Modern Economy, 2011, 2, 383-389
doi:10.4236/me.2011.23041 Published Online July 2011 (http://www.SciRP.org/journal/me)
Computation of the Compensating Variation within a
Random Utility Model Using GAUSS Software
Marilena Locatelli, Steinar Strøm
University of Turin and Child Centre, Turin, Italy,
Frisch Centre, Oslo, Norway
Received February 23, 2011; revised April 10, 2011; accepted April 22, 2011
Abstract
To evaluate a tax reform in terms of change in household welfare one possibility is to estimate the compen-
sating variation using a suitable model to assess the change in the household utility. When a random utility
model is used, the computation of compensating variation is not straightforward, particularly when utility is
not linear in household income. It can be carried out using a methodology recently proposed in the literature.
In this paper we describe a software instrument, implemented using GAUSS programming language for
computing the compensating variation to evaluate the 1991 tax reform introduced in Norway. The program is
flexible and adaptable to different tax systems and different reference years.
JEL Classification: C63, B21
Keywords: Compensating Variation, Gauss Application, Tax System Evaluation
1. Introduction
In 1992 the Norwegian tax system was reformed towards
lower and less progressive tax rates, with a reduction in the
total tax revenue. In the next following years the tax struc-
ture was kept nearly unchanged. To evaluate a tax reform
in terms of change in household welfare one possibility is
to estimate the compensating variation (CV) using a suit-
able model to assess the utility of the households.
The theoretical framework and the empirical results of the
evaluation of the above tax reform are reported in [1] in
which there is also an extensive literature review on this
subject. Novelty of the paper is the use of a random utility
labor supply model, taking into accounting also sectoral
choices (public and private), to assess the impact of tax
reforms on household welfare. Some partial details on the
algorithm implementation were first given in [2].
The computation of CV is not straightforward in a random
utility model, in particular when utility is not linear in
household income. A random utility function implies that
the expenditure function is also random. Until recently, no
analytic formulas have been available for calculating the
distribution of CV. However, [3] have developed analytic
formulas for this purpose, and we apply their methodology
to calculate the distribution of CV and the mean and vari-
ance of this distribution.
What we thus do is to calculate the expected value of
CV for each household and its distribution in the popula-
tion.
The purpose of this note is to describe the software in-
strument, implemented using Gauss software package [4],
to evaluate the above tax reform in terms of change in
household welfare within a random utility model. The pro-
gram is flexible and adaptable to different tax systems and
different reference years.
The paper is organized as follows. In the next Section,
the data used in estimations are described. The model is
explained in Section 3. In Section 4 we analyze the main
steps of the procedure and in Section 5 the program struc-
ture is outlined. The results are reported in Section 6. Sec-
tion 7 draws some conclusions.
The Table upon the estimation is reported in Appendix
A. The interpolation method followed in the computation
of CV is described in Appendix B. In Appendix C we re-
port the GAUSS program flow. The complete program
code can be downloaded [2].
2. The Data
Data on the labor supply of married women in Norway
used in this note consist of a merged sample from “Sur-
vey of Income and Wealth, 1994”, Statistics Norway

M. LOCATELLI ET AL. 384
(1994) and “Level of Living Conditions, 1995”, Statistics
Norway (1995). Data cover married couples as well as
cohabiting couples with common children. The age of
the spouses ranges from 25 to 64. None of the spouses
are self-employed and none of them are on disability or
other type of benefits. All taxes paid are observed and in
the assessment of disposable income, all details of the
tax system are accounted for.
The size of the sample used in estimating the labor
supply model is 810 (referred to as n_record in the pro-
gram).
3. The Model
To evaluate the tax reform of 1992 we calculate the
change in household welfare. One way is to apply the
measure of CV. The calculation of CV is not straight-
forward in a random utility model when utility is not
linear in household income. A random utility function
implies that the expenditure function is also random. A
general treatment of this issue was undertaken by [3]
while in [1] the method was adapted to the change in
household welfare with labor supply random utility
model.
What we do is to calculate the expected value of CV,
E[CV], for each individual and thereafter the distribution
of this value in the population from which we can derive
mean, median etc.
We will assume that the utility function has the struc-
ture

