded using the Electronic Supplementary Material, see footnote at the opening page of this chapter. A2M closures for the BasicArmington and MelitzGE models are at the end of the appendix.

Sections A7.1.1–A7.1.6 is the code for the BasicArmington model, including a welfare decomposition. These sections can be thought of as representing a typical stand-alone Armington model. Sections A7.1.7–A7.1.11 are the additional material required for the conversion from Armington to Melitz. In the case of the BasicArmington model, the inclusion of the additional material requires no alteration to the code of the Armington model.

Being able to convert from Armington to Melitz without altering the Armington model has important practical advantages. When this is possible, it means that we can perform Melitz computations starting from an Armington model, which may be large and complex, without having to undertake time-consuming work in understanding all of the details of the original model or having to reconfigure it. However, in practical applications, alterations of the Armington code may be required. For example, if the original Armington model does not include suitable factory-door taxes [txMel_sd,c], these need to be added. Fortunately, our experience reported in Sect. 7.5 in creating an A2M system for GTAP suggests that the alterations to the original Armington model will usually be minimal.

• GEMPACK code

• Closures

Table 7.9 lists in GEMPACK code two closures (exogenous variables) for the A2M system: one for generating solutions to the BasicArmington model and the other for generating solutions for MelitzGE. The relationship between the two closures is discussed in Sect. 7.2.

Table 7.9 Closures for A2M system: solving the BasicArmington model or MelitzGE^a

Appendix 7.2: Another Decomposition of Welfare for Armington and Melitz

In this appendix we derive the welfare decomposition set out in Eq. (7.29). The derivation uses the equations in Table 7.1 for the BasicArmington model. Notation is given in Table 7.2. The derivation is similar to that in Appendix 6.4. However, there are differences, e.g. new variables, aaMel and txMel, that were not in the earlier decomposition. Then we introduce Melitz equations to derive (7.35), (7.36) and (7.37).

• Derivation of ( 7.29 )

We define the percentage change in welfare for country d as

$$ {\text{welfare}}_{\text{d}} = \sum\limits_{\text{c}} {{\text{MU}}_{\text{d,c}} } * {\text{q}}_{\text{d,c}} $$

(7.38)

where MU_d,c is the share of d’s household expenditure that is devoted to commodity c. We assume fmu_d in (T7.9) of Table 7.1 is zero. Then

$$ {\text{welfare}}_{\text{d}} = \sum\limits_{\text{c}} {{\text{MU}}_{{{\text{d}},{\text{c}}}} } *\left( {{\text{gdp}}_{\text{d}} - {\text{p}}_{{{\text{d}},{\text{c}}}} } \right) $$

(7.39)

Because the MU_d,c’s sum to one over c, (7.39) can be written as

$$ {\text{welfare}}_{\text{d}} = {\text{gdp}}_{\text{d}} - \sum\limits_{\text{c}} {{\text{MU}}_{{{\text{d}},{\text{c}}}} } *{\text{p}}_{{{\text{d}},{\text{c}}}} $$

(7.40)

Combining (7.40), (T7.13) and (T7.7) gives

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\left[ {{\text{w}}_{\text{d}} + \ell {\text{tot}}_{\text{d}} } \right] + 100*\sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{drevm}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + 100*\sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{drevx}}_{{{\text{ds}},{\text{c}}}} } } \\ & \quad - {\text{GDP}}_{\text{d}} *\sum\limits_{\text{c}} {{\text{MU}}_{{{\text{d}},{\text{c}}}} } *\left[ {\sum\limits_{\text{s}} {{\text{SHA}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{p}}_{{ \bullet {\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} } \right]} } \right] \\ \end{aligned} $$

(7.41)

Substituting from (T7.14) and (T7.15) into (7.41) leads to

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\left[ {{\text{w}}_{\text{d}} + \ell {\text{tot}}_{\text{d}} } \right] \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left[ {\begin{array}{*{20}c} {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{qcount}}_{{{\text{sd}},{\text{c}}}} + {\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} + {\text{tm}}_{{{\text{sd}},{\text{c}}}} } \right]} \\ { - {\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{qcount}}_{{{\text{sd}},{\text{c}}}} + {\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } \right]} \\ \end{array} } \right]} } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left[ {\begin{array}{*{20}c} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*\left[ {{\text{qcount}}_{{{\text{ds}},{\text{c}}}} + {\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} + {\text{txMel}}_{{{\text{ds}},{\text{c}}}} } \right]} \\ { - {\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*\left[ {{\text{qcount}}_{{{\text{ds}},{\text{c}}}} + {\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} } \right]} \\ \end{array} } \right]} } \\ & \quad - {\text{GDP}}_{\text{d}} *\sum\limits_{\text{c}} {{\text{MU}}_{{{\text{d}},{\text{c}}}} } *\left[ {\sum\limits_{\text{s}} {{\text{SHA}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{p}}_{{ \bullet {\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} } \right]} } \right] \\ \end{aligned} $$

