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Income effects in quality-quantity models
Economics Letters 3 (1979) 125-131 0 North-Holland Publishing Company
INCOME EFFECTS IN QUALITY-QUANTITY
MODELS
George J. BORJAS University of California, Santa Barbara, CA 92106,
USA
Received 10 September 1979
The theorems on quality-quantity interactions well known in the fertility literature generally hold even when all goods in the utility function have a quality component. Although the results in this paper are presented in terms of the quality-quantity framework, it is clear that they can be applied to any problem in which the constraint is multiplicative in the variables.
1. Introduction
The pioneering work of Theil(1952) and Houthakker (1952) established certain properties of consumer demand models involving quality and quantity of commodities. The structure of the model analyzed was the maximization of a utility function, u= U(n 11. . . .
nk, 41, . . . .
qk) ,
(1)
subject to the budget constraint k
I= C
i=l
@jnj + Ciqj
f 7TfZjqj)
,
(2)
where ni qi ‘lj pi ci
= = = = =
quantity of ith commodity, quality of ith commodity, price of a unit of quality, price of a unit of quantity independent of quality, price of a unit of quality independent of quantity.
One of the important results derived was that an increase in income would lead to an increase in expenditures of the ith commodity (pini + Ciqi + niniqi) under normality conditions, but it could not be established whether ni and qi would both rise. Becker and Lewis (1973) pointed out the underlying reason for this ambiguity: the shadow prices of quality and quantity depend on the levels of quantity and quality, respectively. Thus observed income effects are contaminated by price 125
126
G.J. Borjas / Income effects in quality-quantity
models
changes which could lead to the result that ni or qi is inferior when, in fact, it is ‘truly’ normal. Becker and Lewis derived their theorems in the case where only one commodity in the utility function had a quality component. The present paper will derive a simple hear equation relating observed and true income elasticities in the case where all goods have quality-quantity interactions and the utility function exhibits a special kind of separability property.
2. The model To simplify the mathematical derivations strongly separable across commodities,
The maximization tions,
u; - X[n,qj
tpJ
Ui - h [Tini
+
assume that the utility function
of (1’) subject to (2) leads to the well-known
= 0,
i= 1, . ... k,
is
first-order condi-
(3)
Cl] = 0,
plus the budget constraint. The quantities (yiqi f pi) and (Tini + Ci) are the shadow prices of quantity and quality denoted by sh and ~$2,respectively; A is the marginal utility of income. The shadow prices depend on the levels of quality and quantity purchased. The economic interpretation of this fact is that increasing quality is more costly if the quantity of the commodity is large, while increasing quantity is more expensive for a better quality commodity. Let income rise. The total differentiation of eqs. (3) leads to a:
ai
ai
a:
0
0
0
at
a”2
0
a$
ai
...
0
0
. ..
-S:, 4;
hl
-si -Si
dql h2 dq2
0
=
0
_-s:,4;
-SK
-s;:
...
0
.dh _
(4)
G.J. Borjas /Income
effects in quality-quantity
models
127
where a{= ul;, )
ai, = a{ = vi w - hi
,
ai = UC;, .
Two facts are worth noting about (4). First, the zeros off the diagonal submatrices arise from the assumption of a special kind of strong separability in the utility function and, as can be expected, greatly simplify the derivations. Secondly, the entry of a term containing h in all the a’,‘~ is the source of the quality-quantity interaction. Note that (4) reduces to the traditional consumer choice model with rr = 0. For concreteness, and without loss of generality, let us analyze the observed income effects for the quality and quantity of the first commodity. Using Cramer’s rule yields
where
4 4
a3 4
0
I ’ and A is the bordered Hessian from (4). Note that the second-order conditions ensure that A is positive but are silent about the sign of $r. The observed income effects in (5) and (6) hold prices pi, ci and ni constant but do not hold shadow prices constant. For this reason, they contain two terms. The first term indicates whether quantity (quality) is normal or inferior while the second term is unambiguously negative. Note that if ITS= 0 (so that the second term drops out), a necessary condition to get positive income effects is that @r be positive. We will assume this to be true throughout the paper. Although we cannot say what will happen to nr or qr, as income rises we can show that total expenditures on the commodity will rise. Note that total expenditures on the commodity are given by El = Plnl
Then
+ c1q1 + n1n141 *
(7)
128
G.J. Borjas /Income
effects in quality-quantity
models
where G?l
(VA, - Alrr)
-sfi
WI& -Anr)
qq
-s;
4;
4;
$1 =
.
