Testing separability of public consumption in household decisions
DAVID ARISTEI and LUCA PIERONI
ABSTRACT
In this paper we propose a sequential strategy, based on the microeconomic approach
of the demand theory, in order to test for separability between private and public consumption. The aim
of the present work is to verify, using a conditional Almost Ideal demand system, whether the different
components of public consumption exert conditioning effects on the allocative structure of private
spending. The empirical estimation of the model and the separability tests are developed for both a
demand system in five functional categories of private spending, and for a demand system in six
categories, where the private expenditures on those goods and services which can also be offered by the
public sector are enclosed in a single functional category. The results of the separability tests, obtained
using UK data for the 1974-2000 period, show that public individual consumption plays an important
role in modifying consumer choices, while public collective consumption does not affect private
consumption behaviours. Secondly, the relationships between the different components of private
spending and public individual consumption are both of substitutability and complementarity; in
particular, we find that public individual consumption and the corresponding private expenditures on
“Health, education, recreation and social protection” are complements.
JEL Classification: D1, H3, H4
Key Words: consumer demand, government consumption, non-separability, demand systems
(forthcoming in International Review of Applied Economics, vol. 19)
1. Introduction
Much of the discussion on the effects of government expenditure on private decisions is
dominated by the analysis of the tradeoffs between the goals of efficiency and equity.
The debate on tax and subsidy structure as well as on minimum levels of public goods
has led to analyse the effects of public programmes on the incentive to consume and
save and to consider the complex relationships between private and public consumption.
The investigation has been carried out from a macroeconomic point of view focused
mainly on the evaluation of the crowding-out effects on the private sector, implicitly
assuming the existence of a substitutability relationship between private and
government consumption (Bailey, 1971; Kormendi, 1983; Aschauer, 1985; Bean,
1986). However, Karras (1994) has pointed out that the relationship between
government expenditure and private consumption in the aggregate may be misleading,
since it may hide the possibility that each component of public consumption may exert
different or even opposite effects on the components of private expenditure.
On the other hand, the empirical microeconomic literature has generally disregarded the
impact of government spending on the allocation of private expenditures; the issue of
non-separability of individual preferences between private consumption and public
purchases has received very little attention. Recently, some authors have suggested the
1
need to use a micro-based approach to investigate the relationships between private and
public consumption, asserting that the main shortfall of the previous analyses may have
been due to by the fact that the demand functions were not derived from a utilitymaximization process (Levaggi, 1999; Tridimas,2002). Since the quantities of publicly
provided goods and services are predetermined (i.e. rationed) with respect to consumer
choices, these works use the theory of consumer behaviour under rationing to study the
effects of government spending on the pattern of private demand (Pollak, 1969; Neary
& Roberts, 1980; Deaton, 1981). In this way the non-separability of individual
preferences can be theoretically considered and econometrically tested. Dealing with the
question of how the different components of public spending affect private expenditure,
the estimations have provided evidence of a wide pattern of relationships, both positive
and negative, extracting, in particular, a direct complementarity relationship between
private consumption and government spending on individual - health, education and
social security - government expenditure (Levaggi,1999; Tridimas, 2002).
In the present paper, the micro-based approach is used to analyse the effects of
government consumption spending on private expenditure allocation. With respect to
the previous literature, our analysis differs in some theoretical and methodological
issues, as well as in the representation of public consumption data, which conforms the
new classification of the functions of government.
Theoretically, as suggested by Levaggi (1998), to analyse all the possible interactions
between private and public expenditures, the functional form of the demand functions
should be as flexible as possible. For this reason, the use of an Almost Ideal demand
function seems the most appropriate for the aim of this study since it allows a wide
range of substitution effects (Deaton & Muellbauer,1981). The specification of a
conditional AI model is consistent with the framework adopted, in which publicly
provided goods and services are treated as rationed quantities, the cost of the
government provision is entirely financed by tax payments and a balanced government
budget is postulated. Given these assumptions, the conditional demand for privately
purchased goods depends on the residual after-tax income, which can be allocated to
privately purchased goods at a given level of the government expenditure. Provided that
private and public consumption are non-separable, this structure allows us to investigate
the direct substitution (or complementarity) effects of government expenditures on the
allocation of private spending.
A sequential strategy is adopted in order to efficiently test for the separability condition
between private and public consumption and to estimate the substitutability or
complementary relationships, after checking the non-separability condition. In contrast
to Tridimas (2002), we reduce the inefficiency of the estimates caused by the
simultaneous introduction of government expenditure categories, which could distort
the test and make the estimated parameters not significantly different from zero.
In the empirical section of the paper, we use a UK dataset for the period 1974-2000,
adopting a functional classification of public expenditures in the following five
categories: 1) Public collective consumption (GC); 2) Education (PE); 3) Health (PH);
4) Recreation, culture and religion and social protection (RC+SP). This dataset allows
us to test for separability in a complete dynamic demand system and to compute the
short-run elasticities with respect to public consumption. Moreover, we attempt to
verify whether or not private expenditures on those goods and services (such as health,
education, recreation and social protection) which can also be offered by the public
sector display a closer relationship with government spending.
The empirical results obtained are relevant from a policy perspective, since the nonseparability of government expenditures affects consumer behaviour, modifying the
2
allocation of private consumption. The results can be summed up as follows: (a) public
individual consumption plays an important role in modifying consumer choices, while
public collective consumption (defence, public order and safety, etc.) does not seem to
influence private consumption behaviour; (b) the relationships between the different
components of private spending and public individual consumption are both of
substitutability and complementarity; and (c) we find that public individual
consumption and the corresponding private expenditures on “Health, education,
recreation and social protection” are complementary. Finally, the last section
summarises the results obtained.