(7.42)

Rearranging we obtain

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\left[ {{\text{w}}_{\text{d}} + \ell {\text{tot}}_{\text{d}} } \right] \\ & \quad { + }\sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TM}}_{{{\text{sd}},{\text{c}}}} - 1} \right)*{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{qcount}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TX}}_{{{\text{ds}},{\text{c}}}} - 1} \right)*{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{qcount}}_{{{\text{ds}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*\left[ {{\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} + {\text{txMel}}_{{{\text{ds}},{\text{c}}}} } \right]} } \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} } } \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } } + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left[ {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{tm}}_{{{\text{sd}},{\text{c}}}} } \right]} } \\ & \quad - {\text{GDP}}_{\text{d}} *\sum\limits_{\text{c}} {{\text{MU}}_{{{\text{d}},{\text{c}}}} } *\left[ {\sum\limits_{\text{s}} {{\text{SHA}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{p}}_{{ \bullet {\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} } \right]} } \right] \\ \end{aligned} $$

(7.43)

In BasicArmington there are no transport costs and no intermediate inputs. Consequently,

$$ {\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORY}}({\text{d}},{\text{s}},{\text{c}}) = {\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}}) = {\text{VCIF}}({\text{d}},{\text{s}},{\text{c}}) $$

(7.44)

and

$$ {\text{W}}_{\text{d}} {\text{LTOT}}_{\text{d}} = \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})} } $$

(7.45)

Working with these relationships and using (T7.1)–(T7.4) we reach:

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\ell {\text{tot}}_{\text{d}} \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TM}}_{{{\text{sd}},{\text{c}}}} - 1} \right)*{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{qcount}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TX}}_{{{\text{ds}},{\text{c}}}} - 1} \right)*{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{qcount}}_{{{\text{ds}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} } }* {\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{pfob}}_{{{\text{ds}},{\text{c}}}} \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{aoMel}}_{{{\text{d}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{tm}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad - {\text{GDP}}_{\text{d}} *\sum\limits_{\text{c}} {{\text{MU}}_{{{\text{d}},{\text{c}}}} } *\sum\limits_{\text{s}} {{\text{SHA}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{p}}_{{ \bullet {\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} } \right]} \\ \end{aligned} $$

(7.46)

Now we work on the last term of (7.46). We assume that country d consumes all of its GDP (zero trade deficit). Under this assumption ${\text{GDP}}_{\text{d}} *{\text{MU}}_{{{\text{d}},{\text{c}}}} *{\text{SHA}}({\text{s}},{\text{d}},{\text{c}}) = {\text{VPUR}}({\text{s}},{\text{d}},{\text{c}})$ where VPUR(s,d,c) is household expenditure in d on commodity c sent from s. Then using (T7.5) we obtain:

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\ell {\text{tot}}_{\text{d}} \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TM}}_{{{\text{sd}},{\text{c}}}} - 1} \right)*{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{qcount}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TX}}_{{{\text{ds}},{\text{c}}}} - 1} \right)*{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{qcount}}_{{{\text{ds}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} } }* {\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{pfob}}_{{{\text{ds}},{\text{c}}}} \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{aoMel}}_{{{\text{d}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{VPUR}}({\text{s}},{\text{d}},{\text{c}})*{\text{aaMel}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{tm}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{VPUR}}({\text{s}},{\text{d}},{\text{c}})*\left[ {{\text{pcif}}_{{ \bullet {\text{sd}},{\text{c}}}} + {\text{tm}}_{{{\text{sd}},{\text{c}}}} } \right]} } \\ \end{aligned} $$

(7.47)

Because ${\text{TM}}_{{{\text{sd}},{\text{c}}}} *{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}}) = {\text{VPUR}}({\text{s}},{\text{d}},{\text{c}})$, the last three terms in (7.47) contribute zero. Leaving these terms out and using (T7.8) and (T7.1)–(T7.4), we find that

