. © 1999-2003
Douglas A.Ruby

Permanent Income Hypothesis

Two-Period Consumption

Aggregate Demand

Aggregate Expenditure

Macroeconomic Theory

The Life-Cycle Hypothesis
and the Rate of Time Preference

An extension to the two-period consumption model is that of the Life-Cycle Hypothesis or LCH model. The LCH model defines individual behavior as an attempt to smooth out consumption patterns over one's lifetime somewhat independent of current levels of income. This model states that early in one's life consumption expenditure may very well exceed income as the individual may be making major purchases related to buying a new home, starting a family, and beginning a career. At this stage in life the individual will borrow from the future to support these expenditure needs. In mid-life however, these expenditure patterns begin to level off and are supported or perhaps exceeded by increases in income. At this stage the individual repays any past borrowings and begins to save for her or his retirement. Upon retirement, consumption expenditure may begin to decline however income usually declines dramatically. In this stage of life, the individual dis-saves or lives off past savings until death. 

In the first stage of the life-cycle, the individual will borrow based on expected levels of wealth and income in the future. This wealth is defined as human wealth--the individual's ability to generate or earn income in the future (based on anticipated skills, talents, and initiative) in addition to non-human wealth--ownership of income producing assets. The desire to borrow from one's future will depend on the faith the individual has about his or her ability to repay these debts and to the degree to which an individual discounts future activity. Specifically, a greater faith in the future earning power is consistent with a lower rate of time preference (where the individual discounts the future less and relates future activity to be almost as important as current activity). Less faith in future earning power results in higher rates of time preference and a greater discounting of future activity. In this second case, current consumption depends heavily on current income. 

The Life-Cycle Hypothesis is based on the following model: 

max Ut = ΣL[U(Ct)(1+δ)-t]    "maximize the utility from consumption over time"   


ΣLCt(1+r)-t = ΣNYt(1+r)-t + Wo   "lifetime consumption must equal income"

where U(Ct) is the satisfaction received from consumption in time period 't', Ct is the level of consumption, Yt is income, 'δ' is the rate of time preference ( a measure of individual preference between present and future activity) and Wo is an initial level of income producing assets. 

Given this formulation, the following questions are suggested: 

1. If a particular individual "lives for today", will his/her rate of time preference be higher or lower than someone who "plans for the future"?

(hint: look at the model for only two time periods where t=0 corresponds to the present and t=1 corresponds to the future)

In a two period model the equations would be:

max U(Co)(1+ δ)-0 + U(C1)(1+ δ)-1 (the objective function)
C0(1+r)-0 + C1(1+r)-1 = Y0(1+r)-0 + Y1(1+r)-1 (the constraint)

or the objective function would be:

U(Co) + U(C1)(1+δ)-1   since (1+δ)-0 = 1

as 'δ' ( the rate of time preference) increases, the value of the satisfaction from future consumption 'U(C1)' decreases (is discounted) relative to the value of satisfaction from current consumption. Thus a person with a high rate of time preference discounts the future more or tends to "live for today". 

2. This rate of time preference is like an interest rate for a particular individual. If that individual's rate of time preference is higher than the current market interest rates 'rm', will that individual more likely be a net- saver or net-borrower?

If an individual's rate of time preference is greater than the market interest rate, then this individual discounts the future more than the market (or society as a whole). It is very likely that this individual will borrow funds (at current market interest rates) from those individuals that have a rate of time preference lower than the market rate. This borrowing will come at the expense of future consumption to support current consumption. 

3. A primary result of the life-cycle hypothesis is that current consumption is based on lifetime labor-income (human-wealth) and non- labor income (non-human wealth). This is in contrast to the Keynesian consumption function which states that current consumption is strongly related to current disposable income. Therefore if an individual has a low rate of time preference, is that individual more likely to follow a life-cycle pattern of consumption (current consumption largely unrelated to current income) or a Keynesian pattern of consumption (where current consumption is strongly related to current income)? Why?

An individual with a low rate of time preference will value the future much like the present. Consumption for this person will be based on lifetime wealth and earnings rather than current labor income. This individual's lifetime pattern of consumption would follow that implied by the Life-Cycle hypothesis. Someone with a high rate of time preference would base her/his consumption on current income much like that implied by the Keynesian consumption function. 

4. How do changes in expectations about future wealth affect current consumption behavior?

A short run LCH consumption function can be defined by assuming that the constraint in the above optimization problem is satisfied:

   Co = kWo + k(1+Nα)YL

where Co represents current consumption, Wo represents current levels of income producing assets (non-human wealth) and YL represents the current level of labor income and a proxy for future earnings and earning ability (human wealth). The parameter 'k' represents the marginal propensity to consume and the factor '(1+Nα)' relates future labor income (over 'N' working years) to current consumption. 

The effect of changes in expectations of the non-human wealth component will act as a shift parameter with respect to current consumption as shown in the diagram.