We study the effects of a statutory wage tax sharing rule in a principal - agent framework with moral hazard (à la Holmstrom, 1979) using the approach of Bose, Pal, Sappington (2007) to model the stochastic relationship between the agent's unobserved effort and his observed performance. The analysis indicates that tax sharing with positive legislated contributions from both the employer and employee does not maximize any of the outcomes -- employee effort, wages, profits or welfare. Moreover, a rule which specifies a corner solution, with 100% of the tax statutorily levied on the employer will maximize effort, expected profit and expected welfare while 100% of the tax statutorily levied on the employee will maximize expected wages.moral hazard, taxes, principal-agent model

Barnali Gupta

Christelle Viauroux

English

Research Papers in Economics

IS TAX SHARING OPTIMAL?
AN ANALYSIS IN A PRINCIPAL-AGENT FRAMEWORK
BARNALI GUPTA1 AND CHRISTELLE VIAUROUX2
ABSTRACT. We study the effects of a statutory wage tax sharing rule in a principal - agent
framework with moral hazard (￿ la Holmstrom, 1979) using the approach of Bose, Pal,
Sappington (2007) to model the stochastic relationship between the agent’s unobserved
effort and his observed performance. The analysis indicates that tax sharing with positive
legislated contributions from both the employer and employee does not maximize any of
the outcomes ￿ employee effort, wages, pro￿ts or welfare. Moreover, a rule which speci￿es
a corner solution, with 100% of the tax statutorily levied on the employer will maximize
effort, expected pro￿t and expected welfare while 100% of the tax statutorily levied on the
employee will maximize expected wages.
JEL codes: D8, H2
Keywords: moral hazard, taxes, principal-agent model
1. INTRODUCTION
During the past three decades, the principal-agent framework has become an integral
part of the economics literature with incomplete information. (For a survey, see Sapping-
ton, 1991, and also Laffont and Martimort, 2002). In this paper, we study the effect of a
statutory wage tax sharing rule on wages, effort, pro￿ts and aggregate welfare, in a princi-
pal agent framework with moral hazard.
Our results show that any interior tax sharing rule with positive legislated contributions
from both the employer and employee, will not optimize any outcomes for the principal,
agent or the government. Additionally, a statutory rule with 100% of the tax levied on the
employer, will maximize effort, expected pro￿t and expected welfare while, under some
conditions, a statutory rule of 100% of the tax on the employee will maximize expected
wage.
Thetaxationofwageincomeinvariousforms, iscommonpracticeanditisequallycom-
mon to have sharing rules that spilt the tax burden in some fashion between employer and
employee. In approximately half of all OECD countries, the shares of employer/employee
contributions toward a social security tax, for example, have been stable at approximately
25% of total labor costs. Yet, the distribution of this share between employer/employee
varies across countries. There is a 50:50 split in Germany, Switzerland, United States,
Luxembourg and Japan. In most other countries, employers typically pay the major share.
The exceptions are Denmark and the Netherlands, where employees generally pay the
most. This variation and the lack of formal analysis in the literature, motivate the present
study.
Date: April 20, 2009
1:Department of Economics, Miami University, Oxford, Ohio 45056. (513) 529-2856; (513) 529-8047 (fax);
Email: guptab@muohio.edu. 2: Corresponding author Department of Economics, University of Maryland,
Baltimore County, Baltimore, MD 21250. Email: ckviauro@umbc.edu.
12 BARNALI GUPTA
1 AND CHRISTELLE VIAUROUX
2
There is a substantial literature on the optimal income tax in adverse selection models
(see Diamond, 1998 and Seade, 1977). The focus of our work is on moral hazard and
risk sharing. In the theoretical literature studying the impact of taxes on hours of work,
the typical con￿ict between the substitution effect and income effect has rendered any
conclusion logically indeterminate.
