Voting under Temptation
Monisankar Bishnu
a;y, Min Wang
baEconomics and Planning Unit, Indian Statistical Institute, New Delhi, India
bNational School of Development, Peking University, Beijing, China
Abstract
Within the con…nes of linear tax and complete market, we show that the e¢ ciency force for a negative capital tax may not be strong enough to reverse the politico- economic force for a positive redistributive taxation under temptation and self-control preferences.
Keywords : Consumption, Saving, Temptation, Self-control, Tax, Voting.
JEL Classi…cation: E21, E60, H20, H30, D72
The authors would like to thank the editor Pierre-Daniel Sarte and are grateful to an anonymous referee for extremely useful comments and suggestions. The paper owes its present form to these suggestions and comments. Thanks are also due to Joydeep Bhattacharya, Prabal Roy Chowdhury, Chetan Ghate and the audience of the 11th biannual meeting of the Society for Social Choice and Welfare.
yCorresponding author. Email: [email protected], Phone: +91-11-41493935, Fax: +91-11-41493981.
1 Introduction
The literature on the determinants of the level of (linear) capital tax in an economy is sub- stantially rich. One strand of literature from Representative-agent Ramsey tradition provides normative results against a positive tax, e.g., the celebrated Chamley (1986) and Judd (1985) zero tax result under standard geometric discounting and negative tax result under temp- tation and self-control (Krusell et al. (2010)). At the same time, other competing theories point to a positive redistributive tax under heterogenous - agent environment with geometric discounting, either because of insurance role of capital tax under incomplete markets, or due to the redistributive motive of a poor median voter under political economy frictions (see e.g., Bassetto and Benhabib (2006))1. This paper evaluates an interesting convex combi- nation of aforementioned temptation and political economy stories. It critically examines whether the e¢ ciency force for a negative capital tax under temptation and self-control is strong enough to reverse the politico-economic force for a positive redistributive taxation.
Thus, within the limits of linear capital tax and complete market assumption, this paper focuses on the outcome of a voting process on investment tax when agents’preferences are subject to temptation and self-control.
Temptation and self-control preferences have been introduced by Gul and Pesendorfer (2001, 2004, 2005) to model preference reversals.2 Speci…cally, the utility function under Gul and Pesendorfer preference consists of two parts to formalize the concepts of tempta- tion and self-control: a commitment utility measuring the preference on actual consumption choice and a temptation utility measuring the preference on consumption that would have been chosen had the agent succumbed to temptation. As the agent makes the actual choice, she always incurs the cost of deviating from temptation, i.e. cost of self-control, and thus the actual choice is a compromise between commitment utility and the cost of self-control. Using Gul and Pesendorfer preferences in a homogeneous agents setup, Krusell et al. (2010) show that when consumers are tempted by impatience, under the linear tax-transfer schemes, the optimal policy is to subsidize capital investment3 since subsidy makes surrender to tempta-
1Acemoglu et al. (2008) presented a political-economic analysis of optimal taxation where they focus on the best equilibrium that satis…es the incentive compatibility constraints of politicians.
2Laibson (1997), based on the earlier work by Strotz (1956) and Phelps and Pollak (1968), developed a model where the agents have time inconsistent preferences as time passes. It can explain the phenomenon that as time progresses, consumers exhibit preference reversals. Gul and Pesendorfer (2001, 2004, 2005) present an alternative approach to model preference reversals.
3Subsidizing savings is generally desirable when agents have limited rationality. For example, Amador et al. (2006) study the optimal trade-o¤ between commitment and ‡exibility and show that imposing a minimum level of savings is always a feature of the solution. Also, in a multiple-selves consumption-savings model, Laibson (1996) argued that optimal policy is to subsidize savings. A simultaneous non-optimality of savings and education investment has recently been discussed by Bishnu (2011) in a model of education investment accompanied by consumption externalities.
tions less attractive, thus improving the welfare.4
The present paper has an overlapping generations framework. Agents are heterogenous in the ability to earn income and are allowed to vote for the capital tax rate. Given this framework, the present study reveals a number of interesting results. First, incorporating temptations in the utility representation necessarily increases the political support for sub- sidy on capital investment, i.e., as temptation increases or temptation impatience relative to commitment impatience decreases, fewer people support positive tax on capital investment.