0
Note that J/l is positive by the second-order conditions. Some understanding of the ambiguity in (5) and (6) can be obtained by considering the following experiment: what happens to the consumption of iti and q1 if income is increased while holding shadow prices constant at their initial levels? The consumer’s problem in this experiment is k
max U =C
U'(ni, qi),
(9)
i=l
subject to k
I* = C
(Sini + S6qi).
i=l
In effect, this experiment linearizes the budget constraint by evaluating levels of purchases at their shadow prices. These shadow prices are treated as constant parameters throughout the analysis. Note that by definition k I*
=I+C
i=l
7TlTZiqi.
(10)
The first-order conditions
u; - xs;=0,
for this problem are
i= 1, .. .. k,
(11)
u; - As; = 0,
plus the constraint in (9). The true income effects are defined as the changes in n, or q1 when income (I*) increases holding shadow prices constant. Totally differentiating (11) yields
Gin
Gq
U' nq
U' 44
_-S:,
o
o
...
-S:,
hl
0
4;
dql
0 *
0
v",,
u2,q
4
dn*
0
Uflq
u'gq
-s;:
dqz
-s:,
-s;
-S(:
...
0 _ _dh
=
_
*
*
_-dI*_
Note that unlike the system in (4), (12) does not contain h in the off-diagonal
(12)
129
G.J. Borjas / Income effects in quality-quantity models
terms of the submatrices.
Using Cramer’s rule the ‘true’ income effects are given by (13)
(14) where A* is the bordered Hessian from (12) and is positive, $; follows a similar definition as & and is positive if nr and 4 1 are normal. The notation dn;/dZ is interpreted to mean the change in n 1 when Z* is varied, holding shadow prices constant. To relate eqs. (5) and (6) to the true effects in (13) and (14), it is useful to define E,* =sAnl +siql
.
(1%
Note that E,* measures expenditures on the commodity prices. If income (I*) is changed then
evaluated at the shadow
(16) where $JI follows a similar definition as $ 1 and is positive. Note that the first term in eq. (5) resembles (13). If we convert both these expressions into elasticity form by multiplying by Z/n1 and Z*/nr , respectively, first term in (5) can be written as
the
(17) where $,r is the true income elasticity of quantity. Similarly, the remaining term in (5) can be expressed in terms of (16). First, note that in elasticity form (16) becomes
z* dE; l * + a& 3 -z-=““Enl
,
(18)
where ui’
=silj *> El
j=n,q.
Using eqs. (16) and (18) the remaining
term in (5) can be rewritten
as (19)
Since $1 - +; = -2xlr,s;s:,
= -2kr,s;
u;E;
-
nl
)
130
G.J. Borjas /Income
effects in quality-quantity
the observed income elasticity of quantity
Enl =a&
+a&
models
can be written as
(21)
)
where
Similarly, the observed income elasticity of quality can be written as %l
= Pl&
+P2ci1
(22)
,
where Pl
= a1
>
0,
Eqs. (21) and (22) show that the observed income elasticity of quantity (quality) is a positive function of the true elasticity of quantity (quality) but a negative function of the true elasticity of quality (quantity). Thus, for example, if the true income elasticity of quantity is positive but small (as Becker and Lewis assert in their analysis of fertility) the observed income elasticity could well be negative. In fact, closer inspection of eqs. (21) and (22) reveals that it is not necessary that eir < eir in order for the observed elasticity of quantity (en, ) to be negative. In particular, note that (21) can be written as (23) where ,=@r A* 1 #;AF’ Note that (assuming normality) eq. (16) ensures that the first term in (23) is positive. A necessary (but not sufficient) condition for the observed elasticity to be negative is that 1 * %%,
-c+&
(24)
To understand the meaning of this condition consider the special case where p1 = cr = 0. This is the case in which there is no quality-specific or quantity-specific cost in the consumption of the commodity. If so, it is easy to show that u: = ut = f , In the more general case (24) may and (24) will be satisfied if indeed ei, <
ei,.
G.J. Borjas /Income
effects in quality-quantity
models
131
hold even if E& > e;r. For example, consider the case in which the true elasticities are equal. Eq. (23) holds if oh < ui which requires that pr IZ~ < crqr . This result can be interpreted by considering the ratio of shadow prices, 1 &I
n1q1
+Pl
-i= %
n1n1
+c1
(25)
Even though the true income elasticities are equal (causing equal proportionate increases in CJ1 and nl) the ratio of shadow prices will not remain constant. Thus if the fixed cost component of quality is ‘important’ the relative shadow price of quantity increases causing a price effect away from quantity. Therefore, in general, it is not necessary that true income elasticities be systematically biased in a given direction to obtain observed income elasticities which are opposite in sign to the true income elasticity.
References Becker, Gary S. and H. Gregg Lewis, 1973, Interaction between quantity and quality of children, Journal of Political Economy 81, March/April, S279-S288. Houthakker, Hendrick S., 1952, Compensated changes in quantities and qualities consumed, Review of Economic Studies 19, no. 3,155-161. Theil, Henri, 1952, Qualities, prices, and budget inquiries, Review of Economic Studies 19, no. 3.129-147.
