Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
A Spatial General Equilibrium Model with Discrete Choice of
Differentiated Products
Alexandrina I. Scorbureanu
Athens Institute of Research - ATINER Conference, Greece
20-24 March 2008
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
1
Motivation
2
Literature origins
3
The Model
4
Demand side
5
Supply side
6
The Government
7
Equilibrium conditions and closure
8
Aknowledgements
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
SGE with Discrete Choice of Differentiated Products - Motivation
I
Interpretation and assessment of trade flows and trade policy across
different locations or countries, by making use of the EXACT
AGGREGATION
I
EMPLOYMENT and CONSUMPTION behavior across regions
I
PRODUCTION levels in TRANSPORTS and MANUFACTURING
industries
I
Transportation POLICY assessment across regions (environmental taxes,
congestion, etc.)
I
Evaluation of TRADE across regions.
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
SGE with Discrete Choice of Differentiated Products - Literature origins
Literature origins:
I
Anderson S.P.; De Palma A., Thisse J.F.: Discrete choice theory of
product differentiation. MIT Press. (1992)
I
Bergh Van der J., Nijkamp P., Rietveld editors: Recent Advances in Spatial
Equilibrium Modeling. Methodology and applications. Springer, (1995)
I
Yilmazkuday H.; A general equilibrium spatial model, working paper.
(Jul.2007)
By DIFFERENCE with respect to the current literature:
I
Discrete choice decisions in the consumer’s behavior among goods
produced in different locations - exact aggregation;
I
Cost of transport among the regions;
I
Transport supply and demand.
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Generic set-up and assumptions
I
Exogenous labor supply (simplifying assumption, but easily
implementable);
I
Transport cost applies only to inter-regional (freight) trade flows.
Intra-regional trade has zero-transport cost;
I
No intermediary consumption of commodities (simplifying assumption,
without the lost of generality)
I
No re-sales of imported goods
I
Competitive industries (it is also implementable a situation in which some
sectors are not price-takers)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
A (geo)graphical representation
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Consumer behavior (1)
Consumer computes a two-level choice before deciding his consumption bundle:
A second level choice over varieties in each industry (e.g. for clothing: made in
Italy, made in France, made in USA, etc.) A first-level choice over different
industries (e.g. food, clothing, apparel,etc.)
logit.JPG
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Consumer behavior(2)
The feasible consumption

(G 1, V 1) .

.
.


.
.

.
.
FS = 


.
.


.
.
(G 1, VR) .
set for the consumer in region r is:
.
.
.
.
.
.
.
.
.
.
(Good = j, Variety = r )
.
.
.
.
.
.
.
.
.
.
(GJ, V 1)
.
.
.
.
.
(GJ, VR)










