Technical Appendix Vehicle Currency Use in ... - Google Sites

Technical Appendix

Vehicle Currency Use in International Trade Linda S. Goldberg Federal Reserve Bank of New York and NBER

Cédric Tille Federal Reserve Bank of New York

November 17, 2006

1. Building blocks of the theoretical model 1.1. The profit function The exporter posts a price for her goods before knowing the realization of various shocks affecting the economy. The exporter produces a brand z, is located in country e, and sells her goods to the destination country d. Goods are produced using a technology with decreasing returns to scale:

(1)

Yed  z     H ed  z 

H

1

K ed z 

K

0   H , K  1

~   H   K  1

where Yed  z  is the output of z, H ed  z  is the labor input, K ed  z  is another input, possibly imported, and  H and  K are returns to scale parameter. The firm faces the following demand in country d:

(2)

Yd  z   Ped  z  / Pd  C d -λ

where C d is the total demand for brands of the relevant sector in country d, Ped  z  is the price, in country d currency, of the brand z produced in country e, and Pd is the price index, in country d currency, across all brands of the relevant sector sold in country d. >1 is the elasticity of substitution between the various brands.

1

The exporter can adjust the mix of inputs once the various shocks are known. The factor demands are then given by: K ed  z   Yed z    1 ~

1 ~

We  K     Re  H 

H ~

,

H ed  z   Yed  z    1 ~

1 ~

We  K     Re  H 

 K ~

where We is the nominal wage paid to labor and Re is the price of the other input. The cost of production, as a function of the output and factor prices, is then: H ~ e

We H ed  z   Re K ed  z   W

K ~ e

 R  Yed z    1 ~

1 ~

 K H  ~ ~        K K       H    H  

    

The exporter producing brand z sets its price in currency k before the realization of the shocks affecting the economy. We denote the price by Pedk  z  . Several currencies are available: the currency of the country in which the exporter is located (e), and N other currencies (N>1) that include the currency of the country of destination (d) and N-1 possible vehicle currencies. The invoicing can take place in the producer currency e, the destination currency d, another vehicle currency v, or a combination of the available currencies. In addition of having to set prices in an environment of uncertainty, the exporter faces transaction costs which depend on the currency chosen for invoicing. We denote the exchange rate between currency e and currency k by S ek , in terms of units of currency e per unit of currency k so that an increase corresponds to a depreciation of currency e. When the exporter brings an amount Pedk  z  of currency k to the foreign exchange market for conversion into currency e, she gets only and amount





exp   ek S ek Pedk  z  of currency e, where  ek  0 is a transaction cost.

Conditional on using currency k, the exporter sets its price Pedk  z  to maximize expected discounted profits, which are given by:

2

λ   S ek Pedk  z   k k exp   e S ek Ped z   Cd   S ed Pd   1  edk  z   EDe   K H ~ λ   k   H  K  S P z  1    ~ ~     ek ed K K ~ R  ~  ~         W C       e e d      S ed Pd    H   H    



(3)



        

where De is the state-specific discount factor at which profits are evaluated. The price maximizing (3) is such that: λ   S ek Pedk  z   k k exp   e S ek Ped  z   Cd   S ed Pd   1 0  EDe   K H ~ λ   ~ k     1  H K       K  ~ S ek Ped  z  1   K  ~ ~ ~      We  Re      C d        S ed Pd    H    H    1  H   K  



4



        

1.2. Prices and profits in the absence of uncertainty Consider a situation where there is no uncertainty, as well as no transaction costs in foreign exchange markets (  ek  0 ). We denote variables in this steady state with an upper bar. For simplicity, we consider a symmetric steady state where the exporter brings its price in line with that of her competitors: S ek Pedk  z   S ed Pd . The optimal price (4) is such that: 1~ ~

 

H K λ We  ~ Re  ~ C d S ek Pedk  z   λ 1

 K H  ~   K  ~   K      ~   H   H  

 ~ 1

    

The profits (3) are:

3

 K H   ~   K  ~    K            H    H     K H   1 1    ~ ~      K K ~ C d   ~       H    H   

H K  edk  z   S ek Pedk  z C d  We  ~ Re  ~ C d De

 

H K  λ 1  ~ R  ~   1  W   e e ~  λ 1  

1 ~

1 ~

 

1.3. A second order expansion of the profit function. With express all variables as second-order log expansions around the symmetric steady state described in the previous section. Denoting log deviations by smaller case letters ( x  ln X  ln X ), we use the following relation: 1  2 X a Y b  X a Y b 1  ax  by  ax  by   2  

The first element in the expected discounted profits (3) is written as the following expansion:  S P k z  EDe exp   S ek P  z  ek ed  C d  S ed Pd 



k e



λ

k ed





1  d e   ek  s ek  p edk  z   λ s ek  p edk  z   s ed  p d  c d     De S ek Pedk  z C d  E  1  2 k  d e  s ek  λq ed  c d   2 





In deriving this relation, we use the fact that the difference between the preset price under uncertainty and its value in the steady state, p edk  z  , is of order 2. Similarly, we assume that the transaction cost in the foreign exchange market,  ek , is also of order 2. As we limit our expansion up to terms of order 2, p edk  z  and  ek do not enter the squared term in the expression above.

The term q edk  s ek  s ed  p d in the squared bracket captures the fluctuations in the relative price of brand z in the destination country, vis-a-vis competing brands. Following a shock, the price of brand z

4

is unchanged in currency k. The price paid by a consumer in the destination currency d is then driven by the exchange rate between currency k and currency d: s dk  s ek  s ed . The average price paid by the consumer for competing brands, in currency d, is given by the industry-wide index: p d . An increase in the relative price ( q edk  0 ) indicates that the firms producing brand z looses competitiveness, leading consumers in the destination country to shift their purchases towards other brands. The magnitude of this shift is driven by the elasticity of substitution between brands, . Following a similar procedure, the second element in the expected discounted profits (3) is written as 1

H

K

EDe We  ~ Re  ~

H

K

 De We  ~ Re  ~

  S P k  z   λ  ~   ek ed  C d    S ed Pd       1  d e  ~H we  ~K re      1  1   C d ~ E  ~  λ s ek  pedk  z   s ed  p d  c d     2  1   1    d e  ~H we  ~K re  ~  λq edk  c d    2      

 

 







We



define

the

deviation

of

profits

from

the



symmetric

steady

state

as:



 edk  z    edk  z    edk z  /  edk  z  . Combining our results to we write: (5)



λ λ 1 k  edk z   X ed  ~   E s ek  λq edk e ~ ~ ~    λ1      λ1     2



2





  E s ek  λq edk c d  

2 k k  λ  1~  1   E λq ed   λEq ed c d   ~ 2      λ1  ~   ~    λEq k  w   r 

 1 



2

ed

H

e

K e



where X ed is a term independent of the choice of currency k:

5

 Es ed  Ep d   Ec d  1 X ed  Ed e  ~   λ  1 Ew   Er  ~   λ1     H e K e  2  λ 1 λ  1~ 1  1  2  ~ E d e  c d   E s ed  p d d e  ~ E d e  ~  H we   K re  c d    λ1  ~  2   λ1  ~  2   





Ignoring X ed , (5) can be rewritten as:  k 1    λ  1 k 2 k 2   e  2  E s ek  λq ed  ~ E qed     λ   k  ed  z   ~   ~   λ1      λ 1  λ 1 1  k  k   E  s ek  λ1  q ed c d  ~ E  H we   K re qed   λ ~     



(6)



 

Intuitively, (6) shows that expected profits reflect three components. The first captures the transaction costs in the foreign exchange market, with higher costs reducing profits. The second captures the





2

 

2

volatility of marginal revenue, E s ek  λq edk , and marginal cost, E q edk , with a higher volatility of marginal cost reducing expected profits. The third component reflects the co-movements between competitiveness and the exogenous drivers of costs, namely the strength of demand and factor prices.

