CAPÍTULO III: MARCO METODOLÓGICO
3.4. Técnicas e instrumentos para la recolección de datos
2004 2004.5 2005 2005.5 2006 2006.5 -5 0 5 10 15 20
D: US Exports, broad matches, BC & NC
All dp |dp|>0 |dp|>0.02 Rel. Labor Costs
2004.5 2005 2005.5 2006 2006.5 -15 -10 -5 0 5 10
Quarterly Percentage Change
A: Price changes US Exported Tea
US region 1 US region 2 Can. region 1 2004.5 2005 2005.5 2006 2006.5 -20 -15 -10 -5 0 5 10 15 20
Quarterly Percentage Change
B: Changes in Product-level RER US Exported Tea
Can/US labor costs q within US q Can-US -0.20 -0.1 0 0.1 0.2 0.1 0.2 0.3 0.4 0.5
C: Histogram changes Product-level RER: Broad matches
Quarterly Percentage Change
Frequency
Within Can. Within US Canada-US
Figure 3: Product-Level Real Exchange Rates
-0.20 -0.1 0 0.1 0.2 0.05 0.1 0.15 0.2 0.25
D: Histogram changes Product-level RER: Identical matches
Quarterly Percentage Change
Frequency
Within Can. Within US Canada-US
Appendix
In this appendix we brie‡y describe three widely used demand models that produce the assumed relation between relative prices, demand shifters, and markups represented by (:; :)
in Section 4. Since we focus on markups in a given region and at a point in time, for simplicity we abstract from location (i; r) and time (t) subscripts.
5.1. Non-CES demand
This setting was originally explored in Kimball (1995). A …xed continuum of intermediate goods of measure one, indexed by n, are combined in amounts Cn to produce a …nal good
C (or utility) according to a constant returns to scale technology implicitly de…ned by:
Z C
n
AnC
dn= 1,
The function satis…es the constraints (1) = 1, 0(:)>0and 00(:)<0. Under constant
returns to scale (CES), (:) = (:) 1. Cost minimization (or utility maximization) gives rise to the following …rst-order-condition:
Pn= 0
Cn
AnC AnC
,
where denotes the Lagrange multiplier. Expenditures over all varieties are given by: P C =
Z
PnCndn= D,
where P is the price index,P = R PnCndn =C, and D
R 0 Cn
AnC
Cn
AnCdn. Hence, the
inverse demand function for variety n is:
0 Cn
AnC
= DPn
AnP
, which can we inverted to obtain:
Cn =An
DPn
AnP
C,
where (:) = 0 1(:) > 0 and 0(:) < 0 applying the inverse derivative theorem and
00(:)<0. In logs,
cn =an+ log ( (exp (xn))) +c,
where xn= log (D) an+pn pand an= log (An). The demand elasticity is:
"n = 0(:) (:) DPn AnP ,
which can vary across …rms depending on the shape of (:). It is straightforward to show that dD = 0 up to a …rst-order-approximation around an equilibrium in which "n is equal
across all products. The log change in the aggregate price index (for a …xed aggregate consumptionC = 1) is:28 p = Z P nCn P ( pn+ cn)dn = Z sn pndn+ Z sn cndn. (A1)
To calculate the second term in (A1), we di¤erentiate the aggregator (subject toC = 1):
Z
0Cn
An
( cn an)dn= 0.
Using the FOC from cost minimization and the fact that the expected value of an is equal
to zero, we have that R sn cndn= 0, so p= R
sn pndn.
To put more structure on the dependance of "n on the relative price, Klenow and Willis
(2006) choose a speci…cation that results in a demand function:
log ( (x)) = log [1 x].
The limit of log ( x)as !0 is x as under CES. The demand elasticity is: "n=
@log (x)
@x = 1 xn
which is constant when = 0 and increasing in x when >0. The log markup is:
n= log
1 + xn
and the elasticity of the markup with respect to the relative price and with respect to the demand shifter is:
n;p = n;a =
1 + xn
.
Hence, when >0markups are decreasing in the relative price and increasing in the demand shifter. Moreover, markups are more sentitive to relative prices pin pn (i.e. n;p is higher)
the lower is a …rm’s relative price and the higher is the demand shifter an.
28Depending on parameter values, there may be a choke price above which some products are not consumed
even if …xed costs are zero (see e.g. Arkolakis et. al. 2012). Up to a …rst-order approximation, this margin has no impact on the aggregate price index since entering and exiting products have sales equal to zero.
5.2. Distribution costs
This setting was originally explored in Corsetti and Dedola (2005). Final consumption is given by C = Z C 1 n dn 1 , >1.
In order to deliver an intermediate good to the …nal consumer, a retail (and wholesale) sector in the destination location bundles the domestically produced or imported good with distribution services. Assuming that the retail sector is competitive and combines the good and distribution services at …xed proportions, the retail price (in levels) of the intermediate good n, Pr
n, is given by:
Pnr =Pn+ AnPd
where An denotes the distribution cost per good. We assume that production of one unit
of distribution services uses one unit of the industry bundle, which implies Pd =P.29
Note that under this speci…cation, movements in distribution costs, AnP, across regions
gives rise to movements in product-level RERs at the wholesale and retail level even if producer markups are constant. Given that wholesale distribution margins for the products we consider are on average only 16% in the U.S., it is unlikely that this force by itself can rationalize our empirical …ndings on product and aggregate-RERs.
