Optimality in the Armington, Krugman and Melitz Models

Abstract

Every student of welfare economics is aware of propositions suggesting, perhaps with caveats, that free trade (zero tariffs) in a world of pure competition generates a Pareto optimal or efficient outcome. In this chapter we investigate what can be said about efficiency in the worlds of Krugman and Melitz where industries are monopolistically competitive with prices exceeding marginal costs. We find that the complications introduced by Krugman and Melitz do not prevent free trade from delivering intra-sectoral efficiency: under free trade a Melitz worldwide widget industry satisfies any given levels of widget demands across countries with cost-minimizing worldwide selections of firms, output per firm and trade volumes. However, monopolistic competition in some industries combined with pure competition in others introduces inter-sectoral inefficiency, with possibilities for Pareto improvements by allocating resources away from industries that are purely competitive toward those that are monopolistically competitive. Working with the Dixit–Stiglitz model we show that inter-sectoral welfare costs associated with mixed market structures (pure and monopolistic competition) are likely to be small in an empirical CGE setting. Conclusions reached in this chapter concerning intra and inter-sectoral efficiency are helpful for interpreting results from CGE simulations with Armington, Krugman and Melitz features. For example, Melitz intra-sectoral efficiency with zero tariffs means that envelope theorems are applicable. As we will see in Chaps. 6 and 7, this helps us to understand the circumstances under which Melitz and Armington models produce similar welfare results for the effects of tariff changes.

Appendix 3.1: Equivalence Between Worldwide Cost Minimizing and the AKME Model

Proof of proposition ( 3.22 ): ${\mathbf{Cost}} \, {\mathbf{minimizing}} \, \Rightarrow {\mathbf{AKME}} \, {\mathbf{with}} \, {\mathbf{zero}} \, {\mathbf{tariffs}}$

Let $\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}}$, N_s, Q_ksd and $\Lambda _{\text{d}}$ be a solution to (3.25)–(3.29) for given values of the exogenous variables W_s, and Q_d. Let P_d and P_ksd be defined by (3.30) and (3.31) and define Q_sd, Π_ksd, Πtot_s and L_s as in (T2.4)–(T2.7) of the AKME model. We show that $\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}}$, N_s, Q_ksd, P_d, P_ksd, Q_sd, Π_ksd, Πtot_s and L_s then satisfy the remaining AKME equations, (T2.1)–(T2.3) and (T2.8)–(T2.10), and is therefore an AKME solution.

Equations (T2.8) is satisfied: (T2.8) is the same as (3.26).

Under (3.21) and with zero tariffs (T_sd = 1), (T2.1) is the same as (3.31).

From (3.29)–(3.31) we have

$$ {\text{P}}_{\text{ksd}} = {\text{P}}_{\text{d}} {\text{Q}}_{\text{d}}^{{1/\upsigma}}\updelta_{\text{sd}} {\text{Q}}_{\text{ksd}}^{{ - 1/\upsigma}} ,{\text{k}} \in {\text{S}}({\text{s}},{\text{d}}). $$

(3.32)

Hence

$$ {\text{Q}}_{\text{ksd}} =\updelta_{\text{sd}}^{\upsigma} {\text{Q}}_{\text{d}} \left( {\frac{{{\text{P}}_{\text{d}} }}{{{\text{P}}_{\text{ksd}} }}} \right)^{\upsigma} \quad {\text{k}} \in {\text{S}}({\text{s}},{\text{d}}). $$

(3.33)

Under (3.21) this establishes (T2.3).

From (3.27)

$$ - {\text{W}}_{\text{s}} \left( {\frac{{{\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} }}{{\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} }} + {\text{F}}_{\text{sd}} } \right) +\Lambda _{\text{d}}\updelta_{\text{sd}} {\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}}^{{(\upsigma - 1)/\upsigma}} = 0. $$

(3.34)

Combining (3.30) and (3.33) gives

$$ {\text{P}}_{\text{ksd}} =\Lambda _{\text{d}}\updelta_{\text{sd}} {\text{Q}}_{\text{ksd}}^{{ - 1/\upsigma}} . $$

(3.35)

In particular

$$ {\text{P}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} =\Lambda _{\text{d}}\updelta_{\text{sd}} {\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}}^{{ - 1/\upsigma}} . $$

(3.36)

Putting (3.36) into (3.34) gives

$$ - {\text{W}}_{\text{s}} \left( {\frac{{{\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} }}{{\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} }} + {\text{F}}_{\text{sd}} } \right) + {\text{P}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} {\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} = 0, $$

(3.37)

establishing (T2.10) via (T2.5) with T_sd equals one.