,for 0,1,2,3,UC, h, z=vC,hzz   (1)
where C is the household disposable income, which
equals the sum of the after-tax labor income of husband
and wife plus the after tax capital income and public
transfer like child allowances; h are hours of work. In
details: 0 (not working), 315, 780, 1040, 1560, 1976,
2340, 2600 (both for public and private sector); v(.) is a
deterministic function and ε(z) is a positive random taste
shifter. The taste shifter accounts for unobserved indi-
vidual characteristics and unobserved job-specific attrib-
utes z. {ε(z)}, are independently distributed with c.d.f.
exp(–x–1), x > 0.
From Equation (1) we get the following implicit defi-
nition of the CV when the tax regime of 1991 (prior to
the tax reform) is compared to the tax regime of 1994
(after the tax reform):




,,Taxregime1991
,, Taxregime1994
UCh z
UC CVhz



 (2)
In Equation (1) we have suppressed the subscript of
the individual and we should also keep in mind that the
choice of each individual is to choose to work or not, and
given work, to choose sector and hours of work, given
the job opportunity sets and the budget constraints under
the different tax regimes and CV. In the calculation of
the expected value of CV we take this choice structure
into account. If an individual benefits from the tax re-
form, the expected value of CV for this individual is
positive, meaning that this amount has to be subtracted
from household income under the 1994 tax regime in
order to make the individual indifferent between the two
tax regimes.
To proceed with the calculation we need some nota-
tion. Note first that the deterministic part of the utility
function can be written 

,
iii
vvChhX

(vi referred
to as consumption function in the following), where i
denotes the different job alternatives. In the examined
case the choice alternatives (na) are 15: i = 1 the indi-
vidual is not working, and denote hours of
job in the public sector, while denote hours
of job in the private sector. X is a vector of all exogenous
characteristics.
2,3, ,8i
9, ,1i5
Now let




0Taxregime 1991
ii i
v=vCh,hX (3)
and let




**,,
ii
vyvC hyhXi (4)
where




*,
ii
ChyFhy

,



iii ii
F
hwhTwh
and T(.) is the tax function for 1994, wi is the wage rate.
Let E[CV] be the expected value of the compensating
variation, which can be calculated for each individual as
follows [3]:





15 0
15 0*
0
1
ECV
d
*
max ,
yi
iii
i
iii iii
i
y
Ivgb
vgbvy gb



















(5)
where, according to estimates given in [5] and reported
in Appendix A,


4
6
11
13
1
4
4
1 foralliexceptfor4,6,11,13
exp(0.68)
exp(1.58)
exp(0.80)
exp(1.06)
1
exp4.200.22; for2,3,,8
exp1.140.33;for9,10,,15.
i
i
i
gi
g
g
g
g
b
bXi
bXi








 


and is given by the following equation:
i
y


0*
iii
vvy (6)

M. LOCATELLI ET AL.385
I* (see Equation (5)) equals the sum of the after tax
income of husband earnings and capital income, plus
child allowances. The tax reform of 1992 is a combina-
tion of a change of the tax structure and reduction in tax
revenues.
4. Steps of the Procedure
In 1992 the Norwegian tax system was reformed towards
lower and less progressive tax rates with a reduction in
the total tax revenue. We have thus organized the pro-
gram to allow also the computation of the expected
E[CV] between the 1994 tax system and the flat tax sys-
tem (in our case 29%) that equals tax revenue of year
1994.
4.1. The Algorithm
To calculate E[CV] the following steps are required:
1) Load the matrix file, previously prepared1, with the
values (the household’s deterministic utility (i.e.
consumption function)) for the reference year 1991 and
working hours hi.
0
i
v



0|1991 ,
ii i
vv ChTax functionhX (7)
where is a n_record × na matrix, being n_record
the number of records of the matrix file, and na the
number of alternatives of the choice set (i.e. 15 alterna-
tives).
0
i
v
We also calculate E[CV] with reference to a flat tax
system (in our case the revenue neutral tax rate simulated
on the choice model is 29%). The model under the flat
tax system gives the reference values when the 1994 tax
regime is evaluated against the flat tax system.
Now, load the matrix file with the averaged values
(the household deterministic utility) under the flat tax
system.
0
i
v
2) Load the data sets including:
a) the variables (disposable income, etc.) for the
tax system 1991;
b) the variables (disposable income, etc.) for the
tax system 1994;
3) Load the matrix files that allow us to identify the
deciles associate with poor (first decile), middle (from
second to ninth decile), and rich (tenth decile) of the dis-
tribution of disposable income computed according to
1994 tax system.
4) For each observation and each alternative compute
a matrix Fh (n_record × na) whose elements are:
 
