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1. COBB-DOUGLAS UTILITY AND LOG-LINEAR DEMAND SYSTEMS
Consider a utility function given by
u = v (x) =
∏^ n
i=
xα i i = xα 1 1 xα 2 2 xα 3 3 · · · (1)
We assume that αi > 0. We sometimes assume that Σnk=1 αk = 1. If we maximize utility subject to a budget constraint we obtain
L =
∏^ n
i=
xα i i− λ
[
Σnj=1pj xj − m
]
Differentiating equation 2 we obtain
∂ L
∂xi
αi
∏n i=1 x
αi i xi
− λ pi = 0 (3a)
∂ L ∂ λ
= − Σnj=1pj xj + m = 0 (3b)
If we take the ratio of any of the first n conditions we obtain
αi^ ∏nk=1 xαkk xi αj^ ∏nk=1 xαkk xj
pi pj
αi xj αj xi
pi pj
We can now solve the equation for the jth quantity as a function of the ith quantity and the ith and jth prices. Doing so we obtain
xj =
αj xi pi αi pj =
αj x 1 p 1 α 1 pj
where we treat the first good asymmetrically and solve for each demand as a function of the first. Now substituting in equation 3b we obtain
Date : October 18, 2005. 1
∂ L
∂ λ
= − Σnj=1 pj xj + m = 0
⇒ Σnj=1 pj
αj x 1 p 1 α 1 pj
= m
p 1 x 1 α 1 Σnj=1 αj = m
⇒ x 1 =
α 1 Σnj=1 αj
m p 1
Similarly for the other xk so that we have
xk (p, m) = αk Σnj=1 αj
m pk (7)
This demand equation is clearly homogeneous of degree zero in prices and income. Also note that demand for the kth good only depends on the kth price. Also note that it is linear in income. This implies that the expenditure elasticity is equal to 1. This can be seen as follows
xk = αk Σnj=1 αj
m pk
∂ xk ∂m
m xk
[
αk Σnj=1 αj
pk
] [
m αk Σnj=1 αj
m pk
]
We can obtain the indirect utility function by substituting the optimal xi ’s in the direct utility function.
ψ = v (x (p, m) ) =
∏^ n
i=
xα ii
∏^ n
i=
[
αi Σnj=1 αj
m pi
] (^) αi
∏^ n
i=
[
m Σnj=1 αj
] (^) αi [ αi pi
] (^) αi
[
m Σnj=1 αj
] (^) Σnk=1 αk (^) n ∏
i=
[
αi pi
] (^) αi
We can also compute the partial derivatives of v(x) with respect to income and price. First with respect to income
Taking the ratio of the ith and jth equations we obtain
pi pj
αi xj αj xi
We can now solve the equation for the jth quantity as a function of the ith quantity and the ith and jth prices. Doing so we obtain
xj =
αj xi pi αi pj = αj x 1 p 1 α 1 pj
where we treat the first good asymmetrically and solve for each demand for a good as a function of the first. Now substituting in the utility function we obtain
v =
∏^ n
j=
xα jj
∏^ n
j=
αj x 1 p 1 α 1 pj
)αj (17)
Because x 1 , p 1 and α 1 do not change with j, they can be factored out of the product to obtain
u =
x 1 p 1 α 1
)Σnj=1 αj (^) ∏n
j=
αj pj
)αj (18)
We then solve this expression for x 1 as a function of u and the other x’s. To do so we divide both sides by the product term to obtain
x Σnj=1 αj 1
p 1 α 1
)Σnj=1 αj
u ∏n j=
αj pj
)αj (19)
We now multiply both sides by
α 1 p 1
)Σnj=1 αj to obtain
x
Σnj=1 αj 1 =
α 1 p 1
)Σnj=1 αj u ∏n j=
αj pj
)αj (20)
If we now raise both sides to the power (^) Σn^1 j=1 αj^ we find the value of x 1
x 1 =
α 1 p 1
(^) ∏ u n j=
αj pj
)αj
Σn^1 j=1 αj (21)
Similarly for the other xk so that we have
