A Spatial General Equilibrium with Discrete Choice
of Differentiated Products
Alexandrina I. Scorbureanu
∗
December 2007
Abstract
The aim of this paper is to bring a contribution for a better interpretation of trade flows among different locations, by making use of
the exact aggregation results from discrete choice literature, applied
to product differentiation. By difference with respect to the current
literature, we introduce discrete choice decisions in the consumer’s behavior among goods produced in different distant locations (differentiated goods). We also account for cost of transport among the regions.
In a new version of this model we will also introduce alternatives of
transport (by appropriate cost functions) in order to study policy implications. The two policy evaluation approaches: the one based on
representative agents and the one based on exact aggregation are linked
by the results due to MacFadden (1971) and Anderson, De Palma and
Thisse (1995), results that we will apply to a general spatial GEM
(see Elbers V.(1996),Gazel R.,Hewings J.D. and Sonis M.(1992),etc.),
where individuals choose among two levels of differentiation: products
and varieties of products. They are produced in ’prodeucer’s’ regions
and transported towards the consumer’s regions. This identification
will help us further in analysing pricing policy among different regions
in a country (or in a international context of more than one existing
countries). Conclusions can be reached with regard to environmental
taxation effects and freight transportation flows among regions.
∗
Ph.D. Student,University of Verone (Italy), e-mail: [email protected]
1
1
Introduction
The applied general equilibrium approach lies mostly on the idea of representativeness, that considers individuals as homogeneous as possible in their
choice among bundles of goods, in order to be able to aggregate their preferences into a representative-agent utility function. Nevertheless extracting the “product differentiated preferences” problem from the preferences
of individuals is crucial to understanding how modern economies operate.
Recognizing the “marketing function” and therefore the differentiated characteristics of products that address different preferences of consumers should
be a natural way of describing an economy’s mechanisms. Once it is recognized that consumers have idiosyncratic preferences, it follows that they are
prepared to pay more for variants that are better suited to their preferences.
Furthermore, at the extreme point, the existence of infinitely differentiated
goods imply that we allow for the existence of a huge number of firms that
produce slightly substitutes, and transport them to the final market. Therefore, firms are characterized by their location (region of origin) and their
relationship with the transporter (that could be represented by a branch
of the same firm or a third party logistics operator). In this analysis we
will ignore the increasing returns to scale in research and development and
consider that firms are price-takers in a static model. This is made in order
to allow a such pure as possible policy analysis.
2
Discrete Choice of Individuals
In this study, in order to model discrete choice of individuals among differentiated products, we use the approach similar to one presented in Anderson,
DePalma, Thisse (1995, Discrete choice theory of product differentiation,
Chapter 2, pp.32-33) and apply it into a new framework of “regionalised”
economy, where differentiated products are characterized by “origin” (place
where the firm that produced it is situated) and “destination” (place where
it will be consumed).
In this mainframe, the feasible consumption set for a typical individual
2
will be have the following form:

(G1, V 1) . .
.


.
. .
.



.
. .
.


FS = 
.
. . (Good = j, V ariety = r0 )


.
. .
.



.
. .
.

(G1, V R) . .
.
.
(GJ, V 1)
.
.
.
.
.
.
.
.
.
.
. (GJ, V R)








 (2.1)






where varieties are characterized by the location where they where produced
(r0 ). In order to model that, we will borrow some econometricians’ approach
in considering a population of individuals facing the same choice set J and
aiming to determine the fraction of the population choosing a different alternative. The total population will be divided into subpopulations such that
each subpopulation is homogeneous with respect to certain observable socioeconomic factors (income, age, profession). Each individual is supposed to
have a deterministic utility function U defined on a set of alternatives J.
However, the modeler can only imperfectly observe the characteristics influencing the individual’s choice and has only imperfect knowledge of the
utility function U . The functional U is decomposed in two parts: one being
a function u representing the known part of the utility and defined over
observable characteristics, and the other part, a function e that represents
the difference U − u. For each alternative j = 1, ..., J, the utility derived
from alternative j can be written as Uj = uj + ej . However, individuals are
assumed to be different one from another in subpopulation considered only
with respect to the unobservable characteristics and factors influencing his
choice, so the modeler can at best predict individual’s choice up to a probability function. Therefore the value ej is represented by a random variable j
with mean zero, where Ujh = uj + hj , and the utility of the individual h = j
is Ujh . Here hj takes into consideration the idiosyncratic taste differences of
members of the subpopulation. The probability that an individual chosen at
random selects alternative j then will be defined with the scope of deriving
an aggregation result.
3
3
The model
3.1
3.1.1
Consumer choice over differentiated products
Discrete choice framework
In this framework we consider that individuals (superscripts h stay for indicate individual decisions) have random utility functions of the form:
Ujh = uj + hj .
(3.1)
where uj represents the “objectively” achieved utility, obtained by choosing
alternative j, while hj represents the additional “subjective” utility obtained
by each individual h that cannot be observed. Consider that hj is treated as
a random variable with cumulative function of distribution F (). Give this
draw, the consumer chooses product that gives him the highest utility. We
presume that from the moment that individual h has to decide among goods
first, individuals that had chosen the same good at the first stage, tend to
have correlated values for within the same group. Therefore, it will be
erroneous to consider that are i.i.d. and we will introduce in the following
part of our paper the nested logit model, that is commonly used in literature
in the case of two-level differentiation of goods (see Ben-Akiva, MacFadden,
DePalma and Thisse). Given the utility function in (3.1), the probability
that an individual chooses alternative j over alternative j 0 is given by:
P rjh = P rob[Ujh ≥ Ujh0 , ∀j 0 = 1, ..., J]
= P rob[uj + hj
≥ uj 0 + hj0 , ∀j 0 = 1, ..., J]
= P rob[hj − hj0
≥ uj 0 − uj , ∀j 0 = 1, ..., J]
(3.2)
We can consider the statement (3.2) as being the probability that any individual will choose alternative j, or equivalently, the expected fraction of
individuals that choose that good. The solution for these probabilities depends on the distribution function F (), as well as on the specification of
utility uj .