, 1,2,,15
iii ii
Fh = wh-Twhi (8)
where wihi is the hourly wage multiplied by the hours
associated to each of the 15 alternatives, T(.) is the tax
function for 1994. The choice alternatives are not work-
ing (i = 0), working in the public sector at different hours
(i = 2,3,4,5,6,7,8) and in the private sector (i =
9,10,11,12,13,14,15)
5) Compute a matrix , (n_record × na), whose
elements are
C*


ii
C*h,y= Fh+ y (9)
where y is a matrix (n_record × na) determined by an
iterative procedure so that for y =
y
, at each observa-
tion the following equality holds




01,2, ,15
iii i
v=vFh+y, i=
 (10)
The value of
y
 is determined using an iterative pro-
cedure, described in Appendix B.
4.2. Computation of the Integral
The integral is computed numerically dividing the inte-
gration interval in small steps (a length of NOK 100 was
found sufficient) and then summing up the partial con-
tribution of the integrand function related to each step.
The final result for each observation is obtained by
summing the single integrals, evaluated for each alterna-
tive, over the total number of alternatives.
4.3. E[CV]
Finally, E[CV] is computed subtracting from I* (Equa-
tion (5)) the integral evaluated in the previous step.
5. Program Structure
The program, whose flow-chart is shown in Appendix C,
consists of a main part which resorts to several proce-
dures to accomplish different tasks. They are briefly de-
scribed below.
Main program:
The main program includes the computation of:
1) consumption function using the procedure V and
V_SCALAR
2) disposable income computed with:
a) woman wage before tax using the procedure
W_WAGE
b) woman net wage using the procedure
NETWAGE_W_94
c) men net wage when woman is not working using
the procedure NETWAGE_M_94
d) men net wage when woman is working using the
procedure NETWAGE_MW_94
1Of course these data sets must be previously prepared with the specific
p
rogram that considers the different tax systems to be es
t
imated. 3)
y
: monetary value to be added or subtracted to the

M. LOCATELLI ET AL. 386
woman net wage of 1994 (Fh 94) for which the utility of
1994 equals that of 1991. It is evaluated resorting to the
procedure INCR and the command lines listed in the
main program for the refinement of the interval were the
solution lies.
4) the integral using the procedure INTEGRAND.
5) E[CV] and E[CV] statistics for the total sample and
for deciles of disposable income distribution (poor (first
decile), middle (from second to ninth decile), and rich
(tenth decile).
Procedures (listed in alphabetic order):
HALF: solution refinement through half interval
search
INCR: iterate until an interval is found with function
values of different sign
INTEGRAND: computation of the function to be in-
tegrated and the integral (Equation (5)), for given y, ob-
servation i, and alternative j
INTERP: search a solution via linear interpolation
NETWAGE_M_94: compute man net wage when
woman is not working (1994 tax system).
NETWAGE_MW_94: compute man net wage, when
woman works (1994 tax system).
NETWAGE_W_94: calculation of woman net wage
(i.e. women working) using tax function 1994
V: computation of consumption function
The procedure returns the matrix v (of dimension
n_record × na) of the consumption function for a given
matrix of disposable income disp (of dimension n_record
× na), according to the following equation:




0.64
4
,
,
2
11
0.53
23
0.64
4
,
0.53
1060000 1
exp 1.770.64
115.02 63.619.20
1
3640
1.27 0.970.53
1060000 1
0.12 0.64
11
3640
0.53
ij
ij
ii
j
ii
ij
j
C
v
XX
h
XX
C
h







 




 







 
















 






 


where Ci,j = disposable income of record i and alternative
j, passed as a parameter to the procedure.
The other parameters are passed as global variables
and have the following meanings:
Xi1 is the logarithm of age of the woman, Xi2 is the
number of children aged 0 - 6, and Xi3 is the number of
children aged 7 - 17.
V_SCALAR: computation of the value of consump-
tion function v for a single value of disposable income, a
given sample i and alternative j. This is the scalar version
of procedure V. The procedure returns the value (scalar)
of the consumption function v for a given disposable
income disp, sample i and alternative j according to the
equation reported for procedure V above.
W_WAGE: Woman wage income before tax.
6. Results
The expected CV obtained from the above procedure is
reported in the following Table 1.
From Ta ble 1, we observe that the mean household in
the sample gained NOK 27078 from the 1992 tax reform.
The richest household gained almost 10 times more than
the poorest or 4 times more in relative income terms.
The distribution of expected gain across households is
given in Figure 1, and we observe that most of the
households will benefit from the 1992 tax reform. Thus,
such a reform would have attained support from a clear
majority of households with married and cohabiting
women at an election.
1















