Theories of Consumption Function

Introduction

Following Keynes's short-run consumption-income analysis and his Fundamental Law of Consumption, substantial research has focused on refining the consumption-income relationship.
Later theorists aimed to reconcile the short-run non-proportional and long-run proportional consumption functions through various hypotheses.

Absolute Income Hypothesis

Keynes's absolute income hypothesis stems from his Psychological Law, proposing three generalizations about the income-consumption relationship:
- (i) Current absolute consumption relates to current absolute real income: Ct = f(Yt)
- (ii) The consumption-income relationship is reversible; people decrease consumption at the same rate as income falls.
- (iii) Consumer spending patterns are autonomous, with each consumer group's pattern independent of others.
Keynes was not the first to use consumption and income in an absolute sense. Earlier literature often related current demand for consumer goods to current income in absolute terms.
Keynes's General Theory explicitly stated the absolute income hypothesis, regarding current consumption as a function of current absolute income.
The hypothesis evolved due to debates about automatic full employment equilibrium. Pigou introduced the Real Balance Effect, adding wealth as a determinant of consumption or saving.
While Pigou couldn't conclusively prove full employment equilibrium, wealth or real balances became recognized as influencing consumption, though the quantitative significance of this effect remains debated.

Challenges to the Absolute Income Hypothesis

Duesenberry and others challenged the reversibility of the consumption-income relationship and the autonomous nature of consumer spending.
Some writers found the absolute income hypothesis inadequate because it failed to reconcile short-run and long-run time series data.
Keynes's Law suggests the saving-income ratio increases in the short term, but Kuznets' long-run data (1869-1938) showed a stable APC between 0.84 and 0.891. Goldsmith's findings also indicated stable long-run C-Y and S-Y ratios. He estimated that the APC between 1879 and 1949 in the United States economy as about 892.
Keynes did not clearly distinguish between cyclical and secular consumption functions. He cautiously stated "as a rule and on the average," which aligns with Kuznets' long-run data.
Keynes did not specify the exact percentage of income saved or consumed but indicated that the gap between income and consumption widens as income increases. This saving gap can grow over time, even if the propensity to save remains constant.
Hicks questioned the short-run Y-C relationship, suggesting no convincing theoretical reason for the proportion of income divided between consumption and saving to change with income.
Empirical data and the consensus among business cycle writers generally contradict Hicks's suggestion. Thus, the absolute income hypothesis remains largely accepted, at least in the short run.

Past Income Hypothesis (Duesenberry Hypothesis)

James Duesenberry modified Keynes's absolute income theory with these points:
- (i) Current consumption depends on both current and past income: Ct = f(Yt, Y0), where Ct is current consumption, Yt is current income, and Y0 is the highest previous income level.
- (ii) The consumption-income relationship is not reversible.
- (iii) Short-run and long-run consumption functions can be linked.
Duesenberry argues the basic consumption-income relationship is proportional. Without prior income influence, consumption changes proportionally with income.
Previous income affects current consumption non-proportionally. During expansion, higher current income raises consumption, but the influence of previous lower income moderates this increase, causing the APC to fall.
Conversely, during contraction, declining current income reduces consumption, but prior higher consumption standards limit the reduction, causing the APC to rise.
Once income surpasses previous peak levels, the constraint of prior lower income disappears, and consumption rises steadily proportional to income.
Thus, short-run non-proportional and long-run proportional consumption functions are reconciled.

Graphical Representation

Long-run sustained income increase (Y) leads to proportionate consumption increase (C).
Short-run income path (y) shows fluctuations with corresponding consumption fluctuations (C₁).
Cyclical fluctuations in income and consumption result from lagged adjustments, with consumption lagging behind income.
Consumption peaks and troughs lag one period behind income peaks and troughs due to lagged adjustments between current consumption and past income.