(1)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Utility function and probability of choice. Generic notation (1)
I
Individuals that have random utility functions (objective+subjective part):
Ujh = uj + εjh
I
(2)
The probability of choice then will be:
Prjh = Prob[Ujh
= Prob[uj + εjh
= Prob[εjh − εjh0
≥ Ujh0 , ∀j 0 = 1, ..., J]
≥ uj 0 + εjh0 , ∀j 0 = 1, ..., J]
≥ uj 0 − uj , ∀j 0 = 1, ..., J]
(3)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Utility function and probability of choice. Generic notation (2)
Steps in computing choice probabilities from the above specification
(McFadden 1978,1981):
I
Define a GEV-type distribution function F
(ε):
F (ε1 , ..., εn ) = exp −H e −ε1 , e −ε2 , ..., e −εn
I
Choice probabilities:
Probj = µ ·
∂ ln H (e u1 , ..., e un )
.
∂ uj
(4)
I
p
Where µ = corr (εr ; εk) with r 6= k ∈ Jcell measuring correlation between
error terms within a cell;
I
A particular case of H (ε1 , ..., εn ) = ∑nj=1 εj
1/µ
Probj = µ ·
∂ ln ∑nj0 6=j e uj 0 /µ
∂ uj
=
replaced in F (ε) leads to:
e uj /µ
∑nj0 6=j e uj 0 /µ
.
(5)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
The nested logit demand system. A particularization
I
Following McFadden(1981), we choose a particular form for the H- GEV
function:
h
i1−µ
−εr /1−µ
H e −ε1 , ..., e −εR = ∑G
Jr =1 ∑r ∈Jr e
I
By computing probabilities of choice we obtain:
1−µ
uj /(1−µ)
∂ ln H(e u1 ,...,e un )
Dr
= e Dr
1−µ , ∀j ∈ Jr where
R
∂ uj
∑r =1 Dr
u
/(1−µ)
e k
called ’inclusive value’, measures the sum of utilities
Probj = µ ·
Dr = ∑k∈Jr
obtained from all products in the group Jr
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Utility and choice over varieties
Total population is divided into cells with common characteristics. We consider
individuals that form the industry cell j; each individual from this group derives
a utility from choosing a particular variety r 0 :
I
Estimated utility function:
vrh0 = ln sr 0 − ln pr 0 + εrh0
I
The choice of variety r 0 over other varieties of goods is therefore:
1/µ
Probr 0 =
−1/µ
sr 0 ·pr 0
1/µ −1/µ
∑r 00 sr 00 ·pr 00
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Utility and choice over industries
I
Utility from choosing industry j 0 is:
Vjh0 = ln zj 0 − ln pj 0 + εjh0 ;
I
Probability that an individual chooses to consume goods from industry j 0 :
Probj 0 =
I
exp(Vj 0 /v )
∑j exp(Vj/v )
1/v
=
−1/v
z j 0 pj 0
1/v −1/v
1/µ −1/µ µ/v
zj 0 pj 0 + ∑r 0 sr 0 pr 0
Where the consistency with the second stage decision is fulfilled:
vj = µ ln ∑r 0
exp(ln sr 0 −ln pr 0 )
µ
1/µ −1/µ
= µ ln ∑r 0 sr 0 pr 0
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Aggregated demand
I
Aggregate demand at variety level (K individuals):
Xr ,r 0 (j 0 ) = Kr (j 0 ) · Probr 0 = Kr (j 0 )
I
1/µ
1/µ −1/µ
∑r 00 sr 00 ·pr 00
Aggregate demand at industry level (N individuals):
1/v
Xr (j 0 ) = Nr · Probj 0 = Nr (j 0 )
I
−1/µ
sr 0 ·pr 0
−1/v
zj 0 pj 0
1/v −1/v
1/µ −1/µ µ/v
zj 0 pj 0 + ∑r 0 sr 0 pr 0
Aggregate demand in a country (or region):
Xr = ∑Jj0 =1 Xr (j 0 ) = ∑Jj0 =1 Nr (j 0 )Probj 0
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Suppliers: Production firms (1)
There are two types of (both) competitive firms in the economy: the
production firm (located in region r 0 ) that outcomes variety r 0 and the
transportation firm that supplies freight transport services across regions.
I
The production firms’ bicriterial cost-min program:
dem ,
dem , Λ, l , l
MIN Ldem
,
K
Z
=
∑