2. Theoretical Framework
In empirical studies on private demand, the utility conferred on consumers by publicly
provided goods and services is usually disregarded. Standard microeconomic models
typically represent consumer utility function as defined only on a set of goods and
services freely purchased on the market, so that the resulting demand equations relate
demanded quantities to relative prices and total expenditure. Separability of individual
preferences between private and public consumption items is therefore implicitly
assumed (Tridimas, 2002).
A conditional demand system is derived using an Almost Ideal specification (Deaton &
Muellbauer, 1980), by which we are able to develop an exact test for separability
between private and public consumption. The main advantage of using a conditional
approach in demand analysis is that the utility benefits can be kept separate from the
consumer decision process.
In contrast to private expenditures, the public provision of goods and services is
established a priori and imposed by government; the level and composition of public
spending, as well as the taxes needed to finance those expenditures, are exogenous from
the point of view of the consumer. Given this setting, the theory of consumer behaviour
under quantity constraints (Pollak, 1969; Neary & Roberts,1980; Deaton, 1981) can be
applied to analyse the influence of public consumption on private expenditures.
In order to derive the conditional demand functions, we define, following Pollak (1969),
a class of goods, of which we want to know the m × 1 vector of quantities, q1, as well as
the vector of prices, p1. We then denote another class of goods, whose preferences are
not parameterised and whose consumption is pre-determined in the quantity, q2, at the
fixed p2 price; this class represents rationed goods. The conditional cost function
c * (u, p, q2 ) can be defined as the minimum cost needed to obtain the utility level u,
given the price vector p (which includes both p1 and p2), when the quantity q2 of the
rationed good must be purchased. Formally:
[1]
c * (u , p, q 2 ) = min [ p ⋅ q | u ( q1 , q 2 ) = u ] = p 2 q 2 + min [ p1 ⋅ q1 | u ( q1 , q 2 ) = u ]
q
q1
Applying Shephard’s Lemma to [1], i.e. differentiating c * (u, p, q2 ) with respect to p1,
we obtain Hicksian conditional demand equations, expressed in terms of q1. From [1], it
can be noted that the price of the rationed good (p2) enters the conditional cost function
only through the fixed term p 2 q 2 ; for this reason the Hicksian demand function does
not depend on p2; its expression, in terms of quantities, is the following:
∂c * (u, p, q2 )
= h1 (u, p, q2 ) = q1
[2]
∂p1
3
Inverting the cost function [1] and substituting it into equation [2], we obtain the
Marshallian demand function, which relates q1 to prices p1, to total expenditure x and to
the quantity of the ration q2:
[3]
q1 = g1 ( x, p1 , q 2 )
where q1 represents the m × 1 partitioned quantities of the freely purchased goods.
The presence of quantity constraints exerts two effects on the demand for freely
purchased goods: an income effect, whereby an increase in the ration reduces the
amount of income available to purchase freely chosen items, and a
substitution/complementarity effect, whereby the consumer rearranges his expenditures
on freely chosen goods following a change in the quantity constraint.
Once the theoretical framework is defined, we use Pollak’s solutions to derive an exact
test for separability. Goods are weakly separable if the direct utility function can be
written as:
[4]
u (q1 , q 2 ) = u ′[u (q1 ), u (q 2 )]
and the corresponding cost function becomes:
c * (u, p, q 2 ) = c * [ p1 , g (q 2 , u )]
[5]
Under the null hypothesis of weak separability, the uncompensated demand system
takes the following structure:
[6]
q1 = g1 ( x, p1 )
This restricted specification represents a demand system which includes only the
parameters related to the freely purchased goods. The separability test is used to
statistically verify the information loss implied by the restriction of model [3] into
model [6]. Using the conditional structure, however, nothing can be inferred about
individual preferences on q2, since they are not parameterised and the consumption of
these goods is imposed a priori. We can only determine if the preferences between
freely purchased goods and rationed goods are separable (Browning & Meghir, 1991).
This theoretical framework can be adopted in order to analyse the relationships
between private and public consumption. Solving the constrained maximization
problem, we derive the following uncompensated conditional demand function:
q = g ( x, p, G)
[7]
where p represents the vector of prices of the freely chosen goods, G denotes publicly
provided goods and services and x equals disposable income, since G is assumed to be
entirely financed by tax revenue.
The standard demand function q = g ( x, p) then is simply a special case of the function
[7], which is only correct when public consumption G is separable from private
expenditures q. Our aim is therefore to verify the possibility of restricting the
conditional demand function [7] and to test for separability between private and public
consumption.
In order to derive the separability test, we have to first define a specific functional form
in which public consumption enters the utility function in a non-separable way
(Tridimas, 2002). Otherwise, by construction public consumption will not influence
consumer behaviour and the effect of public purchases will be reduced to an income
effect only, with private expenditures decreasing as government spending, and hence its
financing through taxation, grows. The Almost Ideal Demand System (Deaton &
Muellbauer, 1980) seems to be the appropriate specification, since it provides a first
order approximation of any demand system and allows an exact aggregation over
households (Levaggi, 1998). The static demand equations, in terms of budget shares,
can be written as1:
4
m
wi = α i + ∑ γ ij log p j + βi [ log x − log P ] + θi G
[8]
j
where log P = α 0 + ∑ k (α k + θ k G ) log pk + 1 2 ∑ k ∑ j γ kj log pk log p j is a price index
which can be approximated by the Stone index ( ∑ wk log p k ). Demand theory implies
the following restrictions on the parameters of the conditional AI model:
Adding up:
∑ i αi = 1; ∑ i γ ij = 0; ∑ i βi = 0; ∑ i θi = 0
Homogeneity:
∑ j γ ij = 0
Symmetry:
γ ij = γ ji
In our analysis, provided that the null hypothesis of separability between private
spending and total public consumption is rejected, G will correspond to a particular subaggregation of public expenditures, such as collective or individual consumption. The
introduction of G as an additional element of equation [8] further allows us to measure
the direct (net) effect of public consumption on private spending by computing the
derivative of the demand equation in terms of budget shares with respect to G
( ∂wi ∂G ).
