- Email: [email protected]
A variable-production generalisation of Lerner's theorem
Journal
of Public
Economics
16 (1981) 371-376.
A VARIABLE-PRODUCTION
North-Holland
Publishing
GENERALISATION THEOREM
Company
OF LERNER’S
John BENNETT* University College, CardryCFI IXL, Wales Received August
1980, revised version received January
1981
A government wishes to choose an optimal set of wage rates, but it is uncertain of individual characteristics. All it knows for certain is that each utility function is strictly quasi-concave and that the production function is linear. We assume that it can determine probability distributions, for each individual. of possible utility functions and ability levels. If each of these probability distributions is the same for every individual, expected social welfare is maximised by equalisation of wage rates. But since actual utility functions, and therefore labour supplies, will generally be unequal, incomes will then be unequal.
1. Introduction In The Economics of Control (1944) A.P. Lerner considered how a government should divide a fixed sum of incomes, given that the particular utility functions possessed by individuals are unknown. On the assumptions that each individual has a diminishing marginal utility of income and that, as far as the government knows, each one has the same chance of possessing any (downward-sloping) marginal utility of income curve, Lerner showed that the sum of expected utilities is maximised by paying equal incomes to all. Later writers have generalised Lerner’s theorem in a number of ways,’ but one development which does not seem to have been made is to drop the assumption of a fixed sum of incomes. In this note we shall take output (and thus the sum of incomes) to be a function of the distribution of income. An additional feature is thereby brought into the analysis in that an individual becomes characterised not only by his utility function, but also by his productive ability. In the spirit of Lerner’s theorem we shall therefore assume that the government may not have knowledge of each individual’s ability. The dependence of output on the distribution of income can be formulated in a variety of ways, according to how we specify the nature of the income payment. In the model that we shall develop a conventional wage rate is assumed to be paid to each individual. The government chooses the set of wage rates to maximise an individualistic welfare function of expected *I am grateful to Geoff Heal and a referee for very helpful comments ‘See, in particular,
Sen (1969).
0047-2727/81/000&0000/$02.75
0
1981 North-Holland
utilities. In doing so, it takes into account that the labour input of each individual may depend on the wage rate he is paid. It is important to note how this approach differs from the optimal income tax framework of Mirrlees (1971). In the Mirrlees model, before-tax wage rates are given by ability and then modified by the single tax schedule, \vhiclz is upplied to ereryonr. In our model we are choosing wage rates, which is essentially the same as choosing tax rates, given the before-tax wage rates. However, we are choosing a separate wage rate for each individual; this is the same, in effect, as having before-tax wage rates given by ability and then applying 0 rl~fj-frr-rnt incomr till; .schd~~/e (a different constant marginal tax rate) to each in3ividual.2 Similar models have previously been analysed by Wagstaff (1975) an3 Baumol an3 Fischer (197Y), but in the context of full knowledge by the government of utility functions and abilities. Not surprisingly, such a personally differentiated tax system has a greater redistributive capacity than does Mirrlees-type income taxation, though at the cost of the abandonment of horizontal equity. In the model we shall develop we assume the government is uncertain of both individual abilities and utility functions (which may differ across individuals). Our main result, extending Lerner’s equal income theorem to an equal wage rate theorem, is both cheap to administer and relatively insensitive to parameter changes. Also, little information is required to calculate what the wage rate should be. In section 2 we derive the first-order conditions and in section 3 we discuss the results.
2. The model Assume that the government can identify individuals according to some index i, but that this index only gives the government imperfect information on (a) individual i’s utility function and (b) his ability. However, the government does know the probability distributions of ability and utility function conditional on i. In this framework it wishes to choose wage rates for all i such that, subject to a revenue constraint, an individualistic welfare function of expected utilities is maximised. Let yi be the income received by individual i and /’ be the amount of time he spends working (/’
,$/’
(i= 1,2 >. ., WI1,
where u.’ is the wage rate the government
(1) has determined
should
be paid per
‘WC do not include a lump-sum grant in our model. Nonetheless, in terms of the tax rntcrprctation of the model, redistribution still takes place since some marginal tax rates are negative in the solution.