This observation is similar in spirit to the result in Krusell et al. (2010) where they show in the case of homogenous agents that as temptation increases, optimal capital subsidy should also increase. More importantly, an interesting fact that is revealed in this voting equilibrium is that even if a subsidy is welfare improving for all, since low income consumers save less, they would not be receiving enough subsidy to compensate the tax that they would pay if there is a subsidy on savings. As a result, they might not prefer a subsidy on savings despite its temptation reducing bene…ts. This phenomenon cannot be observed under the standard utility function. Thus the paper shows that the e¢ ciency force for a negative tax may not be strong enough to reverse the politico-economic force for a positive redistributive taxation under temptation and self-control preferences.
From technical point of view, one point in this paper deserves mentioning. When single peakedness property is violated, in a representative agent setup, Bassetto and Benhabib (2006) provided a way so that the median voter can still be shown as the Condorcet winner.
In our model too, single peakedness property is violated but for a di¤erent reason - we allow agents to choose tax rate in the negative domain as well. It is this extension of the domain of tax rate that leads to a violation of the single peakedness property; however, by verifying the order restriction property of policy preference (see Rothstein (1990)), we have been able to show that the median voter is indeed a Condorcet winner.
2 The Model
We consider a two-period overlapping generations model where each generation consists of a continuum of agents with measure one. Each agent is endowed with one unit of labor when young and retires when she is old. We denote (ciy;t; cio;t+1) as the agent i’s consumption in youth and old age respectively, and, kt+1i is the saving or capital investment made by agent i when young. eciy;t, ecio;t+1 and ekt+1i denote the choices that would have been made had the
4When the period utility is logarithmic, they have also shown that as the horizon grows large, the optimal policy recommendation is a constant subsidy which is in contrast to the well known Chamley (1986) and Judd (1985) result.
consumer succumbed to temptation. Agenti has Gul and Pesendorfer preferences with two components of utility u(ciy;t; cio;t+1) and v(ciy;t; cio;t+1) representing commitment utility and temptation utility respectively. Subject to the budget constraints that will be mentioned below, her decision problem is
max
ciy;t;cio;t+1
u ciy;t; cio;t+1 +v ciy;t; cio;t+1 max
e ciy;t;ecio;t+1
v eciy;t;ecio;t+1 . (1)
Agent’s actual choice maximizes the sum u ciy;t; cio;t+1 +v ciy;t; cio;t+1 of the commitment and temptation utilities, and for any choice bundle ciy;t; cio;t+1 , the cost of self-control is max v eciy;t;ecio;t+1 v ciy;t; cio;t+1 .
Agents are identical across generations, however, within a generation they are not iden- tical - they di¤er in their abilities to earn income. Let i represent the ability of the agent i where i has a CDF F i on a compact support [0;1]. Our assumption guarantees that generation-wise the distribution of ability is invariant. Given a wage rate wt, each young agent supplies labor inelastically in competitive labor markets, earns a wage income of wt i. Thus i measures the e¤ective labor supply of agenti. Further, we assume that the mean of i is greater than the median of i, that is E i = > med. At each date t, a single …nal good is produced using a CRS production function F(Kt; Lt) where Kt and Lt denote the capital and labor respectively. Particularly, F(Kt; Lt) = AKtL1t with A > 0 and 2 (0;1): The …nal good can either be consumed or invested. Since measures the total e¤ective labor supply, the wage rate is wt (1 )AKtLt = (1 )AKt and the capital return is Rt AKt 1 1 .5
At each date t, the government collects capital tax from the young which it returns to them in the same period as a lump-sum. Let t and st be the proportional tax imposed on investment and the lump-sum return respectively. The government budget balance condition is st = tKt+1 where Kt+1
Z
kit+1 i dF i . Then as a consumer, agent i maximizes (1) subject to the budget constraints:
ciy;t=wt i+st (1 + t)kit+1; cio;t+1 =Rt+1kt+1i (2)
e
ciy;t=wt i+st (1 + t)ekit+1; ecio;t+1 =Rt+1ekt+1i . (3) Finally, we assume that in this economy the capital tax rate t is decided by voting.