xk =
αk pk
(^) ∏ u n j=
αj pj
)αj
Σn^1 j=1 αj
= u
1 Σnj=1 αj
∏^ n
j=
pj αj
)αj
( 1 Σnj=1 αj
) [ αk pk
]
Now if we substitute for the ith x in the cost expression we obtain
C = Σni=1 pi
αi pi
(^) ∏ u n j=
αj pj
)αj
Σn^1 j=1αj
= ( Σni=1 α (^) i )
(^) ∏ u n j=
αj pj
)αj
1 Σnj=1 αj
= ( Σni=1 α (^) i ) u Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
1 Σnj=1 αj
Now take the derivative of 23 with respect to pk
C = ( Σni=1 α (^) i ) u Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
1 Σnj=1 αj
∂ C
∂ pk
Σnj=1 αj
( Σni=1 α (^) i ) u
1 Σnj=1 αj
∏^ n
j=
pj αj
)αj
( Σn^1 j=1 αj −^1
) [ αk pk
]
∏^ n
j=
pj αj
)αj
= u Σn^1 j=1 αj
∏^ n
j=
pj αj
)αj
( 1 Σnj=1 αj
) [ αk pk
]
This is the same as 22 which verifies Shephard’s lemma.
Note that we can write this as
xk (p, m) =
gk (p) m g(p) where
gk (p) =
pk αk
) (^) γ 1 − 1
and
g (p) = Σnj=1 pj
pj αj
) (^) γ 1 − 1
As with the Cobb-Douglas utility function, demand is proportional to income or expenditure. The indi- rect utility function is of the form
ψ = v (x (p, m) ) =
m [ Σnj=1 α
− 1 γ − 1 j p
γ γ − 1 j
] γ^ − γ 1 (30)
3. KLEIN-RUBIN (STONE-GEARY) UTILITY AND THE LINEAR EXPENDITURE SYSTEM
3.1. Log form of the utility function and Marshallian demand functions. Consider a utility function given by
u = v (x) = Σnk=1 ak log [xk − bk ], ai > 0 , (xi − bi) > 0 , Σnk=1 ak = 1 (31) If we maximize utility subject to a budget constraint we obtain
L = Σnk=1 ak log [xk − bk ] − λ [ Σnj=1 pj xj − m] (32a) ∂ L ∂ xi
ai xi − bi − λ pi = 0 , i = 1, 2 , · · · , n (32b)
∂ L ∂ λ
= − Σnj=1 pj xj + m = 0 (32c)
If we take the ratio any of the first n conditions we obtain ai xi− bi aj xj − bj
pi pj
⇒
a (^) i (xj − bj ) aj (xi − bi )
pi pj
⇒ a (^) i xj − ai bj = aj pi ( xi − bi ) pj
⇒ xj = aj pi ( xi − bi ) ai pj
⇒ xj =
aj p 1 ( x 1 − b 1 ) a 1 pj
where we treat the first good asymmetrically and solve for each demand as a function of the first. Now substituting in equation 32c we obtain
∂ L ∂ λ = − Σnj=1 pj xj + m = 0
⇒ Σnj=1 pj
[
aj p 1 ( x 1 − b 1 ) a 1 pj
]
= m
p 1 a 1
Σnj=
[
aj p 1 x 1 − aj p 1 b 1 a 1
]
⇒ p 1 a 1
Σnj=1 aj x 1 − aj b 1 + Σnj=1 pj bj = m
⇒
p 1 a 1
Σnj=1 aj ( x 1 − b 1 ) + Σnj=1 pj bj = m
⇒
p 1 a 1
( x 1 − b 1 ) Σnj=1 aj + Σnj=1 pj bj = m
⇒ p 1 a 1
x 1 − p 1 a 1
b 1 + Σnj=1 pj bj = m
⇒
p 1 a 1
x 1 = m +
p 1 a 1
b 1 − Σnj=1 pj bj
⇒ x 1 =
a 1 p 1
m + b 1 −
a 1 p 1
Σnj=1 pj bj
3.2. Product form of the utility function, the cost function and Hicksian demand functions. The cost and indirect utility functions for the LES are usually obtained assuming the direct utility function is a transformation (which doesn’t matter) of the function given in 31 , that is
u = v (x) =
∏^ n
k=
[xk − bk]ak^ , ai > 0 , (xi − bi) > 0 , Σnk=1 ak = 1 (40)
First set up the Lagrangian problem
L = Σnk=1 pk xk − λ (
∏^ n
k=
(xk − bk )ak^ − u^0 ) (41)
The first order conditions are as follows
∂L ∂xi = pi − λ
[ ai (x 1 − b 1 )a^1 (x 2 − b 2 )a^2... (xi− 1 − bi− 1 )ai−^1 (xi − bi )ai−^1 (xi+1 − bi+1 )ai+^...
