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A conditional version of working's model
Economics Letters North-Holland
A CONDITIONAt Kenneth
9-l
18 (1985) 97-99
VERSION
OF WORKING’S
MODEL
*
W. CLEMENTS
University of Florida, Gainesville, FL 32611, USA ~~ivers~f~} of Western Australia, Ned~~nds, Western Australia 6009. Awtrulia
Peter S. GOLDSCHMIDT University of Western Australia,
Nedfunds,
Western Ausiralia
6059, Australia
Henri THEIL Universip Received
of Florida, Gainesvifk, 24 September
FL 3261 I, USA
1984
Working’s (1943) model is shown to imply conditional for the group of goods as a whole is about u&y.
Engel curves of the Working
form when the income elasticity
of demand
1. Introduction If the Engel curve for each good is of the Working (1943) form, what is the functional form of the of conditional Engel curves for a group of goods. ? ’ In this paper we show that this is approximately the Working form also when the income elasticity of demand for the group of goods as a whole is not too far from unity. We illustrate the results with the ~nsumption of alcoholic beverages.
2. Working’s model Working’s model specifies that the n budget of total expenditure M, w, = a, + /Ii log M,
i=l
1r.7,
shares w,, , . . , w, are linear functions
of the logarithm
(1)
n,
where CX,and pi are constants satisfying C,cri = 1 and C,& = 0. We choose units such that the geometric mean of total expenditure (‘income’ for short) is unity; * this implies that (Y~= w, at
* Research supported in part by the McKethan-Matherly Eminent Scholar Chair, University of Florida. ’ Conditional Engel curves are concerned with the atlocation of total expenditure on a group of goods to the commodities within the group. ’ We visualize that 7’ observations over time are available and that geometric mean refers to income in the T periods. 0165-1765/85/$3.30
0 1985, Elsevier Science Publishers
B.V. (North-Holland)
geometric
mean income.
The income
elasticity
for good i implied
by (1) is
(2)
17,= I + P/w, 1 so that luxuries
(necessities)
have positive
(negative)
p,‘s.
3. The demand for groups of goods and conditional Engel curves Now let the n goods be divided into G < n groups, S,, . . . , S,, such that each good belongs one group. Summing both sides of (1) over i E S, we obtain the Engel curve for group g:
to only
W,=A,+B,logM=A,(1+C,logM),
(3)
where W, = C, t s w, is the budget share of the group, A, = I, e ,s,a,, B,q = C, t S,P,r and C, = B,/A,Y. The coefficient AK equals W, at geometric mean income. The income elasticity for the group as a whole is 77, = 1 + B,,‘W,.
(4)
Finally, C’ is interpreted as nR - 1 at geometric mean income. As W, = M,/M, where M, is expenditure on sX, it follows from (3) that log M = log M, - log A, - log(1 + C, log M). When C, z 0, which implies v,~ = 1 at geometric mean income, we have log M r log M, - log A, - C, log M, so that (1 + Cq) log M z log M, - log A, or log M = (1 - C,) log( MJA,).
(51
The ratio w,/ W, is the conditional It follows from (1) and (3) that W 2=
a, + P, 1% M
W, A,(1 + C&W)
,$(l
budget
-C<
share of i, the proportion
log M)(a,+P,
of M,, devoted
to good i E &.
log M),
K
for C, = 0. Ignoring the second-order term in log M, which should be a satisfactory approximation the observed income levels are not too far from their geometric mean, this simplifies to -L- =-+a l+k 4 W
P,-G, 4
if
1 - c, log M+‘+(b,-a,(;)-+$. R
R
R
where the second step follows from (5). Thus we have w,/ W, a a: + &’ log MR ,
(6)
where a: = cy,/A, - (jz?, - cy,C,)(l - C,)(log Ag)/AK and p,’ = (p, - (Y,C,)(I - C,)/A,, which satisfy c , t s,a: = 1 and C, E &$ = 0. When C, = 0, LYE z [l - (17, - 1) log W,]w,/ W, and &’ = /3,/A, = (77, - l)w,/ wq at geometric mean income. The conditional income elasticity for i is (7)
11:= I + PL+,/WR). When CR = 0, 7: = 77, at geometric
mean income.