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\ell {\text{tot}}_{\text{d}} + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TM}}_{{{\text{sd}},{\text{c}}}} - 1} \right)*{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{q}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} } }* {\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*[{\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} + {\text{txMel}}_{{{\text{ds}},{\text{c}}}} - {\text{aaMel}}_{{{\text{ds}},{\text{c}}}} ] \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*[{\text{pfactory}}_{{ \bullet {\text{s}},{\text{c}}}} } } + {\text{txMel}}_{{{\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} ] \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{aoMel}}_{{{\text{d}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{aaMel}}_{{{\text{ds}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TX}}_{{{\text{ds}},{\text{c}}}} - 1} \right)*{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{{\text{ds}},{\text{c}}}} } } \\ \end{aligned} $$

(7.48)

Recognizing that the “diagonal” terms in the second and third lines on the RHS of (7.48) cancel and rearranging we obtain

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\ell {\text{tot}}_{\text{d}} + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TM}}_{{{\text{sd}},{\text{c}}}} - 1} \right) * {\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{q}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{{{\text{s}} \ne {\text{d}}}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} } }* {\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*[{\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} + {\text{txMel}}_{{{\text{ds}},{\text{c}}}} - {\text{aaMel}}_{{{\text{ds}},{\text{c}}}} ] \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{{{\text{s}} \ne {\text{d}}}} {{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*[{\text{pfactory}}_{{ \bullet {\text{s}},{\text{c}}}} } } + {\text{txMel}}_{{{\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} ] \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{{\text{ds}},{\text{c}}}} } } \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*[{\text{q}}_{{{\text{ds}},{\text{c}}}} } } - {\text{aaMel}}_{{{\text{ds}},{\text{c}}}} - {\text{aoMel}}_{{{\text{d}},{\text{c}}}} ] \\ \end{aligned} $$

(7.49)

Using (T7.8), (T7.12) and (T7.10) and rearranging gives:

$$ \begin{aligned} {\text{GDP}}_{\text{d}} *{\text{welfare}}_{\text{d}} & = {\text{W}}_{\text{d}} *{\text{LTOT}}_{\text{d}} *\ell {\text{tot}}_{\text{d}} + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TM}}_{{{\text{sd}},{\text{c}}}} - 1} \right)*{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*{\text{q}}_{{{\text{sd}},{\text{c}}}} } } \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{{{\text{s}} \ne {\text{d}}}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} } } *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*[{\text{pfactory}}_{{ \bullet {\text{d}},{\text{c}}}} + {\text{txMel}}_{{{\text{ds}},{\text{c}}}} - {\text{aaMel}}_{{{\text{ds}},{\text{c}}}} ] \\ & \quad - \sum\limits_{\text{c}} {\sum\limits_{{{\text{s}} \ne {\text{d}}}} {{\text{VCIF}}({\text{s}},{\text{d}},{\text{c}})*[{\text{pfactory}}_{{ \bullet {\text{s}},{\text{c}}}} } } + {\text{txMel}}_{{{\text{sd}},{\text{c}}}} - {\text{aaMel}}_{{{\text{sd}},{\text{c}}}} ] \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*[{\text{q}}_{{{\text{ds}},{\text{c}}}} } } - \ell_{{{\text{d}},{\text{c}}}} ] \\ & \quad + \sum\limits_{\text{c}} {\sum\limits_{\text{s}} {\left( {{\text{TX}}_{{{\text{ds}},{\text{c}}}} - 1} \right){\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*} } \ell_{{{\text{d}},{\text{c}}}} \\ \end{aligned} $$

(7.50)

Then using (7.30), (T7.2) and (T7.3) we quickly arrive at (7.29).

• Derivation of ( 7.35 )–( 7.37 )

If industry c is treated as Armington, then under standard assumptions aoMel_d,c and aaMel_sd,c are zero and TX_sd,c equals one. Under these conditions, the c component of the combined last two terms in (7.49), and consequently in (7.50), is zero. Following the notation introduced in (7.35), these two terms can now be written as

$$ \begin{aligned} \Delta {\text{Efficiency}}_{\text{d}} & = \sum\limits_{{{\text{c}} \in {\text{M}}}} {\sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*[{\text{q}}_{{{\text{ds}},{\text{c}}}} - \ell_{{{\text{d}},{\text{c}}}} ]} } \\ & \quad + \sum\limits_{{{\text{c}} \in {\text{M}}}} {\sum\limits_{\text{s}} {\left( {{\text{TX}}_{{{\text{ds}},{\text{c}}}} - 1} \right){\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*\ell_{{{\text{d}},{\text{c}}}} } } \\ \end{aligned} $$