] = 0, i = 1,... , n
(42a)
= pi − ai v xi − bi λ = 0, i = 1,... , n (42b)
∂L ∂λ = −
∏^ n
k=
(xk − bk )ak^ + u = 0 (42c)
Taking the ratio of the ith and jth equations we obtain
pi pj
ai (xj − bj ) aj (xi − bi)
We can now solve the equation for the jth quantity as a function of the ith quantity and the ith and jth prices. Doing so we obtain
xj =
aj pi(xi − bi) aipj
Now treat the first good asymmetrically and solve for each demand for a good as a function of the first to obtain
xj =
aj p 1 (x 1 − b 1 ) a 1 pj
Now substituting in the utility function we obtain
v =
∏^ n
k=
(xj − bk)ak
∏^ n
k=
akp 1 (x 1 − b 1 ) a 1 pk
)ak^ (46)
Because x 1 , p 1 ,a 1 , and b 1 do not change with k, they can be factored out of the product to obtain
u =
(x 1 − b 1 )p 1 a 1
)Σnk=1 ak (^) ∏n
k=
ak pk
)ak (47)
We then solve this expression for x 1 as a function of u and the other x’s. To do so we divide both sides by the product term to obtain
(x 1 − b 1 )Σ
nk=1 ak
p 1 a 1
)Σnk=1 ak
u ∏n k=
ak pk
)ak (48)
We now multiply both sides by
a 1 p 1
)Σnj=1 aj to obtain
(x 1 − b 1 )Σ
nk=1ak
a 1 p 1
)Σnk=1 ak u ∏n k=
ak pk
)ak (49)
If we now raise both sides to the power (^) Σn^1 j=1 aj^ we find the value of x 1
x 1 − b 1 =
a 1 p 1
u ∏n k=
ak pk
)ak
Σn^1 k=1 ak (50)
Similarly for the other xj so that we have
xj − bj =
aj pj
u ∏n k=
ak pk
)ak
Σn^1 k=1 ak
= u Σn^1 k=1 ak
( (^) n ∏
k=
pk ak
)ak^ )
( 1 Σnk=1 ak
) [ aj pj
]
⇒ xj = u Σn^1 k=1 ak
( (^) n ∏
k=
pk ak
)ak^ )
( Σn^1 k=1 ak
) [ aj pj
]
Now note that Σnk=1 ak = 1 by assumption so that we obtain
xj = u
[
aj pj
] (^ ∏n
k=
pk ak
)ak^ )
Now if we substitute for the ith^ x in the cost expression we obtain
Σnk=1 pk xk = Σnk=1 pk
[
fk (p) +
gk (p) g (p)
[ m − f (p) ]
]
= m
⇒ Σnk=1 pk
[
fk (p) −
[
f (p) gk (p) g (p)
] ]
[ [
gk (p) g (p)
]
m
]
= m
⇒ Σnk=1 pk
[
fk (p) −
[
f (p) gk (p) g (p)
] ]
m g (p)
Σnk=1 pk gk (p) = m
⇒ Σnk=1 pk
[
fk (p) −
[
f (p) gk (p) g (p)
] ]
m g (p)
g (p) = m, by homogeneity
⇒ Σnk=1 pk
[
fk (p) −
[
f (p) gk (p) g (p)
] ]
⇒ Σnk=1 pk
[
fk (p) −
[
f (p) gk (p) g (p)
] ]
and nonnegativity of consumption near zero expenditure implies fj − f g g j ≥ 0. This then implies that
fj − f g g j = 0 for all j. Such a demand system comes from an indirect utility function of the form
ψ (p , m) =
m − f (p) g (p)
To see this take the derivatives of 60 with respect to the kth price and income.
