Imperfect competition

The implementation of imperfect competition à la Krugman is one of the distinctive features of the MIRAGE model.

MIRAGE-e and after

MIRAGE-e implementation of imperfect competition à la Krugman (1979) is inspired by two more recent contributions: Balistreri and Rutherford (2013) for theoretical derivation calibration procedure and Bekkers and Francois (2018) for implementation through “generalized marginal cost”.

In a nutshell, we define generalized marginal cost \(GnMC_{i,r}\) as: \[ GnMC_{i,r} = \left\{ \begin{array}{ll} 1 & \text{in perfect competition}\\\\\\ N_{i,r}^\frac{1}{1-\sigma_{VAR}}\frac{\sigma_{VAR}}{\sigma_{VAR}-1}& \text{in imperfect competition} \end{array} \right. \]

Then, the expression of imperfect competition in MIRAGE-e is very similar to the perfect competition framework.

Theoretical setup

The Krugman model is characterized by love of variety, which is materialized in the CES demand functions: \[ \left\{ \begin{array}{ll} D_{i,s}= \left[\displaystyle\int_{\omega\in\Omega} {\left(D^{VAR}_{\omega,i,s}\right)^\frac{\sigma_{VAR}-1}{\sigma_{VAR}}d\omega}\right]^\frac{\sigma_{VAR}}{\sigma_{VAR}-1}\\\\\\ DEM_{i,r,s}= \left[\displaystyle\int_{\omega\in\Omega} {\left(DEM^{VAR}_{\omega,i,r,s}\right)^\frac{\sigma_{VAR}-1}{\sigma_{VAR}}d\omega}\right]^\frac{\sigma_{VAR}}{\sigma_{VAR}-1} \end{array} \right. \]

Let us define production price as \(P_{i,r,s}^{PROD} = m_{i,r,s}(1+tCost_{i,r,s})PY_{i,r}\), where \(m_{i,r,s}\) is the markup over marginal cost. The profit for one firm on one market can be expressed as: \[ \pi^{VAR}_{i,r,s} = P^{PROD}_{i,r,s}DEM^{VAR}_{i,r,s}-PY_{i,r}(1+tCost_{i,r,s})DEM^{VAR}_{i,r,s} \] F.O.C. give: \[ \frac{\partial\pi^{VAR}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}} = P^{PROD}_{i,r,s} + DEM^{VAR}_{i,r,s}\frac{\partial P^{PROD}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}} - PY_{i,r}(1+tCost_{i,r,s})=0\]


If we rewrite this expression from the demand side point of view, we need to use \(PDEM_{i,r,s}^{VAR}\) instead of \(P_{i,r,s}^{PROD}\). The correspondence between both is: \[P^{PROD}_{i,r,s} = \frac{1}{(1+tax^P_{i,r})(1+tax^{EXP}_{i,r,s})} \left[\frac{PDEM^{VAR}_{i,r,s}}{1+Tariff_{i,r,s}}-\mu_{i,r,s}P^{Tr}_{i,r,s}\right]\]

And in derivatives: \[ \frac{\partial P^{PROD}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}} = \frac{1}{(1+Tariff_{i,r,s})(1+tax^P_{i,r})(1+tax^{EXP}_{i,r,s})} \frac{\partial PDEM^{VAR}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}}\]

The CES demand gives the following expression for \(PDEM_{i,r,s}^{VAR}\): \[ PDEM^{VAR}_{i,r,s} = \left(\frac{DEM_{i,r,s}}{DEM^{VAR}_{i,r,s}}\right)^\frac{1}{\sigma_{VAR}} \] and then, under the usual Krugman assumptions (no strategic interactions, hence constant markup) \[ \frac{\partial PDEM^{VAR}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}} = -\frac{1}{\sigma_{VAR}}\left(\frac{PDEM^{VAR}_{i,r,s}}{DEM^{VAR}_{i,r,s}}\right) \]

It follows the full expression: \[ PDEM^{VAR}_{i,r,s} = \frac{\sigma_{VAR}}{\sigma_{VAR}-1}(1+Tariff_{i,r,s})(1+tCost_{i,r,s})\left[(1+tax^P_{i,r,s})(1+tax^{EXP}_{i,r,s})PY_{i,r}+\mu_{i,r,s}P^{Tr}_{i,r,s}\right] \]