0
0
0
1
2
r
r
r
r ,r
h i
dem
dem
dem
Λ wLr 0 + iKr 0
+ (1 − Λ) ∑r Zr 0 ,r · t(j)
α dem 1−α
s.t. : Yr 0 = Krdem
Lr 0
0
1
· Yr 0
=
∑r Zrdem
0 ,r
ω(j)
I
The solution bundle to this program is:
L∗dem
=
r0
Kr∗dem
=
0
Zr∗dem
=
0
L∗dem
≤ Lsup
r0
r0 ;
wα
· L∗dem
r 0 ;
i(1−α)
α
wα
1
= i(1−α)
· ω(j)
· L∗dem
;
∑r Zr∗dem
0 ,r
r0
(6)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Suppliers: Production firms (2)
I
I
I
The resulting production level of firm in region r 0 therefore is:
α
wα
Yr∗0 = i(1−α)
· L∗dem
r0
The resulting total cost function is automatically obtained:
α i
h
t(j)
wα
wΛ
TCrprod
= L∗dem
0
r0
1−α + (1 − Λ) ω(j) i(1−α)
(7)
The competitive production sector sets the factory gate price equal to its
marginal cost:
α
t(j)
w
wα
r0
Pr 0 r 0 (j) = MCprod
= Λ 1−α
− (Λ + 1) ω(j) · i(1−α)
;
(8)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Suppliers: Transport firms (1)
I
The transport firm cost-min program is:
MIN
s.t. :
TTCr 0 = w · LTdem
i · KrTdem
0
r0 +
Tdem
αt Tdem 1−αt
sup
· Lr 0
Zr 0 = Kr 0
(9)
where Zrsup
= ∑j ∑r Zrsup
0
0 r (j).
I
The factor demand bundle obtained and the minimum cost function are:
L∗Tdem
=
r0
Kr∗Tdem
=
0
TTC ∗ =
L∗Tdem
≤ Lsup
r0
r0 ;
αt i ∗Tdem
1−αt w Lr 0 ,
i 2 1−αt
wL + iK = w + w
· αt .
(10)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Suppliers: Transport firms (2)
I
WITHOUT the distorsive environmental tax, the price-taking transport
firm would set the transport price at its marginal cost level:
h
i1−αt h iαt
w
TMCr 0 = 1−α
· αi t
t
I
WITH the environmental tax ψe that the transporter has to pay from its
remuneration for the supplied transport services, the transport price
becomes:
t(j) = ψe +
i(1−αt )(αt +1)/αt +i(1−αt )αt ·w 2 αt
.
(w αt )( αt +1)/αt
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Pricing and Trade cost across regions
I
Trade prices across regions follow an iceberg-typology:
Pr 0 ,r (j) = 1 + τr 0 ,r (j) · Pr 0 ,r 0 (j).
I
In consumer’s utility function, prices are intended as:
pr 0 = Pr ,r 0 (j 0 ) = 1 + τr 0 ,r (j) · Pr 0 ,r 0 (j)
1
pj 0 = ∑r 0 θr 0 ,r Pr ,r 0 (j 0 )1−η 1−η
where τr 0 ,r (j) = t(j)Dr 0 ,r .
(11)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
The Government
I
In a simplified version, revenues from the exogeneous environmental tax
applied to transports are used to pay labor - the only production factor
used by Gov:
i(1−αt ) αt tdem
Gr 0 = ψe Zrsup
= ψe
Lr 0
0
w αt
wLGdem
= Gr 0 ;
r0
I
The labor demand on behalf of the government is therefore derived:
i(1−αt ) αt tdem
LGdem
= w1 ψe
Lr 0 ;
r0
w αt
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Equilibrium conditions
I
Labor market equilibrium in region r 0 :
sup
Tdem =
LGdem
+ ∑j Ldem
∑j Lr 0 ¯ (j)
r0
r 0 (j) + Lr 0
I
Capital market equilibrium in region r 0 :
¯ j K sup
(j) + KrTdem
≤∑
∑j Krdem
0
0
r 0 (j)
I
Goods and transport markets equilibriums across regions are given by the
set of equations:
∑r Zrdem
0 ,r (j) + Xr 0 ,r 0 (j)
∑r Zrdem
0 ,r (j)
dem
∑j Zr 0 (j)
Zrdem
0 ,r (j)
= Yr 0 (j)∀j, ∀r , r 0
= ∑r Zrsup
0 ,r (j)
sup
= ∑j Zr 0 (j)
= Xr ,r 0 (j)
(12)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Some analytical results (0)
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;
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Analytical results(1)
I
The ratio between r 1, r 2 variety demands, specific to a couple of regions is:
−1/µ
1−αt αt
w
i
1+ 1−α
Dr 1r 0
Pr 0 ,r 0 (j)−1/µ
α
t
t
−1/µ
1−αt αt
1/µ
1/µ −1/µ