(7.51)

where M is the set of Melitz industries. Because for Melitz industries the collection of factory-door taxes is fixed on zero, the second term on the RHS of (7.51) is zero. Then using

$$ {\text{W}}_{\text{d}} {\text{L}}_{\text{d,c}} = \sum\limits_{\text{s}} {{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})} $$

(7.52)

we establish (7.35):

$$ \Delta {\text{Efficiency}}_{\text{d}} = \sum\limits_{{{\text{c}} \in {\text{M}}}} {[\{ \sum\limits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{{\text{ds}},{\text{c}}}} } } \} - {\text{W}}_{\text{d}} {\text{L}}_{{{\text{d}},{\text{c}}}} *\ell {}_{{{\text{d}},{\text{c}}}}] $$

(7.53)

In deriving (7.36) and (7.37) we restrict attention to Melitz industries, c ∈ M. Thus, we assume that the shift variables in (T7.19) and (T7.20) are zero. We start the derivation with (T7.20). Writing this equation from the point of view of country d, cancelling out the wage terms, and assuming that h_d,c and f_ds,c are zero, we obtain¹⁸

$$ \begin{aligned} & {\text{W}}_{\text{d}} *{\text{L}}_{{{\text{d}},{\text{c}}}} *\ell_{{{\text{d}},{\text{c}}}} \\ & \quad = \left[ {{{(\upsigma - 1)} \mathord{\left/ {\vphantom {{(\upsigma - 1)}\upsigma}} \right. \kern-0pt}\upsigma}} \right]*\sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{n}}_{{{\text{ds}},{\text{c}}}} } \\ & \quad + \left[ {{{(\upsigma - 1)} \mathord{\left/ {\vphantom {{(\upsigma - 1)} {(\upalpha \upsigma )}}} \right. \kern-0pt} {(\upalpha \upsigma )}}} \right]*\sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{n}}_{{{\text{d}},{\text{c}}}} } \\ & \quad + \left[ {{{(\upalpha - (\upsigma - 1))} \mathord{\left/ {\vphantom {{(\upalpha - (\upsigma - 1))} {(\upalpha \upsigma )}}} \right. \kern-0pt} {(\upalpha \upsigma )}}} \right]*\sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{n}}_{{{\text{ds}},{\text{c}}}} } \\ \end{aligned} $$

(7.54)

From here, (7.36) and (7.37) can be derived directly from Table 7.1 by a series of substitutions into (7.54). However, an alternative and instructive approach is to use a key result from Appendix 4.1: namely that costs in industry d,c are split in fixed proportions between production, set up of firms and setup on links. Noting that $\sum\nolimits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})} = \sum\nolimits_{\text{s}} {{\text{TX}}_{{{\text{ds}},{\text{c}}}} *{\text{FACTORYV}}({\text{d}},{\text{s}},{\text{c}})} = {\text{W}}_{\text{d}} {\text{L}}_{{{\text{d}},{\text{c}}}}$, and using the fixed-split result we see from (7.54) that

$$ \ell_{{{\text{d}},{\text{c}}}} { = }\sum\limits_{\text{s}} {\frac{{{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})}}{{{\text{W}}_{\text{d}} {\text{L}}_{{{\text{d}},{\text{c}}}} }}*{\text{n}}_{{{\text{ds}},{\text{c}}}} } = {\text{n}}_{{{\text{d}},{\text{c}}}} $$

(7.55)

Using (T7.21) in (7.55) together with (T7.22)–(T7.24) gives

$$ 0 = \sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*\upphi_{{ \bullet {\text{ds}},{\text{c}}}} } = \sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{ \bullet {\text{ds}},{\text{c}}}} } $$

(7.56)

Via (T7.17), (T7.8) and (7.56),

$$ \begin{aligned} \sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{n}}_{{{\text{ds}},{\text{c}}}} } & = \sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{{\text{ds}},{\text{c}}}} } \\ & \quad - \sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{aaMel}}_{{{\text{ds}},{\text{c}}}} } \\ \end{aligned} $$

(7.57)

Substituting (T7.19) into (7.57) gives

$$ \frac{\upsigma}{{\upsigma - 1}}\sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{n}}_{{{\text{ds}},{\text{c}}}} } = \sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{{\text{ds}},{\text{c}}}} } $$

(7.58)

and then from (7.55)

$$ {\text{W}}_{\text{d}} {\text{L}}_{{{\text{d}},{\text{c}}}} *\ell_{{{\text{d}},{\text{c}}}} { = }\frac{{\upsigma - 1}}{\upsigma}\sum\limits_{\text{s}} {{\text{MARKETV}}({\text{d}},{\text{s}},{\text{c}})*{\text{q}}_{{{\text{ds}},{\text{c}}}} } $$

(7.59)

We obtain (7.36) and (7.37) by substituting from (7.59) into (7.35).