Computing choice probabilities from the above specification may be computationally intensive since they require integration over various subsets of
the domain of the distribution function. However, McFadden(1978,1981)
has shown that for some class of distribution functions F (), known as “the
4
generalized extreme value functions” from which the multinomial logit is a
special case (see Appendix for a detailed exposure), for which the expected
demands can be easily obtained by differentiation.
Define the generalized extreme value distribution function:
F (1 , ..., n ) = exp −H e−1 , e−2 , ..., e−n
(3.3)
with H a nonnegative function defined over <N
+ satisfying the following
properties:
(i) H is homogeneous of degree 1/µ;
(ii) limxi ∞ H (x1 , ..., xn ) = ∞∀i = 1, ...n (conditions needed to ensure that
F () is a cumulative probability distribution function), then the choice probabilities P robj may be determined as:
P robj = µ ·
∂ln H (eu1 , ..., eun )
.
∂uj
(3.4)
For example, by using the particularization of H like:
H (1 , ..., n ) =
n
X
1/µ
j .
(3.5)
j=1
We can verify that the properties above are respected and obtain the
appropriate cumulative distribution function:
h P
i
1/µ
F (1 , .., n ) = exp [−H (e−1 , e−2 , .., e−n )] = exp − nj=1 exp −j
1/µ Qn
−
= j exp −e j
We can conclude therefore that the product of n i.i.d. double exponential distributions characterizes the stochastic behavior of utilities uhj and it
follows from the theorem (see appendix) that:
P robj = µ ·
∂ln
Pn
j 0 6=j
euj 0 /µ
∂uj
euj /µ
= Pn
.
uj 0 /µ
j 0 6=j e
(3.6)
which are the choice probabilities derived from a multinomial logit population, characterized by the dispersion parameter µ (positive constant). It
also provides a good approximation to the normal distribution as well (see
Ben-Akiva and Lerman (1985,p.128)). Furthermore, we can compute the
5
cross-elasticities of probability with respect to the utility from (3.6):
P robj P robj 0
;
µ
P robj 0 ·uj 0
P robj ; uj 0 = −
µ
j 0 and derivatives that
∂P robj
∂uj 0
Elast
where
j, j 0
= 1, ...n, j 6=
=−
are independent of j.
Therefore, any change in the “objective” utility level associated with alternative j will affect in a symmetric way the choice probabilities of all other
alternatives. This property usually found in the literature under denomination of independence of irrelevant alternatives can be bypassed by using a
nested multinomial logit, as suggested by Ben-Akiva(1973). We introduce a
nested multinomial logit system of demand, built on two levels of decision
on consumer’s behalf, a choice among types of goods and a successive choice
among varieties of goods.
3.1.2
Nested Logit Demand System
Suppose now that we are in a case where consumers have to choose among
two levels of product differentiation (good and variety). A priori and in
general, each variety of good j is produced in one single region r0 (by the
producer firm), and consumed in region r. Therefore, Xr,r0 (j) represents the
demanded variety r0 of good j that is produced in region r and consumed in
region r. Therefore in the first stage, consumers choose in which industry
they want to buy (a choice between js, where j is the index for goods)
and in the second stage they decide varieties they prefer among the goods
form industry chosen at stage one. Technically speaking, they first make
a choice among a set of goods J that groups varieties r that have several
observable characteristics in common: Jr ; r = 1, ..., R.It is assumed that
individual selects with a certain probability the subset Jr ∈ J, from which
he will then choose a particular variety r ∈ Jr according to a probability
that depends on the utility of that alternative ur∈Jr . The multinomial logit
model is traditionally used for each of these two stages (see Ben-Akiva).
Suppose that individual had chosen Jr from the set J at the first stage. We
count for the components of each subset in the following way: Jr = 1, ..., R
and we have G classes of goods J = 1, ..., G. MacFadden(1981) suggests to
6
use for this type of choice a nested logit demand system. Therefore, like in
(3.5) we choose a particular form for the function H:
H e
−1
, ..., e
−R
=
G
X
#1−µ
"
Jr =1
X
−r /1−µ
e
.
(3.7)
r∈Jr
where 0 ≤ µ < 1 Using this choice of H , we compute the probabilities as
in (3.4):
euj /(1−µ)
Dr1−µ
∂ln H (eu1 , ..., eun )
=
· PR
P robj = µ ·
1−µ , ∀j ∈ Jr .
∂uj
Dr
r=1 Dr
where the term Dr =
P
k∈Jr
(3.8)
euk /(1−µ) and is called “inclusive value” 1 , since
it measures the sum of utilities obtained from all products in the group Jr .
In the same equation (3.10), is due to Johnson and Kotz (1972,256) to report
p
that the parameter µ = corr(r , k ) for r, k ∈ Jr , r 6= k roughly measures
the correlation between random terms r within a group2 .
Let us consider a cell of individuals that are consumers of varieties that
form industry j. Let us consider that the each individual’s utility in this
cell, derived from choosing a particular variety r0 depends on the variety’s
quality characteristics sr0 and its price pr0 and let us furthermore account
for all other varieties with the notations sr00 and respectively pr00 .
vrh0 = ln sr0 − ln pr0 + hr0 ;
(3.9)
Note that ln sr0 and ln pr0 terms are common to all h individuals in this
industry cell. By substituting (3.9) into (3.10), and after some simple logarithmic computation we obtain the probability that an individual chooses
in the second stage variety r0 :
1/µ
−1/µ
s 0 ·p 0
P robr0 = P r 1/µ r −1/µ .
r00 sr00 · pr00
(3.10)
Let us consider that individual h obtains utility Vjh0 from choosing industry j 0 , that is Vjh0 = ln zj 0 − ln pj 0 + hj0 ;. Like in the previous case, we obtain
1
see R. Feenstra, Advanced International Trade - Theory and evidence, Princeton
University Press,1995,p234
2
Under the additional conditions that corr(r , k ) > 0 for µ > 0
7
the probability that an individual chooses to consume goods from industry
j0:
1/v −1/v
z j 0 pj 0
P
.