Non-Reversibility of Income-Consumption Relationship

Duesenberry disputes Keynes's reversibility, noting asymmetrical contraction and expansion effects.
During downswings, pressure to cut consumption proportionally to income is countered by the force of habit from previously higher consumption standards.
Families sacrifice saving to maintain living standards, leading to a smaller rate of consumption reduction.
During recovery, consumption increases slowly due to:
- The influence of prior low-income levels.
- The desire to compensate for previous dissaving.
Once past dissavings are recovered, consumption increases more rapidly, aligning with income.
Consumers find it easier to increase consumption than to decrease it. This is known as the Ratchet Effect.

Ratchet Effect

The Ratchet Effect stabilizes the economy by preventing it from losing all income gains from the preceding boom.

Saving Function

Duesenberry proposes a saving function:
- St / Yt = a + b(Yt / Y0)
- Where St and Yt are current saving and income, Y_0 is the previous peak income, and a and b are constants with b < 0.
- The average propensity to save (APS) is a function of the ratio of current income to the highest previous income.
If (Yt / Y0) remains constant, (St / Yt) will also be constant.
If Yt falls below Y0, (St / Yt) declines, implying (Ct / Yt) rises during recession.
During recovery and expansion, Yt > Y0, or (Yt / Y0) > 1, and (St / Yt) rises.

Empirical Evidence

For 1929-1940 in the US, a = 0.25 and b = -0.196.
- St / Yt = .25 - .196(Yt / Y0)
- Ct / Yt = 1 - .25 + .196(Yt / Y0)
- Ct = 1.196Yt - .25(Yt^2 / Y0)
Example: If Yt = 200 billion and Y0 = 150 billion:
- St / Yt = 0.25 - 0.196(200 / 150) = 0.137
- C_t = 1.196 * 200 - 0.25(200^2 / 150) = 172.53 billion
Prof. Duesenberry notes that during 1923-1940, a = 0.165 and b = -0.066. \frac{St}{Yt} was computed as 102 which was very close to Kuznets' estimate of (stable) saving ratio in the period 1879-1919.

Improvements on Consumption Function Views

Duesenberry's hypothesis improves on earlier views by:
- (i) Recognizing that current consumption spending is influenced by previous income, not just current income.
- (ii) Asserting that the consumption-income relationship is not reversible.
- (iii) Linking proportional secular and non-proportional cyclical consumption functions.

Implications of Duesenberry's Consumption Function

(a) Turning Points of Cycles: The failure of consumption to keep pace with income changes is key in explaining business cycle turning points. The decreasing APC during prosperity and increasing APC during depression logically explain why prosperity turns into depression and vice versa.
(b) Link between Business Cycle and Economic Development: Duesenberry links business cycle theory and economic development theory. The long-run consumption function's linearity challenges stagnation theories that assume the saving gap increases more than proportionally with income. This linearity implies that stagnation phenomenon may occur in the short period and not in the long period.
(c) Compulsion to Employ Women and Children: It explains why families put non-working members (women and children) to work during recessions. Consumption is a function of income, but income can also become a function of consumption.
Families aim to maintain a consumption standard, necessitating supplementary income from non-working members during economic downturns.

Relative Income Hypothesis

Also associated with James Duesenberry, this hypothesis disputes two Keynesian contentions:
- Consumption isn't solely based on absolute income levels.
- Consumer behavior is interdependent, with the APC depending on an individual's position on the income scale.
- Keynes suggested consumption was reversible over time, but Duesenberry argued it is not.
Underlying the hypothesis is the idea that humans are social and influenced by their neighbors' consumption. Income disparities can signal social status, leading to dissatisfaction among those at the lower end of the income scale.
Those in the community are urged for consumption in order to 'keep up with the Joneses.'
If all incomes increase equally, relative positions remain unchanged, and so does the consumption-income ratio (C/Y), even with rising disposable incomes.
The relative income hypothesis defines APS as:
- S/Y = a + b(Y/Y_1)
- C/Y = 1 - a - b(Y/Y_1)
- Where: Y = individual or group average income, Y1 = community average income, a = saving coefficient independent of income, b = saving coefficient dependent on Y and Y1
An individual's APC changes based on the relative changes in Y and Y_1. If income rises faster, slower, or identically to the community, APC falls, rises, or remains constant, respectively.
Greater income equality reduces the consumption-income ratio for lower-income groups by alleviating the pressure to 'keep up with the Joneses.' But with more equitable income distribution, it does not reduce APC.
- There may simultaneously be a trend among the richer sections of society to keep ahead of the Smiths. So the effects are determined by the relative strengths of the Jones Effect and the Smiths Effect, there isn't any possibility of cleat cut conclusion of reliable empirical data.
Consumption responses vary between expansion and contraction, and Jones and Smiths effects differ, making the consumption-income relation unlikely to be reversible, unlike the absolute income theory's assumption.