w
i
sr 0 1+ 1−α
Dr 1r 0
Pr 0 ,r 0 (j)−1/µ +∑r 00 sr 00 pr 00
α
1/µ
sr 0
Xr 1,r 0 (j 0 )
k1
=
Xr 2,r 0 (j 0 )
k2
t
t
I
t
−1/µ
1−αt αt
1/µ
w
i
sr 0 1+ 1−α
Dr 2r 0
Pr 0 ,r 0 (j)−1/µ
α
t
t
−1/µ
1−αt αt
1/µ
1/µ −1/µ
i
w
Dr 2r 0
Pr 0 ,r 0 (j)−1/µ +∑r 00 sr 00 pr 00
sr 0 1+ 1−α
α
t
(13)
The aggregate industry and variety demand ratio between two regions:
Xr 1
Xr 2
=
N1
N2
−1
1/v
z 0 [∑r 0 θrr 0 Pr 1r 0 (j)1−η ] v (1−η)
j
−1
1−αt
1/v
1/µ
w
i
z 0 [∑r 0 θrr 0 Pr 1r 0 (j)1−η ] v (1−η) + ∑r 0 s 0
1+ 1−α
αt
j
r
t
−1
1/v
z 0 [∑r 0 θrr 0 Pr 2r 0 (j)1−η ] v (1−η)
j
( )
αt
−1
1/v
1/µ
w
z 0 [∑r 0 θrr 0 Pr 2r 0 (j)1−η ] v (1−η) + ∑r 0 s 0
1+ 1−α
j
r
t
( αit )
αt
(
(
)
)
1−αt
Dr 0 r 1
!µ/v
−1/µ
−1/µ
P 0 0 (j)
r r
Dr 0 r 2
!µ/v
−1/µ
−1/µ
P 0 0 (j)
r r
(14)
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Analytical results(2)
I
The ratio between aggregated consumption values levels obtained is:
1
1
1−ε
∑j βr 1 (j)[∑r 0 θrr 0 Prr 0 (j)1−η ] (1−η)(1−ε)
Pr 1 X r 1
Pr 2 X r 2
I
=
N1
1
N2 1
1−ε
∑j βr 2 (j)[∑r 0 θr 0 r Prr 0 (j)1−η ] (1−η)(1−ε)
1/v −1/v
z0 p0
j
j
1/v −1/v
1/µ −1/µ µ/v
z0 p0
+ ∑r 0 s 0 ·p 0
j
j
r
r
1/v −1/v
z0 p 0
j
∗j
1/v −1/v
1/µ −1/µ µ/v
z 0 p 0 + ∑r 0 s 0 ·p 0
j
∗j
r
r
(15)
The ratio of total labor demand in production/ transport activities across
all regions is:
sup
L∗dem
(j)
(1 − α)
r0
=
sup 4Kr 0 w α(3α − 1) + 2L(2α − 1)
∗Tdem
Lr 0
(3α − 1)(2L − wKr 0 )
(16)
where L = Lsup − LGdem
.
0
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Further developments:
I
Estimation method
I
Calibration and numerical results from simulation: 2-reg X 8-industry X
2-variety model
I
Intermediary consumption
I
Region-specific transport technologies
I
Endogenous labor supply
I
Consider different transport alternatives.
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Aknowledgements (a)1
EUROMIDBRIDGE is a logistic corridor that provides direct links to Middle
Asia geographical area, by using existing infrastructures. Its gravitational and
administrative center is the Verona’s Freight Village (in Italy) and it is linked
with ports and logistic infrastructures from the North of Europe and Italy. It
continues through the Mediterranean Sea and enters Israel through the port of
Haifa. Furthermore, it continues in West Bank region with the industrial area
of Jenin and passes through Jordan.
1
Person of Contact: Prof. Michela Sironi, [email protected], Director of Euromidbridge
Project, Quadrante Servizi S.R.L., Consorzio ZAI, Dipartimento di Scienze Economiche Universita di Verona
Motivation
Literature origins
The Model
Demand side
Supply side
The Government
Equilibrium conditions and closure
Aknowledgements
Aknowledgements (b)2
A proposal has been made to assess new trade patterns among the
triangular-economy Israel-Palestine-Jordan under the Euromidbridge Project
scheme for 2007/2008. Requirements impose the evaluation of actual trade
potential in this particular geographical area, in order to compare the actual
situation with a scenario of investments in transports and logistics. This paper
is a part of the preliminary analysis of trade patterns within the Is-Pa-Jo area,
whose final objective is to evaluate tolling schemes for potential investments in
road and plant-type infrastructures.
2
Person of Contact: Prof. Michela Sironi, [email protected], Director of Euromidbridge
Project, Quadrante Servizi S.R.L., Consorzio ZAI, Dipartimento di Scienze Economiche Universita di Verona