1/v −1/v
1/µ
−1/µ µ/v
zj 0 pj 0 +
r0 sr0 · pr0
(3.11)
P 1/µ −1/µ
= µ ln r0 sr0 · pr0
reflects the con-
exp(Vj 0 /v )
P robj 0 =
=
exp(Vj 0 /v ) + exp(Vj/v )
where Vj = µ ln
P
r0
exp(ln sr0 −ln pr0 )
µ
sistency with the second stage decision (individual’s utility from choosing
industry j is composed by the partial utilities obtained by consuming varieties from this industry).
3.2
Aggregate demand
As the aggregate level, as MacFadden has noted, “intra-individual and interindividual variations in tastes are indistinguishable in their effect on the
observed distribution of demand” (c.1981). If we consider an industry cell
population of N individuals (N consumers for an industrial branch) who are
statistically and identical independent, and for simplicity consider that demand is uniformly distributed among industries (industrial cells of the same
dimension), the former assumption means that choices are governed by the
same probability distribution, although the actual choices will generally differ across individuals. The latter implies that the probability of an individual choosing a particular alternative is independent of the choices of other
individuals. Under these assumptions, the distribution of choices across individuals is multinomial with mean X¯j 0 that is given by: X¯j 0 = N · P robj 0
with j 0 = 1, ..., J which is the expected demand in the industry cell j 0 (see
Anderson, DePalma, Thisse, 1995. Discrete choice theory of product differentiation, Chapter 2, pp.34).
If we suppose that there are k individuals that choose sub-cell variety r0
and that demand is also uniformly distributed among varieties (variety subcells having the same dimension), we may compute the aggregate demand
at the variety level:
1/µ
X
r,r0
−1/µ
s 0 ·p 0
(j ) = k · P rob = k P r 1/µ r −1/µ .
r00 sr00 · pr00
0
r0
8
(3.12)
Integrating the results obtained in (3.10) we obtain the demand for goods
making part of industry j 0 in the consumer’s residential region (or country)
r:
1/v −1/v
z j 0 pj 0
Xr (j ) = N · P robj 0 = N ·
P
.
1/v −1/v
1/µ
−1/µ µ/v
zj 0 pj 0 +
r0 sr0 · pr0
0
(3.13)
Furthermore, we can easily compute the aggregate demand in a country (or
region):
Xr =
J
X
0
Xr (j ) = N
j 0 =1
J
X
P robj 0 .
(3.14)
j 0 =1
The prices pr0 and pj 0 from now on will be referred as: pr0 = Pr,r0
#
"
pj 0 = pj =
X
1
1−η
1−η
θr,r0 Pr,r
0
(3.15)
r0
The first price is the price of variety produced in region r0 and consumed in
region r, while the second definition refers to price index of good j which is
composed of different varieties.
To reassume, we have obtained now aggregate demand functions for all
levels of detail, variety, industry, country with the equation (3.12), equation
(3.13) and respectively, equation (3.16).
At regional level, there is nevertheless the budget constraint that must
be fulfilled (given by an initial total wealth of individuals Υ):
Pr Xr ≤ Υ ⇔ Pr N
J
X
P robj 0 ≤ Υ.
(3.16)
j 0 =1
4
Trade costs and firms behavior
We consider two competitive industries that coexist in the economy: a production sector, and a transport one. These two sectors are represented by
homogeneous firms, such that aggregation for each sector into a representative agent is possible. Firms although are placed in different regions and
produce slightly differentiated goods.
9
4.1
The trade cost
We consider that there is a trade cost associated with the flows of goods
between regions, whereas the cost of trade within the same region (or country) is zero. At a regional level, trade between regions is subject to a trade
price Pr,r0 (j) defined as:
Pr0 ,r (j) = 1 + τr0 ,r (j) · Pr0 ,r0 (j).
(4.1)
where τr0 ,r (j) > 0 is the transport tariff t(j) (that we will derive later on)
multiplied by distance between the two regions, or in other words, τr0 ,r (j) =
t(j)Dr0 ,r . The notations above state simply that the price of a good j that is
produced by country (or region) r0 is more expensive in the country r than
in country r0 or in other words, transport costs inflate f.o.b. prices of goods
(the “iceberg costs approach” frequently used in transportation and gravity
literature, see Anderson and van Wincoop 2003,2004).
4.2
The production sector
We consider that firms are placed in determined regions and each of them
produces a specific variety of a product. All over this section we will only
consider the case for a firm that produces a variety of j. Therefore, the
aggregate production of a region among all industries will have the value
P
j,r0 ,r Pr0 ,r (j) · Yr0 (j) where Pr0 ,r (j) represents the consumer price of the
variety r0 produced in the region with the same name and consumed in region
r. Recall from the previous section that Pr0 ,r (j) = 1 + τr0 ,r (j) ·Pr0 ,r0 (j).For
simplicity let us consider that intermediary marketed goods do not exist,
and that firms only employ capital and labor (that do not require to be
transported) in their production processes. Each production firm demands
a fixed quantity of transport services per each destination r, that we call
Zrdem
0 ,r (j) from the transportation firm. In this situation, the representative
firm in region r0 aims to maximize a multicriterial objective function, formed
on one hand, by the minimization of its external expenditure in transport
services (but under the constraint of serving all its external market) and on
the other hand, seeking to minimize its internal production costs (under the
technological constraint). We also consider prices of the production factors
10
and transportation cost as being exogenously taken in a context of perfect
competition. Therefore the dual-minimization plan of the representative
production firm in region r0 for a particular good j is composed as follows
(we eliminated the subscript(j) from the following program, in order to
simplify notation):
dem
dem
dem
M IN IM IZE ΛFprod (Ldem
r0 ; Kr0 ) + (1 − Λ) Ftransp (Lr0 ; Kr0 )
where
:
Fprod =
M IN IM IZE wLdem
+ iKrdem
0
r0
α dem 1−α
Yr ≥ Krdem
· L
0
nPr0
o
dem t(j)
M IN IM IZE
Z
0
r r ,r
P dem
¯
Yr ≥ r Zr0 ,r
P dem
X¯r,r0 ≥
Z0
SU BJ.T O :
Ftransp =
SU BJ.T O :
r
Zrdem
0 ,r
r ,r
≥ 0;
(4.2)
where
dem
Fprod (Ldem
r0 (j); Kr0 (j)
represents the optimum-value function ob-
tained after the cost-minimizing program of the firm regarding the prodem
ductive plan, and Ftransp (Ldem
r0 (j); Kr0 (j) represents the optimum-value
function obtained after the cost-minimizing program of the firm regarding
the transportation plan of excessive production (versus consumer’s region).