Empirical Support

Data on US White and Negro income-consumption patterns support this hypothesis. A Negro is sensitive to his position within his own group and relative to White groups' incomes.
In segregated areas, Negro consumption patterns are mainly influenced by other Negroes' spending. In integrated areas, the demonstration effect of Whites' higher incomes prompts Negroes to emulate their consumption, reducing inter-racial APC gaps.
Differences in APC between white people and negroes were smaller in New York City than in Atlanta. Segregation is less stringent at the former place and, the Jones effect being strong, the gap between the APC of different groups is reduced.

Resolution of a Statistical Puzzle

Time series data show a fairly constant APC, but budget studies (at a point of time) show that APC decreases as income rises
Budget studies underestimate APC because they don't account for relative income changes.

Implications for Savings in Advanced Countries

In some advanced countries, almost 75% of the families virtually save nothing
When all incomes change proportionally, APC remains constant. A group's APC doesn't decline despite rising disposable incomes.
Many families in advanced nations save almost nothing because intense social pressure and superior expenditure patterns leave them with virtually no savings.

Permanent Income Hypothesis

Developed by Milton Friedman, this theory posits that expected or future incomes influence current consumption.
It departs from Keynes's absolute income theory, which correlated current consumption directly with current absolute income.
Friedman and others believe expectations aren't always offsetting; expected or future incomes can influence consumption.
- Measured or actual income can be derived as; Y = Yp + Yt

Components of Income

Permanent Income (Y_p):
- The amount a consumer can consume while maintaining their wealth intact.
- Wealth includes both non-human and human wealth, with non-human wealth being more significant.
- Y_p = iW, where W = Wealth and i = Rate of return on wealth.
Transitory Income (Y_t):
- Unexpected additions to or subtractions from income (windfall gains or losses).

Components of Consumption

C = Cp + Ct
Permanent Consumption (C_p):
- The planned value of services consumed during a period.
Transitory Consumption (C_t):
- Unanticipated additions or subtractions in consumption.

Basic Assumptions

(i) The transitory component of income is uncorrelated with permanent income (p = 0).
(ii) The transitory component of consumption is uncorrelated with permanent consumption (1_{pc} = 0).
(iii) There is no correlation between the transitory income and transitory consumption (c = 0).

Key Implications

Windfalls or temporary income variations have little effect on consumption, primarily affecting saving. Conversely, unexpected drops in transitory income reduce saving rather than consumption.
Permanent consumption is related only to permanent income, as unexpected income and consumption changes cancel out over time.
- Cp = KYp, where K is the proportionality coefficient between Cp and Yp.
With a homogeneous utility function, planned or permanent consumption is directly proportional to wealth.
- C = qw, where q is dependent on consumer's tastes and rate of interest (i)
The long-run APC for the community equals the long-run MPC.

Factors Influencing the Coefficient K

Age structure of the population.
Interest rates for borrowing or lending.
Ratio of non-human wealth to permanent income.

Modified Consumption-Income Relationship

C_t = K(i, W, μ)Y
- Where: i = rate of interest, W = current wealth (human and non-human), μ = propensity to consume rather than accumulate wealth (determined by age structure and income variability).