In addition, it is proper to analyse the statistical time-series properties of the variables of
the demand system. Many empirical studies have shown that both private consumption
variables – budget shares, relative prices and total income – (Lewbel & Ng, 2000) and
public consumption expenditures (Sturm, 1998; Mittnick & Thorsten, 2001) are nonstationary. A first difference specification of the AI model is therefore necessary to
avoid the risk of spurious relations (Ng, 1995).
For these reasons, following Deaton & Muellbauer (1980), Keller (1984) and Keller &
Van Driel (1985), we propose a dynamic version of the conditional AI model (DAI,
Dynamic Almost Ideal), obtained by differentiating both sides of the static demand
functions [9]; formally:
m
dwi = α i + ∑ γ ij d log p j + β i d log( x P) + θ i dG
[9]
j
where the rates of change of each budget share is the dependent variable, expressed as a
function of the rate of change of relative prices, real total expenditure and public
consumption.
In conclusion, even if the first difference specification [9] reduces the risk of spurious
regressions, it does not allow the long-run relationships among quantities, relative prices
and total expenditure to be evaluated. Consequently, the estimate parameters must only
be interpreted as short-run relationships.
3. Data
In order to obtain a functional representation of both private and public consumption
expenditures the data used in the present paper are taken from annual time series of UK
National Accounts, published by the Office for National Statistics (ONS)2.
Private expenditures are represented according to the COICOP (Classification Of
Individual Consumption By Purpose) guidelines. In particular, the functional
classification used in this study consists in a simpler version of the two-digit level
COICOP scheme and provides a representation of household consumption in the
following six spending categories: 1) Food, beverages and tobacco (FBT); 2) Clothing
and footwear (CF); 3) Housing, fuels and furnishings (HF); 4) Transport and
5
communications (TC); 5) Health, education, recreation and social protection (HER); and
5) Other goods and services (OGS).
The main feature of this functional classification consists in the aggregation of the
private expenditures on health, education, recreation and culture and social protection in
a single category. The definition of this particular category allows us to aggregate all the
expenditures on those goods and services which can also be offered by the public sector.
The series of private consumption expenditures are expressed both at current and 1995
constant prices; total private expenditure is transformed into per capita terms, by
dividing the original series by the resident population. Private spending is net of durable
consumption expenditures, such as the expenditures on furnishings, appliances,
vehicles, audio-visual equipment and other durables for recreation and culture, which
are included in some non-food spending components. The presence of durable
components within private consumption, in fact, may be a problem for a correct
specification of the demand system. The temporal gap between the moment in which
those expenditures are made and when they transfer their utility implies an
intertemporal choice process that cannot be correctly explained by the model of
analysis, since it is essentially based on a single-period decision3.
The adoption of a functional classification of private expenditures according to
COICOP guidelines requires that a public spending representation be used that
corresponds to the new version of the COFOG (Classification Of the Functions Of
Government) scheme, introduced by the 1993 System of National Accounts (SNA93).
According to the aims of the present analysis, the main use of COFOG is to identify
public individual consumption expenditures on health, education, social protection,
recreation and culture (GI), which benefit individual households. Those expenditures
are also reported in Division 14 of COICOP in order to obtain a measure of household
actual final consumption4; Table 1 shows the links between the two functional
classifications.
Table 1 – Public individual consumption according to COICOP and COFOG
Components
COICOP
COFOG
Housing
Group 14.1
Group 10.6
Health
Group 14.2
Groups 07.1-07.4
Recreation and culture
Group 14.3
Groups 08.1-08.2
Education
Group 14.4
Groups 09.1-09.6
Social protection
Group 14.5
Groups 10.1-10.5 e 10.7
On the other hand, public collective consumption (GC) includes all government
expenditures on defence, public order and safety, general public services, environmental
protection, economic affairs and housing, which are directly associated with providing
those services destined to satisfy collective needs, which are connected to the existence
of the government itself.
The introduction of SNA93 and the definition of the new functional classification of
government expenditures has required a complete revision of the National Account
aggregates. The empirical analysis proposed in this paper is focused on the UK, where
the public consumption time series are sufficiently broad. However, a complete
representation of public consumption expenditures is only available for the 1986-2000
period and so there is a trade-off between the use of a disaggregated representation of
the data and the sample size. Since our aim is to test for the existence of
6
substitutability/complementarity relationships between private spending and some of
the main components of public consumption (health, education and social protection),
which are the object of fiscal policies, we accept a simplification of the representation
structure in order to extend the time series. From Table 11.2 of the Blue Book, we
obtain the annual time series of final public consumption spending on “Health” and
“Education” for the 1974-2000 period, while the time series for the categories “Social
protection” and “Recreation, culture and religion” only cover the years from 1980
onward. Therefore, we decided to aggregate these two latter components into a single
category in order to obtain the 1974-2000 time series as the difference between public
individual consumption and public expenditures on health and education.
The functional representation of public consumption, used in the present work, consists
in the following four spending categories: 1) Public collective consumption (GC); 2)
Education (GE); 3) Health (GH); and 4) “Recreation, culture and religion and Social
protection” (GR). The sample covers the period 1974-2000; as the data are only
available at current prices, we obtain the series at constant prices by deflating the
original series through the public consumption deflator at 1995 prices (OECD, 2001).
The categories of public consumption expenditures are then expressed in per capita
terms.