Therefore, productive firms’ objective-program is shared among two plans:
choosing production factors such as to minimize the production costs and
minimizing the cost of optimal networking allocation of excess production
(linear programming problem, originated in Dantzig 1958). Furthermore,
we divide the firm’s program into two sub-programs, corresponding to the
two sub-objectives, that we will solve one by one in the following.
Sub-program 1 The cost-minimizing regional factor choices on behalf of
the productive firm, respectively Krdem
(j); Ldem
(j) are the solutions to the
0
r
following dual-cost minimization program:
dem
= M IN IM IZE w · Ldem
r0 (j) + i · Kr0 (j)
subj.to : Yr0 (j) ≥ α ln Krdem
(j) + (1 − α) ln Ldem
0
r0 (j)
Fprod
11
(4.3)
dem
where Fprod ; Ldem
r0 ; Kr0 (j) is the minimum cost function obtained
from the cost-minimization problem, by demanding the optimal quantities
of production factors, Kr∗dem
(j); L∗dem
(j).
0
r0
Sub-program 2 The cost-minimizing network allocations of variety r’, according to the firm’s demand for transport quantity Zrdem
0 ,r (j), are the solutions of the following dual-cost minimization program:
nP o
dem (j) · t(j)
Ftransp =
M IN IM IZE
Z
0
r
r ,r
P dem
¯
SU BJ.T O : Yr0 ≥ r Zr0 ,r (j)
P dem
X¯r,r0 ≥
Z 0 (j)
r
(4.4)
r ,r
Zrdem
0 ,r (j) ≥ 0;
In order to simplify the problem, let us consider an equivalent and simplified problem (for all j, that will be omitted from subscripts), and let us
assume that all constraints are binding with equality, like in the following:
hP i
dem + iK dem + (1 − Λ)
dem · t(j)
P
M IN IM IZELdem
=
Λ
wL
Z
,Krdem
, r Zrdem
r
r0
r0
r0 ,r
0
0 ,r ,Λ,l1 ,l2
r0
dem α dem 1−α
.
Lr0
SU BJ.T O :
Yr0 = Kr0
P dem
1
r Zr0 ,r = ω(j) · Yr0 .
(4.5)
where ω(j) represents the proportion between the production and the quantity that is exported by the firm and we consider implicitly all variables
nonnegative. By substituting the production into last constraint of (4.5) we
dem α dem 1−α
P
1
. The equation (4.5) aims
Lr0
obtain that r Zrdem
0 ,r = ω(j) · Kr 0
to maximize a convex function under a concave set of constraints, and we
aim to minimize both sub-programs, by choosing Λ > 0, therefore we can
compute in a traditional manner the first order conditions, and obtain:
L∗dem
= L∗dem
≤ Lsup
r0
r0
r0 ;
Kr∗dem
=
0
Zr∗dem
=
0
Yr∗0 =
wα
∗dem
i(1−α) · Lr0 ;
α
P ∗dem
wα
Z
=
0
i(1−α)
r r ,rα
wα
∗dem
· Lr0 .
i(1−α)
12
·
1
ω(j)
· L∗dem
;
r0
(4.6)
where L∗dem
; Kr∗dem
represent the optimal factor demands for the produc0
r0
P ∗dem
tion firm whereas r Zr0 ,r = Z ∗dem (j) represents the sum by destinations
of the quantities of goods exported by the representative firm from industry j (therefore that form the transport demand). By l1 ; l2 we denoted
the Lagrange multipliers associated respectively to the first and the second
constraint in (4.5).
Claim 1 Export decisions of the firms are taken as below:
1. If the wage rate slightly increases, the firm will export more, at increasing marginal rates;
2. If the marginal transport cost slightly increases, the firm will export
less, at decreasing marginal rates;
3. If the capital price slightly increases,the firm will export more, at increasing marginal rates;
Proof 1 See the appendix.
Let us rewrite now the total cost function in (4.5), by pooling in all
information from (4.6) and by rearranging terms we obtain:
α i
h
wα
∗dem wΛ + (1 − Λ) t(j)
=
L
T Crprod
0
0
r
1−α
ω(j) i(1−α)
(4.7)
By totally differentiating (4.7), we obtain the marginal cost function of
the representative production firm in region r0 :
0
t(j)
w
r
M Cprod
= Λ 1−α
− (Λ + 1) ω(j)
·
wα
i(1−α)
α
;
(4.8)
The total revenue obtained in production activities in industry j in region r0 is then:
Revrprod
(j)
0
=
ω(j) − 1
ω(j)
Yr0 (j)Pr0 r0 (j) +
13
1
Yr0 (j)Prr0 (j)
ω(j)
(4.9)
where by replacing the values for Pr0 r0 (j), Prr0 (j) and Yr0 (j) we obtain a
function of revenue that depends on distances between locations of customers
and the location of the production firm.
The productive firm that is competitive, fix its price Pr0 r0 (j) at the level
of its marginal cost, and the factory gate price is equal for all clients (whatever would be their location). Therefore the price at the factory gate will
be
α
t(j)
wα
w
.
(4.10)
− (Λ + 1)
·
Pr0 r0 (j) = Λ
1−α
ω(j)
i(1 − α)
At industry level, the price index for industry j good, across all regions
is the composite from the first equation below, whereas the second equation
represents the cost of living index in the consumer’s region r:
1−η
r0 θr0 r Pr0 r (j)
Pr (j) =
P
Pr =
hP
j
βr (j)Pr
(j)1−
i
1
1−η
1
1−
.