J. Bennett,
Generulisation
c$ Lerner’s
unit of Pi and m is the number of individuals. Subject maximise the strictly quasi-concave utility function U’ = U’ (y’, F’),
to (l), i chooses
Ul, < 0.
where U; > 0,
373
theorem
t’ to
(2)
Although the government does not know the particular utility function possessed by any individual, we suppose that it has determined the set of utility functions which each one may possibly possess, and that all of these functions are strictly quasi-concave. We denote by piJ the probability, in the eyes of the government, that individual i has utility function UiJ(yiJ,/iJ); the superscript J (= 1,2,. .,N) will be used throughout on the values of variables which would result from the possession of function J. If, for example, the government assigned the same uniform distribution of functions to each individual, we would have a close parallel to Lerner’s own (‘equal ignorance’) assumption.3 The government knows that i will maximise his utility subject to (1). If it knew that i possessed function J, it would therefore be able to calculate i’s labour supply for any vvi; for it knows that i would then satisfy the firstorder condition M’iul;J + u;J = 0.
(3)
Thus, for each utility function, J, that it believes i may possess, the government can calculate PiJ. Doing this for all J, it can therefore calculate i’s expected labour supply, his expected income and so his expected utility level : E(ui)=
:
piJuiJ(yiJ,/iJ),
(4)
.I=1
Calculating (4) for all i at a given welfare, G, is found:
G= G[E(U’),
set of wage rates,
the level of expected
E(U2), . . ., E(U’“)].
(5)
It is assumed that G is concave and anonymous (in the sense that G is invariant with respect to who is expected to possess a particular utility level).
3This approach is used ugument’ (1969, p. 214).
by Sen, who
regards
it as ‘a close
approximation
to the Lerner
374
J. Bennett, Generalisation of Lerner’s theorem
We assume that output,
Q, is a linear function
of labour
inputs:
Q = f niLi.
(6)
i=l
The constant, ni, here is a measure of i’s ‘true’ ability. The government does not necessarily know n’ for any i; but we assume that it has determined a probability distribution of ability for each i, qiK being the probability, as the government sees it, that i has ability niK. It is assumed that there is an amount of output which the government wishes to retain to cover its own expenses. We call this ‘desired revenue’, RD. But with C’ and n’ possibly unknown, for each i, the government cannot be sure that actual revenue, R ( = Q -xi y’), will equal RD for any particular set of wage rates. We must therefore assume that the constraint the government uses in calculating optimal wage rates is put in terms of expectations: R~
=
11
c
qiKniKpiJ[iJ
iKJ
_
piJwipJ,
12 i
(7)
J
Desired revenue is made equal to expected revenue (expected output minus the expected wage bi11).4 Using the set of wage rates as tools, the government maximises G, subject to (7). The first-order conditions are
+
j,
,iJ
=o
(i=l,2
,...) m), (8)
where Gi=dG/?[E(Ui)] and i is the Lagrange multiplier But for each individual and each possible utility function from (3) to simplify (8) to obtain GiE[U’;Pi]
.
(i’
1,2,. . .) m).
on the constraint. we can substitute
(9)
“As the government discovers that its expectations are mistaken, actual revenue not equalling desired revenue, it may be able to use the information gained to improve estimates for the next period. If the government envisaged from the beginning using the information it acquires to improve its estimates in the future, then our first-order conditions would no longer apply. For, with Bayesian optimisation over time, it might pay to disperse wage rates to get better estimates sooner.
J. Bennett, Generalisation
of Lerner’s theorem
For brevity, we use the expectation operator here, the earlier having been fully written out in order to clarify the derivation.5
315
equations
3. Discussion One of the uses to which eq. (9) can be put is to obtain what can be regarded as a variable-production generalisation of Lerner’s theorem. Consider two individuals, 1 and 2, to whom the government has assigned the same distributions of ability and utility function.‘j It is then found that (9) is satisfied by setting w1 = w2. For it is known from eq. (3) that if 1 and 2 each possess utility function J, equality of wage rates will lead them to respond such that /1J=/2J, ((‘/‘J/~,c1)=(~/2Jli~~,2) and U~“=U~“: this is true for all 1. Also, expected abilities are equal. Thus, when w1 =w2, the left-hand side of (9) is the same for both individuals. Thus, when we introduce variable production into Lerner’s framework, retaining the spirit of his theorem by assuming that abilities are unknown, the solution is to set wage rates equal. And as Sen (1969) has shown for the Lerner result, this solution is independent of the choice of (concave) welfare function. There is a significant difference from Lerner’s result, however, in that with wage rates equal, but ‘true’ utility functions presumably differing in general, individuals such as 1 and 2 will generally earn different incomes, since they will work for different lengths of time. The level at which the common wage rate is set in the solution is given by the constraint (7). Putting identical probability distributions of ability and utility function into (7), writing w for the common wage rate and E(f) and E(n), respectively, for the expected labour supply and expected ability of any individual, we have RD = mE(t)[E(n)-
~1.