99
K. W. Clemenrs et al. / Conditional version of Workrng’s model
Table 1 (asymptotic
standard
Beer Wine Spirits
errors in parentheses).
a
P,’
Conditional income elasticity 11:
-0.196 (0.034) - 0.023 (0.033) 0.219 (0.039)
0.73 0.81 2.39
a These estimates are homogeneityand symmetry-constrained and have constant the second part of the sample period; see Goldschmidt (n.d.) for full details.
terms in the wine and spirits equations
for
Eq. (6) is the conditional Engel curve for i E S, and it has the same functional form as (1). Thus Working’s model implies that the conditional budget shares are approximately linear functions of the logarithm of group expenditure when the income elasticity for the group is approximately unity.
4. Illustrative example We use (6) to analyze the conditional demand for three alcoholic beverages in Australia, beer, wine and spirits. Clements and Johnson (1983) find that the income elasticity for the group is about unity, so that the approximations leading to (6) will be satisfactory. The data are annual, 1955/56-1976/77, and come from Clements and Johnson (1983) who used these to estimate the absolute price version of the Rotterdam model. To apply (6) to these time series data we replace the constant conditional marginal shares in the Rotterdam model with p,’ + W,,/w’,, where W,, is the arithmetic average of w, marginal share over the years t - 1 and t and W,, = C, E s,W,,; &’ + W,,/ W,, is the variable conditional implied by (6). 3 Changes in relative prices are accounted for by using constant conditional Slutsky coefficients; see Goldschmidt (n.d.) for full details. The ML estimates of the coefficients fi,’ and the conditional income elasticities at sample means are found in table 1. Hence, within alcohol, beer is a necessity, $ for wine is not significantly different from unity, while spirits are a strong luxury.
References Clements, K.W. and L.W. Johnson, 1983, The demand for beer, wine and spirits: A system-wide analysis, 56, 273-304. Goldschmidt, P.S., n.d., Economic aspects of alcohol consumption in Australia, Master’s dissertation Western Australia, Nedlands) forthcoming. Working, H., 1943, Statistical laws of family expenditure, Journal of the American Statistical Association
3 The conditional
marginal
share is the change
in expenditure
This share is equal to s:w, / W, = P,’ + w, / WK. from (7).
on
i for a one-dollar
increase
Journal
of Business
(The University
of
38, 43-56.
in MR, prices remaining
constant
A CONDITIONAt Kenneth
9-l
18 (1985) 97-99
VERSION
OF WORKING’S
MODEL
*
W. CLEMENTS
University of Florida, Gainesville, FL 32611, USA ~~ivers~f~} of Western Australia, Ned~~nds, Western Australia 6009. Awtrulia
Peter S. GOLDSCHMIDT University of Western Australia,
Nedfunds,
Western Ausiralia
6059, Australia
Henri THEIL Universip Received
of Florida, Gainesvifk, 24 September
FL 3261 I, USA
1984
Working’s (1943) model is shown to imply conditional for the group of goods as a whole is about u&y.
Engel curves of the Working
form when the income elasticity
of demand
1. Introduction If the Engel curve for each good is of the Working (1943) form, what is the functional form of the of conditional Engel curves for a group of goods. ? ’ In this paper we show that this is approximately the Working form also when the income elasticity of demand for the group of goods as a whole is not too far from unity. We illustrate the results with the ~nsumption of alcoholic beverages.
2. Working’s model Working’s model specifies that the n budget of total expenditure M, w, = a, + /Ii log M,
i=l
1r.7,
shares w,, , . . , w, are linear functions
of the logarithm
(1)
n,
where CX,and pi are constants satisfying C,cri = 1 and C,& = 0. We choose units such that the geometric mean of total expenditure (‘income’ for short) is unity; * this implies that (Y~= w, at
* Research supported in part by the McKethan-Matherly Eminent Scholar Chair, University of Florida. ’ Conditional Engel curves are concerned with the atlocation of total expenditure on a group of goods to the commodities within the group. ’ We visualize that 7’ observations over time are available and that geometric mean refers to income in the T periods. 0165-1765/85/$3.30
0 1985, Elsevier Science Publishers
B.V. (North-Holland)
geometric
mean income.
The income
elasticity
for good i implied
by (1) is
(2)
17,= I + P/w, 1 so that luxuries
(necessities)
have positive
(negative)
p,‘s.