The presence of additive distribution costs, however, leads to variable markups at the producer level. The elasticity of demand faced by an intermediate good producer is
"n= @logCn @logPn = 1 sdn where sd n = An P
Pn+An P denotes the share of distribution services in the retail price. The
distribution share and the elasticity of demand are both decreasing in the ratio of the …rm’s producer price to the local cost componentPn=P. The optimal mark-up for a monopolistic
price-setter is: n = log " 1 sd n (1 sd n) 1 # = log 1 exp (an (pn p)) . The elasticity of the markup with respect to the relative pricepn pis
p;n= 1 1 Anexp( (pn p)) 1 = 1 ( 1)1 sdn sd n 1 , (A2)
and the markup elasticity relative toanis a;n = p;n. Clearly p;n= 0ifsdn= 0and p;n >0
if sd
n>0.30 Note also the the markup elasticity is increasing in sdn (and hence is decreasing 29An alternative assumption that gives similar results is that distribution services are produced using local
labor instead of the industry bundle. In such case, the markup is a decreasing function of the price relative to the wage, pn w, instead of pn p. Markups in this case respond to changes in local wages and not
directly to changes in the local aggregate price.
30A necessary condition for
p;n > 0 in this model is that the elasticity of substitution in the retail
in relative price and increasing inan). Finally, under the assumption that distribution costs
are produced using the …nal good, it is straightforward to show that changes in the …nal price index are given by p=R sn pndnas in the previous model.
5.3. Strategic complementarities in pricing with CES demand
This setting was originally studied in Dornbusch (1987) and more recently in Atkeson and Burstein (2008). Final sector output is modeled as a CES of the output of a continuum of sectors m with elasticity of substitution and sector output is CES over a…nite number of di¤erentiated products with elasticity , where1 :
C= Z C 1 m dn 1 and Cm = " X A 1 nC 1 n # 1 :
Firms own single products within each sector and compete in prices (Bertrand). Taking as given prices of other …rms in its sector, the elasticity of demand for goodiselling in country n in any given sector is:
"n= sn+ (1 sn),
where sn = exp (an+ (1 ) (pn p)) represents the expenditure share of product i with
taste parameteran in that sector andp= 11 log (Pian+ (1 ) (pn p))is the log of the
aggregate sector price. Note that, if < , "n is decreasing in the expenditure share of the
…rm in that sector. The optimal markup that results from choosing price to maximize pro…ts taking prices of other …rms in the sector as given is:
n= log "n "n 1 = log sn+ (1 sn) sn+ (1 sn) 1 ,
which is increasing in sn (hence, decreasing in pn p for a …xedan and increasing inain for
a …xed pn p) if > . The elasticity of the markup with respect to relative price is:
n;p= ( ) ( 1)
sn
[ sn+ (1 sn)] [ sn+ (1 sn) 1]
, (A3)
and the markup elasiticty of the markup with respect to the demand shifter is n;a = n;p=( 1), both of which are positive if 1 . That is …rms with lower relative
pricepn p, higher demand shifteran, and higher expenditure sharesn set higher markups.
Markup elasticities n;p and n;a are higher the higher is a …rm’s market share sn (e.g. the
lower is its relative price pn p and the higher is the demand shifter an). These results are
qualitatively unchanged if …rms compete in quantities (Cournot).31 Moreover, the pricing 31Under Cournot competition,"
n = sn +1 sn
1
and n = ( 1) 1 1 nsn, wheresn andpare
given by the same expressions as under Bertrand competition, and n ="n=("n 1). Once again, n;p is
implications of this model are continuous in the elasticity of substitution between products within a sector, . Hence, with competition in prices and ! 1 there is limit pricing (prices are equal to the minimum between the monopoly price and the cost of the latent competitor) as discussed in Atkeson and Burstein (2007).
Note that with a …nite number of positive-mass …rms per sector, any change in a product’s pricepn has a non-zero e¤ect on the sectoral pricep. Taking this into account, the change in
price of goodn given by equation (4.5) becomes (abstracting from region and time indices): pn = zn+ wn+ en+ (1 s1n) p;n an 1 + (1 sn) p;n + (1 sn) p;n 1 + (1 sn) p;n p n,
where en denotes the change in the exchange rate between the production and destination
locations for productn, and p nis an expenditure-weighted average of price changes in the
sector exclusive of product n (in models with a continuum of products, sn = 0). Markups
are constant if sn = 0 or if sn = 1. Hence, in this case movements in markups and in
product-level RERs across …rms are non-monotonic in relative marginal costs or in demand shifters as in the models with a continuum of …rms. The markup elasticity is decreasing in relative price and increasing in the demand shifter only in the region in which(1 sn) p;nis
increasing insn. This assumption is required for our discussion in the paper on how markup
elasticities vary across location.
The change in aggregate price in a given sector is: p= P sn 1+ p;n zn an 1 + wn+ en P sn 1+ p;n .
If the number of products in this sector is small, idiosyncratic shocks do not wash-out when calculating the change in the sector aggregate price. However, idiosyncratic shocks do cancel-out then calculating our statistics over a continuum of ex-ante symmetric sectors and we obtain the expressions for the change in the aggregate prices as in our previous models.