1 While this paper is related to the topic of taxation
under uncertainty (see Eaton and Rosen 1980 a, b), it is fundamentally different in the
model being used and the implication of a statutory tax sharing rule that is being studied.
It is useful to see the work by Feldstein (1995) for discussion on the extent to which taxable
income as a whole, and not just labor supply, responds to changes in marginal tax rates.
It is clearly illustrated that regardless of the tax distribution chosen as a policy matter,
an interior solution with positive shares for both principal and agent, does not optimize any
of the relevant outcomes. In fact, in each case, a corner solution does exactly that. From
this perspective, it is dif￿cult to justify an interior distribution of tax shares. The next
section describes the model. Section 3 presents the analysis while Section 4 concludes and
discusses some extensions.
2. MODEL
The model is based on Holmstrom (1979). However, while Holmstrom uses an expo-
nential distribution we adopt the approach of Bose, Pal and Sappington (2007) and rep-
resent the stochastic relationship between the agent’s effort and the output produced, by a
Gamma distribution
2.
3 An advantage of this distribution is to facilitate the identi￿cation of
conditions under which the "￿rst-order approach" can be employed to solve the principal’s
problem (see Jewitt, 1988). The basic framework is the familiar one of a risk neutral em-
ployer (or principal) and risk averse employee (or agent) who works for wages. Employee
wages are taxed by the government with a statutorily mandated distribution of the tax be-
tween the employer and employee. The employer-employee relationship involves moral
hazard, where the agent’s effort is unobservable by the principal, but it affects the expected
outcome as well as the riskiness of outcomes. The realized output is a noisy signal of the
agent’s effort. Therefore the principal wants to use the contract to induce the agent to exert
optimal effort.
More speci￿cally, the principal observes a realized output x, x 2 [x0;1) and pays the
agent an amount w(x): The principal’s payoff function is UP(x;w) = ￿ (x;w) = x ￿ w.
The agent, on the basis of the agreed payment schedule w(x) chooses an action a (effort
say)andhasaseparablevonNeumann-MorgensternutilityUA(w;a) =2(w)
￿￿a2:Weset
￿ = 1=2 to satisfy the conditions associated to the ￿rst order approach to principal-agent
problems (Jewitt, 1988)4. Note that the usual conditions that 2(w)
￿ be increasing concave
and a2 be increasing convex are also satis￿ed. In a different line of business, the agent
could receive expected utility U; so a constraint on the principal’s choice of w is that the
agent’s maximized expected utility must not be less than U: The technology which is com-
mon knowledge is represented by the distribution of output depending on effort, F (xja)
where F (xja) is absolutely continuous with respect to the same nonnegative measure for
each a: Hence, F (xja) has a density f (xja): For its ￿exibility and general properties,
1Eaton and Rosen (1980 a) summarize the extensive econometric research as suggestive of very small re-
sponses in hours of work to changes in net wage for prime male earners. However, other groups, such as married
women, have considerably higher labor supply response rates.
2We use a slightly more general gamma distribution, allowing the minimum output to be strictly positive.
3The exponential distribution is a special case of the Gamma distribution, with p = 1.
4Note that ￿ ￿ 1
2; will satisfy Jewitt’s conditions. We use ￿ = 1=2;to get explicit solutions.IS TAX SHARING OPTIMAL? 3
we assume that the output conditional on effort is Gamma distributed xja ￿ ￿(p;a) 5 ,
where parameter p is a shape parameter allowing ￿exibility in the distribution of the output,
while effort a simply scales the output distribution horizontally or vertically (see Bose, Pal
and Sappington, 2007). We further assume that the government impose a wage tax t, of
which share ￿ is paid by the employee. Finally, welfare W(x) can be written as W(x) =
x ￿ (1 ￿ t￿)w(x) + 2(1 ￿ t￿)1=2w(x)1=2 ￿ a2:
We use the standard principal agent framework with moral hazard. The Principal’s
problem [P] can be written as:
Maximizew;a
Z 1
x0
[x ￿ w(x) ￿ (1 ￿ ￿)tw(x)]f(xja;x0)dx
subject to the Participation Constraint
(2.1)
Z 1
x0
2
p
w(x) ￿ t￿w(x)f(xja;x0)dx ￿ a2 = U;
and the Incentive Compatibility Constraint
(2.2)
Z 1
x0
2
p
w(x) ￿ t￿w(x)fa(xja;x0)dx ￿ 2a = 0:
The following proposition summarizes the results of the above computations. We as-
sume that U is suf￿ciently large such that w0(x) > 0 for all x ￿ 0:
Proposition 1. The solution to the Principal’s Problem [P] (second best solution) is char-
acterized by the following set of equations:
￿ =
1 + t ￿ t￿
2(1 ￿ t￿)
(a2 + U); (2.3)
￿ =
(1 + t ￿ t￿)a3
p(1 ￿ t￿)
; (2.4)
a =
3
q
￿L +
p
L2 + K3 +
3
q
￿L ￿
p
L2 + K3; (2.5)
w(x) =
1
4(1 ￿ t￿)
￿
2a
p
(x ￿ x0) + U ￿ a2
￿2
; (2.6)
where
(2.7) K :=
pU
3(p + 4)
and L := ￿
p2(1 ￿ t￿)
2(p + 4)(1 + t ￿ t￿)
:
Given the solution to [P], we study the relationship between the employee’s tax share ￿
and his optimal effort a, expected wage E(w), actual wage w(x), the principal’s expected
pro￿t E(￿) and the expected aggregate welfare, E(W). These results are presented in
Theorem 2 below.
5The density function for the gamma distribution is given by:f (xja;x0) = f(x;p;a;x0) =
1
ap￿(p) (x ￿ x0)p￿1 e￿(x￿x0)=a; for x 2 [x0;1); where ￿(p) =
R 1
x0 e￿ttp￿1dt:4 BARNALI GUPTA
1 AND CHRISTELLE VIAUROUX
2
Theorem 2. The following relations hold for all tax shares ￿ 2 [0;1]:
(a) @a
@￿ < 0; i.e.,employee effort is a decreasing function of ￿.
(b) If 1 ￿ t > (2 + ￿)t; then
@E(w)
@￿ > 0;i.e. expected wage is an incrasing function of
￿. If a
2￿U
2a2 ap < x ￿ x0 < ap,
@w(x)
@￿ > 0, i.e. for low levels of output, the actual wage is
an increasing function of ￿.
(c)
@E(￿)
@￿ < 0; i.e. the principal’s expected pro￿t is a decreasing function of ￿.
(d)
@E(W)
@￿ < 0; i.e. expected welfare is a decreasing function of ￿.
From Theorem 2 (a), we see that the higher the employee’s mandated tax share, the
lower his effort. Since he is risk averse, ceteris paribus, the agent exerts less effort as his
expected post tax wage falls.
If the share of net wages (1 ￿ t), is greater than twice the tax rate t, then expected
wage increases with ￿ (Theorem 2 (b)). Therefore, if there is an upper bound on the tax
rate t, we see that as the employee’s share of the tax increases, his expected wage will
also increase. This is because the risk averse employee has to be compensated with higher
expected wage as his share of the wage tax increases. The mandates on tax distribution,
place limits on the employer’s ability to trade-off risk sharing versus incentives. This result
is further substantiated in the second part of theorem 1(b), which shows that the result is
valid not only on average but for any given wage, provided that the output is low (between
average and its half). If U ￿ a2, then the conditions imposed in the second part of theorem
1(b) are satis￿ed provided that the output is smaller than average. Computation results in
the next section show that this result is quite robust.)
As the employer’s expected wage increases with ￿, so also the principal’s expected
pro￿t falls with ￿; from Theorem 2 (c). The constraint on the principal’s ability to tradeoff
risk sharing versus incentives, lowers her expected pro￿t.
Expected welfare is a strictly decreasing function of the agent’s share of the wage tax
(Theorem 2 (d)). So, while we expect welfare to decrease with a (wage) tax, we see that
the decrease in expected pro￿t is larger than the higher expected wage and the expected
government revenue from the tax. Hence, while expected wage is maximized if the agent
is legislatively mandated to pay 100% of the wage tax, the agent’s effort, expected pro￿t
and in aggregate, expected welfare, are all minimized. What is abundantly clear is that
regardlessofthetaxdistributionchosenasapolicymatter, aninteriorsolutionwithpositive
shares for both principal and agent, does not optimize any of these outcomes. From this
perspective, it is dif￿cult to justify an interior distribution of tax shares.
We explore further, the theoretical ￿ndings in Theorem 2 (b) with numerical computa-