2. Optimal invoicing in a currency basket 2.1. Description of currency shares The exporter chooses weights on the available currencies (e, d and the N-1 available vehicles v) in her invoicing currency basket k. Specifically, the weight of a currency i that is not the exporter’s ( i  e ) in the invoicing of exports to country d is  di . The set of currencies indexed by i includes the destination currency d and the N-1 available vehicle currencies v’s. The invoicing the weight of currency e is the residual: 1 



i  d , v 's

i d

. The exchange rate between the exporter's currency, e, and the composite

currency in which she invoices, k, is then a linear combination of the exchange rates between currency e and the other currencies with the weights reflecting the composition of the invoicing basket:

6

s ek 

(7)



i  d ,v ' s

i d

s ei

Producer currency stability (PCS) corresponds to low value of  di ’s, while local currency stability (LCS) and a vehicle currency stability (VCS) correspond to high values of  dd and the corresponding

 dv respectively. The cases of pricing in one currency only are given by setting the weights to 0 or 1. Specifically, Producer currency pricing (PCP) corresponds to  di  0, i  d , v , while local currency pricing (LCP) corresponds to  dd  1,  dv  0 v , and a vehicle currency pricing (VCP) is the given by

 dv '  1,  di  0 i  d , v  v' . The relative price between brand z and the competing brands, q edk , is a central component of the invoicing decisions. We now describe to the sensitivity of the price index of competing brands, p d , to exchange rate movements. Some brands are invoiced in currency d, so the price paid by the consumers for these brands is unaffected by exchange rate movements. Other brands are invoiced in currency e, and the consumer price in currency d moves with the exchange rate between the two currencies, s ed , with consumer paying a higher price when currency e appreciates (i.e s ed  0 ). Other of brands are invoiced in the various vehicle currencies v’s, so the price paid by consumers is higher when the corresponding currency v appreciates (i.e. s ed  s ev  0 ). We denote the share of brands invoiced in currency i by  di , with the shares summing to one. The impact of exchange rate movements on price index of competing brands is then: (8)

  p d   de s ed   di s ed  s ei    de   di  s ed   di s ei   1   dd s ed   di s ei i  v 's i v 's i  v 's i  v 's  





Combining (7) and (8), we write the relative price q edk as:

7

(9)

q edk  s ek  s ed  p d 



   d d

d d

s

ed



i  v 's







     s ei  i v 's



s   dd s ed  s ed  1   dd s ed   di s ei

i d ei i d

i d

 

i  d ,v ' s

i d



i  v 's

  s ei i d

(9) shows that a full stabilization of the relative price requires the exporter to choose weights on the different currencies that exactly correspond to their shares in the industry wide price index:  di   di i . In terms of the transaction cost on the foreign exchange market, we consider that it is a weighted average of the transaction costs for the various currencies, weighted by their shares in the invoicing basket:

(10)

   ek  1    di  ee    di  ei   ee    di  ei   ee  i  d ,v ' s i  d ,v ' s  i  d ,v ' s 

2.2. Derivatives of the profit function. The firm chooses the invoicing weights  di ’s to maximize the expected profits (6), under the constraint that  di ’s and



i  d , v 's

i d

do not fall outside the [0,1] interval. For a concise illustration of the

determinants of the optimal weights, we focus on an interior solution where both weights are given by setting the first derivatives to zero. The derivatives of (6) with respect to a particular  di , with i = d, v’s are:   ek  s q k   λ  1 k q edk  i  E s ek  λq edk  eki  λ edi   Eq ed  ~  d   di  edk  z  λ   d   d  ~    λ1  ~    s ek  di qedk  λ  1 1  q edk λ 1  E   i  λ  i 1  λ ~  c d  ~ E  H we   K re   i d d  d  





      

Using the definition of the exchange rate (7), the relative price (9), and the transaction cost (10) we write:

8

 ek   ei   ee i  d

s ek q edk   s ei  di  di

i  d , v' s

Using these results, we derive:  edk  z  λ 1  ~   i e        λ  1  E  s s    λ  1Eqedk s ei  λ  1Emed s ei   e e ek ei i ~ ~ ~   λ1       d 





where med summarizes the impact of movements in factor costs and demand in the destination market:

med 

 H we   K re 1   H   K   cd H K H K

Using (7) and (9) to substitute for s ek and q edk , we obtain:  edk  z  1  ~ j  j i e        λ  1  Em s   λ  1      d   dj  e e ed ei  d j  d ,v 's  ~  di



(11)









 E s 

s

ej ei





1  i  e   ee  1   Emed s ei    dj   dj E s ej s ei  j  d ,v 's λ 1





where  denotes a relation of proportionality and  is defined as: λ1  ~   ~  0,1   λ1  ~  The intuitive interpretation of (11) is as follows. Consider the marginal impact of increasing the invoicing role of currency d (an increase in  dd ). The first term shows that reduces profits are reduced if dealing in currency d is relatively costly in the foreign exchange market (  ed   ee  0 ). The second term shows that profits are increased if currency e tends to be weak against currency d when marginal costs are high. This makes invoicing in currency d appealing as a given revenue in currency d, generates a larger revenue in currency e precisely when costs are high. The final term in (11) captures the impact on the volatility of marginal cost and marginal



revenue. The corresponding term in (6), E s ek  λq edk



2



2  λ  1 E q edk , is maximized by setting H K

 

9

 dj   dj  0 i . Around that point, a marginal increase in the invoicing role of currency d has no impact. If  dd   dd , an additional increase in  dd makes the marginal cost more volatile and reduces profits.

2.3. First order conditions. With a destination currency d and N-1 vehicle currencies, there are N conditions (11). Setting them to zero leads to a system of N equations:



















1  d  e   ee j  d ,v 's λ 1 1   v1 j j  e   ee   d   d E s ej s ev1   1   Emed s ev1  j  d ,v 's λ 1 1   v2 j j  e   ee   d   d E s ej s ev 2   1   Emed s ev 2  j  d ,v 's λ 1 ... j j   d   d E s ej s ed   1   Emed s ed 



 

We put this system in the following matrix form: Es ed s ed



d d

  dd

 Es

ed

s ev1

Es ed s ev 2 ...

Emed s ed  1   

Emed s ev1 Emed s ev 2 ...

Es ev1 s ed



  dv1   dv1

 Es

s

ev1 ev1

Es ev1 s ev 2 ...

Es ev 2 s ed



  dv 2   dv 2

 Es

s

ev 2 ev1

Es ev 2 s ev 2 ...

 ed   ee 1    ev1   ee  λ  1  ev 2   ee ...

Consider the results that one would get from regressing the marginal cost on the N-1 exchange rates between currency e and currency i=d, v’s, with all variables measured as deviations from the steady states so their averages are zero. The matrices of dependent and independent variables are:

 Y  med

 X  s ed

 s ev1

 s ev 2

...

10

The regressions coefficients, denoted as ’s, are computed as follows (considering that the sample moments are equal to the expected moments):

 med , s ed  E s ed2   med , s ev1  Es ed s ev1 X'X  X 'Y   med , s ev 2  Es ed s ev 2 ...

Es ed s ev1

Es ed s ev 2

  2 ev1

Es Es ev1 s ev 2 ....

Es ev1 s ev 2 E s ev2 2

 

 med , s ed  Emed s ed ...  med , s ev1  Emed s ev1   med , s ev 2  Emed s ev 2 ... ... ...

