Taste Heterogeneity, Elasticity of Substitution and
Green Growth
Jan Witajewski-Baltvilks
EEA-ESEM 2016
Witajewski-Baltvilks
Overview of the Theoretical Results
In the presence of consumers' taste heterogeneity, price
elasticity of demand can be expressed as a function of
elasticity of substitution between goods for individual
consumer and
dispersion of consumers' relative valuation of goods
Witajewski-Baltvilks
Application to green growth theory:
if the dispersion of consumers' tastes is large enough,
technological progress in clean industries increases the
consumption of dirty goods and and emissions.
This is because green growth increases real income of all
consumers, including those who do not value clean goods.
for any given level of taste dispersion, aggregate elasticity of
substitution between clean and dirty goods increases with the
number of varieties of the clean good.
Witajewski-Baltvilks
Related Literature
Relating elasticity of substitution and elasticity of demand to
taste heterogeneity of consumers
Hotelling (1929), Salop (1979), Perlo and Salop (1985),
Anderson, de Palma and Thisse (1988, 1989)
Interpretation of the Dixit and Stiglitz framework
Chamberlin (1950), Dixit and Stiglitz (1975, 1977,1979,
1993), Pettingel (1979), Yang and Heijdra (1993)
Dependence of green growth on heterogeneity of taste and
number of varieties
Acemoglu et al. (2012, 2014) , Aghion et al. (2014), Andree
and Smulders (2014), Hassler et al. (2014)
Witajewski-Baltvilks
Dixit and Stiglitz (1977)
Representative consumer with preferences described by the
CES utility function:
Nt
U = ∑ (xj )ρ
! ρ1
j =1
The demand faced by the producer given by:
Qj =
ρ
− 1−ρ
pj
ρ
− 1−ρ
∑k pk
pj−1 Y
and price elasticity of demand is
dQj pj
dpjt Qj
Witajewski-Baltvilks
== −
1
1−ρ
Dixit and Stiglitz (1977)
Representative consumer with preferences described by the
CES utility function:
Nt
U = ∑ (xj )ρ
! ρ1
j =1
The demand faced by the producer given by:
Qj =
ρ
− 1−ρ
pj
ρ
− 1−ρ
∑k pk
pj−1 Y
and price elasticity of demand is
dQj pj
dpjt Qj
== −
Witajewski-Baltvilks
1
1−ρ
(1 − ρφj )
Dixit and Stiglitz with Taste Heterogeneity
Consumer with preferences described by the CES utility
function:
Nt
Ui = ∑ (θij xij )ρ
! ρ1
j =1
xij is the quantity of product j consumed by individual i
θ is the idiosyncratic taste parameter
ij
Taste heterogeneity: each consumer might have dierent
valuation of product j
The agent chooses optimal consumption basket, x given his
income, y and set of prices p
Witajewski-Baltvilks
Individual and Aggregate Demand
The demand faced by the producer if ρ < 1 is given by:
ρ
Qj =
Z Z
Z
...
(θij/pj ) 1−ρ
∑k
ρ
(θik/pk ) 1−ρ
pj−1 yg (θ ) d θ
and price elasticity of demand is
dQj pj
dpj Qj
=−
1
RR
1−ρ R R
... φij2 yg (θ ) d θ
R
... φij yg (θ ) d θ


E
φj2
1 

=−
1−ρ
1−ρ
E (φj )
1−ρ
Witajewski-Baltvilks
R
!
=
Meassure of taste heterogeneity
Symetric equilibrium exists if the distribution of tastes is
symmetric
in thesense that E (ψj ) = E (ψk ),
E ψj2 = E ψk2 andCov (ψj , ψk ) = Cov (ψj , ψh ) for any j ,
and h and if all goods have the same supply curve.
Let θ 1−ρ ∼ Gamma Dµ , D1 , then, in symmetric equilibrium,
ρ
1−ρ
φ = ψ = θ ρ ∼ Dirichlet Dµ , Dµ , ..., Dµ and
ρ
∑k θ 1−ρ
E φ2
E (φ )
Witajewski-Baltvilks
=
µ +D
Nµ + D
k
Representative Consumer
At the point of symmetric equilibrium, the demand faced by
rm is given by
1 ρ
log (Qj ) = −
1 − ρ D (ψ) −
∗ log (pj )
1−ρ
N
+
ρ
1−ρ
−D (ψ) 1
+
N −1 N
∑ log (pk ) + log
k 6=j
This is exactly the Walrasian demand of the
consumer with U
y N
representative
1
1
t (x )η η where η = ρ 1− N −D
= ∑N
j
1
j =1
1− −ρ D
N
Witajewski-Baltvilks
Application: Green Growth
Consumer i derive utility from two types of goods: clean and
dirty:
ui = θci xcρ + xdρ
ρ1
The clean and dirty goods produced with labour and the range
of machines:
Qj = lj1−α
Z 1
0
A1vj−α zvjα dv
Machines supplied by monopolists charging the mark-up µ = α1
over unit cost of production, ϕ
Production of dirty good is associated with pollution (or CO2
emission), P = ϑ Qd
Witajewski-Baltvilks
Emissions Path
P = ϑ Qd
In equilibrium
d log (P ) d log (Qd ) d log (Ac )
=
dt
d log (Ac ) dt
E φid2
d log (Qd )
ρ
=α−
(1 − α) 1 −
d log (Ac )
1−ρ
E [φid ]
Witajewski-Baltvilks
!
Variety of Clean Goods
Suppose there is a variaty of clean goods and for individual
consumer they are perfectly substitutable:
ui =
n
∑ θcik xcik
k =1
!ρ
+ xid
! ρ1
ρ
Suppose also that θcik are iid with cdf given by G (θ ).
Each consumer chooses one favourite clean good. Let
ϕci = maxk {θcik }
Witajewski-Baltvilks
Eect of variety on Emission Path
d log (P )
=
dt
!!
E φid2
ρ
d log (Ac )
α−
(1 − α) 1 −
1−ρ
E [φid ]
dt
E φid2
E [φid ]
φ 2 G (x )n−1 g (x ) dx
=R d
φd G (x )n−1 g (x ) dx
R
which is a decreasing function of n, number of varieties of clean
good.
Witajewski-Baltvilks
Application to green growth theory:
Elasticity of substitution between goods at the aggregate level
can be expressed as a function of the dispersion of consumers'
relative valuation of goods
if the dispersion of consumers' tastes is large enough,
technological progress in clean industries increases the
consumption of dirty goods leading to more emissions
for any given level of taste dispersion, aggregate elasticity of
substitution between clean and dirty goods increases with the
number of varieties of the clean good.
Witajewski-Baltvilks
The research leading to these results has received funding from the European
Union Horizon2020 under Grant Agreement No 642260
Witajewski-Baltvilks