# Imperfect competition (1.1)

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

MIRAGE-e implementation of imperfect competition à la (Paul R Krugman, 1979) is inspired by two more recent contribution to the New Quantitative Trade Models literature: (Edward J Balistreri, Thomas F Rutherford, 2013) for theoretical derivation calibration procedure and (Eddy Bekkers, Joseph 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 demand CES 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 $P^{PROD}_{i,r,s} = m_{i,r,s}(1+tCost_{i,r,s})PY_{i,r}$, 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 $PDEM^{VAR}_{i,r,s}$ instead of $P^{PROD}_{i,r,s}$. The correspondance 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 demand CES gives the following expression for $PDEM^{VAR}_{i,r,s}$: $$PDEM^{VAR}_{i,r,s} = \left(\frac{DEM_{i,r,s}}{DEM^{VAR}_{i,r,s}}\right)^\frac{1}{\sigma_{VAR}}$$ 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 (4), 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 loosing 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 derivated as: $$fc_{i,r} = {NB^0}_{i,r}^\frac{\sigma_{VAR}}{1-\sigma_{VAR}} \frac{Y^0_{i,r}}{\sigma_{VAR}-1}$$

1. ^ Paul R Krugman, 1979. Increasing returns, monopolistic competition, and international trade. Journal of international Economics, 9, Elsevier, pp.469–479.
2. ^ Edward J Balistreri, Thomas F Rutherford, 2013. Computing general equilibrium theories of monopolistic competition and heterogeneous firms. Handbook of computable general equilibrium modeling, Elsevier, pp.1513–1570.
3. ^ Eddy Bekkers, Joseph Francois, 2018. A Parsimonious Approach to Incorporate Firm Heterogeneity in CGE-Models. Journal of Global Economic Analysis, 3, pp.1–68.