Using this result, we re-write our matrix system as:













X ' X  dd   dd  X ' X  dv1   dv1  X ' X  dv 2   dv 2  ...

column 1

column 2

column 3

 med , s ed   ed   ee  med , s ev1  1    ev1   ee  1   X ' X   med , s ev 2  λ  1  ev 2   ee ... ...

 med , s ed   dd   dd  ed   ee  v1   dv1  1   X ' X  med , s ev1   1    ev1   ee  X'X d  med , s ev 2  λ  1  ev 2   ee  dv 2   dv 2 ... ... ... We assume that  ed   ee , as invoicing in the destination or exporter’s currency involves a direct exchange of currency d for currency e. If the transaction costs are the same for the consumer and the exporter, it does not matter who undertakes the currency exchange. Re-arranging, we express the optimal invoicing weights as:

(12)

(13)

 med , s ed  0  dd  dd v1 v1 v1  d    d  1     med , s ev1   1    X ' X 1  e   ed  med , s ev 2  λ  1  ev 2   ed  dv 2  dv 2 ... ... ... ...  de  1 



i  d ,v 's

i d

The term  solely reflects the structural parameters of the model, namely the elasticity of substitution between goods, , and the degree of returns to scale, ~ .  is large in industries where

11

goods are more substitutable ( is large), as movements in relative prices then leads to large fluctuations in quantities sold. The effect is also stronger the more the technology exhibits decreasing returns to scale ( ~ is small), because fluctuations in output generate large movements in marginal cost.

If there is only one invoicing currency (N=2), we get:









(14)

 ev   ed E s ed s ev  1      1    med , s ed     1 E s ed 2 E s ev 2  E s ed s ev E s ed s ev 

(15)

 ev   ed E s ed  1      1    med , s ev     1 E s ed 2 E s ev 2  E s ed s ev E s ed s ev 

(16)

 de  1   dd   dv

d d

v d

d d

v d

2

where :

 med , s ed    med , s ev  

E s ev  E s ed med   E s ev med E s ed s ev  2

E s ev  E s ed   E s ed s ev E s ed s ev  2

2

E s ed  E s ev med   E s ed med E s ed s ev  2

E s ev  E s ed   E s ed s ev E s ed s ev  2

2

2.4. A numerical illustration. 2.4.1. General parametrization. Consider that ~  0.65 , λ  6 , implying   0.76 . There are independent exogenous shocks in all three countries, with f i being the shock in country i. The variances of all shocks are  2 . Exchange rates are driven by shocks differentials, and the marginal cost is driven only by the domestic shock: med  f e

s ei  f e  f i

This implies that E s ei   2 2 , E s ei s ej i    2 and Emed s ei   2 for any i and j. 2

Consider a case where there are two vehicle currencies (N=3). We write the following matrices:

12

 

E s ed2 X ' X  Es ed s ev1 Es ed s ev 2  X ' X 

1

Es ed s ev1 E s ev2 1 Es ev1 s ev 2

 

Es ed s ev 2 2 1 1 Es ev1 s ev 2  1 2 1  2 E s ev2 2 1 1 2

 

3 1 1 3 1 1 1 1   4 1 3 1  1 3 1 4 6 4 2 1 1 3 1 1 3

Emed s ed 1  med , s ed  1 1 2 X ' Y  Emed s ev1  1    med , s ev1   1  4 Emed s ev 2 1  med , s ev 2  1 In which case (12)-(13) become:

 dd  dd 1 3 1 1 0 1 v1 v1 v1  d  0.76  d  0.06 1  0.012 2  1 3  1  e   ed   dv 2  dv 2 1  1  1 3  ev 2   ed  de  1 



i  d ,v ' s

i d

2.4.2. Identical exporters If all exporters are located in the same countries, the ‘s and ‘s of a given currency will be equal in equilibrium. Considering that =0.5 we write:

 dd 0.125 3 1 1 0 1 v1 v1  d  0.125  0.05 2  1 3  1  e   ed   dv 2 0.125  1  1 3  ev 2   ed  de  1 



i  d ,v 's

i d

In the absence of transaction costs, most invoicing is done in the exporter’s currency:  de  62.5% , with only 12.5% of invoicing going through each of the destination and vehicle currencies. Consider now that transacting through all currencies involves equal transaction costs, except that currency v2 entails lower costs:  ed   ev1   ev 2 . We parametrize the cost based on Hau and al. (2002),

