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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 2019/06/03 08:13 jfoure created2019/05/29 15:36 jfoure ↷ Page name changed from imperfect_competition to imperfect_competition_1.1 2019/06/03 08:13 jfoure created2019/05/29 15:36 jfoure ↷ Page name changed from imperfect_competition to imperfect_competition_1.1 Line 1: Line 1: + ====== 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: + $$+ fc_{i,r} = {NB^0}_{i,​r}^\frac{\sigma_{VAR}}{1-\sigma_{VAR}} \frac{Y^0_{i,​r}}{\sigma_{VAR}-1} +$$