APPENDIX. INCOME DISTRIBUTION TRENDS AND FUTURE FOOD
DEMAND
Estimation and simulation of food Engel curves
The empirical estimation of Engel curves can be performed in many ways. Brown and
Deaton (1972) and Sadoulet and de Janvry (1995) offer excellent reviews. Prais and
Houthakker (1971) wrote a comprehensive review and performed estimations of the
following forms; linear, hyperbolic, semilogarithmic, double logarithmic, and logarithmic
reciprocal. All these forms were shown to have some advantages over the other forms for
some of the goods or for part of the range of the relationship. Prais and Houthakker
(1971) concluded that the widely used double logarithmic and the semi-logarithmic forms
performed better than the others in terms of goodness of fit.
A major concern in the estimation of Engel curves is that the functional form used should
be consistent with observed consumer behaviour. Engel curves should be able to
represent luxuries (commodities whose consumption increases more than proportionally
with income), necessities (whose consumption increases less than proportionally with
income), and inferior goods (commodities whose consumption decreases as income
increases). In addition, Engel curves should allow the same commodity to be a luxury for
the poor but a necessity for the rich. Based on these criteria, the semi-logarithmic form
performs quite well. This form is able to represent necessities, luxuries, and inferior
goods, and allows the income elasticity to vary with income levels:
qi  a  b ln yi
(A.1)
The choice of the functional form should not only be based on practical criteria of
goodness of fit, but also on principles of demand theory. One of these principles in
particular, the adding-up condition, is violated by all the forms listed above including the
semi-logarithmic. Adding-up requires that consumers do not spend more than their
income. This principle places some restrictions on the demand elasticities of each goods,
known as Engel’s and Cournot’s equations. Simply put, these equations state that changes
in income and prices determine changes in the composition of the budget constraint but
leave its value unchanged. One functional form that satisfies adding-up, and that is able
to represent closely consumer behaviour was originally proposed by Working (1943),
elaborated by Leser (1963), and widely used afterwards Deaton and Muellbauer (1980).
This form is known as the Working-Leser, and relates the commodity budget shares to
the logarithm of per capita expenditure:
wi  ai  bi ln y
(A.2)
This form satisfies the adding-up condition provided that the sum of the parameters ai
estimated over all commodities in the household budget is equal to one, and that the sum
of the parameters bi is equal to zero. It allows for luxuries, necessities and inferior goods,
and for elasticities to vary with income. Finally, the form is linear in the logarithm of
1
expenditure, and is easily estimated by ordinary least square (OLS) equation by equation,
with the adding-up restrictions being automatically satisfied.
One disadvantage of the Working-Leser form is that necessities and luxuries are
represented by different curves, which means that the same good, for example, food,
cannot be a luxury for some households and a necessity for others. Each commodity can
only have an elasticity that is either above or below one. A second shortcoming of this
form is that although it allows for varying elasticities, these are bound to vary always in
the same direction. Elasticities can only decrease as income increases. If the good is a
necessity, the elasticity decreases down to minus infinity as income increases, while if the
good is a luxury the elasticity decreases down to 1+b.
More recent research on Engel curves has focused on the estimation of the WorkingLeser equation using polynomials in the expenditure term (by adding quadratic or cubic
terms), and non-parametric or semi-parametric methods. Unlike the standard parametric
Working-Leser form, these methods allow expenditure elasticities to vary in any direction
for the same good over the entire range of household expenditure. Most studies have
found evidence of non-linear budget shares and elasticities for a large number of goods.
Studies in developed economies include (Atkinson et al. 1990, Banks et al. 1997), while
studies of developing countries include (Bhalotra and Attfield 1998, Gong et al. 2005,
Kedir and Girma 2007).
In line with the developments in the empirical literature of demand systems, the
modelling literature has tried in recent years to integrate more realistic and flexible
demand systems to account for some degree of Engel flexibility. Until very recently one
problem of many simulation models was the use of simple functional demand forms
based on their tractability during estimation, but at the cost of unrealistic predictions,
failing to capture changes on income elasticities of demand. For example, the simple
Homothetic Cobb-Douglas (HCD) system used in many CGE models, while easy to
calibrate, shows constant average budget shares, which is in contradiction with Engel’s
law.
Lewbel (1991) develops a useful term for demand systems and Engel curves, the rank of
a demand system. This is the maximum dimension of the function space contained by the
Engel curve. In practical terms, rank one demand systems generate constant elasticities
independent of income/expenditure. Rank two systems generate linear Engel curves,
while rank three systems produce non-linear Engel curves. The author estimates nonparametrically the rank of the demand system using expenditure data without price
variation from the US and the UK, and finds rank three demand systems for these
countries to be the best demand system representation.
This issue has been further explored by Yu et al. (2004) and Cranfield et al. (2003), who
compare different demand systems used for projecting world food demand that are rank
two, with the ‘optimal’ rank three demand systems: mainly An implicit Additive Demand
System (AIDADS) and the Quadratic Almost Ideal Demand System (QUAIDS). Some of
the properties that a desired demand system should have are: non-negativity of budget
2
shares, predicted shares on the interval [0,1] and Engel flexibility. Table A.1 summarises
the comparison and different properties of these demand systems.
Yu et al. (2004) compare different rank two demand systems to the AIDADS system
using a multiregional CGE model. They calibrate the four CGE models with the same
data in 1985 and simulate income and population growth to 2020 assuming that factors
adjust in order to maintain prices unchanged. The authors find that rank two demand
systems tend to overestimate consumption and food income demand elasticities with
respect to the AIDADS system.
Cranfield et al. (2003) estimate different demand systems for 64 countries and 113
commodities and evaluate their in sample and out of sample predictions. They find AIDS
and QUAIDS to better fit the data, and to a lesser extent QES. Concretely, they find that
QUAIDS systems are better in the context of simulations where price variation may be
important. On the other hand, AIDADS systems have less price parameters to estimate
and are better when price variation may not be that important. Also, the advantage of
QUAIDS is that is easier to estimate.
The conclusion of this literature is, therefore, the need to integrate rank three demand
systems in model simulations if one wants to take into account the observed dynamic
effects of falling marginal food budget shares.
3
Table A.1. Comparison of different demand systems
Demand system
Budget share form
Linear Expenditure
pit  it
pt' 
w