(11)
We have also calculated the expected value of com-
pensating variation of a flat tax reform. In the calcula-
tions, the tax-revenue-neutral flat tax reform of 29% is
used as a reference. Negative values mean that the nu-
merical values have to be subtracted from household
incomes under the flat tax regime in order to make the
households indifferent in welfare terms between the 1994
regime and the flat tax regime. The expected CV is re-
ported in the following Table 2. This Table then says
that, on average, the households will gain NOK 51528 if
there is a shift from the 1994 tax regime to a flat tax re-
gime.
The richest households gain around 8 times more than
the poorest. Thus, in a distributional sense, the richest
household benefited more from having the 1991 regime
replaced with the 1994 tax regime than they would have
in the case of a shift from the 1994 tax regime to a flat
tax regime. In Figure 2, we show the population density
of the individual mean CV. We observe that a vast ma-
jority will benefit from the replacement of the 1994 tax
regime with a flat tax regime.
More details of the results are reported in [1].

M. LOCATELLI ET AL.387
Table 1. Expected value of compensating variation (in NOK
1994) for the 1992 tax reform . T he 1991 ta x sys tem is us ed as
a reference against the 1994 tax system.
E[CV] E[CV] in percent of observed
disposable income*
All 27,078 11.46
Deciles in the distribution
of household disposable
income*:
1 (poor) 6,761 4.32
2 – 9 (middle) 24,896 11.11
10 (rich) 64,150 16.66
*Decile(s) refers to the deciles in the distribution of disposable income,
1994.
E[CV]
Figure 1. Population density of expected Compensating
Variation. Distribution of E[CV], comparing the 1991 tax
regime against the 1994 tax regime.
Table 2. Exp ected valu e of co mpensating variation (in NOK
1994) for a flat tax reform. A flat tax regime is used as a
reference against the 1994 tax system.
E[CV]
All –51,437
Deciles in the distribution of household disposable
income, flat tax:
1 (poor) –17,155
2 – 9 (middle) –53,093
10 (rich) –146,966
7. Conclusions
In this note we have described a GAUSS software in-
strument, to compute the value of expected compen-
E[CV]
Figure 2. Population density of expected Compensating
Variation. Distribution of E[CV], with the flat tax system of
29% used as a reference against the 1994 tax regime.
sating variation within the discrete choice setting sug-
gested in [1].
The program refers to the following tax reforms: a) tax
systems in force in 1991 and in 1994 using 1991 as ref-
erence year, b) 1994 tax system and a flat tax system
taking 1994 as reference year. The program is flexible
and can be easily modified to take into account different
tax systems and different reference years.
The results show the different impact of the tax re-
form:
1) from the 1992 tax reform we observe that most of
the households will benefit from the 1992 tax reform; the
richest household gained almost 10 times more than the
poorest or 4 times more in relative income terms. Thus,
such a reform would have attained support from a clear
majority of households with married and cohabiting
women at an election.
2) from the 1994 tax system and a flat tax system tak-
ing 1994 as reference year we observe that the richest
households gain around 8 times more than the poorest.
We observe that a vast majority would benefit from the
replacement of the 1994 tax regime with a flat tax re-
gime.
8. Acknowledgements
M. Locatelli gratefully acknowledges financial support
by Frisch Centre, Oslo, Norway.
9. References
[1] J. K. Dagsvik, M. Locatelli and S. Strøm, “Tax Reform,
Sector-Specific Labor Supply and Welfare Effect,” Scan-
dinavian Journal of Economics, Vol. 111, No. 2, 2009, pp.
265-287. doi:10.1111/j.1467-9442.2009.01565.x