Appendix 7.3: The Theory of GTAP-HET

Akgul, Villoria and Hertel (2016), hereafter AVH, describe what they call GTAP-HET or GTAP with heterogeneous firms. Their aim is to incorporate the Melitz specification of trade with firm heterogeneity into a readily accessible version of the GTAP model. In our view, they fall a little short of the mark.

The most obvious problem is their version of market prices. In Melitz, the price of good c delivered from region s to region d by the typical firm on the sd-link reflects this firm’s productivity. In terms of our BasicMelitz model (the Melitz model derived from the BasicArmington-A2M system) Melitz pricing requires

$$ {\text{pmarket}}_{{ \bullet {\text{sd}},{\text{c}}}} = {\text{w}}_{\text{s}} -\upphi_{{{\text{sd}},{\text{c}}}} \quad {\text{for}}\;{\text{all}}\;{\text{c}},{\text{s}},{\text{d}} , $$

(7.60)

where

${\text{pmarket}}_{{ \bullet {\text{sd}},{\text{c}}}}$ :

is the percentage change in the market price (price just beyond the factory door) of good c from country s destined for country d;

w_s:

is the percentage change in the wage rate in country s (which, in our A2M system, is the cost of a unit of input to production of c in country s); and

ϕ_sd,c:

is the percentage change in the marginal productivity of the typical c-producing firm operating on the sd-link.

Instead of (7.60), AVH assume that¹⁹

$$ {\text{pmarket}}_{{ \bullet {\text{sd}},{\text{c}}}} {\text{ = w}}_{\text{s}} - \left( {\frac{{\upsigma - 1}}{\upsigma}} \right)*\upphi{\text{ave}}_{{{\text{s}},{\text{c}}}} \quad {\text{for}}\;{\text{all}}\;{\text{c}},{\text{s}},{\text{d}} $$

(7.61)

where ϕave_s,c is the percentage change in the marginal productivity of the average c-producing firm in country s. This is defined more precisely in what follows. For now, the most important point is that the right hand side of (7.61) doesn’t depend on the destination d. The coefficient (σ − 1)/σ is the share of variable costs in the total costs of industry s,c.²⁰ It appears on the right hand side of (7.61) presumably because changes in ϕave_s,c operate on variable costs per unit of output, not total costs.

Equation (7.61) is inconsistent with Melitz theory. But does this matter? To investigate this issue we produced a version of BasicMelitz in which AVH’s equation (7.61) replaces (7.60). To do this we started from the Melitz closure in Table 7.2. In this closure the percentage changes in the powers of export taxes [txMel_sd,c] are endogenously determined by equation (T7.18), while the shift variables [ftxMel_sd,c] in that equation are exogenous. Now, to move to the AVH version of the determination of market prices, we switched the closure so that txMel_sd,c becomes exogenous and ftxMel_sd,c becomes endogenous. With txMel_sd, set on zero, equations (T7.1) and (T7.2) in the A2M system collapse to

$$ {\text{pmarket}}_{{ \bullet {\text{sd}},{\text{c}}}} = {\text{w}}_{\text{s}} - {\text{aoMel}}_{{{\text{s}},{\text{c}}}} \quad {\text{for}}\,{\text{all}}\,{\text{c}},{\text{s}},{\text{d}}, $$

(7.62)

By setting aoMel_s,c according to

$$ {\text{aoMel}}_{{{\text{s}},{\text{c}}}} = \left( {\frac{{\upsigma - 1}}{\upsigma}} \right) *\upphi{\text{ave}}_{{{\text{s}},{\text{c}}}} \quad {\text{for}}\,{\text{all}}\,{\text{c}},{\text{s}} $$

(7.63)

we can reproduce AVH pricing.