4. Strategy of analysis and time-series properties
The hypothesis of separability of individual preferences between private and public
consumption items is tested using a conditional demand system in which the different
components of public expenditures are treated as rationed quantities. Firstly, the
strategy of analysis adopted in the present paper foresees the specification of an Almost
Ideal model conditioned to total public consumption (G). Secondly, public consumption
is divided into the categories “Public collective consumption” (GC) and “Public
individual consumption” (GI). This representation of public expenditures allows us to
separate those goods and services which are non-rival and non-excludible in
consumption from those which were more akin to private goods (Fiorito & Kollintzas
2002).
Generally, in testing the hypothesis of separability between private consumption and
these two wide public expenditure categories, we expect that public individual
consumption, given the expenditures it contains, would exert a much more evident and
significant influence on the allocation of private spending than the collective public
goods category (Karras, 1994; Levaggi, 1999).
Under the assumption of non-separability between private expenditures and public
individual consumption (GI), we attempt to verify whether the different components of
GI affect consumer behaviour, by specifying three demand systems conditioned to
government spending on “Education”, “Health” and “Recreation, culture and religion
and Social protection”.
Figure 1 shows a schematic drawing of the strategy used to test for separability. The
strategy is developed both for the complete demand model in six categories (DAI6) and
for a restricted demand system in five categories (DAI5), in which the functional
category, including private expenditures on those goods and services also provided by
the public sector (education, health, social protection, etc.), is omitted. Therefore, by
subtracting this category from total private consumption, we attempt to verify how the
presence of such expenditures affects the separability test. These private expenditures,
being akin to public individual consumption, are expected to display a statistically
significant relationship with government spending (Kuehlwein, 1998).
7
Figure 1 – Structure of the analysis
Total public
consumption
(G)
Public collective
consumption
(GC)
Public individual
consumption
(GI)
Health
(GH)
Recreation, culture, religion
and social protection
(GRCP+PS)
Education
(GE)
The first-difference conditional specification [9] requires that the variables of the
demand system (budget shares, relative prices, real total expenditure and public
consumption) are non-stationary, i.e. I(1).
Table 2 summarises the results of the Augmented Dickey-Fuller tests (ADF(t) e ADF(F))
(Dickey, Fuller, 1978, 1981)5 for the conditional AI model. The critical values of the t
and F statistics are tabulated by MacKinnon (1992).
The results of the univariate test show that the null hypothesis of a unit root in the datagenerating process of the variables cannot be rejected at the 5% significance level; a
value close to the critical level is only obtained for the public spending category
“Recreation, culture and religion and Social protection” (GR).
From the ADF(F) test results, it can be noted that the time-series of budget shares, except
w3 (“Housing”), and of total public consumption (G) and public health (GH) do not
contain a deterministic time trend.
In conclusion, the ADF(t) e ADF(F) tests point out a non-stationary behaviour of all the
variables in the conditional AI demand system, so that a dynamic specification of the
model is statistically correct.
Table 2 – Unit root tests
Test
w1
w2
w3
w4
w5
w6
lp1
lp2
lp3
lp4
ADF(t)
-1.493
[.789]*
-1.946
[.604]
-2.790
[.223]
-1.773
[.682]
-1.553
[.999]
-1.884
[.632]
-2.154
[.502]
-1.586
[.757]
-1.878
[.635]
-2.586
[.300]
ADF(F)
2.549
[.105]
2.385
[.119]
3.954
[.037]
1.609
[.226]
1.265
[.305]
1.885
[.179]
4.573
[.024]
3.237
[.062]
6.786
[.006]
5.686
[.012]
Test
lp5
lp6
GC
GE
GH
GR
GI
G
Income
ADF(t)
-1.762
[.687]
-2.689
[.259]
-2.868
[.197]
0.086
[.988]
-2.179
[.490]
-3.419
[.078]
-0.247
[.978]
-2.415
[.376]
-2.654
[.273]
ADF(F)
3.835
[.040]
5.994
[.010]
6.123
[.009]
6.917
[.006]
2.970
[.075]
6.383
[.009]
5.824
[.011]
2.973
[.075]
3.612
[.047]
* p-values in brackets
8
Given that the variables are I(1), we cannot however exclude that the demand system
could be cointegrated; a differential AI specification as in [9] will then lead us to
incorrect results. To test the statistical long-run specification, we extend the previous
model [9] to a conditional model in ECMs (Boswjik, 1995):
p −1
Γ0 ∆yt = B0 ∆z t + λβ ' xt −1 + ∑ (Γ j ∆yt − j + B j ∆z t − j ) + ε t
t = 1,...,T
[10]
j =1
where x ' t is the vector of partitioned variables ( y t′ , z t′ ), y t are g × 1 budget shares, zt
are (k − g ) ×1 exogenous variables regarding the log prices, real income and public
consumption, Γ0 is a g × g non-singular matrix, y t − j and z t − j are lagged variables, λ
is a structural error correction matrix and ε t is the error term.
The specification [10] must satisfy the identifying restrictions of the system parameters
[9] in a long-run demand system specification. We impose the matrix Γ0 to be unitary,
restricting the matrix Γ j and B j to zero. The restricted dynamic demand system can be
tested for cointegration. Using the suggestion of Urbain (1995), a multi-equation model
is cointegrated if the rank (r) of matrix λ is equal to the number of equations. Since the
additivity constraint reduces the rank of the demand system to n − 1 non-singular
equations, the condition for cointegration requires r = g − 1 = n − 1 , while the demand
system is not cointegrated when r = g − 1 < n − 1 .
The modified Wald test is used under the null hypothesis of non-cointegration6. Table 3
shows the estimation results, with the 10% and 5% critical values tabulated by Boswjik
(1994) reported in the note. The rank condition, which requires five significant
cointegrating vectors, given the presence of six endogenous variables in each equation,
is not accepted, leading to the rejection of the long-run specification of the demand
system.