13

taking a cost of 5 basis points. We set exchange rate volatility based on the variance of the G3 exchange rate, which is equal to 9 basis points: E s ed   0.09%   2  0.045%. the weights then becomes: 2

 dd 0.125 3 1 1 0 0.125 1.11 0.07 0 . 05  dv1  0.125  1 3 1 0  0.125  0.05 1.11  0.07 0.045% v2 d 0.125  1  1 3  0.05% 0.125  3.33 0.29  de  1 



i  d ,v 's

i d

 0.57

The share of invoicing going through the low-cost currency increased by 16.5 percentage points. Most of this comes at the expense of invoicing in the destination and other vehicle currency (11 points). There is a more moderate reduction of the role of the exporter’s currency (5.5 points), which remains the lion share of invoicing.

2.4.3. Exporters in different countries Consider now that exporters selling in country d are located in two other countries, a and b, each accounting for half the exporters. The available currencies are a, b, d and a third vehicle currency c. We now index the ‘s depending on the location of the exporters. The ‘s are then given as follows:









1 a  d a    da b  2 1  db   db a    db b  2

 da 





1 d  d a    dd b  2 1  dc   dc a    dc b  2

 dd 





The ‘s for the exporter in country a are computed as:

 dd a   dd 0.03 3 1 1 0 1 b b  d a   0.76  d  0.03  0.012 2  1 3  1  ab   ad   dc a   dc 0.03  1  1 3  ac   ad  da a   1   dd a    db a    dc a  The ‘s for the exporter in country b are computed as:

14

 dd b   dd 0.03 3 1 1 0 1 a a  d b   0.76  d  0.03  0.012 2  1 3  1  ba   bd   dc b   dc 0.03  1  1 3  bc   bd  db b   1   dd b    da b    dc b  The system is solved by:

 dd a  0.125 1 0 0 0 0 1.52 2.48 b a d b  d a   0.315   1  1  1 D b   b  D  a   ad 0.96 0.96  dc a  0.125 0 0 1  bc   bd  ac   ad  da a   1   dd a    db a    dc a   dd b  0.125 1 0 0 0 0 1.52 2.48 a b d a  d b   0.315  1 1 1 D a  a  D  b   bd 0 . 96 0 . 96  dc b  0.125 0 0 1  ac   ad  bc   bd  db b   1   dd b    da b    dc b  3 1 1 1 D  0.012 2  1 3  1  1 1 3 In the absence of transaction costs, we get:

 da a   0.435

 db a   0.315

 da b   0.315  da  0.375

 db b   0.435  db  0.375

 dc a   0.125

 dc b   0.125

 dc  0.125

 dd a   0.125

 dd b   0.125

 dd  0.125

The exporters invoice predominantly in their own currency, followed by the currency of their competitors. The local and pure vehicle currencies get small weights. Consider now that all transactions are equally costly, except for one currency. In a first case, consider that the pure vehicle currency c is the low cost currency. We take the same parametrization of costs as in the previous section, implying the following values for the transaction costs: 0

    a b c b

0 d b d b



0  0.05%

0

    b a c a

0 d a d a



0  0.05%

15

The invoicing shares are then:

 da a   0.378

 db a   0.260

 da b   0.260  da  0.319

 db b   0.378  db  0.319

 dc a   0.292

 dc b   0.292

 dc  0.292

 dd a   0.070

 dd b   0.070

 dd  0.070

Compared to the case without transaction costs, the pure vehicle currency gains 16.7 percentage points in invoicing, with all other three currencies equally loosing ground. Consider now that the low cost currency is also the currency of one of the exporting countries, namely b. The transaction costs are: 0

    a b c b

0 d b d b

0

   

 0.05% 0.05%

b a c a

0 d a d a

  0.05% 0

The invoicing shares are then:

 da a   0.358  da b   0.225  da  0.292

 db a   0.461

 db b   0.567  db  0.5 14

 dc a   0.048  dc b   0.035

 dc  0.041

 dd a   0.133

 dd b   0.173

 dd  0.153

The pattern is very similar as in the previous example, with the low cost currency gaining 13.9 percentage points, with the other currencies loosing ground. As currency b now combines the advantages of being a lost cost currency, as well as the home currency of half the exporters, it plays a dominant role in world invoicing.