(10)
Thus, w SE(n) as RDzO. Differentiating (10) it is found that an increase in E(n) raises w, given that the restriction noted in footnote 5 is met.7 But although w is dependent on the expected level of ability, an advantage in 51n using eq. (9) we assume that the denominator on the left-hand side is positive for all i. We return to what this means for the ‘Lernerian’ case in footnote 7. “The parallel with Lerner’s original theorem would be to have the same unifornl distributions for 1 and 2. However, the argument is more general heie, applying whenever the distributions are the same. ‘In the present case the restriction can be written (ZE(t)/?w)(w/E(C))[l
-E(n)/w]
> - 1.
If RD=O the square bracket is zero and so the restriction is never violated. If RD#O the restriction is violated by a sufficiently positive (negative) RD and a suff%iently negative (positive) elasticity of expected labour supply.
implementation is that it is never affected by a change in uncertainty about ability. Furthermore, it can be seen from (10) that w can be calculated without knowledge of the probability distribution of utility function. If RD =0 the solution is simply to set \I‘= E(n). If R $0 the government has to be able to estimate the expected labour supply curve in the region of the solution: but, again, it does not need knowledge of the underlying utility function distribution. Interpreting the model as a taxation scheme we can define I?/~ as i’s ‘before-tax’ income and NY” as his ‘after-tax’ income. Taxation paid by i as a proportion of his before-tax income is then (1 - IV!.H~).Since this proportion increases with 17~, the tax scheme is progressive. Finally, we note that eq. (9) can be used under a variety of other informational assumptions. For example. the government might be able to establish a hierarchy of expected abilities; or it may know all abilities and utility functions for certain, in which case eq. (9) describes the solution to the analysis of Baumol an3 Fischer. One extreme case worth mentioning is that Lerner’s theorem itself follows from (9) if it can be assumed that i/‘/i\~’ is zero in the region of the solution and if /’ is known for all i. Our framework has an advantage over Lerner’s, however, in that each individual sees that his income depends on his own labour input. The set of individuals together is thereby given the incentive to product what is then distributed amongst them. In Lerner’s model of lump-sum payments there is no such incentive. References
Baumol,
W.J. and Fischer, D., 1979. The output distribution frontier: Alternatives to income t;Lxcs and transrcr\ for strong ‘qLI;LIIt). r“o;ll\. ~~nlcl-lcan Fcon0mIc I
of Public
Economics
16 (1981) 371-376.
A VARIABLE-PRODUCTION
North-Holland
Publishing
GENERALISATION THEOREM
Company
OF LERNER’S
John BENNETT* University College, CardryCFI IXL, Wales Received August
1980, revised version received January
1981
A government wishes to choose an optimal set of wage rates, but it is uncertain of individual characteristics. All it knows for certain is that each utility function is strictly quasi-concave and that the production function is linear. We assume that it can determine probability distributions, for each individual. of possible utility functions and ability levels. If each of these probability distributions is the same for every individual, expected social welfare is maximised by equalisation of wage rates. But since actual utility functions, and therefore labour supplies, will generally be unequal, incomes will then be unequal.
1. Introduction In The Economics of Control (1944) A.P. Lerner considered how a government should divide a fixed sum of incomes, given that the particular utility functions possessed by individuals are unknown. On the assumptions that each individual has a diminishing marginal utility of income and that, as far as the government knows, each one has the same chance of possessing any (downward-sloping) marginal utility of income curve, Lerner showed that the sum of expected utilities is maximised by paying equal incomes to all. Later writers have generalised Lerner’s theorem in a number of ways,’ but one development which does not seem to have been made is to drop the assumption of a fixed sum of incomes. In this note we shall take output (and thus the sum of incomes) to be a function of the distribution of income. An additional feature is thereby brought into the analysis in that an individual becomes characterised not only by his utility function, but also by his productive ability. In the spirit of Lerner’s theorem we shall therefore assume that the government may not have knowledge of each individual’s ability. The dependence of output on the distribution of income can be formulated in a variety of ways, according to how we specify the nature of the income payment. In the model that we shall develop a conventional wage rate is assumed to be paid to each individual. The government chooses the set of wage rates to maximise an individualistic welfare function of expected *I am grateful to Geoff Heal and a referee for very helpful comments ‘See, in particular,
Sen (1969).