Table 3 – Cointegration tests
Food, beverages and
tobacco
Separable specification
Conditional specifications:
- Total public consumption
- Public collective consumption
- Public individual consumption
Housing, fuels and
furnishings
standard
constant
standard
constant
standard
constant
55.022
51.783
15.692
16.707
31.108
27.766
39.509
52.336
38.583
39.680
57.237
36.372
29.672
20.749
22.770
26.526
18.648
21.245
28.533
30.460
28.802
25.383
27.12480
25.63273
Transport and
communications
Separable specification
Conditional specifications:
- Total public consumption
- Public collective consumption
- Public individual consumption
Clothing and
footwear
Health, education,
recreation and social
protection
Other goods and
services
standard
constant
standard
constant
standard
constant
4.471
4.652
17.454
46.023
16.721
14.877
4.757
3.718
5.013
4.604
6.630
4.698
22.009
29.161
17.164
46.406
45.266
47.139
23.076
23.466
17.824
22.790
26.753
15.845
The critical values at the 5% and 10% level for seven exogenous variables are approximately equal to 24.77 and 22.29 for
the model without constant. For the model with constant they are equal to 27.27 and 24.62, respectively.
9
5. Results
5.1. Testing theoretical restrictions
The first-difference specification of the model implies a one-period lag, so that the
sample provides twenty-six observations (1975-2000). Moreover, since the data add up,
the covariance matrix is implicitly singular and an arbitrary equation must be omitted
from the system. In particular, we chose to omit the residual category “Other goods and
services”, specifying a demand model in five independent equations with a total of 130
observations. Since each equation of the demand system contains the same variables,
we have Σε it ε jt ≠ 0 for each i ≠ j ; for this reason, the symmetry constraint, which
implies cross-equation restrictions, requires the use of a full-information maximum
likelihood (FIML) method of estimation.
The empirical validity of the theoretical constraints of homogeneity and symmetry is
tested using a likelihood ratio (LR) test, based on the log-likelihood values of the
restricted and unrestricted models. The test statistic, which is asymptotically χ (2k ) under
2
2
the null7, is defined as LR = T (σˆ R2 σˆ UR
) = 2(log LUR − log L R ) , where σ̂ R = e′R eR T and
2
′ eUR T are the residual variances of the restricted and unrestricted models,
= eUR
σˆ UR
respectively. Many empirical studies have shown that, in finite samples, standard
asymptotic results can be very misleading, since they are biased towards over-rejection
when the number of equations and parameters of the models is large with respect to the
sample size (Dufour & Khalaf, 2002). In particular, testing homogeneity and symmetry
in demand systems is shown to be very sensitive to this problem, especially when the
models are heavily parameterised (Laitinen, 1978; Pudney, 1981; Theil & Fiebig, 1985).
It is therefore worth implementing a small-sample adjustment to correct the tendency of
the LR test to over-reject in large demand systems. Following Pudney (1981), we define
the adjusted statistics as follows:
2
LR ∗ = T (log σˆ R2 T0 ) (log σˆ UR
T1 )
[11]
= LR + log{(nT − p1 nT − p0 }
{
}
where n is the number of equations, p0 and p1 are the numbers of parameters of the
restricted and unrestricted models, respectively, and Ti = T − pi n . An analogous
correction is also carried out for the critical values, which take the form
K = nT log 1 + dFnTd − p1 nT − p1 , where d = p1 − p0 and FnTd − p1 is the 5% critical value
of the F distribution.
The results for the tests of homogeneity and symmetry, with and without the small
sample correction, are presented in Table 4. As is common in demand analysis, the joint
asymptotic test for the theoretical constraints leads to a general rejection of the null
hypothesis; however, in the present analysis, this result is mainly attributable to the
homogeneity restriction, as symmetry is accepted in almost all the specifications. This
result is quite interesting since, in empirical demand studies, it is the symmetry
constraint which is usually inconsistent with the data. Specifically, the hypothesis of
homogeneity (Section 1 of Table 4) is rejected at the 5% significance level for each
conditional specification, except for the demand system conditioned to “Education”. So,
the theoretical hypothesis of the absence of money illusion in consumer decisions
cannot be accepted. The results of the asymptotic tests concerning the hypothesis of the
symmetry of price effects, reported in Section 2 of Table 4, show that the theoretical
constraint cannot be rejected in almost all the specifications, with the calculated values
[
]
10
close to the 5% critical level only when the model is conditioned to public collective
consumption and to public expenditures on education. As can be noted, even after
applying the small-sample correction, the test results do not change much; there is only
a slight improvement in the statistical significance of the homogeneity constraint
concerning the separable specification and the model conditioned to “Recreation,
culture and religion and Social protection”.
The results of the asymptotic joint test of homogeneity and symmetry (Section 3 of
Table 4) show that the restrictions of the demand theory do not hold. The null
hypothesis cannot be rejected at the 5% level only when the model is conditioned to
public expenditures on “Recreation, culture and religion and Social protection”.
However, it is important to underline that, when the small-sample adjustment is made,
the joint test for homogeneity and symmetry is not rejected in any of the specifications,
with the exception of the models conditioned to “Total public consumption” and to
“Public collective consumption”.
In conclusion, these results underline that the consistency of the theoretical restrictions
with the data appears to be questionable; hence, we decided to impose a priori
homogeneity and symmetry.
