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# Theory of Consumption Function

## Theory of Consumption Function

Keynes’ Consumption Function: The Absolute Income Hypothesis
Keynes in his General Theory postulated that aggregate consumption is a function of aggregate current disposal income. The relation between consumption and income is based on his fundamental psychological law of consumption which states that when income increases consumption expenditure also increases but by a smaller amount.
The Keynesian consumption function is written as
C = a + cY                  a > 0                0 < c < 1
Where a is the intercept, a constant which measures consumption at a zero level of disposal income, c is the marginal propensity to consume MPC and Y is the disposal income.
The above relation that consumption is a function of current disposable income whether linear or non-linear is called the absolute income hypothesis. This consumption function has the following properties:
1. As income increases, average propensity to consume (APC = C / Y) falls.
1. The marginal propensity to consume MPC is positive but less than unity (0 < C < 1) so that higher income leads to higher consumption.
1. The consumption expenditure increases or decreases with increase or decrease in income but non-proportionally. This non-proportional consumption function implies that in the short run average and marginal propensities do not coincide (APC > MPC).
1. This consumption function is stable both in the short run and the long run. This consumption function is expressed in diagram 1, where C = a + cY is the consumption function. At point E on the C curve the income level is OY1. At this point, APC > MPC where APC = OC1 / OY1 and MPC = ΔC / ΔY = ER / RE0. This shows disproportional consumption function. The intercept ‘a’ shows the level of consumption corresponding to a zero level of income.
At income level OY0 where the curve C intersects the 45 degree line, point E0 represents APC (=OC0 / OY0). Below the income level OY0, consumption is more than income. In this range, APC is greater than 1. Above the income level OY0 consumption increases less than proportionately with income so that APC declines and it is less than one.

The Permanent Income Hypothesis of Consumption

Early Keynesian models of the consumption function related current consumption expenditure to current levels of income or disposable income. These models took the form of:
C = a + bYd
where
C = Consumption Expenditure
a = Autonomous consumption
consumption expenditure independent of the level of income.
b = the Marginal Propensity to Consume 'MPC'
which represents the fraction of each additional dollar of income
devoted to consumption expenditure.
and
Yd = Current Disposable Income.
Several theoretical implications can be developed by taking the ratio of consumption expenditure to the level of disposable income. This ratio known as the 'APC' the average propensity to consume eliminates the need to convert nominal values into their real counterpart in that changes in the price level cancel out:
APC  =  Real Consumption
Real Income =  Nominal Consumption / P =   Nominal Consumption
Nominal Income/P                 Nominal Income
Thus the APC can be computed by dividing both sides the the Keynesian consumption function by disposable income:
APC = C/Yd = a/Yd + b(Yd/Yd)
or
APC = a/Y+  MPC. Given this final result we can look at the theoretical implications of the Keynesian consumption function over a different income groups (the cross- section) and over time (a time series). For the cross-section we would expect that lower-income groups would consume a greater proportion of their income relative to high-income groups:
APClow income > APChigh income
With time series data we would expect that over time and as disposable income increases the APC should decline:
APCt-1 > APC> APCt+1
It is in this latter case that this particular consumption function fails to explain real world behavior. In empirical studies, the APC is observed to be smaller for higher income groups relative to low income groups. However, over time the APC is observed to be constant independent of growth in aggregate measures of income. This failure led to the development of alternative theories of the consumption function one of which is the Permanent Income Hypothesis or 'PIH'.

The PIH begins to explain consumption behavior by first redefining measures of income. Observed values of aggregate income 'Y' can be divided up into two separate components: 'YP' Permanent (or projected levels of) Income and 'YT ' Transitory (or unexpected changes in) Income. Thus:
Y = YP + YT.
The transitory component has an expected value of zero (E[YTt] = 0) reflecting the notion that over time transitory gains are offset by future transitory losses and vice-versa.  Thus in the long run observed levels of income 'Y' are equal to permanent income 'YP'.
Finally, according to the PIH consumption expenditure is proportional to permanent income:
C = kYP
such that the parameter 'k', a constant, represents both the average propensity to consume and the marginal propensity to consume. This consumption function (as shown with the blue line below) is described more accurately as a long run consumption function consistent with the observed long run results of consumption behavior.
Observed short run behavior is explained through the value of transitory income for different income groups. Specifically, transitory income for low income groups is assumed to be negative reflecting the notion that over time transitory losses exceed transitory gains for this group of individuals:
YTL < 0  => YL < YPL
For middle income groups the value of transitory income is equal to zero over time such that observed and permanent income take the same value:
YTM = 0  => YM = YP Finally, for high income groups, transitory gains exceed transitory losses such that transitory income is on average positive over time or:
YTH > 0  => YH > YP
The impact of this transitory component can be used to develop a short run consumption function (the red line) as shown in the diagram.