0047-2727/81/000&0000/$02.75
0
1981 North-Holland
utilities. In doing so, it takes into account that the labour input of each individual may depend on the wage rate he is paid. It is important to note how this approach differs from the optimal income tax framework of Mirrlees (1971). In the Mirrlees model, before-tax wage rates are given by ability and then modified by the single tax schedule, \vhiclz is upplied to ereryonr. In our model we are choosing wage rates, which is essentially the same as choosing tax rates, given the before-tax wage rates. However, we are choosing a separate wage rate for each individual; this is the same, in effect, as having before-tax wage rates given by ability and then applying 0 rl~fj-frr-rnt incomr till; .schd~~/e (a different constant marginal tax rate) to each in3ividual.2 Similar models have previously been analysed by Wagstaff (1975) an3 Baumol an3 Fischer (197Y), but in the context of full knowledge by the government of utility functions and abilities. Not surprisingly, such a personally differentiated tax system has a greater redistributive capacity than does Mirrlees-type income taxation, though at the cost of the abandonment of horizontal equity. In the model we shall develop we assume the government is uncertain of both individual abilities and utility functions (which may differ across individuals). Our main result, extending Lerner’s equal income theorem to an equal wage rate theorem, is both cheap to administer and relatively insensitive to parameter changes. Also, little information is required to calculate what the wage rate should be. In section 2 we derive the first-order conditions and in section 3 we discuss the results.
2. The model Assume that the government can identify individuals according to some index i, but that this index only gives the government imperfect information on (a) individual i’s utility function and (b) his ability. However, the government does know the probability distributions of ability and utility function conditional on i. In this framework it wishes to choose wage rates for all i such that, subject to a revenue constraint, an individualistic welfare function of expected utilities is maximised. Let yi be the income received by individual i and /’ be the amount of time he spends working (/’
,$/’
(i= 1,2 >. ., WI1,
where u.’ is the wage rate the government
(1) has determined
should
be paid per
‘WC do not include a lump-sum grant in our model. Nonetheless, in terms of the tax rntcrprctation of the model, redistribution still takes place since some marginal tax rates are negative in the solution.
J. Bennett,
Generulisation
c$ Lerner’s
unit of Pi and m is the number of individuals. Subject maximise the strictly quasi-concave utility function U’ = U’ (y’, F’),
to (l), i chooses
Ul, < 0.
where U; > 0,
373
theorem
t’ to
(2)
Although the government does not know the particular utility function possessed by any individual, we suppose that it has determined the set of utility functions which each one may possibly possess, and that all of these functions are strictly quasi-concave. We denote by piJ the probability, in the eyes of the government, that individual i has utility function UiJ(yiJ,/iJ); the superscript J (= 1,2,. .,N) will be used throughout on the values of variables which would result from the possession of function J. If, for example, the government assigned the same uniform distribution of functions to each individual, we would have a close parallel to Lerner’s own (‘equal ignorance’) assumption.3 The government knows that i will maximise his utility subject to (1). If it knew that i possessed function J, it would therefore be able to calculate i’s labour supply for any vvi; for it knows that i would then satisfy the firstorder condition M’iul;J + u;J = 0.
(3)
Thus, for each utility function, J, that it believes i may possess, the government can calculate PiJ. Doing this for all J, it can therefore calculate i’s expected labour supply, his expected income and so his expected utility level : E(ui)=
:
piJuiJ(yiJ,/iJ),
(4)
.I=1
Calculating (4) for all i at a given welfare, G, is found:
G= G[E(U’),
set of wage rates,
the level of expected
E(U2), . . ., E(U’“)].
(5)
It is assumed that G is concave and anonymous (in the sense that G is invariant with respect to who is expected to possess a particular utility level).
3This approach is used ugument’ (1969, p. 214).
by Sen, who
regards
it as ‘a close
approximation
to the Lerner
374
J. Bennett, Generalisation of Lerner’s theorem
We assume that output,
Q, is a linear function
of labour
inputs:
Q = f niLi.
(6)
i=l
The constant, ni, here is a measure of i’s ‘true’ ability. The government does not necessarily know n’ for any i; but we assume that it has determined a probability distribution of ability for each i, qiK being the probability, as the government sees it, that i has ability niK. It is assumed that there is an amount of output which the government wishes to retain to cover its own expenses. We call this ‘desired revenue’, RD. But with C’ and n’ possibly unknown, for each i, the government cannot be sure that actual revenue, R ( = Q -xi y’), will equal RD for any particular set of wage rates. We must therefore assume that the constraint the government uses in calculating optimal wage rates is put in terms of expectations: R~
=
11
c
qiKniKpiJ[iJ
iKJ
_
piJwipJ,
12 i
(7)
J
Desired revenue is made equal to expected revenue (expected output minus the expected wage bi11).4 Using the set of wage rates as tools, the government maximises G, subject to (7). The first-order conditions are
+
j,
,iJ
=o
(i=l,2
,...) m), (8)
where Gi=dG/?[E(Ui)] and i is the Lagrange multiplier But for each individual and each possible utility function from (3) to simplify (8) to obtain GiE[U’;Pi]
.
