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.
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- 1.
For Dixit–Stiglitz, R is leisure or rest. It can be thought of as being created under constant returns to scale by using one unit of the labor endowment per unit of its production.
- 2.
In deriving the planner’s solution, we set up the Lagrangian and derived three first order conditions by differentiating with respect to \( {\text{Q}}_{ \bullet } \), N and R. Then we divided the first of these conditions by the second, which gave us (3.14). Next we divided the second condition by the third, which gave us an expression for N * \( {\text{Q}}_{ \bullet } \) + N * H in terms of R. By using this in (3.7) we obtained (3.12), and eventually (3.13).
- 3.
In the Dixit–Stiglitz model, the value of the marginal product of labor in thingamajigs is (σ − 1)/σ times that in widgets.
- 4.
In solving this problem, we set up the Lagrangian and derive two first order conditions by differentiating with respect to N and \( {\text{Q}}_{ \bullet } \). The second of these conditions gives an expression for the Lagrangian multiplier in terms of N. Substituting this into the first condition shows that \( {\text{Q}}_{ \bullet } = \left( {\upsigma - 1} \right)*{\text{H}} = {\text{Q}}_{ \bullet }^{{m}} \). Then (3.18) and (3.19) give N = Nm.
- 5.
Dhingra and Morrow (2012) reach similar conclusions via a different mathematical approach to the one adopted here.
- 6.
If there is a firm in s that is not trading on the sd-link but has productivity greater than or equal to Φmin(s,d), then it is easy to show that costs can be reduced by allowing this firm to trade on the sd-link and reducing the trade flow for a firm with equal or lower productivity.
- 7.
In deriving (3.27) we treat Φk as a continuous variable.
References
Balistreri, E., & Rutherford T. (2013). Computing general equilibrium theories of monopolistic competition and heterogeneous firms (Chap. 23). In P. B. Dixon & D. W. Jorgenson (Eds.), Handbook of Computable General Equilibrium Modeling (pp. 1513–1570). Amsterdam: Elsevier.
Debreu, G. (1959). Theory of value: an axiomatic analysis of economic equilibrium (pp. xi + 114). New Haven: Cowles Foundation, Yale University Press.
Dhingra, S. & Morrow, J. (2012). The impact of integration on productivity and welfare distortions under monopolistic competition (pp. 50), Centre for Economic Performance Discussion Paper No. 1130, London School of Economics, February.
Dixit, A. K., & Stiglitz, J. E. (1977). Monopolistic Competition and optimum product diversity. American Economic Review, 297–308.
Lanclos, D. K., & Hertel, T. W. (1995). Endogenous product differentiation and trade policy: implications for the U.S. food industry. American Journal of Agricultural Economics, 77(3), 591–601.
Negishi, T. (1960). Welfare economics and the existence of an equilibrium for a competitive economy. Metroeconomica, 12(2–3), 92–97.
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Appendix 3.1: Equivalence Between Worldwide Cost Minimizing and the AKME Model
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 \( \Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} \), Ns, Qksd and \( \Lambda _{\text{d}} \) be a solution to (3.25)–(3.29) for given values of the exogenous variables Ws, and Qd. Let Pd and Pksd be defined by (3.30) and (3.31) and define Qsd, Πksd, Πtots and Ls as in (T2.4)–(T2.7) of the AKME model. We show that \( \Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} \), Ns, Qksd, Pd, Pksd, Qsd, Πksd, Πtots and Ls 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 (Tsd = 1), (T2.1) is the same as (3.31).
Hence
Under (3.21) this establishes (T2.3).
From (3.27)
Combining (3.30) and (3.33) gives
In particular
Putting (3.36) into (3.34) gives
establishing (T2.10) via (T2.5) with Tsd equals one.
From (3.28) and (3.35) we obtain
Simplifying, rearranging and multiplying through by Ns gives
Via (T2.5) with Tsd 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
Then multiplying through by Nsgs(Φk), aggregating over s and k, and using (3.25) we obtain (T2.2) under assumption (3.21).
Proof of proposition ( 3.23 ):
Let \( \Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} \), Ns, Qksd, Pd, Pksd, Qsd, Πksd, Πtots and Ls satisfy (T2.1) to (T2.10) for given values of the exogenous variables Ws and Qd and with Tsd equals one for all s,d. Define \( \Lambda _{\text{d}} \) by (3.30). We show that \( \Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} \), Ns, Qksd 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
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 Tsd = 1 and (T2.10) give
To establish (3.27) we need to eliminate Pmin(s,d) and introduce \( \Lambda _{\text{d}} \). We do this via (T2.3), (3.21) and (3.30) which give
and, in particular
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 Tsd = 1, (T2.6) and (T2.9) we obtain
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 Tsd = 1 and (3.21).
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Dixon, P.B., Jerie, M., Rimmer, M.T. (2018). Optimality in the Armington, Krugman and Melitz Models. In: Trade Theory in Computable General Equilibrium Models. Advances in Applied General Equilibrium Modeling. Springer, Singapore. https://doi.org/10.1007/978-981-10-8325-9_3
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