3. Optimal invoicing in a single currency 3.1. Adjusted invoicing weights. Our setup allows for the possibility of partial pass-through by letting exporters to invoice their sales in a basket of currencies. An alternative is to constrain exporters to set their price entirely in one of the three currencies, choosing the option leading to the highest expected profits. While these two setups 16

can appear substantially different at first, there is a simple relation between them, similar to the one documented by Engel (2005) in a two-country model. For convenience, we define the following adjusted invoicing weights: (16)

~  di   di 



j  d ,v  i

 dj

Es ej s ei

E s ei 

2

2 ~ 2  E s ei   di  E s ei   di 



j  d ,v  i

j d

Es ej s ei In a matrix form, we

write these adjusted weights as a simple transformation of the optimal invoicing weights: 2 2 ~ Es ed  Es ed   dd 2 ~ E s ev1   dv1  Es ev1 s ed 2 ~ Es ev 2 s ed E s ev 2   dv 2 ... ....

Es ev1 s ed 2 Es ev1 

Es ev 2 s ed Es ev1 s ev 2

Es ev1 s ev 2 ...

2

E s ev 2  ...

...  dd ...  dv1  X'X ...  dv 2 ... ...

 dd  dv1  dv 2 ...

The adjusted weights (16) capture the sensitivity of the price, expressed in the exporter’s currency e, to movements in exchange rates. Consider the case of a 1 percent depreciation of currency e vis-à-vis a currency i ( s ei  1 ). With a share  di of the price set in currency i, the depreciation leads to a direct increase in the exporter’s unit revenue, in her currency, by  di percent. In addition, a depreciation of currency e vis-à-vis currency i may be associated with a depreciation vis-à-vis the other currencies j. The magnitude of the second effect for a particular currency j is captured by the regression coefficient of s ej on s ej , namely Es ej s ei / E s ei  . Taking this second channel into account across all the available 2

currencies, the depreciation translates of an increase in the exporter’s unit revenue, in her currency, by ~  di percent. ~ While  di in (16) does not exactly correspond to  di , its interpretation is similar. In particular, ~ stabilizing the price in currency i (a high value of  di ) also translates into a high value of  di .

17

3.2. Expected profits. We re-write the expected profits (6) as follows: 2   k 1  λ  1 k 2 E q edk   λ   e   E s ek  λq ed  ~ 2   z   ~     λ1  ~   k k   E s ek  q ed c d  λ  1Emed q ed  k k   e   λ  1Emed q ed    λ  ~  1 λ 2~    λ  1 2 ~ k k 2    λ1      E s ek   2 λE s ek qed  E q ed   ~   2 λ  ~ E s ed  p d c d   λ1  ~ 



k ed





 







 

The last term is not affected by the invoicing decision and can be omitted. We also can abstract from the 1 constant scaling factor λ~  λ1  ~  . Using (7), (9) and (10) to substitute for s ek , q edk and  ek we

write:

 edk  z   

  

i  d ,v 's

i d

i e



  ee   λ  1

 

i  d ,v ' s

i d



  di Emed s ei

2      i  E    d s ei   2 λE    di s ei   di   di s ei   i  d ,v ' s 1  i  d ,v ' s  i  d ,v 's     2 2  λ 2~    λ  1    i i E    s   d d ei    ~ i  d ,v 's   









Note that under complete producer currency pricing (  di  0 i  d , v' s ) the profits are:

 1 λ 2~    λ  1    z, PCP    λ  1  Emed s ei  E   di s ei  ~ 2  i  d ,v 's i  d ,v 's  k ed

2

i d

We make extensive use of the following algebraic step: bd   E   a i s ei  b i s ei   a d i  d ,v 's i  d , v ' s 

a v1

a v2

... X ' X

b v1 b v2 ...