Table 4 – Tests for theoretical restrictions
1. Test for homogeneity
Specification of the model
Separable specification
Conditional specifications:
- Total public consumption
- Public collective consumption
- Public individual consumption
- Education
- Health
- Recreation, culture, religion and social protection
Standard asymptotic results
LR statistics (χ2(5)) P-value
Small-sample correction
Test statistics Critical value
11.704
0.039
11.655
14.182
26.156
17.253
16.767
8.552
21.952
12.305
0.0001
0.004
0.005
0.128
0.0005
0.031
26.105
17.202
16.716
8.501
21.901
12.254
14.917
14.917
14.917
14.917
14.917
14.917
2. Test for symmetry
Specification of the model
Separable specification
Conditional specifications:
- Total public consumption
- Public collective consumption
- Public individual consumption
- Education
- Health
- Recreation, culture, religion and social protection
Standard asymptotic results
Small-sample correction
LR statistics (χ2(10))
P-value
15.318
0.121
15.213
25.347
15.639
18.679
12.777
17.788
13.282
10.390
0.110
0.045
0.236
0.059
0.208
0.407
15.591
18.631
12.729
17.740
13.234
10.342
26.775
26.775
26.775
26.775
26.775
26.775
Test statistics Critical value
3. Test for homogeneity and symmetry
Specification of the model
Separable specification
Conditional specifications:
- Total public consumption
- Public collective consumption
- Public individual consumption
- Education
- Health
- Recreation, culture, religion and social protection
Standard asymptotic results
Small-sample correction
LR statistics (χ2(15))
P-value
27.022
0.029
26.868
33.756
41.796
35.932
29.545
26.340
35.234
22.695
0.0002
0.002
0.014
0.035
0.002
0.091
41.633
35.769
29.382
26.177
35.071
22.532
35.610
35.610
35.610
35.610
35.610
35.610
11
Test statistics Critical value
The homogeneity of the estimated share equations and the symmetry of the estimated
Slutsky matrix guarantee the consistency of consumer choices and, therefore, the
validity of the weak axiom of revealed preferences. However, in order to satisfy the
strong axiom of revealed preferences, the concavity of the cost function must also hold
(Pendakur, 2001). Concavity is satisfied provided that the estimated Slutsky Matrix is a
negative semidefinite matrix (negativity constraint). In the AI specification, concavity
requires that the matrix K, with elements:
∂w ∂w
k ij = i + i w j + wi w j − wi δ ij
∂x
∂p j
[12]
= γ ij + β i β j log( x P ) + wi w j − wi δ ij
is negative semidefinite; so, to empirically check negativity, it is sufficient to verify that
the eigenvalues of K are non-positive, as they have the same signs of those of the
Slutsky matrix. Table 5 presents the eigenvalues of the K matrix estimated at the mean
point, for both the separable and the conditional (with total public consumption inserted
as a rationed quantity) specifications of the model.
Table 5 – Test for concavity
Eigenvalues*
Separable specification
Conditional specification (G)
1
0.002187
0.002352
2
-0.003218
-0.003197
3
-0.025241
-0.021732
4
-0.046689
-0.043896
5
-0.148878
-0.179369
*Eigenvalues of the matrix K with elements: kij = γ ij + βi β j log(x P) + wi wj − wiδij
As can be noted, all the eigenvalues are negative, with the exception of the first ones of
both specifications, which are, however, not significantly different from zero. This
result shows that the homogeneity and symmetry-restricted estimates satisfy (local)
concavity, ensuring that the preference ordering is consistent with the axioms of rational
choice and that the estimated demand functions can be integrated to recover the
underlying cost function (Perali, 2002).
5.2. Testing separability between private and public consumption
The presence of substitutability or complementarity relations between private and public
goods can be statistically verified by using a LR test, based on the log-likelihood values
of the conditional and separable specifications8. Whenever the null hypothesis of
separability is accepted, the substitutability/complementarity effects disappear and
consumer demand depends only on relative prices and on total private expenditure, and
the effect of public consumption is limited to an income effect only. The LR values,
obtained for the different specifications of the two demand systems (DAI5 and DAI6),
are reported in Table 6 9.
12
Table 6 – Separability tests
A. Dynamic demand system for five spending categories (DAI5)
H0: Separability between private and public consumption*
Type of public consumption
Total public consumption
Public collective consumption
Public individual consumption
Education
Health
Recreation, culture, religion and social protection
LR statistics (χ2(4))
P-value
8.469118
1.147690
15.04866
10.96269
3.731316
9.514895
0.07583
0.88663
0.00460
0.02699
0.44359
0.04944
* With homogeneity and symmetry imposed
B. Dynamic demand system for six spending categories (DAI6)
H0: Separability between private and public consumption*
Type of public consumption
Total public consumption
Public collective consumption
Public individual consumption
Education
Health
Recreation, culture, religion and social protection
LR statistics (χ2(5))
P-value
10.36336
1.118578
24.47964
10.95327
9.598791
11.37277
0.06557
0.95245
0.00018
0.05231
0.08744
0.04447
* With homogeneity and symmetry imposed
The results for the DAI5 model are presented in Section A. In the specification
conditioned to total public consumption, the chi-squared ratio is slightly lower than the
5% critical value, with a p-value of 0.075; so, the validity of the null hypothesis of
separability between private and public consumption may be criticised. By splitting
total public consumption expenditures into collective and individual consumption, we
try to verify the presence of substitutability/complementarity relationships between
these components of government expenditures and private spending. The chi-squared
ratios obtained show that the public provision of collective services does not affect the
allocation of private expenditures (LR=1.15, p-value=0.885); on the other hand, public
individual consumption (GI) exerts an important influence on the structure of private
sector spending (LR=15.05, p-value=0.005). These results confirm the hypothesis,
sustained by Karras (1994), Kuehlwein (1998) and Levaggi (1999), according to which
public individual expenditures, such as health, education and social protection,
remarkably affect consumer behaviour, while public spending on defence, justice and
public order and safety does not modify the structure of private demand.
In order to verify the presence of substitutability/complementarity relationships between
private spending and the components of public individual consumption, three different
demand systems are specified, conditioned respectively to “Education”, “Health” and
“Recreation, culture and religion and Social protection”. The hypothesis of separability
is rejected for “Education” and “Recreation, culture and religion and Social protection”,
with p-values equal to 0.026 and 0.049, respectively. On the contrary, public
expenditures on health do not affect the allocation of private spending.