INCOME EFFECTS IN QUALITY-QUANTITY
MODELS
George J. BORJAS University of California, Santa Barbara, CA 92106,
USA
Received 10 September 1979
The theorems on quality-quantity interactions well known in the fertility literature generally hold even when all goods in the utility function have a quality component. Although the results in this paper are presented in terms of the quality-quantity framework, it is clear that they can be applied to any problem in which the constraint is multiplicative in the variables.
1. Introduction
The pioneering work of Theil(1952) and Houthakker (1952) established certain properties of consumer demand models involving quality and quantity of commodities. The structure of the model analyzed was the maximization of a utility function, u= U(n 11. . . .
nk, 41, . . . .
qk) ,
(1)
subject to the budget constraint k
I= C
i=l
@jnj + Ciqj
f 7TfZjqj)
,
(2)
where ni qi ‘lj pi ci
= = = = =
quantity of ith commodity, quality of ith commodity, price of a unit of quality, price of a unit of quantity independent of quality, price of a unit of quality independent of quantity.
One of the important results derived was that an increase in income would lead to an increase in expenditures of the ith commodity (pini + Ciqi + niniqi) under normality conditions, but it could not be established whether ni and qi would both rise. Becker and Lewis (1973) pointed out the underlying reason for this ambiguity: the shadow prices of quality and quantity depend on the levels of quantity and quality, respectively. Thus observed income effects are contaminated by price 125
126
G.J. Borjas / Income effects in quality-quantity
models
changes which could lead to the result that ni or qi is inferior when, in fact, it is ‘truly’ normal. Becker and Lewis derived their theorems in the case where only one commodity in the utility function had a quality component. The present paper will derive a simple hear equation relating observed and true income elasticities in the case where all goods have quality-quantity interactions and the utility function exhibits a special kind of separability property.
2. The model To simplify the mathematical derivations strongly separable across commodities,
The maximization tions,
u; - X[n,qj
tpJ
Ui - h [Tini
+
assume that the utility function
of (1’) subject to (2) leads to the well-known
= 0,
i= 1, . ... k,
is
first-order condi-
(3)
Cl] = 0,
plus the budget constraint. The quantities (yiqi f pi) and (Tini + Ci) are the shadow prices of quantity and quality denoted by sh and ~$2,respectively; A is the marginal utility of income. The shadow prices depend on the levels of quality and quantity purchased. The economic interpretation of this fact is that increasing quality is more costly if the quantity of the commodity is large, while increasing quantity is more expensive for a better quality commodity. Let income rise. The total differentiation of eqs. (3) leads to a:
ai
ai
a:
0
0
0
at
a”2
0
a$
ai
...
0
0
. ..
-S:, 4;
hl
-si -Si
dql h2 dq2
0
=
0
_-s:,4;
-SK
-s;:
...
0
.dh _
(4)
G.J. Borjas /Income
effects in quality-quantity
models
127
where a{= ul;, )
ai, = a{ = vi w - hi
,
ai = UC;, .
Two facts are worth noting about (4). First, the zeros off the diagonal submatrices arise from the assumption of a special kind of strong separability in the utility function and, as can be expected, greatly simplify the derivations. Secondly, the entry of a term containing h in all the a’,‘~ is the source of the quality-quantity interaction. Note that (4) reduces to the traditional consumer choice model with rr = 0. For concreteness, and without loss of generality, let us analyze the observed income effects for the quality and quantity of the first commodity. Using Cramer’s rule yields
where
4 4
a3 4
0
I ’ and A is the bordered Hessian from (4). Note that the second-order conditions ensure that A is positive but are silent about the sign of $r. The observed income effects in (5) and (6) hold prices pi, ci and ni constant but do not hold shadow prices constant. For this reason, they contain two terms. The first term indicates whether quantity (quality) is normal or inferior while the second term is unambiguously negative. Note that if ITS= 0 (so that the second term drops out), a necessary condition to get positive income effects is that @r be positive. We will assume this to be true throughout the paper. Although we cannot say what will happen to nr or qr, as income rises we can show that total expenditures on the commodity will rise. Note that total expenditures on the commodity are given by El = Plnl
Then
+ c1q1 + n1n141 *
(7)
128
G.J. Borjas /Income
effects in quality-quantity
models
where G?l
(VA, - Alrr)
-sfi
WI& -Anr)
-s;
4;
4;
$1 =
.
0
Note that J/l is positive by the second-order conditions. Some understanding of the ambiguity in (5) and (6) can be obtained by considering the following experiment: what happens to the consumption of iti and q1 if income is increased while holding shadow prices constant at their initial levels? The consumer’s problem in this experiment is k
max U =C
U'(ni, qi),
(9)
i=l
subject to k
I* = C
(Sini + S6qi).
i=l
In effect, this experiment linearizes the budget constraint by evaluating levels of purchases at their shadow prices. These shadow prices are treated as constant parameters throughout the analysis. Note that by definition k I*
=I+C
i=l
7TlTZiqi.
