Income Elasticities at different price vectors

Economics Letters North-Holland

24 (1987) 261-265

INCOME ELASTICITIES Henri

THEIL

AT DIFFERENT

and James L. SEALE,

Universiiy of Florida,

Received Accepted

261

PRICE VECTORS

*

Jr.

Gainesville, FL 32611, USA

27 January 1987 7 April 1987

Cross-country data are used to evaluate the income elasticities of the demand for ten broad groups of goods at two different price vectors: the observed prices of separate countries and the geometric means of these prices across all countries. The major effect is on the income elasticity of the demand for food; using observed rather than geometric mean prices results in lower elasticity values for the more affluent countries.

Working’s model and the consumption data of the International Comparison Project have been used by several authors for the tabulation of income and price elasticities of demand, including Finke et al. (1983, 1984) Theil and Clements (1987, ch. 2), Theil and Suhm (1981, ch. 4) and Theil, Suhm and Meisner (1980). These elasticities were all evaluated at geometric mean prices, not at the observed prices in the countries represented in the sample. The choice of geometric mean prices is convenient (see the next paragraph); also, Theil and Seale (1987) showed that this choice can be justified by means of a minimum mean-squared distance criterion. Nevertheless, it is appropriate to ask whether we can tabulate income elasticities at prices different from geometric mean prices. For ten goods (see the top row of table 1) let w,, be the budget share of good i in country c and let Q, be per capita real income of that country (see column 2). Working’s model amounts to w,, = a, + P, log

Q, >

and the implied

income

1+++ tc

(1) elasticity

is

P,

(2)

a, + P, log Q, .

To take account

of different

relative

prices in different

countries

we extend

(1) to

(3)

where

+ = income

* Research supported 0165-1765/87/$3.50

flexibility

and

z,, = log( p,,/j,)

in part by the McKethan-Matherly

0 1987, Elsevier Science Publishers

Eminent

with

P,~ = price

Scholar

B.V. (North-Holland)

Chair,

of good

University

i in country

of Florida.

and

0.52

0.51

0.140

0.141

0.190

0.206

0.246

0.255

0.261

0.272

0.290

0.346

0.371

0.435

0.451

0.464

0.467

0.532

0.588

0.606

0.641

0.686

0.750

0.751

0.756

0.756

0.785

0.794

1 .ooo

Philippines

Thailand

Malaysia

Korea

Brazil

Colombia

Syria

Iran

Romania

Yugoslavia

Mexico

Poland

Uruguay

Ireland

Hungary

Italy

Japan

Spain

U.K.

Netherlands

Belgium

Austria

Germany

France

Denmark

Luxembourg

U.S.

0.20

0.32

0.33

0.35

0.35

0.35

0.35

0.38

0.41

0.08

0.27

0.28

0.27

0.28

0.30

0.25

0.28

0.31

0.41

0.43

0.44

0.43

0.47

0.50

0.42

0.42

0.53

0.56

0.58

0.61

0.61

0.47

0.50

0.50

0.55

0.57

0.60

0.61

0.61

0.63

0.62

0.61

0.59

0.69

0.66

0.68

0.72

0.62

0.65

0.66

0.69

0.69

0.74

0.74

0.73

0.103

0.72

0.089

Pakistan

Sri Lanka

(3b) 0.75

(3a)

0.75

(2)

Observed

0.068

GM

Food

tabulated

India

income

capita

Per

income elasticities of demand

(1)

Country

Cross-country

Table 1

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

(4a)

GM

tobacco

Beverages,

at geometric

0.94

0.93

0.98

0.98

0.98

0.98

0.98

0.98

0.9x

0.98

0.98

0.98

0.98

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.98 0.98

0.93

0.93

0.93

0.99

0.98

0.98

0.93

0.93 0.93

0.98 0.98 0.98

0.93

0.93

0.93

0.98

0.98

0.98

0.93

0.98 0.98

0.94

0.98

0.94 0.94

0.98

0.94

0.94

0.94

(5a)

GM

footwear

0.98

0.98

0.98

0.98

(4b)

Observed

0.92

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.93

0.92

0.93

0.94

0.93

0.94

0.94

0.93

0.93

0.94

0.92

0.94

0.93

0.94

0.92

0.93

0.93

0.93

0.93

0.93

0.94

(5b)

Observed

prices, 1975.