To obtain analytical results, we consider the instantaneous utility function that takes a
5In order to make our calculation analytically tractable, we assume that capital between the periods depreciates fully.
logarithmic form.6 Speci…cally we assumeu ciy;t; cio;t+1 = lnciy;t+ lncio;t+1andv ciy;t; cio;t+1 =
lnciy;t+ lncio;t+1 , where the parameters in this representation have the same interpre-
tations as in Krusell et al. (2010). While represents the long-run discount rate, is the discount rate in the short run, where < 1 measures the temptation impatience rela- tive to commitment impatience. The parameter represents the strength of temptation; in particular, the consumer completely surrenders to temptation when ! 1 while ! 0 represents the full commitment case. The cost of self-control is expressed by the term lneciy;t+ lnecio;t+1 lnciy;t+ lncio;t+1 . The problem involving commitment utility is to maximize
max
ciy;t;cio;t+1
(1 + ) lnciy;t+ (1 + ) lncio;t+1 (4) subject to (2),and the temptation utility maximization problem is given by
max
e ciy;t;ecio;t+1
lneciy;t+ lnecio;t+1 (5)
subject to (3). If we denote the capital investment calculated from the above two optimiza- tion problems by ki;t+1 and ekt+1i; respectively, then we have the following Lemma.
Lemma 1 For any individual i,ki;t+1 >ekt+1i; . Further, ki;t+1 ekt+1i; increases with income.
Proof. Solving commitment and temptation utility maximization gives the agent’s capital investment levels as follows:
kt+1i; = (1 + ) wt i+st
(1 + t) [1 + + (1 + )] (6)
and
ekt+1i; = wt i+st
(1 + t) (1 + ). (7)
Now, consider the function
fi(x) = x wt i+st
(1 + t) (1 + x) where x= 1 +
1 + ; .
6Our analysis can easily be extended to a case with more general utility function, for e.g., CRRA. In fact, it can be shown that the Lemma 1 of this paper holds independent of the elasticity of intertemporal substitution. However, to show the remaining results, we have to rely on simulations. A point to note here is that dealing with log utility function shuts o¤ one general equilibrium e¤ect, more particularly, the future voting results are independent of current savings.
Note that fi(x= (1 + )=(1 + )) =ki;t+1 and fi(x= ) =ekt+1i; . It can easily be checked that dfi(x)=dx > 0. Since (1 + )=(1 + )> , dfi(x)=dx > 0 implies that ki;t+1 >eki;t+1. Further, it is straightforward to show that ki;t+1 ekt+1i; is increasing in wt i. Hence proved.
Lemma 1 indicates that agents are tempted to save less. Krusell et al. (2010) show that when the agents are homogeneous, the optimal policy is to subsidize the capital investment so that temptation becomes less costly for agents, thus increasing agents’ welfare. Given this welfare feature, since we are interested to see whether a capital subsidy survives in a political outcome, we solve the voting problem. Since tax and subsidy scheme in each period have no e¤ects on the old living in the same period, we assume the voters consist of only young agents. Taking government budget balance into account, each young agent chooses an optimal tax rate it to maximize her value function
V( it) = (1 + ) ln wt i+ itKt+1 1 + it kt+1i; + (1 + ) ln Rt+1kt+1i; (8) lnh
wt i+ itKt+1 1 + it ekt+1i; i
ln Rt+1eki;t+1 .
Since we are interested in the sign of the capital tax rate, we …rst look for the political support for positive tax rate. Under this general equilibrium framework, the political support is as follows:
Lemma 2 Let (1 + ) (1 + )
(1 + ) + (1 + ) . Then if < , the optimal feasible i;t chosen by i has the following property:
i;
t
( 0, if i
<0, if i > :
Proof. First we substitute the value ofkt+1i; (as in (6)) in the expressionKt+1 Z
ki;t+1 i dF i and use the balanced budget condition st = tKt+1 to get the …nal expression for Kt+1 in terms of the parameters. Precisely, it is given by
Kt+1 = (1 + )wt =B( t) (9)
where B( t) = (1 + )(1 + t) + (1 + ). By substituting the equilibrium values of Kt+1, wtandRtinto the value function and then eliminating the constant parts, the e¤ective value functionVe (ein the subscript denotes ‘e¤ective’that is the value function that contains it)
becomes
Ve( it) = (1 + ) ln
i
+
i
t (1 + )
B( it) + (1 ) ln B it ln 1 + it . (10) For Ve( it) to be well de…ned, both the two terms under the log function should be strictly positive, that is, it> 1 and
i t>
i( + ) + i(1 + )
+ + i(1 + ) . (11)
Note that when i , the right hand side of (11) is larger than 1 and the choice set for voter i is de…ned by (11). On the other hand, if i > , the right hand side of (11) is less than 1 and it > 1 becomes the choice set for voter i. We prove our proposition in two parts.