Using this transformation, the profits under producer currency pricing are:

18

'

 dd  dd 1 λ 2~    λ  1  dv1  dv1  edk  z , PCP    λ  1  di Emed s ei  X ' X 2 ~ i  d ,v ' s  dv 2  dv 2 ... ...

It is convenient to express the expected profits under a particular invoicing strategy relative to their value under producer currency pricing. The expression can be written as:

ˆ edk z    edk z    edk z , PCP  '

'

 dd  ed   ee  dd Emed s ed  v1  ev1   ee  λ  1  dv1 Emed s ev1  d  dv 2  ev 2   ee  dv 2 Emed s ev 2 ... ... ... ... '  d '   d  dd   dd  dd d d  v1    v 1   d X ' X  dv1  2 λE  dv1 X ' X   dv1   d    v2 v2   v2  v2 v2 d  d d  d  d   ...  ... ... ... ...     1    ' d d d ' d  2 d  d         d d d d  2~  dv1  dv1 v1  v1  v1 v1  2~  λ     λ  1   d   d  X ' X   d   d   λ     λ  1  d X ' X  d   v2   v2   ~ ~  dv 2   dv 2   dv 2  dv 2      d d    ...   ...  ... ... ... ...      

We re-arrange terms further to get: '

'

 dd  ed   ee  dd Emed s ed  v1  ev1   ee   v1 Emed s ev1 1 λ  1 ˆ edk z    d  λ  1 d   dv 2  ev 2   ee  dv 2 Emed s ev 2 2 1   ... ... ... ...

'

 dd  dv1 X'X  dv 2 ...

 dd   dv1   λ  1 v2 1  d ...

 dd  dd  dv1  dv1 X ' X v2 d  dv 2 ... ... '

Consider an invoicing that puts all the weight on a particular currency. We denote the profits when invoicing is entirely in currency i by ˆ edk  z , i  . For instance, we write:

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ˆ edk  z , d    ed   ee    λ  1Emed s ed 

ˆ edk z , v1   ev1   ee   λ  1Emed s ev1 

1 λ 1 2 E s ed  2 1 

 dd  v1    λ  1 X ' X dv 2 1   row 1  d ...

1 λ 1 2 E s ev1  2 1 

 dd  v1    λ  1 X ' X dv 2 1   row 1  d ...

We can write the various profits in the following matrix form: E s ed2   dd Emed s ed ˆ edk  z , d   ed   ee 2  dv1  Emed s ev1 1 λ  1 E s ev1  ˆ edk  z , v1  ev1   ee    λ  1    λ  1 X ' X v2 2 1  d Emed s ev 2 2 1   E s ev 2  ˆ edk  z , v 2   ev 2   ee ... ... ... ... ... From the first order conditions of invoicing in a basket of currencies, recall that: Emed s ed Emed s ev1 Emed s ev 2 ...

 med , s ed   dd  dd  ed   ee v1 v1 v1 e 1  med , s ev1   X ' X   dv 2    X ' X  dv 2  1  ev 2   ee  X'X  1  λ 1  e  e  med , s ev 2  1   d d ... ... ... ...

The profits then become: 2 ~ E s ed2  E s ed2  ˆ edk  z , d   dd Es ed   dd 2 2 ~ ˆ edk  z , v1  λ  1  X ' X   dv1  1 λ  1 E s ev1   λ  1 E s ev1 2  dv1  1 λ  1 E s ev1  2 2 ~ ˆ edk  z , v 2  1    dv 2 2 1   E s ev 2  1   E s ev 2 2  dv 2 2 1   E s ev 2  ... ... ... ... ...

1 2 ~ E s ed    dd   2  ~ 1 λ  1 Es ev1 2   dv1    2  1  1 ~ 2 Es ev 2    dv 2   2  ... There is a clear parallel between invoicing in a currency basket and invoicing in a single currency. For instance, entirely invoicing in currency i is more profitable than fully invoicing in

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currency e precisely when the exporter would choose a high weight for currency i in a basket invoicing:

~  di  0.5 .

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