Clothing,

mean and observed

1.17

1.18

1.18

1.18

1.18

1.18

1.18

1.19

1.19

1.19

1.19

1.20

1.20

1.20

1.20

1.20

1.21

1.21

1.22

1.23

1.23

1.23

1.23

1.24

1.25

1.26

1.26

1.29

1.30

1.33

(6a)

GM

fuel

Gross rent,

1.17

1.17

1.18

1.18

1.18

1.19

1.17

1.18

1.18

1.19

1.18

1.19

1.20

1.19

1.20

1.23

1.22

1.23

1.27

1.23

1.23

1.23

1.21

1.25

1.25

1.25

1.28

1.33

1.28

1.36

(6b)

Observed

Household

1.20

1.21

1.21

1.22

1.22

1.22

1.22

1.22

1.22

1.23

1.23

1.23

1.24

I.24

1.24

1.25

1.26

1.26

1.27

1.28

1.28

1.28

1.29

1.30

1.31

1.34

1.34

1.38

1.40

1.45

(7a)

GM

operations 3

1.21

1.22

1.23

1.22

1.22

1.22

1.22

1.23

1.22

1.21

1.23

1.24 1.24

1.26 1.25 1.21 1.24

1.26

1.26

8 3 2 < 3. 2 E 2 0 a

z % E 2. 0. 2 @ E?

1.28 1.27 1.27

2 a

3 \

; ti g “2

Y .$.

1.32 1.27

1.33 1.31

1.36

1.48 1.42 1.43

(7b)

Observed

furnishings,

1 (continued)

1.40

1.33

1.63

1.54

1 so

1.44

1.43

1.38

1.37

India

Pakistan

Sri Lanka

Philippines

Thailand

Malaysia

Korea

1.31

1.29

Mexico

Poland

1.25

1.23

Luxembourg

U.S.

1.25

Germany

1.25

1.25

Austria

1.25

1.25

Belgium

Denmark

1.26

Netherlands

France

1.27

1.26

U.K.

1.29

1.23

1.21

1.25

1.26

1.24 1.29 1.27

1.27

1.28

1.29

1.28

1.28

1.29

1.27 1.29

1.29

1.29

1.24

1.29

1.29

1.30

1.29

1.24

1.23

1.30

1.31 1.29

1.32

1.31

1.33

1.33

1.25

1.27

1.32

1.31

1.34 1.33

1.34

1.37

1.27

1.27

Spain

1.39 1.37

1.34

1.36

1.37

I .40

1.41 1.41

1.32

Japan

1.41 1.41

1.42

1.40

1.42 1.42

1.49

1.48

1.53

1.58

1.75

1.46

1.48

1.55

1.56

1.67

1.70

1.90

1.74

1.93

Observed (9b)

(9a)

1.32

1.28

1.32

1.29

1.28

Italy

1.28

1.28

1.32

1.29

1.33

1.34

Hungary

1.29

1.31

1.29

1.33

Romania

Yugoslavia

Ireland

1.36

1.34

Iran

Uruguay

1.31

1.34

Syria

1.33

1.35

1.35

Brazil

Colombia

1.39

1.41

1.47

1.55

1.57

1.66

(8b)

GM

Observed

@a)

communications

GM

Transport,

Medical

care

(1)

Country

Table

1.49 1.41

1.35

1.27

1.29

1.29

1.29

1.29

1.29

1.29

1.30

1.31

1.31

1.31

1.32

1.34

1.34

1.34

1.27

1.29

1.29

1.29

1.29

1.28

1.29

1.30

1.31

1.31

1.32

1.32

1.35

I .32 I .34

1.36

1.37

1.41

1.40

1.42

1.42

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

? i?s :! =. 6 i?

1.31 1.31 1.29

2 ii m $ s 2

1.21

1.28 1.27 1.25 1.25 1.25

1.28 1.27 1.27 1.26

1.05 1.04

1.05

1.26 1.26

1.04

1.24

1.26 1.05 1.05 1.05

1.25

1.26

1.05

1.23

1.26

1.25

1.25

1.26

1.05

1.05

1.27

1.28 1.06

1.31 1.28

1.30

1.06 1.05

1.29

2 0”

1.35 1.33

1.30

1.30

1.30

1.32

> \

s .c

;

;! .s

3

1.36 1.35

1.35 1.36

1.05

1.06

1.06

1.32

1.06 1 .Oh

1.34

1.07

1.06

1.35

1.06

1.36

1.36

1.07 1.07

1.36

1.40 1.39

1.07 1.07

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.06

1.07

1.06

1.46

1.58

1.46

1.49 1.44 1.41 1.42

1.60

1.70

1.53

1.56 1.50

Observed (12b)

1.10

1.69

(12a)

GM

Other

1.58

1.10 1.09 1.08

1.39

1.35

Observed (llb)