(i’
1,2,. . .) m).
on the constraint. we can substitute
(9)
“As the government discovers that its expectations are mistaken, actual revenue not equalling desired revenue, it may be able to use the information gained to improve estimates for the next period. If the government envisaged from the beginning using the information it acquires to improve its estimates in the future, then our first-order conditions would no longer apply. For, with Bayesian optimisation over time, it might pay to disperse wage rates to get better estimates sooner.
J. Bennett, Generalisation
of Lerner’s theorem
For brevity, we use the expectation operator here, the earlier having been fully written out in order to clarify the derivation.5
315
equations
3. Discussion One of the uses to which eq. (9) can be put is to obtain what can be regarded as a variable-production generalisation of Lerner’s theorem. Consider two individuals, 1 and 2, to whom the government has assigned the same distributions of ability and utility function.‘j It is then found that (9) is satisfied by setting w1 = w2. For it is known from eq. (3) that if 1 and 2 each possess utility function J, equality of wage rates will lead them to respond such that /1J=/2J, ((‘/‘J/~,c1)=(~/2Jli~~,2) and U~“=U~“: this is true for all 1. Also, expected abilities are equal. Thus, when w1 =w2, the left-hand side of (9) is the same for both individuals. Thus, when we introduce variable production into Lerner’s framework, retaining the spirit of his theorem by assuming that abilities are unknown, the solution is to set wage rates equal. And as Sen (1969) has shown for the Lerner result, this solution is independent of the choice of (concave) welfare function. There is a significant difference from Lerner’s result, however, in that with wage rates equal, but ‘true’ utility functions presumably differing in general, individuals such as 1 and 2 will generally earn different incomes, since they will work for different lengths of time. The level at which the common wage rate is set in the solution is given by the constraint (7). Putting identical probability distributions of ability and utility function into (7), writing w for the common wage rate and E(f) and E(n), respectively, for the expected labour supply and expected ability of any individual, we have RD = mE(t)[E(n)-
~1.
(10)
Thus, w SE(n) as RDzO. Differentiating (10) it is found that an increase in E(n) raises w, given that the restriction noted in footnote 5 is met.7 But although w is dependent on the expected level of ability, an advantage in 51n using eq. (9) we assume that the denominator on the left-hand side is positive for all i. We return to what this means for the ‘Lernerian’ case in footnote 7. “The parallel with Lerner’s original theorem would be to have the same unifornl distributions for 1 and 2. However, the argument is more general heie, applying whenever the distributions are the same. ‘In the present case the restriction can be written (ZE(t)/?w)(w/E(C))[l
-E(n)/w]
> - 1.
If RD=O the square bracket is zero and so the restriction is never violated. If RD#O the restriction is violated by a sufficiently positive (negative) RD and a suff%iently negative (positive) elasticity of expected labour supply.
implementation is that it is never affected by a change in uncertainty about ability. Furthermore, it can be seen from (10) that w can be calculated without knowledge of the probability distribution of utility function. If RD =0 the solution is simply to set \I‘= E(n). If R $0 the government has to be able to estimate the expected labour supply curve in the region of the solution: but, again, it does not need knowledge of the underlying utility function distribution. Interpreting the model as a taxation scheme we can define I?/~ as i’s ‘before-tax’ income and NY” as his ‘after-tax’ income. Taxation paid by i as a proportion of his before-tax income is then (1 - IV!.H~).Since this proportion increases with 17~, the tax scheme is progressive. Finally, we note that eq. (9) can be used under a variety of other informational assumptions. For example. the government might be able to establish a hierarchy of expected abilities; or it may know all abilities and utility functions for certain, in which case eq. (9) describes the solution to the analysis of Baumol an3 Fischer. One extreme case worth mentioning is that Lerner’s theorem itself follows from (9) if it can be assumed that i/‘/i\~’ is zero in the region of the solution and if /’ is known for all i. Our framework has an advantage over Lerner’s, however, in that each individual sees that his income depends on his own labour input. The set of individuals together is thereby given the incentive to product what is then distributed amongst them. In Lerner’s model of lump-sum payments there is no such incentive. References
Baumol,
W.J. and Fischer, D., 1979. The output distribution frontier: Alternatives to income t;Lxcs and transrcr\ for strong ‘qLI;LIIt). r“o;ll\. ~~nlcl-lcan Fcon0mIc I