AVH specify equations to determine their ϕave_s,c. In terms of the variables and equations in Table 7.1, their specification is²¹:

$$ \begin{aligned}\upphi{\text{ave}}_{{{\text{s}},{\text{c}}}} & = \sum\limits_{\text{d}} {\text{SHRSMD}} ({\text{s}},{\text{d}},{\text{c}})*\upphi_{{{\text{sd}},{\text{c}}}} \\ & \quad + \left( {\frac{1}{\sigma - 1}} \right)*\sum\limits_{\text{d}} {{\text{SHRSMD}}({\text{s}},{\text{d}},{\text{c}})*} [{\text{n}}_{{{\text{sd}},{\text{c}}}} - {\text{nt}}_{{{\text{s}},{\text{c}}}} ]\quad {\text{for}}\;{\text{all}}\;{\text{c,s}} \\ \end{aligned} $$

(7.64)

where

SHRSMD(c,s,d):

is the share of destination d in the market value of sales by industry s,c; and

nt_s,c:

is the percentage change in the total number of links being operated by firms in industry s,c.

This is given by:

$$ {\text{nt}}_{{{\text{s}},{\text{c}}}} = \sum\limits_{\text{d}} {\text{SHARE}} ({\text{c}},{\text{s}},{\text{d}})*{\text{n}}_{{{\text{sd}},{\text{c}}}} \quad {\text{for}}\;{\text{all}}\;{\text{c}},{\text{s}} $$

(7.65)

where

SHARE(s,d,c):

is the number of c-producing firms on the sd-link [N(s,d,c)] divided by the total number of links being operated by firms in s,c.

That is,

$$ {\text{SHARE}}({\text{s}},{\text{d}},{\text{c}}) = \frac{{{\text{N}}({\text{s}},{\text{d}},{\text{c}})}}{{\sum\limits_{\text{dd}} {{\text{N}}({\text{s}},{\text{dd}},{\text{c}})} }}\quad {\text{for}}\;{\text{all}}\;{\text{c}},{\text{s}},{\text{d}} $$

(7.66)

Substituting from (7.64) and (7.65) into (7.63) we see that AVH’s model can be implemented in the A2M system by specifying aoMel_s,c according to

$$ \begin{aligned} {\text{aoMel}}_{{{\text{s}},{\text{c}}}} & = \frac{{(\upsigma - 1)}}{\upsigma }*\sum\limits_{\text{d}} {\text{SHRSMD}} ({\text{s}},{\text{d}},{\text{c}})*\upphi_{{{\text{sd}},{\text{c}}}} \\ & \quad + \frac{1}{\upsigma}*\sum\limits_{\text{d}} {{\text{SHRSMD}}({\text{s}},{\text{d}},{\text{c}})} \\&\quad *[{\text{n}}_{{{\text{sd}},{\text{c}}}} - \sum\limits_{\text{k}} {\text{SHARE}} ({\text{c}},{\text{s}},{\text{k}})*{\text{n}}_{{{\text{sk}},{\text{c}}}} ]\quad {\text{for}}\;{\text{all}}\;{\text{c,s}} \\ \end{aligned} $$

(7.67)

In the Melitz closure in Table 7.2, aoMel_s,c was already endogenous. Adding another equation to determine it requires endogenization of a currently exogenous similarly dimensioned variable. This variable has to be dcolrevx_s,c: with all the rates, txMel_sd,c, exogenous as required for the AVH specification of market prices, collection of revenue from destination specific factory-door taxes on s,c firms must be endogenous.

We are now ready to answer the question we posed earlier: Does omitting destination-specific pricing from the Melitz model make much difference? In Table 7.10 we compare Melitz welfare results from Table 7.4 with welfare results computed under AVH assumptions, that is with: txMel_sd,c exogenous on zero; ftxMel_sd,c endogenous; aoMel_s,c specified according to (7.67); dcolrevx_sd,c endogenous; and the rest of the closure the same as Melitz in Table 7.2. The comparison in Table 7.10 strongly suggests that the answer to our question is yes.

Table 7.10 BasicArmington-A2M simulations: Melitz and AVH results for the effects of a 10% tariff imposed by country 2 on all imports from country 1 (percentage changes)

On further investigation we found, under AVH’s treatment of market prices (no destination specificity), that variations in the specification of aoMel_s,c (and hence ϕave_s,c) have no effect on welfare or other variables of economic significance such as q_d,c, gdp_d, and w_s. In fact aoMel_s,c can be treated as exogenous and given any value. Thus, AVH’s use of the variable ϕave_s,c is not only inconsistent with Melitz but its specification via (7.64)–(7.66) has no relevance for AVH’s results.