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"
s.t.
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)
s.t.
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
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.

## Relative Theory of Consumption

On the first attempts to reconcile the short run and long run consumption functions was by Arhur Smithies and James Tobin. They tested Keynes absolute income hypothesis in separate studies and came to the conclusion that the short run relationship between consumption and income is non-proportional but the time series data show the long run relationship to be proportional.
The latter consumption income behaviour results through an upward shift or drift in the short run non proportional consumption function due to factors other than income. Smithies and Tobin discuss the following factors:
1. Asset Holdings Tobin introduced asset holdings in the budget studies of negro and white families to test this hypothesis. He came to the conclusion that the increase in the asset holdings of families tends to increase their propensity to consume thereby leading to an upward shift in their consumption function.
1. New Products Since the end of the second world war, a variety of new household consumer goods have come into existence at a rapid rate. The introduction of new products tends to shift the consumption function upward.
1. Urbanisation Since the post world war there has been an increased tendency toward urbanisation. This movement of population from rural to urban areas has tended to shift the consumption function upward for the reason that the propensity to consume of the urban wage earners is higher than that of the farm workers.
1. Age Distribution There has been a continuous increase in the percentage of old people in the total population over the long run. Though the old people do not earn but they consume commodities. Consequently, the increase in their numbers has tended to shift the consumption function upward.
1. Decline in Saving Motive The growth of social security system makes automatic saving and guarantees income during illness. Redundancy disability and old age has increased the propensity to consume.
1. Consumer Credit The increasing availability and convenience of short term consumer credit shifts the consumption function upward. The greater case of buying consumer goods with credit cards, debit cards, use of ATMs and cheques and availability of instalment buying causes an upward shift in the consumption function.
1. Expectation of income increasing Average real wages of workers have increased and they expect them to rise in the future. These cause an upward shift in the consumption function. Those who expect higher future earnings tend to reduce their savings or even borrow to increase their present consumption. The consumption drift theory is explained in the diagram 3 where CL is the long run consumption function which shows the proportional relationship between consumption and income as we move along it. CS1 and CS2 are the short run consumption functions which cut the long run consumption function CL at points A and B. but due to the factors mentioned above, they tend to drift upward from point A to point B along the curve CL curve.
Each point such as A and B on the CL curve represents an average of all the values of factors included in the corresponding short run functions, CS1 and CS2 respectively and long run function, CL connecting all the average values. But the movement along the dotted portion of the short run consumption functions, CS1 and CS2 would cause consumption not to increase in proportion to the increase in income.
Its Criticisms
The great merit of this theory is that it lays stress on factors other than in income which affect the consumer behaviour. In this sense, it represents a major advance in the theory of the consumption function. However it has its short comings.
1. The theory does not tell the rate of upward drift along the CL curve. It appears to be a matter of chance.
1. It is just a coincidence if the factors explained above cause the consumption function to increase proportionately with increase in income so that the average of the values in the short run consumption function equals a fixed proportion of income.
1. According to Duesenberry all the factors mentioned as causes of the upward shift are not likely to have sufficient force to change the consumption savings relationship to such an extent as to cause the drift.
1. Duesenberry also points out that many of the factors such as decline in saving motive would lead to a secular fall in the consumption function. Such saving plans as life insurance and pension programs tend to increase savings and decrease the consumption function. Moreover, people want more supplementary savings to meet post retirement needs which tend to decrease their current consumption.
The Relative Income Hypothesis
The relative income hypothesis of James Duesenberry is based on the rejection of the two fundamental postulations of the consumption theory of Keynes. Duesenberry states that
(1) Every individual’s consumption behaviour is not independent but interdependent of the behaviour of every other individual and
(2) That consumption relations are irreversible and not reversible in time.
In formulating theory of the consumption function, Duesenberry writes “A real understanding of the problem of consumer behaviour must begin with a full recognition of the social character of consumption patterns.” By the “Social character of consumption patterns” he means the tendency in human beings not only “to keep up with the Joneses” but also to surpass the Joneses. Joneses refers to rich neighbours.
In other words the tendency is to strive constantly toward a higher consumption level and to emulate the consumption patterns of one’s rich neighbours and associates. Thus consumer’s preferences are interdependent. It is however differences in relative incomes that determine the consumption expenditures in a community.