First we consider the case when i . Taking the …rst order condition of Ve( it) with respect to it gives
(1 + ) (1 + ) [1 + + (1 + )]= B i;t 2
i + i;t (1 + )= B i;t 1 + i;t +(1 + ) (1 )
B i;t = 0
where i;t denotes the individuali’s optimally chosen tax. Simplifying the above, we get the following equation
i
(1 + ) (1 + i;t ) +
i
+ i;t (1 + ) (1 + ) (1 + i;t ) + 2(1 + )
= (1 + ) (1 + ) [1 + + (1 + )] 1 + i;t . (12)
We denote the left and right hand side by J( it) and H( it) respectively where i;t is the solution to J i;t = H i;t . Let i;mint be the minimum value of it. While it can be checked thatH( it)is linear and increasing in it withH( 1) = 0andH i;mint >0, it can be veri…ed that J00( it) > 0 and J i;mint = 0 < H i;mint . Figure 1 has been presented below to understand the proof well. The bottom most convex line is J( it) that is e¤ective
from i;mint . Evidently, i;t is the unique solution and i;t 0 if and only if H(0) J(0),
i.e.,
(1 + ) (1 + )> i 2(1 + ) + (1 + ) ,
Figure 1: Optimal tax rate preferred by agent i:
which, in turn, gives
i 6 (1 + ) (1 + )
(1 + ) + (1 + ) :
It remains to prove that for any i > , the agent prefers a negative tax rate. Under the situation i > , the choice set for voter i is it > 1 and J( 1) > H( 1) 0. We, thus, can have two possible scenarios as follows: case (a) when i is su¢ ciently large such that J( it)> H( it)for all it, i.e.,Ve0( it)<0for all itand case (b) when i is su¢ ciently low such that J( it) intersects H( it) twice, i.e., there exists two solutions of i;t where Ve0 i;t = 0 and if < , we have H(0)< J(0) such that both solutions are negative. These two cases are incorporated in Figure 1 below. Let bi;t and ei;t be those two solutions and say w.l.g.
that bi;t <ei;t .
It is straightforward to verify that for case (a), a tax rate i;t = (1 ")with any small
" > 0 is optimal, however, for (b), if there exist two solutions, then the smaller one bi;t
is the local minimum and larger one ei;t is the local maximum. To see this, we evaluate the second derivative Ve00( it) at it = i;t . It can be veri…ed that Ve00( it) i
t=bi;t > 0 and Ve00( it) i
t=ei;t < 0. Therefore, Ve( it) is …rst convex and then concave in it and for all
i
t>ei;t , it is monotonically decreasing. As a consequence, Ve( it) has two local maxima at
i
t= (1 ") and it=ei;t , both of which are negative tax rate.7 Hence the proof.
When a subsidy is welfare improving, notionally, people from comparatively lower ability group prefer a positive tax rate so that the gain from redistribution can improve their welfare by dominating the e¤ect of a negative tax that was required to improve welfare in the presence of temptations. Note that both the redistributive and the e¢ ciency e¤ect will be in favor of those agents who have relatively high income and prefer a negative tax rate. But for the agents who prefer a positive tax, while the redistributive e¤ect will favor them, they will be adversely a¤ected because of the welfare loss originating from temptation and self-control.
Before we present our result on the outcome of the voting equilibrium, we summarize how the choice of tax di¤ers with respect to the crucial parameters of the economy.
Proposition 1 As increases or decreases, decreases, that is, the support for a subsidy increases.
Proof. The proof is direct from Lemma 2.
The implication of the above result is very clear. Incorporating temptations necessarily reduces the political support for a positive tax. This implies that under full commitment, the threshold is higher compared to when temptation is present. When self-control and temptation is introduced in the model, the marginal agents who chose a positive tax in the absence of temptation, will optimally prefer a negative tax because the temptation e¤ect now dominates the redistribution e¤ect from the tax program. As the strength of the temptation increases, the temptation e¤ect becomes larger and the support of a positive tax thus decreases. A similar reasoning is also true for a decrease in the value of so that the temptation impatience relative to commitment impatience decreases and hence the temptation grows larger. We observe a link between the above proposition and the result in Krusell et al. (2010). There optimal subsidy on saving increases as temptation grows larger. In our heterogeneous agents case, the support of the negative tax rate increases as temptation grows larger.