From (3.28) and (3.35) we obtain

$$ \left[ {{\text{W}}_{\text{s}} \sum\limits_{\text{d}} {\sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {{\text{g}}_{\text{s}} (\Phi _{\text{k}} )*\left( {\frac{{{\text{Q}}_{\text{ksd}} }}{{\Phi _{\text{k}} }} + {\text{F}}_{\text{sd}} } \right)} } } \right] + {\text{W}}_{\text{s}} {\text{H}}_{\text{s}} - \sum\limits_{\text{d}} {\sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {\frac{{{\text{P}}_{\text{ksd}} {\text{Q}}_{\text{ksd}}^{{1/\upsigma}} }}{{\updelta_{\text{sd}} }}{\text{g}}_{\text{s}} (\Phi _{\text{k}} )\updelta_{\text{sd}} {\text{Q}}_{\text{ksd}}^{{(\upsigma - 1)/\upsigma}} } } = 0. $$

(3.38)

Simplifying, rearranging and multiplying through by N_s gives

$$ \left[ {{\text{W}}_{\text{s}} \sum\limits_{\text{d}} {\sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {{\text{N}}_{\text{s}} {\text{g}}_{\text{s}} (\Phi _{\text{k}} )*\left( {\frac{{{\text{Q}}_{\text{ksd}} }}{{\Phi _{\text{k}} }} + {\text{F}}_{\text{sd}} } \right)} } } \right] + {\text{N}}_{\text{s}} {\text{W}}_{\text{s}} {\text{H}}_{\text{s}} = \sum\limits_{\text{d}} {\sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {{\text{N}}_{\text{s}} {\text{g}}_{\text{s}} (\Phi _{\text{k}} ){\text{P}}_{\text{ksd}} {\text{Q}}_{\text{ksd}} .} } $$

(3.39)

Via (T2.5) with T_sd equals one and (T2.6), (3.39) leads to (T2.9).

Now all that remains is to establish (T2.2). We start by rearranging (3.33) as

$$ \updelta_{\text{sd}}^{\upsigma} {\text{P}}_{\text{ksd}}^{{1 -\upsigma}} = {\text{Q}}_{\text{ksd}}^{{(\upsigma - 1)/\upsigma}}\updelta_{\text{sd}} {\text{Q}}_{\text{d}}^{{(1 -\upsigma)/\upsigma}} {\text{P}}_{\text{d}}^{{1 -\upsigma}} . $$

(3.40)

Then multiplying through by N_sg_s(Φ_k), aggregating over s and k, and using (3.25) we obtain (T2.2) under assumption (3.21).

Proof of proposition ( 3.23 ):

$$ {\mathbf{AKME}}\;{\mathbf{with}} \, {\mathbf{zero}} \, {\mathbf{tariffs}} \Rightarrow {\mathbf{first}}{ - }{\mathbf{order}} \, {\mathbf{optimality}} \, {\mathbf{conditions}} \, {\mathbf{for}} \, {\mathbf{cost}} \, {\mathbf{minimizing}} $$

Let $\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}}$, N_s, Q_ksd, P_d, P_ksd, Q_sd, Π_ksd, Πtot_s and L_s satisfy (T2.1) to (T2.10) for given values of the exogenous variables W_s and Q_d and with T_sd equals one for all s,d. Define $\Lambda _{\text{d}}$ by (3.30). We show that $\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}}$, N_s, Q_ksd and $\Lambda _{\text{d}}$ is a solution to (3.25)–(3.29).

Condition (3.26) is the same as (T2.8).

Under (3.21), (T2.3) gives

$$ \updelta_{\text{sd}} {\text{Q}}_{\text{ksd}}^{{(\upsigma - 1)/\upsigma}} = {\text{Q}}_{\text{d}}^{{(\upsigma - 1)/\upsigma}}\updelta_{\text{sd}}^{\upsigma} {\text{P}}_{\text{d}}^{{\upsigma - 1}} {\text{P}}{}_{\text{ksd}}^{{1 -\upsigma}} . $$

(3.41)

Multiplying through by ${\text{N}}_{\text{s}} {\text{g}}_{\text{s}} (\Phi _{\text{k}} )$, summing over all s and all ${\text{k}} \in {\text{S}}({\text{s}},{\text{d}})$ and using (T2.2) and (3.21) gives (3.25).

Equation (T2.5) with T_sd = 1 and (T2.10) give

$$ {\text{P}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} {\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} - \left( {\frac{{{\text{W}}_{\text{s}} }}{{\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} }}} \right){\text{Q}}_{\text{min(s,d)}} - {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} = 0. $$

(3.42)

To establish (3.27) we need to eliminate P_min(s,d) and introduce $\Lambda _{\text{d}}$. We do this via (T2.3), (3.21) and (3.30) which give

$$ {\text{P}}_{\text{ksd}} =\updelta_{\text{sd}}\Lambda _{\text{d}} {\text{Q}}_{\text{ksd}}^{{ - 1/\upsigma}} $$

(3.43)

and, in particular

$$ {\text{P}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} =\updelta_{\text{sd}}\Lambda _{\text{d}} {\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}}^{{ - 1/\upsigma}} . $$

(3.44)

Multiplying (3.42) through by ${\text{N}}_{\text{s}} {\text{g}}_{\text{s}} (\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} )$ and using (3.44) quickly leads to (3.27).

From (T2.5) with T_sd = 1, (T2.6) and (T2.9) we obtain

$$ \sum\limits_{\text{d}} {\sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {{\text{g}}_{\text{s}} (\Phi _{\text{k}} )\left[ {{\text{P}}_{\text{ksd}} {\text{Q}}_{\text{ksd}} - \left( {\frac{{{\text{W}}_{\text{s}} }}{{\Phi _{\text{k}} }}} \right){\text{Q}}_{\text{ksd}} - {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} } \right]} } - {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} = 0. $$

(3.45)

Then, substituting from (3.43) gives (3.28).

To obtain (3.29), we start from (3.43), multiply through by ${\text{N}}_{\text{s}} {\text{g}}_{\text{s}} (\Phi _{\text{k}} )$ and use (T2.1) with T_sd = 1 and (3.21).