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imperfect_competition [2019/05/29 15:36]
jfoure ↷ Page name changed from imperfect_competition to imperfect_competition_1.1
imperfect_competition [2019/06/03 08:13] (current)
jfoure created
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 +====== Imperfect competition ======
  
 +The implementation of imperfect competition à la Krugman is one of the distinctive features of the MIRAGE model.
 +
 +===== MIRAGE-e 1.1 and after =====
 +MIRAGE-e implementation of imperfect competition à la [(:​harvard:​Krug79)] is inspired by two more recent contribution to the New Quantitative Trade Models literature: [(:​harvard:​Bali13)] for theoretical derivation calibration procedure and [(:​harvard:​Bekk17)] 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 [(:​harvard:​Bekk17)],​ 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:
 +$$
 +fc_{i,r} = {NB^0}_{i,​r}^\frac{\sigma_{VAR}}{1-\sigma_{VAR}} \frac{Y^0_{i,​r}}{\sigma_{VAR}-1}
 +$$