M. LOCATELLI ET AL. 388
[4] Aptech System, Inc., “GAUSS Software (Version 10),”
Black Diamond, Ridgefield, 2009.
[2] M. Locatelli and S. Strøm, “Computation of the Com-
pensating Variation within a Random Utility Model Us-
ing GAUSS Software,” Child Working Paper No.
02/2006. [5] J. K. Dagsvik and S. Strøm, “Sectoral Labor Supply,
Choice Restrictions and Functional Form,” Journal of
Applied Econometrics, Vol. 21, No. 6, 2005, pp. 803-826.
doi:10.1002/jae.866
[3] J. K. Dagsvik and A. Karlstrøm, “Compensating Varia-
tion and Hicksian Choice Probabilities in Random Utility
Models That are Non-Linear in Income,” Review of Eco-
nomic Studies, Vol. 72, No. 1, 2005, pp. 57-76.
doi:10.1111/0034-6527.00324
[6] S. S. Kuo, “Computer Applications of Numerical Meth-
ods,” Addison-Wesley, Boston, 1972.
Appendix A
Table A1. Estimation results for the parameters of the labor supply probabil i t i e s .
Uniformly distributed offered hours with part-time and fulltime peaks
Variables Parameters Estimate t-values
Preferences:
Consumption:
Exponent α1 0.64 7.6
Scale 10–4 α2 1.77 4.2
Subsistence level C0 in NOK per year 60 000
Leisure:
Exponent α3 –0.53 –2.1
Constant α4 115.02 3.2
Log age α5 –63.61 –3.2
(log age)2 α6 9.20 3.3
# children 0 - 6 α7 1.27 4.0
# children 7 - 17 α8 0.97 4.1
Consumption and Leisure, interaction α9 –0.12 –2.7
Subsistence level of leisure in hours per year 5120
The parameters b1 and b2;
j1 j2
log =+
j
bffS
Constant, public sector (sector 1) f11 –4.20 –4.7
Constant, private sector (sector 2) f21 1.14 1.0
Education, public sector (sector 1) f12 0.22 2.9
Education, private sector (sector 2) f22 –0.34 –3.3
Opportunity density of Offered hours, gk2(h), k = 1,2
Full-time peak, public sector (sector 1)*



11
log Full 0
g
hgh
1.58 11.8
Full-time peak, private sector (sector 2) 


22
log Full 0
g
hgh
1.06 7.4
Part-time peak, public Sector



11
log Part 0
g
hgh
0.68 4.4
Part-time peak, private Sector 


22
log Part 0
g
hgh 0.80 5.2
# observations 810
Log likelihood –1760.9

M. LOCATELLI ET AL. 389
Appendix B based both on an absolute and relative tolerance.
Considering the last interval x, x where the solution
To determine the value
y
 we must solve the following
equation (for each record and each alternative) v0 =
v(Fh94 + ).
y

That means that we must find the zero of the function

f
x defined as:
 
0, with 94
f
xvvxxFh y
4
,
where y is a generic amount of income to be added to the
1994 woman net wage .
Calling x* the value for which , e have

0fx
*9
y
xFh

To determine the value of x* the following steps are
done:
1) iterate until an interval is found where the solution
lies, using procedure INCR;
2) refine the interval iterating until an approximate
solution is found. For the very first iterations (NITER ≤
5), the new value is searched using linear interpolation
(see Figure 1), implemented by the procedure INTERP.
Then (NITER > 5) the solution is refined using a
half-interval search method, implemented by the proce-
dure HALF. Fore more details on these methods see, for
example, Ch. 6 of [6].
The exit test is performed only after the solution has
been refined using HALF (i.e. only if NITER > 5) and is
x
1
x
2
f(x
1
)
f(x
2
)
Solution
x
app
Figure 1. The linear interpolation method: 1 =
app
x
x





 
121
12
f
xx x
f
xfx

.
1 2
is sought, the function value

m
f
x at the mean value


12
0.5
m
x
xx is computed. Furtermore the relative
error on x, rel_err =
h
211
x
xx,valuated. The solu-
tion is accepted
is e
if


m
f
x ≤ abs_tol or rel_err ≤ rel_tol.
A satisfactory trade-off between speed and accuracy
has been found assumbs_tol = 10 and rel_tol = 1e-8.
Th *
ing a
en the procedure exits assuming x = xm .
Appendix C: the GAUSS Program Flow
Load Set dy
0
=100
Compute Fh94 (disposable income, year 1994)
Call: w_wage, netwage_W_94, netwage_m_94, netwage_mW_94
Initial values:
Call V: computation of consumption function
where y
0
is the male disposable income for 1994
Give an increment to y
0
to find an interval where fx
1
and fx
2
have different
signs:
Call: INCR
Refine the solution and call it x*
Call: INTERP, HALF , and V_scalar
is the solution found. Set
Compute the integral Call: INTEGRAND
Compute E[CV] statistics
END
N
O
YES
YES
Are tolerance limits
satisfied?
N
O
YES
N
O
10 1
f
xVV(x)


0
Fh94
2
xy


1
Fh 9 4x
20 2
f
xVV(x)


1
0
f
x?

Fh 94x


12
0
f
xfx ?


00 0
y
ydy


Fh 94yx*

x

E[CV] Integral
*
I
00 00 0
( V_91 or V29)VV V