(
1

)
it
i
System (LES)
yt
yt
Quadratic Expenditure
System (QES)
(Howe et al.1979),
Almost Ideal Demand
System (AIDS)
(Deaton and
Muellbauer 1980).
Quadratic Almost Ideal
Demand System
(QUAIDS)
Banks et al. 1997)
 p'   p 
p
p '  n  p 
wit  it it   i 1  t    it it   i t   it 
yt
yt  i 1  yt 
 yt   yt
2 1
 pt'  
1 

 yt 
Properties
- Rank two demand system.
- Limited Engel flexibility- constants marginal
budget shares over all income levels (as income
increases without a bound, average budget shares
converge monotonically towards one, and food
demand increases monotonically with income).
- Special case that nests at AIDS.
- Rank two demand system.
- Limited Engel flexibility- produces Linear Engel
curves.
- Nested at LES.
- Rank two demand system.
- Limited Engel flexibility- produces Linear Engel
curves.
- Budget share predictions may lie above unity.
2
n
y 
wit   i    it ln( pit )  i ln  t* 
i 1
 Pt 
y 
 n

wit   i    it ln( pit )   i ln  t*   i   plt1 
i 1
 i 1

 Pt 
n
An Implicit Direct
pit  it  i   i exp( u t )
pt' 
w


(
1

)
it
Additive Demand
yt
1  exp( u t )
yt
System (AIDADS)
(Rimmer and Powell
1996)
Source: Based on Yu et al. (2004) and Cranfield et al. (2003)
4
1
  y t 
ln  * 
  Pt 
2
- Rank three demand system.
- High Engel flexibility – allows non-linear Engel
curves and for goods to be luxury at low income
levels and normal at high expenditure level.
- Budget shares predictions may lie outside the
[0,1] interval.
- Rank three demand system.
- High Engel flexibility- marginal budget shares
are non-linear on income/expenditure,
- Budget shares constrained to the interval [0,1]
- Nested at LES.
Household composition and aggregation
Correct estimation of Engel curves should be performed using expenditure per adult
equivalents rather than per capita expenditure, because the use of the latter has the effect
of overestimating food expenditure by households of larger size (Deaton, 1997). There
are two reasons for why per capita expenditure is inappropriate for the estimation of
Engel curves. First, children in developing countries consume proportionally more food
than adults. For example, consumption by infants is likely to consist mainly of food
among poor families. As a result, for the same level of per capita expenditure, larger
households spend more on food. Households of different size have different Engel curves
and the shape of the curve may be biased if household size is correlated with income. The
solution to this problem is to divide household expenditure by units of adult equivalents
using appropriate equivalence scales, rather than by household size. Unfortunately, there
is no agreement on how equivalence scales should be built, and the adjustment is
normally performed using standard caloric requirements by age and sex provided by the
World Health Organization. This ad hoc procedure is considered superior to the practice
of simply using per capita expenditure. Second, there are economies of scale in the
consumption of goods including food. For example, a TV set is used by all household
members and adding a new member to the family adds in terms of the welfare enjoyed by
the family but not in terms of costs. Similarly, the fire used for cooking consumes the
same fuel whether the food is prepared for four or for five people. Again, there is no
standard methodology for estimating economies of scale in consumption and the practice
is to use arbitrary measures. For example, one device adopted in OECD statistics consists
of dividing household expenditure by the square root of household size, rather than by
household size.
The number of children in the household and economies of scale in consumption have
implications for the estimation of Engel curves across countries as well. Because children
consume proportionally more food than adults, food consumption in countries with high
fertility rate will be higher for the same level of per capita expenditure compared to other
countries. For example, the Engel curve of India will lie above the Engel curve of China
for the same levels of per capita expenditure, because average household size in India is
larger than in China. Today’s fertility rates in developing countries are no longer as high
as they were, for example, 50 years ago. However, the existing differences are likely to
bias the estimate of the world Engel curve. One possible way to proceed in order to
correct this bias would be dividing the country GDP by the number of adult equivalents
rather than by population size. The adult equivalents could be calculated using
demographic data on average household size and number of children in total population.
Changes in the demographic structure of the household over time also have implications
for changes in shape of the Engel curve over time. For example, a decline in fertility
would have the effect of reducing household food consumption for the same level of per
capita household expenditure. In this case the shape of each country Engel curve would
change over time as fertility rates change, and the curve estimated from a single crosssection may not be reliably used for predicting future food consumption. Fertility falling
worldwide may have the effect of decreasing the amount of per capita food consumption.
5
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