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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: | ||
+ | $$ | ||
+ | fc_{i,r} = {NB^0}_{i,r}^\frac{\sigma_{VAR}}{1-\sigma_{VAR}} \frac{Y^0_{i,r}}{\sigma_{VAR}-1} | ||
+ | $$ |