3. The demand for groups of goods and conditional Engel curves Now let the n goods be divided into G < n groups, S,, . . . , S,, such that each good belongs one group. Summing both sides of (1) over i E S, we obtain the Engel curve for group g:
to only
W,=A,+B,logM=A,(1+C,logM),
(3)
where W, = C, t s w, is the budget share of the group, A, = I, e ,s,a,, B,q = C, t S,P,r and C, = B,/A,Y. The coefficient AK equals W, at geometric mean income. The income elasticity for the group as a whole is 77, = 1 + B,,‘W,.
(4)
Finally, C’ is interpreted as nR - 1 at geometric mean income. As W, = M,/M, where M, is expenditure on sX, it follows from (3) that log M = log M, - log A, - log(1 + C, log M). When C, z 0, which implies v,~ = 1 at geometric mean income, we have log M r log M, - log A, - C, log M, so that (1 + Cq) log M z log M, - log A, or log M = (1 - C,) log( MJA,).
(51
The ratio w,/ W, is the conditional It follows from (1) and (3) that W 2=
a, + P, 1% M
W, A,(1 + C&W)
,$(l
budget
-C<
share of i, the proportion
log M)(a,+P,
of M,, devoted
to good i E &.
log M),
K
for C, = 0. Ignoring the second-order term in log M, which should be a satisfactory approximation the observed income levels are not too far from their geometric mean, this simplifies to -L- =-+a l+k 4 W
P,-G, 4
if
1 - c, log M+‘+(b,-a,(;)-+$. R
R
R
where the second step follows from (5). Thus we have w,/ W, a a: + &’ log MR ,
(6)
where a: = cy,/A, - (jz?, - cy,C,)(l - C,)(log Ag)/AK and p,’ = (p, - (Y,C,)(I - C,)/A,, which satisfy c , t s,a: = 1 and C, E &$ = 0. When C, = 0, LYE z [l - (17, - 1) log W,]w,/ W, and &’ = /3,/A, = (77, - l)w,/ wq at geometric mean income. The conditional income elasticity for i is (7)
11:= I + PL+,/WR). When CR = 0, 7: = 77, at geometric
mean income.
99
K. W. Clemenrs et al. / Conditional version of Workrng’s model
Table 1 (asymptotic
standard
Beer Wine Spirits
errors in parentheses).
a
P,’
Conditional income elasticity 11:
-0.196 (0.034) - 0.023 (0.033) 0.219 (0.039)
0.73 0.81 2.39
a These estimates are homogeneityand symmetry-constrained and have constant the second part of the sample period; see Goldschmidt (n.d.) for full details.
terms in the wine and spirits equations
for
Eq. (6) is the conditional Engel curve for i E S, and it has the same functional form as (1). Thus Working’s model implies that the conditional budget shares are approximately linear functions of the logarithm of group expenditure when the income elasticity for the group is approximately unity.
4. Illustrative example We use (6) to analyze the conditional demand for three alcoholic beverages in Australia, beer, wine and spirits. Clements and Johnson (1983) find that the income elasticity for the group is about unity, so that the approximations leading to (6) will be satisfactory. The data are annual, 1955/56-1976/77, and come from Clements and Johnson (1983) who used these to estimate the absolute price version of the Rotterdam model. To apply (6) to these time series data we replace the constant conditional marginal shares in the Rotterdam model with p,’ + W,,/w’,, where W,, is the arithmetic average of w, marginal share over the years t - 1 and t and W,, = C, E s,W,,; &’ + W,,/ W,, is the variable conditional implied by (6). 3 Changes in relative prices are accounted for by using constant conditional Slutsky coefficients; see Goldschmidt (n.d.) for full details. The ML estimates of the coefficients fi,’ and the conditional income elasticities at sample means are found in table 1. Hence, within alcohol, beer is a necessity, $ for wine is not significantly different from unity, while spirits are a strong luxury.
References Clements, K.W. and L.W. Johnson, 1983, The demand for beer, wine and spirits: A system-wide analysis, 56, 273-304. Goldschmidt, P.S., n.d., Economic aspects of alcohol consumption in Australia, Master’s dissertation Western Australia, Nedlands) forthcoming. Working, H., 1943, Statistical laws of family expenditure, Journal of the American Statistical Association
3 The conditional
marginal
share is the change
in expenditure
This share is equal to s:w, / W, = P,’ + w, / WK. from (7).
on
i for a one-dollar
increase
Journal
of Business
(The University
of
38, 43-56.
in MR, prices remaining
constant