(4.11)
where θrR0 ,r0 represents the preference coefficient for the region r0 and βr (j)
represents the preference coefficient for the industry j. Firms producing the
same good across regions are homogeneous, therefore the price at the factory
gate will be the same for all firms that form the same industry. A further
development will be to consider difference in technologies endowments of
companies across regions.
4.3
The transport sector
In this sector the firms are competitive and they transport variety produced
in region r0 to any region r by employing capital and labor force. We consider
that prices of production factors, the labor wage and the capital remuneration are exogenous. In analogy we use the notation KrT0 dem that represents
the capital-input demanded by the transport firm whereas LTr0dem represents
the region-specific quantity of labor demanded by the transportation firm.
Additionally, we must specify that αt represents the elasticity of substitution between the two production factors for the transportation firm. The
total cost that a transport firm has to face therefore is:
T T C = w · LTt dem + i · KrT0 dem .
14
(4.12)
Program 1 The cost-minimizing factor demands of the transporter t, re
spectively KrT0¯dem ; LTr0¯dem are the solutions to the following dual-cost minimization program:
M IN IM IZE T T Cr0 = w · LTr0dem + i · KrT0 dem
α 1−αt
SU BJ.T O : Zrsup
= KrT0 dem t · LTr0dem
0
where Zrsup
=
0
P P
j
r
(4.13)
Zrsup
0 r (j) represents the total quantity transported by
the representative firm that takes orders from all industries in region r0 and
delivers versus all destinations r.
This implies that the marginal cost of transporting good j from region
r0 to region r is given by:
w
T M Cr0 =
1 − αt
1−αt αt
i
·
αt
(4.14)
where T M C = t(j) would represent the price per unit of distance transported by the transportation firm for a unit of currency worth of good j,in
the absence of any distorsive tax, therefore:
w
t(j) =
1 − αt
1−αt αt
i
·
αt
(4.15)
Let us revisit the trade cost part in (4.1) and it yields that: if t(j) =
i1−αt h iαt
h
w
then τr0 ,r (j) = 1−α
· αit
· Dr0 ,r . From this new statement
t
τr0 ,r (j)
Dr0 ,r ,
we obtain:
"
Pr0 ,r (j) = 1 +
w
1 − αt
#
1−αt αt
i
·
· Dr0 ,r Pr0 ,r0 (j)
αt
(4.16)
This new formula will allow us to reach comparative statics conclusions
regarding the evolution and pattern of trade flows in the following section.
Now let us consider that ψe represents the environmental tax, that the
transport firm has to pay on its transport services supply. In the transport
sector, profits are calculated by taking into consideration the environmental
tax 3 , therefore the transportation firm will implement the following profit
3
For a similar treating, see also van den Bergh and Rietveld. Modeling the economic
effects of environmental policy measures applied to transports, Springer 1995
15
maximizing program:
M AXΠr0 = (t(j) − ψe ) Zrsup
− T T¯Cr0
0
1−αt
α
SU BJ.T O : Zrsup
= KrT0 dem t · LTr0dem
.
0
(4.17)
By solving this program we automatically obtain the following optimum
bundle solution regarding the factors demand on behalf of the transportation
firm and the total cost fumction:
dem = L∗T dem ≤ Lsup ;
L∗T
r0
r0
r0
dem =
Kr∗T
0
αt i ∗T dem
,
1−αt w Lr0
T T C ∗ = wL + iK = w +
(4.18)
i2
w
·
1−αt
αt
.
By replacing the last equation from (4.18) in the zero/profit condition
found in (4.17) we obtain the cost of transport t(j):
t(j) = ψe +
i(1−αt )(αt +1)/αt +i(1−αt )αt ·w2 αt
.
(wαt )( αt +1)/αt
(4.19)
The supply of transport services must to fulfill the constraint condition:
h
iα h
i1−αt i(1 − α ) αt
t
sup
tdem t
tdem
Zr 0 = K
=
L
Ltdem
.
(4.20)
r0
wαt
therefore, by deriving furthermore this condition we obtain:
(α+αt ) X Ldem
i
αt (1 − α) α
r0 (j)
=
.
ω(j)
w
α(1 − αt )
(4.21)
j
This result may be interpreted as follows: the more labor-intensive production industries and capital-intensive transport industries are in one region,
the more production will be exported in other regions (in order to see this
clearly, pose for example αt = 2α).
At this point we are able to compute the total gross revenue in the
transport sector in region r0 :
Revrtransp
= Zrsup
· t(j)
0
0
P sup
=
r Zr0 r (j)
tdem
αt tdem 1−αt
= K
L
α t
i(1−αt )
Ltdem
· ψe +
=
r0
wαt
i(1−αt )(αt +1)/αt +i(1−αt )αt ·w2 αt
;
(wαt )( αt +1)/αt
(4.22)
It results from (4.22) and (4.9) that the total revenue in region r0 is
T OT REV = Revrprod
+ Revrtransp
.
0
0
16
4.4
The government
The environmental policy brings us immediately to the government sector,
even if we assume that the environmental tax is exogenous. Revenues and
outlays are balanced, while all outlays are used to pay wages at the national rate to labor, which is the only production factor of the government.
therefore we have to model the government sector:
α t
t)
Ltdem
Gr0 = ψe Zrsup
= ψe i(1−α
0
r0
wαt
wLGdem
= Gr 0 ;
r0
(4.23)
It results that the labor demand on behalf of the government is then:
i(1 − αt ) αt tdem
1
Gdem
Lr0 ;
(4.24)
Lr0
= ψe
w
wαt
5
Equilibrium conditions and price set-up
The market clearing conditions for all regional and national markets are
formulated in the following. They state that supply through production
and transport have to satisfy demands for each good in each region.
Labour market equilibrium for the economy (summed over all industries
in region r0 ) has to fulfill the equilibrium condition:
LGdem
+
r0
X
T dem
Ldem
=
r0 (j) + Lr0
j
X
¯
Lsup
r0 (j).
(5.1)
j
¯
0
where Lsup
r0 represents the initial given endowment in labor force of region r
(once we have assumed that labor offer is exogenous). By replacing (4.24),
(4.18) and (4.6) into (5.1) we will find the price of labor (wage) in the
economy (the same all over the regions).