tion to check the robustness of the suf￿cient conditions.
3. RESULTS
The simulations results below verify the robustness of the solutions from Theorem 2 .
Conclusion 1. Employee effort a; expected ￿rm pro￿t E(￿) and expected welfare E(W)
are maximized when the statutorily mandated employee’s share of the tax ￿ is zero; em-
ployee expected wage E(w) is maximized when the statutorily mandated employee’s share
of the tax is one.IS TAX SHARING OPTIMAL? 5
We verify the robustness of our suf￿cient conditions in Theorem 2 (b)6 using simulation.
Tables 1￿2 report the numerical results7 for p = 3, U = 1:5, x0 = 0; t and ￿ varying from
0.1 to 0.9 in increments of 0.1. We see that given t; the expected wage increases with ￿: As
stated in the second part of Theorem 2 (b), computation results verify that wage increases
with ￿ for a given t for x =
ap
2
h
a
2￿U
2a2 ￿ 1
i
(center of the interval speci￿ed).
4. CONCLUSION-EXTENSIONS
In the presence of incomplete information, the statutory liability of a tax has very clear
implications for effort, pro￿ts, wages and aggregate welfare. The theoretical and numerical
results do not ￿nd any justi￿cation for distributing the burden of a wage tax between em-
ployer and employee. While the results are derived using speci￿c functions, the point we
wish to make is quite general￿an interior distribution of tax shares does not maximize any
outcomes for any of the parties. Clearly the moral hazard intrinsic in the second best case
is critically important to the results obtained here. In the ￿rst best case, it can be shown
relatively easily that pro￿t and welfare are maximized when the employee’s statutory tax
share is 100%, while simultaneously wages and tax revenue are minimized.
Finally, we can reasonably wonder whether the agent’s reservation utility might depend
on the prevailing tax regime. We explored this extension to allow the agent’s opportunity
wage to depend on the tax environment. For example, if we let U = U0(1 ￿ t￿)￿, (￿ > 0)
such that ￿ is the elasticity of the agent’s utility with respect to post tax share of wage,
we ￿nd suf￿cient conditions on ￿ such that as long as the agent’s reservation utility is not
"too responsive" to changes in the share of wages that must be paid in taxes, the results
from Theorem 2 , with ￿xed reservation utilty, are generally robust. The conclusions in
this paper suggest that further study of the connection between mandated tax liability, its
implications for employer and employee earnings and optimal policy in this context, is
warranted.
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No. R5/2007, Statistics-Mathematics Unit, Indian Statistical Institute, Kolkata.
[2] Diamond, P., 1998, Optimal Income Taxation: An Example with a U-Shaped Pattern of Optimal Marginal
Tax Rates, American Economic Review, 88:1, 83-95.
[3] Eaton, J. and H.R. Rosen, 1980, Taxation, Human Capital and Uncertainty, American Economic Review,
70:4, 705￿715.
[4] Eaton, J. and H.R. Rosen, 1980, Labor Supply, Uncertainty, and Ef￿cient Taxation, Journal of Public Eco-
nomics, 14, 365-374.
[5] Holmstrom, B., 1979, Moral Hazard and Observability, Bell Journal of Economics, 10:1, 74￿91.
[6] Jewitt, I., 1988, Justifying the First-Order Approach to Principal￿Agent Problems, Econometrica, 56:5,
1177￿1190.
[7] Feldstein, M., 1995, The Effect of Marginal Tax rates on Taxable Income: A Panel Study of the 1986 Tax
Reform Act, The Journal of Political Economy, 103:3, 551-572.
[8] Laffont, J-J.andD.Martimort, 2002, TheTheoryofIncentives: ThePrincipalAgentModel, 2002, Princeton
University Press.
[9] Sappington, David E.M., 1991, Incentives in Principal Agent Relationships, The Journal of Economic Per-
spectives, 5:21, 45-66.
[10] Seade, J.K., 1977, On The Shape of Optimal Tax Schedules, Journal of Public Economics, 7:2, 203-235.
6Robustnmess results of other parts of Theorem 2 are available on request.
7Simulations performed for alternative values of p did not affect the results.

Is Tax sharing Optimal? An Analysis in a Principal-Agent Framework

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