1.06

1.07

1.07

1.07

(lla)

GM

Education

1.61

1.74

1.76

2.03

1.38

1.40

1.42

1.42

1.43

1.43

1.47

1.49

1.57

1.57

1.70

1.77

1.98

Observed (lob)

GM (lOa)

Recreation

264

H. Theil, J. L. Se&,

Jr. / Income elasticities at different price uectors

p, = geometric mean of p,,, plz, . . . .If the relative prices in c are all equal to those of the geometric means, i.e., if p,,./p,,. = PI/>, holds for all pairs (i, j), then z,,. = 0 for each i and (3) reduces to (1) so that the income elasticity of i in c is still of the form (2). These elasticities, at the per capita real income of c and at the geometric mean prices, are shown in columns 3a to 12a. ’ In columns 3b to 12b we show the elasticities at the prices of country c; their derivation is our next concern. Our strategy will be to write such elasticities in the form 1 + /~Jw,~ with w,, specified as the right side of (3). The problem is, of course, that this right side involves w,,.. To handle this we apply a series of approximations by letting prices move along a log-linear path, in vector form as Bz, where B is a scalar increasing from 0 to 1 and z, is the vector [zjC]. At 8 = 0 all prices are equal to the geometric means; at 0 = 1 they are all equal to the prices of country c. Specifically, write w,‘:’ = (Y,+ p, log Q,. for the budget share at 0 = 0 and consider two successive steps: from 0 = 0 to 19= i and then from 13= i to 8 = 1. We apply a linear Taylor expansion to conclude from the last two terms in (3) that the first step yields the following change in w,,.:

(4) When we add this to WI?, we obtain an approximation to the budget w(“‘*). The second step w ill then yield this change in w,,.: *c

+ $ ( w IJ’2’ I

9

-

;

share at 0 = i, to be written

(wj;‘*) + p,) $

(5)

J=l

When we add this to wjt”‘, we obtain an approximation to the budget share at 6 = 1. However, we should expect a closer approximation to w,, at B = 1 when we take a larger number of smaller steps, e.g., the following four: from t? = 0 to 0 = a to 6 = : to B = + to 8 = 1. The first of these yields a change in w,, equal to expression (4) except that z,,/2 becomes z,,./4. When we add the expression thus modified to wl:) we obtain an approximation to w$/~); we use this to obtain an this series of approximations until the interval approximation to w if12’, and so on. We continued (0, 1) was divided into 8192 equal steps, at which stage we had obtained stability of w,,. at B = 1 in six decimal places for each i and c. The income elasticities in columns 3b to 12b are obtained by substituting these converged values for w,, in the elasticity expression 1 + &/wIL.. A comparison of these elasticities with the corresponding values in columns 3a through 12a reveals three things. First, using observed rather than geometric mean prices destroys the regular behavior of the income elasticities (their monotonic declines as income increases) but not by much. Second, the major impact is on food; there is only one case outside food (transport and communications in Sri Lanka) in which the impact on the elasticity value exceeds 0.05 in absolute value. Third, the impact on the income elasticity of the demand for food tends to be downward in the more affluent countries. This results from the fact that food is relatively less expensive than other goods and services in these countries; this fact yields a negative value of the first price term in (3) that dominates the positive value of the second (substitution) term.

with w,, specified ’ All elasticities of table 1 are of the form 1+ p,/w,, as the pooled estimates in table E-l of Theil and Clements (1987).

as excluding

the residual

and $I, the a,s and the ,l?,s

H. Theil, J.L. Scale, Jr. / Income elasticities at different pxe

vectors

265

References Finke, R., W.-H. Lu, and H. Theil, 1984, A cross-country tabulation of own-price elasticities of demand, Economics Letters 14, 137-142. Finke, R., M.C. Rosalsky and H. Theil, 1983, A new cross-country tabulation of income elasticities of demand, Economics Letters 12, 391-396. Theil, H. and K.W. Clements, 1987, Applied demand analysis: Results from system-wide approaches (Ballinger. Cambridge, MA). Theil, H. and R. Finke, 1985, Income and price elasticities of demand at low levels of real income, Economics Letters 18, l-5. Theil, H. and J.L. Seale, Jr., 1987, Measuring the distance between relative price vectors of different countries, Economics Letters 23, 371-374. Theil, H. and F.E. Suhm, 1981. International consumption comparisons: A system-wide approach (North-Holland, Amsterdam). Theil. H., F.E. Suhm, and J.F. Meisner. 1980, Statistical inference in cross-country demand systems, Economics Letters 5, 383-387.