Capital market equilibrium in the economy (region) is given by:
X
Krdem
(j) + KrT0 dem ≤
0
j
X
¯
j
Krsup
0 (j).
(5.2)
¯
where Krsup
0 (j) represents the initial given endowment in capital in each industry from region r0 . For now let us just assume that capital and labor are
immobile across regions, therefore their markets clear at the initial endowment level.
17
Goods and transport markets price-equilibriums are given by the following system of equations (assuming that there exist no re-sales activity):
P
r
Zrdem
= Yr0 (j)∀j, ∀r, r0
0 ,r (j) + Xr 0 ,r 0 (j)
P dem
P sup
r Zr0 ,r (j) =
r Zr0 ,r (j)
P dem
P sup
j Zr0 (j) =
j Zr0 (j)
(5.3)
Zrdem
= Xr,r0 (j)
0 ,r (j)
where Zrdem
(j) =
0
P
r
sup
Zrdem
0 ,r (j) and Zr 0 (j) =
P
r
Zrsup
0 ,r (j) are the sums by
destination of demanded and respectively offered quantity of transport services (quantities of demand to be transported). The last condition implies
that the imported (transported) quantity of variety r0 into the consumer’s
region r must correspond to the consumption of the same variety in the
consumer’s region (we assume that there is no re-export of goods). Remark that Xr,r0 (j) 6= Xr0 ,r0 (j) (the first term indicates the demand in the
consumer region r of variety produced in region r0 in terms of the same industry j, whereas the second term states for the domestic demand in region
r0 of the variety produced in home region). For an easier interpretation we
might see the transported quantity outside the region as an export and the
imported quantity inside a region as an import.
The first part of equation (5.3) suggests that from the point of view
of the region where the producer of that variety has its establishment (r0 ),
the total quantity that he decides to export in all the other regions (that
is the quantity for which he demands transport services, Zrdem
0 ,r ) plus the
total quantity of variety r0 consumed locally, must be at most equal to its
production capacity Yr0 (j). It would be useful here to imagine each producer
like a monopolist of his own variety (the only one that produces variety r0
of good j). The second condition in (5.3) states that the transport demand
by all destinations r must not exceed the transport supply offered (summed
over all destinations).
6
Closing the model and explicit solutions
Calibration of the model in the case of two regions is made by using the two
given static social accounting matrices corresponding to the two regions.
18
We use the derived formulas in order to calibrate parameters, as described
subsequently. Following the notation in the first part of this paper, the
variety demand in region r is calculated as:
1/µ −1/µ
s r 0 pr 0
P 1/µ −1/µ .
1/µ −1/µ
sr0 pr0
+ r00 sr00 · pr00
Xr,r0 (j 0 ) = k · P robr0 = k ·
(6.1)
Furthermore, the industry demand is:
1/v −1/v
Xr (j 0 ) = N · P robj 0 = N ·
zj 0 pj 0
P
.
1/v −1/v
1/µ
−1/µ µ/v
zj 0 p j 0 +
r0 sr0 · pr0
(6.2)
and the aggreegated country (region) demand is:
1/v −1/v
Xr = N ·
P
j0
P robj 0 = N
zj 0 pj 0
P
1/v −1/v
1/µ −1/µ µ/v
zj 0 pj 0
+
r 0 sr 0 ·pr 0
(6.3)
In these formulas, we must specify that the relation between prices are intended to be compatible with the results obtained in the supply side sections:
1−αt αt
w
i
pr0 = Pr0 r (j) = 1 + 1−αt
· αt
· Dr0 ,r Pr0 ,r0 (j)
1
P
1−η 1−η
(6.4)
pj 0 = Pr (j) =
r0 θr0 r Pr0 r (j)
hP
i 1
1− 1− .
Pr =
j βr (j)Pr (j)
We may compute the ratio between two regions variety demands and obtain:
−1/µ
1−αt αt
w
i
1+ 1−α
D
Pr0 ,r0 (j)−1/µ
0
r1,r
αt
t
−1/µ
1−αt αt
P
1/µ −1/µ
1/µ
w
i
Dr1,r0
Pr0 ,r0 (j)−1/µ + r00 sr00 pr00
sr0 1+ 1−α
α
1/µ
sr 0
Xr1,r0 (j 0 )
k1
=
·
Xr2,r0 (j 0 )
k2
t
t
−1/µ
1−αt αt
w
i
Pr0 ,r0 (j)−1/µ
1+ 1−α
Dr2r0
α
t
t
−1/µ
1−αt αt
P
1/µ
1/µ −1/µ
w
i
sr0 1+ 1−α
Dr2r0
Pr0 ,r0 (j)−1/µ + r00 sr00 pr00
α
1/µ
sr 0
t
t
(6.5)
We may observe that it depends on the number of consumers of the variety
in each region and it also depends on the distance at which each consuming
region is from the producer’s center.
At the industry level, we can compute the aggregate demand ratio be-
19
tween two regions:
−1
1/v P
1−η v(1−η)
z 0
r 0 θrr 0 Pr1r 0 (j)
j
!µ/v
−1/µ
−1
1−αt
−1/µ
1/v P
w
i αt D
1−η v(1−η) + P s1/µ 1+
P 0 0 (j)
z 0
αt
1−αt
r 0 r1
r0 r0
r 0 θrr 0 Pr1,r 0 (j)
r r
j
−1
1/v P
1−η v(1−η)
z 0
r 0 θrr 0 Pr2,r 0 (j)
j
!µ/v
−1/µ
−1
1−αt
1/v P
−1/µ
w
i αt D
1−η v(1−η) + P s1/µ 1+
z 0
P 0 0 (j)
αt
1−αt
r 0 θrr 0 Pr2,r 0 (j)
r0 r0
r 0 r2
j
r r
[
Xr1
Xr2
=
N1
N2
·
[
]
(
]
[
[
)
( )
]
]
(
)
( )
(6.6)
We can observe that this ratio depends on the number of abitants in each region, and it depends on the distance of each region (explicitly and implicitly
through Prr0 (j) that contains transport costs that are proportional to the
distance from the producer center) from the producer region r0 , provided
that we consider identical taste parameters across regions at variety level
θr1r0 = θr2r0 = θrr0 and identical elasticities.