(10)
The first-order conditions
u; - xs;=0,
for this problem are
i= 1, .. .. k,
(11)
u; - As; = 0,
plus the constraint in (9). The true income effects are defined as the changes in n, or q1 when income (I*) increases holding shadow prices constant. Totally differentiating (11) yields
Gin
Gq
U' nq
U' 44
_-S:,
o
o
...
-S:,
hl
0
4;
dql
0 *
0
v",,
u2,q
4
dn*
0
Uflq
u'gq
-s;:
dqz
-s:,
-s;
-S(:
...
0 _ _dh
=
_
*
*
_-dI*_
Note that unlike the system in (4), (12) does not contain h in the off-diagonal
(12)
129
G.J. Borjas / Income effects in quality-quantity models
terms of the submatrices.
Using Cramer’s rule the ‘true’ income effects are given by (13)
(14) where A* is the bordered Hessian from (12) and is positive, $; follows a similar definition as & and is positive if nr and 4 1 are normal. The notation dn;/dZ is interpreted to mean the change in n 1 when Z* is varied, holding shadow prices constant. To relate eqs. (5) and (6) to the true effects in (13) and (14), it is useful to define E,* =sAnl +siql
.
(1%
Note that E,* measures expenditures on the commodity prices. If income (I*) is changed then
evaluated at the shadow
(16) where $JI follows a similar definition as $ 1 and is positive. Note that the first term in eq. (5) resembles (13). If we convert both these expressions into elasticity form by multiplying by Z/n1 and Z*/nr , respectively, first term in (5) can be written as
the
(17) where $,r is the true income elasticity of quantity. Similarly, the remaining term in (5) can be expressed in terms of (16). First, note that in elasticity form (16) becomes
z* dE; l * + a& 3 -z-=““Enl
,
(18)
where ui’
=silj *> El
j=n,q.
Using eqs. (16) and (18) the remaining
term in (5) can be rewritten
as (19)
Since $1 - +; = -2xlr,s;s:,
= -2kr,s;
u;E;
-
nl
)
130
G.J. Borjas /Income
effects in quality-quantity
the observed income elasticity of quantity
Enl =a&
+a&
models
can be written as
(21)
)
where
Similarly, the observed income elasticity of quality can be written as %l
= Pl&
+P2ci1
(22)
,
where Pl
= a1
>
0,
Eqs. (21) and (22) show that the observed income elasticity of quantity (quality) is a positive function of the true elasticity of quantity (quality) but a negative function of the true elasticity of quality (quantity). Thus, for example, if the true income elasticity of quantity is positive but small (as Becker and Lewis assert in their analysis of fertility) the observed income elasticity could well be negative. In fact, closer inspection of eqs. (21) and (22) reveals that it is not necessary that eir < eir in order for the observed elasticity of quantity (en, ) to be negative. In particular, note that (21) can be written as (23) where ,=@r A* 1 #;AF’ Note that (assuming normality) eq. (16) ensures that the first term in (23) is positive. A necessary (but not sufficient) condition for the observed elasticity to be negative is that 1 * %%,
-c+&
(24)
To understand the meaning of this condition consider the special case where p1 = cr = 0. This is the case in which there is no quality-specific or quantity-specific cost in the consumption of the commodity. If so, it is easy to show that u: = ut = f , In the more general case (24) may and (24) will be satisfied if indeed ei, <
ei,.
G.J. Borjas /Income
effects in quality-quantity
models
131
hold even if E& > e;r. For example, consider the case in which the true elasticities are equal. Eq. (23) holds if oh < ui which requires that pr IZ~ < crqr . This result can be interpreted by considering the ratio of shadow prices, 1 &I
n1q1
+Pl
-i= %
n1n1
+c1
(25)
Even though the true income elasticities are equal (causing equal proportionate increases in CJ1 and nl) the ratio of shadow prices will not remain constant. Thus if the fixed cost component of quality is ‘important’ the relative shadow price of quantity increases causing a price effect away from quantity. Therefore, in general, it is not necessary that true income elasticities be systematically biased in a given direction to obtain observed income elasticities which are opposite in sign to the true income elasticity.
References Becker, Gary S. and H. Gregg Lewis, 1973, Interaction between quantity and quality of children, Journal of Political Economy 81, March/April, S279-S288. Houthakker, Hendrick S., 1952, Compensated changes in quantities and qualities consumed, Review of Economic Studies 19, no. 3,155-161. Theil, Henri, 1952, Qualities, prices, and budget inquiries, Review of Economic Studies 19, no. 3.129-147.