#
Imperfect competition (1.1)
#

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

##
MIRAGE-e 1.1 and after
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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:Bekk18)]
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 $P_{i,r,s}^{PROD} =
m_{i,r,s}(1+tCost_{i,r,s})PY_{i,r}$, 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 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
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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
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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:

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

##
Priori to MIRAGE-e 1.1
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See
Imperfect competition (Outdated)