Now we can write the formulas for aggregated consumption values and
see comparisions between regions (take for example regions r1 and r2). The
aggregated consumption value in a region (or country) r1 with N 1 abitants,
across all industries j 0 will have the following form:
P
Pr1 Xr1 = Pr1 N 1 · j 0 P robj 0
z
1/v −1/v
0 p 0
j
= Pr1 N 1 j1/v −1/v
zj 0 p j 0
P
−1/µ µ/v
1/µ
+
r0 sr0 · pr0
1
i 1−
hP
1
P
1−η
(1−η)(1−)
=
N1
r0 θr0 r Pr0 r (j)
j βr1 (j)
1/v −1/v
zj 0 pj 0
.
P
1/v −1/v
1/µ −1/µ µ/v
zj 0 p j 0
+
r 0 sr 0 pr 0
(6.7)
Proceeding in the same manner with the region r2 and performing the ratio
between aggregated consumption values levels we obtain:
P
Pr1 Xr1
Pr2 Xr2
=
N1
N2
·
P
j
1
1−
1/v −1/v
z 0 p 0
j
j
P
1/v −1/v
1/µ −1/µ µ/v
z 0 p 0
+
r 0 sr 0 ·pr 0
j
j
1/v −1/v
1
1
z 0 p 0
1−
P
j
∗j
1−η (1−η)(1−)
θ
P
(j)
0
0
0
r r r rr
P
1/v −1/v
1/µ −1/µ µ/v
z 0 p 0 +
r 0 sr 0 ·p∗r 0
j
∗j
1
P
1−η (1−η)(1−)
]
j βr1 (j)[ r 0 θrr 0 Prr 0 (j)
βr2 (j)[
]
(6.8)
where the ‘*’ subscripts attention us to the fact that variables depend on
region characteristics (for example price p∗j 0 depends on distance of the
consumer region from the factory gate). For this reason we cannot simplify
terms provenient from multinomial forms in (6.8). This ratio depends on
20
the industry preference parameters and variety taste parameters and the
number of abitants of the two regions.
The ratio of total labor demand (in all regions) used in production and
respectively transport activities can be easily computed by summing up over
all regions and industries the following ratio:
sup
L∗dem
(j)
(1 − α)
r0
4Kr0 wα(3α − 1) + 2L(2α − 1)
=
sup
∗T
dem
Lr0
(3α − 1)(2L − wKr0 )
(6.9)
Gdem .
where L = Lsup
r0 − Lr0
It is interesting now to verify that the value of consumption in one region
equals the gross regional product (value of transport income plus the value
of production income):
(6.10)
Pr Xr = GRPr0
where the gross regional product can be obtained by summing up the
outcome from the production across all industries and transports activities
respectively from (4.22) and (4.9):
GRPr0 =
7
P
j
(j)Pr0 (j) +
Yrprod
0
P
j
Zrsup
0 t(j)
(6.11)
Conclusions
We developed a trade flow pattern based on the assumption that we consider
products that are localised sufficiently distant, are differentiable in terms of
consumer’s choice. Future developments of the actual model will include
the introduction of endogeneous labor supply and comparative statics. Also
considering intermediate goods trade within CET production functions will
be among our objectives for a future work. Nevertheless, considering cost
functions that distinguish among various alternatives of transport (road,
rail, maritime, etc) with different taxation systems and technologies, will
be considered in order to evaluate the effects of changes from a system to
another on the social welfare, employment and international trade. Adding
dynamics to the model, under its simplified version, is also feasible and it
may lead to interesting results.
21
8
Appendices
A
The generalized extreme value specification
The generic specification of the generalized extreme value function has the
cumulative distribution function:
#
" x − µ −1/ξ
F (x, µ, σ, ξ) = exp − 1 + ξ
σ
with 1 + ξ
x−µ σ
(A.1)
> 0 where µ ∈ < is the location parameter,σ > 0 is the
scale parameter and ξ ∈ < is the shape parameter. The density function is
consequently
" #
1
x − µ −1/ξ−1
x − µ −1/ξ
f (x, µ, σ, ξ) = · 1 + ξ
· exp − 1 + ξ
σ
σ
σ
(A.2)
x−µ > 0. A variable X distributed like
under the same condition 1 + ξ σ
GEV will be E(X) = µ −
σ
ξ
+ σξ g1 and the variance V ar(X) =
σ2
(g
ξ2 2
− g12 )
where gk = Γ(1 − kξ) with k = 1, 2, 3, 4 and Γ(t) is the Gamma function.
Credit for the extremal types theorem (or convergence to types theorem)
is given to Gnedenko (1948), previous versions were stated by Fisher and
Tippett in 1928 and Fréchet in 1927.
B
Theorems of derivation of Extreme value distributions
Theorem 1 Extremal types or convergence to types theorem, Gnedenko 1948 Let X1 , X2 ... be a sequence of independent and identically distributed random variables, say Mn = max X1 , ..., Xn . If two sequences of
Mn−bn
≤
x
real numbers an , bn exist such that an > 0 and liminf
P
= F (x)
n
an
then if F is a non degenerate distribution function, it belongs to either the
Gumbel, the Fréchet or the Weibull family.
Clearly, the theorem can be reformulated saying that F is a member of the
GEV family. Thus the role of extremal types theorem for maxima is similar
22
to that of the central limit theorem for averages. The latter states that the
limit distribution of arithmetic mean of a sequence Xn of random variable
is the normal distribution no matter what the distribution of the Xn , The
extremal types theorem is similar in scope where maxima is substituted for
average and GEV distribution is substituted for normal distribution.
Theorem 2 (McFadden 1978,1981) Let H be a nonnegative function deN that satisfies the following properties:
fined on R+
(i) H is homogeneous of degree one;
(ii)H → inf as any of its argument approaches infinity;
(iii) The mixed partial derivatives of H with respect to k variables exist
and are continuous, nonnegative if k is odd and nonpositive if k is even,
K=1,...,N.