Comparing the results of the DAI5 with those of the complete demand model in six
categories, DAI6, it is possible to note that when the expenditures on education, health,
recreation and social protection, which have a direct correspondence with the public
provision of individual goods and services, are included within private spending the
substitutability/complementarity relationships are much more evident. The results of the
13
LR tests for the different conditional specifications of the DAI6 model are presented in
Section B of Table 6. As regards the test of separability between private and total public
consumption, it can be noted that the calculated LR statistic increases from 10.36 to
11.07 and come closer to the 5% critical level. Moreover, the LR test for the model
conditioned to public individual consumption clearly rejects the null hypothesis of
separability, with a very high value of the chi-square ratio (LR=24.48). The empirical
evidence obtained confirms the presence of a significant relationship between private
spending and individual consumption, which is much more evident when private
expenditures on education, health and social protection are included in the demand
system; this result implicitly confirms the intuition developed by Kuehlwein (1998).
The same evidence is obtained when we test for separability between private
consumption and the different components of public individual spending; in all three
cases (GE, GH and GR), the statistic values are very close to the 5% significance level,
bringing into discussion the separability hypothesis and supporting a non-separable
specification of the demand model.
5.3. Parameter estimates
We focus attention on the estimated parameters of the DAI6 model only, since it
provides a better and more complete representation of private spending and its
relationship to public consumption.
Firstly, in order to evaluate the efficiency of the estimates, we analyse the correlation
coefficient between observed and estimated data. The values, presented in Table 7,
show a good predictive capability of the model; on the average, the correlation
coefficients are higher than 0.75, with a substantial improvement of the predictive
performances when public consumption is taken into account. This improvement is
particularly evident when private spending is conditioned to the public provision of
individual goods and services. These results further confirm the results obtained in the
separability tests.
Table 7 – Correlation coefficients between observed and fitted values
G
Conditional specifications:
GC
GI
0.8158
0.6808
0.9289
0.8981
0.6472
0.7539
0.6826
0.9260
0.8980
0.6398
0.8305
0.6887
0.9348
0.9068
0.6656
Separable specification
FBT
CF
HF
TC
HER
0.7550
0.6863
0.9268
0.8944
0.6380
The estimated parameters are presented in Table 8; in particular, we report only the
estimates for the separable specification (where θ = 0 ) and for the demand systems
conditioned to public individual consumption10. Analysing the results, it is possible to
notice that the estimated prices and total expenditure coefficients are significant in both
specifications.
The estimated coefficients of public consumption provide evidence of significant
relationships between private expenditure and public individual consumption. The
existence of both substitutability and complementarity relationships is confirmed by the
signs of the estimated θ coefficients; moreover, the significance of the estimated θ
implies that all the categories of private spending, except for “Clothing and footwear”,
are affected by public individual consumption.
14
Table 8 – Parameter estimates with homogeneity and symmetry imposed
A. Separable specification
FBT
FBT
CF
HF
TC
HER
OGS
0.1350
[.000]*
0.0014
[.945]
-0.0610
[.001]
-0.0190
[.139]
-0.0490
[.010]
-0.0074
[.859]
0.0160
[.548]
-0.0180
[.297]
-0.0310
[.013]
0.0560
[.004]
-0.0244
[.477]
0.1660
[.000]
-0.0280
[.011]
-0.0092
[.614]
-0.0498
[.075]
0.0970
[.000]
-0.0290
[.012]
0.0100
[.597]
0.0310
[.340]
0.0002
[.988]
CF
HF
TC
HER
OGS
0.0713
[.091]
Income
-0.0840
[.005]
0.0280
[.352]
-0.1270
[.000]
0.0036
[.860]
-0.0650
[.005]
0.2444
[.000]
Constant
-0.0016
[.220]
-0.0010
[.521]
0.0010
[.362]
-0.0004
[.641]
0.0047
[.000]
-0.0027
[.199]
* p-values in brackets
B. Demand system conditioned to Public individual consumption
FBT
FBT
CF
HF
TC
HER
OGS
0.1459
[.000]*
0.0177
[.373]
-0.0644
[.000]
-0.0191
[.105]
-0.0499
[.005]
-0.0302
[.351]
-0.0008
[.978]
-0.0333
[.048]
-0.0403
[.001]
0.0745
[.000]
-0.0178
[.627]
0.1751
[.000]
-0.0280
[.007]
-0.0199
[.208]
-0.0296
[.263]
0.0967
[.000]
-0.0288
[.007]
0.0196
[.361]
0.0437
[.117]
-0.0196
[.445]
CF
HF
TC
HER
OGS
0.0776
[.082]
Income
-0.0906
[.001]
0.0281
[.370]
-0.1147
[.000]
0.0076
[.697]
-0.0815
[.000]
0.2512
[.000]
Constant
-0.0014
[.225]
-0.0013
[.412]
0.0005
[.636]
-0.0004
[.587]
0.0055
[.000]
-0.0029
[.185]
GI
0.0429
[.003]
-0.0029
[.864]
-0.0287
[.028]
-0.0212
[.050]
0.0221
[.061]
-0.0121
[.651]
* p-values in brackets
15
In order to obtain a quantitative measure of the relationships between private and public
consumption, the values of the public consumption elasticities are calculated; formally,
the elasticity of private expenditure to public consumption is expressed as ei = θ i wi .
The estimated values are presented in Table 9; in particular, it should be noted that a
negative value of the public consumption elasticity characterises a substitutability
relationship between private and public consumption, while a positive value of ei
suggests the presence of a complementarity relationship.
