# 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: \[ \frac{\partial\pi^{VAR}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}} = P^{PROD}_{i,r,s} + DEM^{VAR}_{i,r,s}\frac{\partial P^{PROD}_{i,r,s}}{\partial DEM^{VAR}_{i,r,s}} - PY_{i,r}(1+tCost_{i,r,s})=0\]

#### Markup

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:

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

#### Aggregation

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.

#### Identification

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

### Calibration

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: \[ fc_{i,r} = {NB^0}_{i,r}^\frac{\sigma_{VAR}}{1-\sigma_{VAR}} \frac{Y^0_{i,r}}{\sigma_{VAR}-1} \]

## References

*Handbook of Computable General Equilibrium Modeling*, edited by Peter B. Dixon and Dale W. Jorgenson, 1:1513–70. Elsevier. https://doi.org/10.1016/b978-0-444-59568-3.00023-7.

*Journal of Global Economic Analysis*3 (2): 1–68. https://doi.org/10.21642/jgea.030201af.

*Journal of International Economics*9 (4): 469–79. https://doi.org/10.1016/0022-1996(79)90017-5.