C
Proofs
C.1
Proof of Claim (3)
P dem
1
Yr0 , that is if ω(j) rises,
= ω(j)
We know (by definition) that
r Zr 0 r
P dem
r Zr0 r decreases proportionally. Furthermore, we will derive conditions
for ω(j). We solve the multi-criterial minimization problem, that we will
report here for simplicity:
M inimizeF Ldem ,K dem ,P
r0
r
Zrdem
0 ,r ,Λ,l1 ,l2
subj.to :
hP i
dem + (1 − Λ)
dem · t(j)
Λ wLdem
+
iK
Z
r
r0
r0 ,r
dem α dem 1−α
Yr0 = K
Lr0
.
P dem
1
r Zr0 ,r = ω(j) · Yr0 .
(C.1)
Therefore the Lagrangian for this problem is:
P dem
dem ,
dem + iK dem
L
Ldem
r Zr0 ,r , Λ, l1 , l2 = Λ wLr0
r0 , K
dem α dem 1−α P
0
+ (1 − Λ)t(j) r Zrdem
Y
−
K
Lr0
+
l
0 ,r
1
r
h
i
P
α
1−α
1
Ldem
− r Zrdem
+ l2 ω(j)
K dem
.
0 ,r
r0
23
(C.2)
From this we derive the first order conditions:
α dem −α
∂L
= Λw − l1 (1 − α) K dem
Lr 0
+
∂Ldem
r0
l2
ω(j) (1
α dem −α
− α) K dem
Lr0
= 0;
(C.3)
∂L
∂K dem
= Λi − l1 α
α−1 dem 1−α
K dem
Lr0
+
l2
ω(j) α
α−1 dem 1−α
K dem
Lr0
= 0;
(C.4)
P∂L
∂ r Zrdem
0 ,r
= (1 − Λ)t(j) − l2 = 0;
(C.5)
!
X
∂L
= (wL + iK) − t(j)
Zrdem
= 0;
(C.6)
0 ,r
∂Λ
r
h
iα h
i1−α ∂L
dem
dem
= Yr0 − K
Lr0
= 0;
(C.7)
∂l1
!
1 h dem iα h dem i1−α X dem
∂L
Lr0
−
Zr0 ,r = 0;
(C.8)
K
=
∂l2
ω(j)
r
α dem −α
By factorizing K dem
Lr0
, simplifying and making the ratio between
equations (C.3) and (C.4)we obtain:
K dem =
wα
· Ldem
r0 ;
(1 − α)i
(C.9)
Now use equations (C.6) and (C.8) and substituting for (C.9) we obtain:
α
w
t(j)
wα
=
;
(C.10)
1−α
ω(j) i(1 − α)
By rearranging this expression for ω(j) and taking logs, we obtain:
w
1
i
ln ω(j) = (α − 1) ln
+ ln t(j) − ln ;
(C.11)
1−α
α α
Taking first order derivatives of (C.11) with respect to wage rate w, transport cost t(j) and capital price i respectively we obtain the claimed positive/negative correlations:
∂ ln ω(j)
∂w
∂ ln ω(j)
∂t(j)
∂ ln ω(j)
∂i
=
=
=
α−1
w < 0, ∀w > 0, 0 <
1
t(j) > 0, ∀t(j) > 0;
1
− iα
< 0, ∀i, α > 0;
24
α < 1;
(C.12)
The first equation in (C.12) denotes a negative correlation between the evolution of wage rate and the ratio between production and exported quantity,
that is, at higher wage rates we expect ω(j) to decrease, therefore firms will
export more. Furthermore, the second relation tells us that if transport costs
become higher, ω(j) gets higher also and firms will export less (this is an expected result, given that the sales abroad cost more due to distance between
regions, whereas domestic sales have zero-transport costs, by assumptions).
Nevertheless this result is interesting because it was obtained without taking
into consideration the distance between regions. From the last equation we
observe that the higher capital price, the higher exported quantity toward
other regions. We must take into consideration that our simplified model
here assumes that capital price is uniform across the regions, therefore there
is no incentive to do arbitrage in terms of capital movement.
Finally, take the second order derivatives of the equations in (C.12) with
respect to the same variables, and we obtain the claimed convexity/concavity
relations:
∂ 2 ln ω(j)
∂w2
∂ 2 ln ω(j)
∂t(j)2
∂ 2 ln ω(j)
∂i2
=
1−α
w2
> 0, ∀w > 0, α > 0, convexity;
= −t(j)−2 > 0, ∀t(j) < 0, concavity;
= α−1 i−2 > 0, ∀i, α > 0, convexity.
The Claim (3) is prooved.
A
List of notations
25
(C.13)
r, r0 ∈ R
regions (origin and destination)
j∈J
numerator for goods
generic Xr0 ,r (j)
states for the variable X where:
r0 is related to the region of origin(production) therefore, variety of good j
r is related to the region of destination (consumption)
j is related to the good (industry)
Xr (j)
demand index of varieties of good j
elasticity of substitution across goods j
η>1
elasticity of substitution across varieties
µr (j), θr,r0
taste parameters of individual choice
Pr (j)
price index of good j
Pr
cost of living in region r
w
hourly nominal wage for production labors
τr0 ,r > 0
net transportation cost for variety r0 from region of origin r0 to region of destina
Dr0 ,r
distance between region r0 and r
Kr0 (j)
good specific technology for the productive firm
Ldem
r0 (j)
LTr0dem
KrT0 dem
Lsup
r0
LGdem
r0
Zrdem
0 r (j)
Ldem
r0 (j)
Labor demand in the productive sector
Λ
weights for the multicriterial decision
l1 , l2 , λ
Lagrange multipliers
α, αt
production specific and respectively
labor demand in the transport sector
capital demand in the transport sector
Labor supply in region r0
labor demand on behalf of the government
Quantity demanded by the firm in r0 to be transported in r, per industry j
Labor demand in the productive sector
transport specific preference parameters of the factor demands
t(j) > 0
price of transporting any variety of industry j, from factory gate r0 to consumer
26
References
[1] Anderson J.E.; Wincoop E.V.2003. Gravity with gravitas: a solution
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