By identification, the markup is:

\[m_{i,r,s} = \frac{\sigma_{VAR}}{\sigma_{VAR}-1} \]


From the F.O.C, it also follows that:

\[ DEM_{i,r,s} = DEM^{VAR}_{i,r,s}N_{i,r}^\frac{\sigma_{VAR}}{\sigma_{VAR}-1} \quad \text{and} \quad PDEM_{i,r,s} = PDEM^{VAR}_{i,r,s}N_{i,r}^\frac{1}{1-\sigma_{VAR}} \]

Hence, we can write:

\[ PDEM_{i,r,s} = N_{i,r}^\frac{1}{1-\sigma_{VAR}}\frac{\sigma_{VAR}}{\sigma_{VAR}-1}(1+Tariff_{i,r,s})(1+tCost_{i,r,s})\left[(1+tax^P_{i,r,s})(1+tax^{EXP}_{i,r,s})PY_{i,r}+\mu_{i,r,s}P^{Tr}_{i,r,s}\right] \]

Generalized marginal cost

We define \(c_{i,s}\) after Bekkers and Francois (2018), as the generalized marginal costs of producing good \(i\) in region \(s\). As such, domestic prices can be written: \[PD_{i,s} = c_{i,s} PY_{i,s} \left(1+tax^P_{i,s}\right). \]

We also define \(t_{i,r,s}\) as the generalized trade cost. Export price can be written: \[ PDEM_{i,r,s} = c_{i,r} \left(1+Tariff_{i,r,s}\right) t_{i,r,s} \left[\left(1+tax^P_{i,r}\right)\left(1+tax^{EXP}_{i,r,s}\right)PY_{i,r} + \mu_{i,r,s}P^{Tr}_{i,r,s}\right] \] where \(PY_{i,s}\) is the marginal cost of producing good \(i\), and other notations follow usual MIRAGE notations.


As a consequence, in the Krugman case, we can identify

\[c_{i,r} = N_{i,r}^\frac{1}{1-\sigma_{VAR}}\frac{\sigma_{VAR}}{\sigma_{VAR}-1} \]

Subcase of the Armington economy

In the perfect competition Armington specification:

\[ \left\{ \begin{array}{ll} PD_{i,s} &= 1 . PY_{i,s} \left(1+tax^P_{i,s}\right)\\\\\\ PDEM_{i,r,s} &= 1. \left(1+Tariff_{i,r,s}\right)\left(1+tCost_{i,r,s}\right)\left[\left(1+tax^P_{i,r}\right)\left(1+tax^{EXP}_{i,r,s}\right)PY_{i,r} + \mu_{i,r,s}P^{Tr}_{i,r,s}\right] \end{array} \right. \]

Hence, \(c_{i,r}=1\).


In this theoretical framework, the markup is constant and does not depend on the number of firms so we can choose \(NB_{i,r,}=100\) without losing generality, but only on the elasticity of substitution between varieties. In this regard, we stick to MIRAGE traditional rule of \(\sqrt{2}\) for elasticities of substitution: \[\sigma_{VAR} -1 = \sqrt{2} \left(\sigma_{IMP}-1\right) \]

Fixed costs are then derived as: \[ fc_{i,r} = {NB^0}_{i,r}^\frac{\sigma_{VAR}}{1-\sigma_{VAR}} \frac{Y^0_{i,r}}{\sigma_{VAR}-1} \]


Balistreri, Edward J., and Thomas F. Rutherford. 2013. “Computing General Equilibrium Theories of Monopolistic Competition and Heterogeneous Firms.” In Handbook of Computable General Equilibrium Modeling, edited by Peter B. Dixon and Dale W. Jorgenson, 1:1513–70. Elsevier.
Bekkers, Eddy, and Joseph Francois. 2018. “A Parsimonious Approach to Incorporate Firm Heterogeneity in CGE-Models.” Journal of Global Economic Analysis 3 (2): 1–68.
Krugman, Paul R. 1979. “Increasing Returns, Monopolistic Competition, and International Trade.” Journal of International Economics 9 (4): 469–79.