As shown in the Proof of Lemma 2, the single peakedness assumption is violated for some i higher than but close to and median voter theorem cannot be directly applied to our model. This occurs because of the fact that it can be negative. If the choice set of it is positive, all agents would have single peaked preferences. However, though the median voter theorem can not be directly applied to our model, we obtain the voting equilibrium by pair-wise comparing the tax rate preferred by each voter .
Proposition 2 The tax preferred by the agent with median income is a Condorcet winner.
7Note that in this case, though agenti’s optimal choice is still negative tax, the value function has two peaks. That is why median voter theorem can not be applied to this model.
Proof. We have already mentioned that the two curvesJ i;t andH i;t intersect at i;t . From (12), we can also observe that J( it) is monotonically increasing in i, while H( it) is independent of i. Therefore the tax rate preferred by agent i, i;t (orei;t if agent has two local maxima), is monotonically decreasing in i. Moreover, by the above discussion, the voters whose single peakedness feature is violated are those with is higher than but close to . Their value function is …rst convex (and decreasing) and then concave in itand for all
i
t > ei;t , it is monotonically decreasing. A diagram is presented below to understand the proof well.
Figure 2: The e¤ective value function of agent i:
With the conditions stated above, we can easily check that when the median voter com- petes with any voter with i < med, all voters with i med will support the median voter, and, when the median voter competes with any voter with i > med, she will be supported by all the voters with i med. Hence, the median voter will remain as the Condorcet winner, even if the single peakedness is violated for some i.
3 Conclusion
This paper has examined the outcome of capital taxation in a political economy when agents are tempted to save less. Additionally, our study may also help explain the empirically observed desire of democracies for capital taxation even if the welfare of individuals can be improved via a capital subsidy which reduces their cost of self-control.
References
[1] Acemoglu, D., Golosov, M., Tsyvinski, A., 2008. Political Economy of Mechanisms.
Econometrica 76, 619–641.
[2] Amador, M., Werning, I., Angeletos, G.-M., 2006. Commitment vs Flexibility. Econo- metrica 74, 365–396.
[3] Bassetto, M., Benhabib, J., 2006. Redistribution, Taxes and the Median Voter. Review of Economic Dynamics 9, 211-223.
[4] Bishnu, M., 2011. Linking consumption externalities with optimal accumulation of hu- man and physical capital and intergenerational transfers. Forthcoming in Journal of Economic Theory.
[5] Chamley, C., 1986. Optimal Taxation of Capital Income in General Equilibrium With In…nite Lives. Econometrica, 54, 607–622.
[6] Gul, F., Pesendorfer, W., 2001. Temptation and Self-Control. Econometrica 69, 1403–
1435.
[7] Gul, F., Pesendorfer, W., 2004. Self-Control and the Theory of Consumption. Econo- metrica 72, 119–158.
[8] Gul, F., Pesendorfer, W., 2005. The Revealed Preference Theory of Changing Tastes.
Review of Economic Studies 72, 429–448.
[9] Judd, K. L., 1985. Redistributive Taxation in a Simple Perfect Foresight Model. Journal of Public Economics 28, 59–83.
[10] Krusell, P., Kuru¸sçu, B., Smith, Jr., A. A., 2010. Temptation and Taxation. Economet- rica 78, 2063-2084.
[11] Laibson, D., 1996. Hyperbolic Discount Functions, Undersaving, and Savings Policy.
Working Paper 5635, National Bureau of Economic Research.
[12] Laibson, D., 1997. Golden Eggs and Hyperbolic Discounting. Quarterly Journal of Eco- nomics 62, 443–477.
[13] Phelps, E. S., Pollak, R. A., 1968. On Second-Best National Saving and Game- Equilibrium Growth. Review of Economic Studies 35, 185–199.
[14] Rothstein, P., 1990. Order Restricted Preferences and Majority Rule. Social Choice and Welfare 7, 331-342.
[15] Strotz, R. H., 1956. Myopia and Inconsistency in Dynamic Utility Maximization. Review of Economic Studies 23, 165–180.