Table 9 – Public individual consumption elasticities
FBT
CF
HF
TC
HER
OGS
1975
0.148
-0.029
-0.141
-0.185
0.192
-0.069
1980
0.158
-0.032
-0.138
-0.169
0.181
-0.066
1990
0.218
-0.039
-0.132
-0.171
0.155
-0.049
2000
0.267
-0.043
-0.121
-0.164
0.139
-0.049
Mean
0.196
-0.037
-0.128
-0.169
0.161
-0.056
The presence of both substitutability and complementarity relationships can be pointed
out; specifically, significant substitutability relationships between public individual
consumption and private expenditures on “Housing, fuels and furnishings” and
“Transports and communications” are recorded, with average values of -0.128 and 0.169, respectively. The substitutability relationships for the “Clothing and footwear”
and “Other goods and services” categories are less evident. On the other hand, the
functional category “Food, beverage and tobacco” displays an evident and timeincreasing complementarity relationship with public individual consumption; this result
suggests that the public provision of individual goods and services exerts an expansive
effect on food consumption. Finally, evidence of a complementarity relationship
between public individual consumption and private expenditures on “Education, health,
recreational services and social protection” is found; this result is in line with the
findings of Karras (1994), Kuehlwein (1998) and Fiorito & Kollintzas (2002) and
suggests that, in the short period, an increase in government expenditures on education,
health and social protection causes an increase of private spending on the corresponding
functional category.
6. Conclusions
In empirical demand analyses, individual preferences are generally considered to be
separable from the public provision of goods and services. Standard demand systems
assume that the real components of economy do not affect consumer behaviour. This
hypothesis is often contradicted by the pervasive presence of market failures, which
implies the necessity of consistent public interventions. One of the most important
consequences of this interaction between the private and public sectors is the alteration
of consumer utility function, which may be caused by substitutability/ complementarity
relationships.
In this paper, we have empirically analysed the issue of non-separability between
private and public consumption by a differential Almost Ideal demand system,
conditioned to the quantities of goods and services provided by the government. The
16
estimation has been carried out using UK data. The results of the tests bring the validity
of the null hypothesis of separability of individual preferences into question. In
particular, public individual consumption exerts significant conditional effects on the
allocative structure of private spending, while consumer decisions appear to be
independent of the public provision of collective services such as defence, public order
and safety.
Substitution elasticities have been used to obtain a quantitative measure of the
relationships between the different components of private spending and public
individual consumption. Significant substitutability relationships have been pointed out
for the categories “Housing, fuels and furnishings” and “Transports and
communications”, while the same relationships were less evident for “Clothing and
footwear” and “Other goods and services”. On the contrary, the spending categories
“Food, beverage and tobacco” and “Education, health, recreational services and social
protection” showed an evident complementarity relationship with public individual
consumption; this result supports the empirical evidence found by Karras (1994),
Kuehlwein (1998) and Fiorito & Kollintzas (2002).
The identification of the relationships between private and public consumption not only
provides a better representation of consumer behaviour, but also allows an improved
definition of government fiscal policy. The efficiency of public intervention strictly
depends on a correct use of this instrument in choosing alternative economic policies.
For these reasons, the substitutability and complementarity relationships between
private spending and public consumption become important policy variables, which
should be taken into account when the optimal level of government expenditure on
goods and services is being determined.
Notes
1
2
3
4
5
The derivation of the demand system from the conditional AI cost function is not presented here, but
it is available from the authors upon request.
Precisely, private expenditure data are taken from “Household final consumption expenditure:
classified by purpose” (Blue Book, table 6.4 and 6.5), while the data concerning public consumption
spending are taken from the annual series “General Government: analysis of total outlays by
classification of function of government” (Blue Book, table 11.2).
Many supporting arguments for this methodology can be found in the literature. A similar approach
was adopted by Aschauer (1985) who, in a macroeconomic analysis of the effects of fiscal policy on
private consumption, used consumer spending on non-durables and services as a measure of private
consumption in each period. Another justification for this methodology can be found in Marrinan
(1998) who verifies that the proportion of durable to non-durable spending is stable over time, so that
the exclusion of non-durable expenditures from total private consumption does not affect the results.
On the contrary, some authors, such as Graham & Himarios (1991) and Ni (1995), show that the
estimated relationship between private and public consumption depends on whether durable
expenditures are omitted.
The definition is given in paragraph 9.72 of SNA93: “Household actual final consumption consists of
the consumption goods or services acquired by individual households by expenditures or through
social transfers in kind received from government units or non-profit institutions serving households
NPISHs” (OECD, 1998).
In particular, two different tests are implemented. The first (ADF(t)) is a univariate test for the null
hypothesis of a unit root, with both an intercept and a time trend, in the data-generating process of the
variables of the model. The second (ADF(F)) is a joint test to verify the simultaneous presence of a
unit root and a deterministic time trend, with an intercept, in the variables.
17
6
7
8
9
10
The estimation of [10], imposing the diagonality restriction on
Γ0 , implies that the OLS have the
same statistical properties as those based on a two-stage least squares (2SLS) estimation.
This statistic is asymptotically distributed as a χ 2 with the number of degrees of freedom equal to
the difference between the number of parameters of the restricted and unrestricted demand system. In
the present analysis, the homogeneity constraint implies five degrees of freedom, while symmetry
generates ten degrees of freedom. The joint imposition of both theoretical properties implies fifteen
degrees of freedom.
The statistic is asymptotically distributed as a χ 2 with the number of degrees of freedom equal to the
number of non-singular equations of the demand system. So, for the DAI5 and DAI6 models, there
are four and five degrees of freedom, respectively.
The test was also carried out applying the small-sample correction [11]. However, since the adjusted
test outcomes are close to the asymptotic results, they are not presented here.
The estimated parameters for the remaining conditional demand systems are not presented in the text,
but are available upon request.
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19