_{1}

This paper investigates time-consistent plans by incorporating the reference-dependent consumption model with endogenous labor supply. To what extent these plans can help the consumer overcome the double self-control problems of over-consumption and late-retirement is discussed. The consumption level and labor supply of the preferred personal equilibrium solution are compared with those of ex-ante optimal solution and present-biased solution. For all preferred personal equilibrium plans, both consumption level and labor supply are lower than those of the consumer who is present-biased but without reference-dependent preference. Although the preferred personal equilibrium solutions do not include the ex-ante optimal level, they help mitigate the welfare loss caused by present bias.

People usually desire immediate gratification but underrate the wellbeing in the future. This present-biased preference causes self-control problems. To overcome the self-control problem, internal commitment devices are often employed, such as goals and self-rewards (Hsiaw, 2013 [

An individual with present-biased preference suffers from double self-control problems of over-consumption and late-retirement (Diamond and Köszegi, 2003 [

In this model, quasi-hyperbolic discounting is employed to approximate present-biased preference. The ex-ante perspective in the first period is optimal in the long run. In order to identify the time-consistent plan solution, referred as the preferred personal equilibrium (PPE) by Köszegi and Rabin (2009) [

The main novel point which makes the results of this paper different from Köszegi and Rabin (2009) [

Furthermore, the following novel points are also made. First, for a time-consistent PPE plan, the reference- dependent preference is a factor which helps the consumer overcome her self-control problem. In this model, the consumer is facing double problems, but her instrument (the plan, or the “goal” as in Hsiaw, 2013 [

Consider a representative consumer who lives for three periods

consume or work, but she can form a consumption plan for the next periods. The consumer decides labor supply l in period 1 with wage rate

The consumer has present-biased preference. Following Laibson (1997) [

For simplicity, the long-run discount factor

Assumption. The consumer has the reference-dependent preference on consumption. The instantaneous utility of consumption

where

The first term

oss utility from comparing with the reference level

In period 0, the consumer evaluates the consumption of period 1 and period 2 equally, and makes the optimal allocation of consumption from the ex-ante perspective. However, this ex-ante optimal consumption allocation may not be preferred by self-1 because of present-biased preference. As the interest of this paper, we consider the time-consistent plans that will be implemented eventually, which is consistent with preferred personal equilibrium PPE by Köszegi and Rabin (2009) [

For self-0, the optimal consumption level

Note that

For a consumption plan

where

There could be four scenarios where period-1 consumption and labor supply deviate from the ex-ante optimal level (^{1}. In other words, neither will the period-1 consumption decrease from the ex-ante optimal level, nor will the period-2 consumption increase.

When

In the neighborhood of the ex-ante optimal level

Since

Lemma 1. In the case of

Self-0 prefers a smooth allocation

tion plan because her loss aversion is strong. However, when

Lemma 2. In the case of

The consumer with present-biased preference has to work longer to get higher income. Working longer leads to higher income level for the whole life, but also higher immediate disutility. When

In period 1, the present-biased preference causes the consumer to confront with two self-control problems: (1) how to allocate income; and (2) whether to work longer. Note that we consider endogenous labor supply, so consumption level and labor supply are simultaneously determined. Therefore, we have to jointly consider Lemma 1 and Lemma 2.

When

for the consumer in period 1. If the consumer in period 0 adopts it as the consumption plan, it will be implemented eventually, referred as the PPE consumption plan. With the same method, the other PPE consumption

plans under each condition (

Proposition 1.

(1) when

(2) when

(3) when

Remark 1. In this model with endogenous labor supply and reference-dependent preference, the ex-ante optimal level

Associating with different levels of present-bias discount factor

To investigate whether these could mitigate the two self-control problems, consider the case without reference-dependent consumption utility. In the case without gain-loss utility, the first-order conditions in period 1 read

Compare Equation (12) with the first-order conditions in Proposition 1, and since

Proposition 2. For all the PPE consumption plans, consumption level

By taking derivatives of first-order conditions under each case of

Proposition 3. For the PPE consumption plans, consumption level (

(1) when

(2) when

(3) when

According to Proposition 3, it is easy to show that for all PPE consumption plans, there exists a minimum value for period-1 consumption and labor supply when

Under the conditions of Equation (13), the consumption and labor supply are the closest to the ex-ante optimal level. And when

This paper incorporates endogenous labor supply into the model of reference-dependent consumption, and derives the consumption level and labor supply in the preferred personal equilibrium. It is shown that the PPE consumption plans can attenuate the double self-control problems caused by present-biased preference. The smooth consumption pattern which is optimal for the ex-ante utility is not a time-consistent plan.

The author acknowledges the financial support from Research Subsidy of Kanazawa Seiryo University.

LinZhang, (2015) Reference-Dependent Consumption with Endogenous Labor Supply. Theoretical Economics Letters,05,503-508. doi: 10.4236/tel.2015.54059

When

Taking derivatives of

Equations (A2) and (A3) imply that a decrease in