∂ ψ (p , m) ∂ pk
g (p) (− fk (p) ) + ( f (p) − m ) gk (p) g (p)^2
= f (p) gk (p) − g (p) fk (p) − m gk (p) g (p)^2 ∂ ψ (p , m) ∂ m
g (p)
Then take the ratio of the derivatives and simplify
∂ ψ (p , m) ∂ pk ∂ ψ (p , m) ∂ m
f (p) gk (p) − g (p) fk (p) − m gk (p) g (p)
= − fk (p) +
f (p) gk (p) g (p)
gk (p) g (p)
m
= − xk (p , m)
by Roy’s identity. The form of the indirect utility function in 60 is known as the ”Gorman polar form”. As an example consider f(p) = Σnk=1pk bk and g(p) = Πnk=1pa kk with Σnk=1 ak = 1.
Then fk = bk, gk = ak^ pgk( p)and (^) gg(kp) = a pkk. This gives as a demand system
xi (m, p) = fi (p) −
gi (p) g (p)
f (p) +
gi (p) g (p)
m
= bi −
ai pi
Σnj=1 pj bj +
ai pi
m
which is the linear expenditure system. While the complete class of utility functions leading to demand systems linear in expenditure is not fully characterized, one important class leading to linear systems is
v (x) = T [ h ( x − b ) ] (^) (64)
where T’(·) > 0 and h is linearly homogeneous. A function v that satisfies equation 64 is said to be homothetic to the point b. The indifference curves are scaled up versions of a base indifference curve and the expansion path radiates in a straight line from a translated origin. The Gorman polar form corresponding to equation 64 has g(·) dual to h(·) with f (p) = Σnk=1 pk bk. In this sense the LES system is a translation of the Cobb-Douglas system. We can similarly define translations of CES systems.
- DEMAND SYSTEMS QUADRATIC IN EXPENDITURE Demand systems quadratic in expenditure are of the form
xi (m, p) = ci (p) + bi (p) m + ai (p) m^2 (65)
This specification allows for non-linear relationships between income and consumption as is more com- monly thought to occur. Van Daal and Merkies commenting on a paper by Hoew, Pollak and Wales show that theoretically consistent demand system that are quadratic in expenditure must be of the form
xi (m, p) =
γ
gi g
γi γ
m^2 +
[
gi g
2 f γ
gi g
γi γ
) ]
m +
f^2 γ
gi g
γi γ
gi g f + fi + χ
g γ
g^2 γ
gi g
γi γ
We can also write this as
xi (m , p) =
γ
gi g
γi γ
( m − f )^2 + gi g
( m − f ) + fi + χ
g γ
g^2 γ
gi g
γi γ
In 66 and 67 we assume that f, g, and γ are homogeneous of degree 1 and that the function χ(·) is a
function of one variable. An alternative is to let α(p) = g(p)
2 γ(p) and then write^67 as
xi (m , p) =
g
αi −
gi g
α
( m − f )^2 +
gi g
( m − f ) + fi + χ
α g
αi −
gi g
α
When γ(·) = 0, these quadratic systems are characterized by the indirect utility functions
ψ (p, m) = −
g (p) m − f (p)
g (p) γ (p)
or
ψ (p, m) = −
g (p) m − f (p)
α (p) g (p)
If we write the Gorman polar form in equivalent form as
ψ (p , m) =
− g (p) m − f (p)
xk (p , m ) =
− ∂ ψ ∂ p^ (p mk ) ∂ ψ (p m) ∂ m
− (^) ∂ p∂^ Ωk ∂ Ω ∂ m
ˆxk (p , m ) =
− ∂^
ψˆ (p m) ∂ pk ∂ ψˆ (p m) ∂ m
[
∂ Ω ∂ pk −^
∂ Ω ∂ m fi^ (p)
]
∂ Ω ∂ m
=
− (^) ∂ p∂^ Ωk ∂ Ω ∂ m
= xk ( p , m − f (p) ) + fi (p)
