Imagine that there lives a representative agent having utility speci ed as equation (1), in periods of discrete time over an in nite-horizon, with a linear production technology:
yt= tht; (6)
where htis hours devoted to working; ytis the nal real output and t is a random shock to production, with mean = 1. Capital is assumed xed in this simple model for simplicity. Government levies tax rate t on labor income and rebates them to household each period. The transfer payment then is:
t= tyt: (7)
Household aims to maximize their expected life-time utility beginning at period
3Pintea (2006) adopts a similar approach but assumes consumption and leisure to be nonseparable.
0:
E0 X1
t=0
tU (ct; ht; Ht) ; (8)
subject to the budget constraint:
ct= (1 t) yt+ t; 8t; (9)
along with the time endowment as equation (2) and the production technology as equation (6). Since individuals do not take into consideration the e ect of own leisure choice on the others, taking economy-wide level of hours worked as given when they maximize their utility, the existence of leisure externalities can potentially render suboptimal outcome to the economy as a whole and thus allow room for bene cial government intervention to restore the e ciency.
Ljungqvist and Uhlig (2000) study the business cycle property of such opti-mal tax in the case of consumption externalities, and nd that the optimal tax moves procyclically if externalities enter into utility with a one-period lag (which they refer to as \catching up with the Joneses"). So a traditional Keynesian style demand-management policy appears, though for rather unconventional reason: the economy is optimally cooling downed with higher taxes when it is overheating in booms and is optimally stimulated with lower taxes in recessions to keep consump-tion up. In the following secconsump-tion, an analogous analysis is rst conducted for the case of leisure externalities.
3 Optimal taxation
3.1 Contemporaneous externalities
Assume leisure externalities enter into utility with no lags, so that individual's period utility depends not only on current own consumption and own leisure times but also on current level of economy-wide average leisure times. Given model structure already described in section 2, the mathematical maximization problem
for individual can be depicted as follows:
A Lagrangian is formed (with multiplier t on constraint equation (10)) which gives the following rst order conditions:
w.r.t. ct : ct = t; (11)
w.r.t. ht : A (ht+ Ht)! = (1 t) t t: (12) Along with market clearing condition requiring that ct = yt(= tht) and symmetric equilibrium requiring that ht= Ht, we can solve for optimal labor supply to be:
ht= (1 t) 1t A (1 + )!
1
!+
: (13)
Notice that in this simple model we have labor supply that responses to tech-nology shock negatively (or positively) as long as > 1 (or < 1):
@ht
When = 1 the wealth e ect exactly o sets the substitution e ect and thus labor supply will not response to technology shock at all. We will show later that this amount of labor supply is not socially optimal. We seek for potentially rst-best solution by considering a central planner solution, where the government internalizes leisure externalities for the economy. That is, the government take into consideration Ht= ht and t= t tht. Its optimization problem then can be
subject to
ct= yt(= tht) ; 8t (16)
with relevant rst order conditions:
w.r.t. ct : ct = t; (17)
w.r.t. ht : A (1 + )1+!h!t = t t: (18) Using market clearing condition we can again solve for optimal labor supply to be:
ht =
Compared to equation (13), we can nd that the tax that supports the rst-best optimality must be:
t =
1 + : (20)
This result suggests that the government should subsidize household for the case of < 0 (jealousy together with leisure coordination), and tax household for the case of > 0 (admiration together with congestion e ect). The intuition goes as follows. When people are jealous, we have @U ( )@L < 0 @U ( )@H > 0 , the economy-wide average leisure has a negative e ect on individual's utility. But each agent does not recognize such negative e ect of their own choice on other people. As a result, the economy as a whole ends up consuming too much leisure, and thus too little labor hours. In addition, for < 0 we also have @(@U ( )@h )
@H > 0, lower average labor hours lead to higher marginal utility of leisure since people are easy to nd time enjoyed together, such leisure coordination e ect further ampli es the ine cient over-consuming of leisure.
Mathematically, when tax is set at zero, the optimal labor hours from equation (13) can be solved to be ht=h 1
t
A(1+ )!
i!+1
, while the rst-best solution is described by equation (19). For < 0, It is readily seem that the rst-best solution renders higher equilibrium quantity of labor supply (notice that 1+1 > 1 for 1 < < 0).
As this is the case, it is desirable for government to subsidize labor income so that each individual will face the correct marginal trade-o between consumption and leisure. This is described by the rst order condition with respect to labor (under symmetric equilibrium condition): The \true" marginal disutility of labor is the left hand side of equation (18), which is lower than the \perceived" marginal disutility of labor when < 0.
People are thus under-working. When individuals' labor incomes are (optimally) subsidized, they choose to work more, in such a way that the externalities can be fully internalized via a negative tax.
Since the government can e ectively correct this distortion period by period, the optimal tax does not have any cyclical property at all. Optimal tax t is a constant not subject to changes in technology shock t. Such result is in line with Ljungqvist and Uhlig (2000), where they nd contemporaneous consumption externalities do not generate any tax-induced cyclical consequence.4