What this says is that we can add a constant term (not depending on income) to the demand function by subtracting some function of price from the measure of income and then adding back the derivative of that function as the constant. Consider a couple of examples. First consider the simple proportional system
xi (m, p) = bi (p) m (^) (78)
The modified system is given by
x ˆi (m, p) = bi (p) ( m − f (p) ) + fi (p) (79)
This is the same as the Gorman polar form in 57 where bi (p) = gi (p)/g(p). Next consider the PIGL system in 72 where σ = 2. We obtain
ψ (p , m) =
− m −^1 − γ (p) −^1 g (p) −^1
− m −^1 g (p)−^1
γ (p) −^1 g (p) −^1
g m
g γ
Now transform it by introducing f(p). We obtain
ψ^ ˆ (p , m) = − g m − f (p)
g γ (81) which is the same as the quadratic system in 69. This could also be done for a PIGL system 73 where σ =
- This would give
xi (m, p) =
gi g
γi γ
m 3 γ 2
gi g m
x ˆi (m, p) =
gi g
γi γ
( m − f (p) ) 3 γ 2
gi g
( m − f (p) ) + fi (p)
When expanded this will give a system with linear, quadratic and cubic terms in expenditure and a term independent of expenditure. But this system is not completely general in that quadratic and cubic terms will be proportional. The same is true of all polynomial demand systems.
6.2. PIGLOG demand systems. Muellbauer also introduced a class of demand systems related to the poly- nomial PIGL class. The PIGLOG class of demand systems has the form
xi (m, p) = bi (p) m + di (p) m log[ m ] (83) Expenditure thus enters linearly and as a log function of m. In share form equation 83 can be written
ωi (m, p) =
pi xi m
pi bi (p) m + pi di (p) m log[ m ] m
=
pi bi (p) m m
pi di (p) m log[ m ] m = pi bi (p) + pi di (p) log[ m ] =ˆbi(p) + dˆi(p) log[m]
Thus the share form has a term that is independent of income along with a term that is linear in log m. Muellbauer has shown that theoretically plausible systems of this form must be written as
xi (m, p) =
gi g
m −
Gi (p) G (p)
[ log [ m ] − log [ g (p) ] ] m (85)
Here G(p) is homogeneous of degree zero and g(p) is homogeneous of degree 1 and the subscript i as usual denotes a derivative with respect to the ith price. The indirect utility function associated with 85 is
ψ^ ˆ (p , m) = G (p) [ log [ m ] − log [ g (p) ] ] (86)
Roy’s identity holds as is obvious from
xk (p , m ) =
− ∂ ψ^ ∂ p(p , mk ) ∂ ψ (p , m) ∂ m
G (P )
− gi (p) g (p)
- Gi (p) [ log [ m ] − log [g (p) ] ] G (p) m = m
gi (p) g (p)
Gi (p) G (p)
[ log [ m ] − log [g (p) ] ] m
We will show later that some forms of the translog and the AIDS demand systems are special cases of the PIGLOG system. The class can be extended by adding terms that are quadratic in log m. More general systems are discussed in a series of papers by Lewbel in the late 1980’s and early 1990’s.
7. LINEAR TRANSLOG DEMAND SYSTEM
This is usually written in share form as
ωi(m, p) =
pi xi(m, p) m
pi bi m
[
αi + Σnj=1 βij log [pj ]
] [
1 − Σnk=1 bk
pk m
]
βij = βji, Σni=1 βij = 0 , Σnk=1 αk = 0
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