Empirical Findings

For non-war years between 1905 and 1951:
- Ct = 0.88Yp, i.e., the average propensity to consume out of Y_p is 88%.
Adaptive Mechanism for Expressing Relationship Between Actual and Permanent Income:
- Y{pt} = Y{p(t-1)} + α[Yt - Y{p(t-1)}] where α is the adjustment coefficient.

Reconciliation of Short-Run and Long-Run Consumption Functions

Friedman's theory reconciles long-run proportional consumption function and short-run non-proportional consumption function suggested by Keynes' absolute income theory.

Criticisms of the Permanent Income Hypothesis

The assumption of zero correlation between Yt and Ct (MPC out of Y_t as zero) has been heavily debated.
Contradictory evidence exists regarding the impact of unexpected changes in Y_t on consumption versus saving.
There is conflicting evidence about purchases of consumer durables as investments.
Statistical studies have variably supported and contradicted Friedman's contentions.
The zero correlation between Yp and Yt has also been disputed.
Lumping together human and non-human wealth is questionable.
The time span for determining permanent income has led to controversy.

Support for the Theory

The highest APC among 18-24-year-olds supports the hypothesis, as young people often incur debt anticipating future income.

Policy Implications

Monetary and fiscal policies should manage adverse expectational patterns.
Tax reductions during recessions may not boost consumption if seen as temporary or indicative of future economic problems.

Life Cycle Hypothesis

Developed by F. Modigliani, R. Brumberg, and A. Ando, this aims to reconcile long-run proportional and short-run cyclical consumption behavior.

Short-Run Non-Proportionality

Cyclical non-proportionality is attributed to:
- Extent of unemployment.
- Redistribution of income.

Recession Dynamics

During recessions, unemployment increases, and income is redistributed against low-consuming richer sections.
Consumption remains high due to these factors while income falls, increasing the C-Y ratio or APC.

Expansion Dynamics

During expansion, unemployment falls, and income redistributes in favor of the rich.
Consumption rises slowly relative to income growth, causing the consumption-income ratio or APC to fall.

Long-Run Consumption Behavior

This hypothesis stresses that individuals plan consumption over their lifetime rather than linking it to short term income.

Lifetime Planning

Consumption plans extend over an individual's life, with income in any period being just one factor shaping such plans.

Graphical Representation

Over a lifetime, income is low initially, peaks during full-time employment, and declines in later years.
Consumption follows a smoother path, with individuals dissaving early and late in life and saving during their career's middle part.

Formulation

Ct = KVt, where Ct is current consumption, K is the proportionality factor, and Vt is the present value of lifetime resources.

Resources Available

The sum of an individual's net worth at the end of the preceding period, current non-property income, and the discounted values of expected future non-property incomes.

Aggregate Consumption Function

Ct = aYt^n + bA_{t-1}
- Where: Ct = current consumption, Yt^n = aggregate non-property income, A_{t-1} = aggregate net worth at the end of t-1.
Total income is the sum of non-property income (Yt^n) and property income (\gamma A{t-1}), where \gamma is the rate of return on net worth.
Short run consumption function is non-proportional, the intercept term being given by (b-a\gamma)A_{t-1}. As the net worth increases, the short run consumption function shifts up.

Warren Smith's Observations

The ratio A{t-1} to Yt is around 5.
Substituting this value, we find that the long run consumption = 0.86.
The long-run consumption is a constant proportion of current income, represented by a straight line from the origin.

Similarities with the Permanent Income Hypothesis

(i) Indirect Effect of Current Income: Both theories state income influences current consumption only through a key variable. These are permanent income in Hhiedman's hypothesis
(ii) Proportionality Postulate: Both theories use postualtes although it is taken only as a convenient assumption rather than a vital assumption.
(iii) Reconciliation between short and long run consumption functions: Both suggest proportionality for long run and non-proportionality for short run.

Differences from the Permanent Income Hypothesis

(i)Time span: Considers end of life. By contrast, Friedmans takes perpetual annuity on total wealth. It implies that the consumer plans over an infinite time.
(ii) Non-human assets: Friedman's consumption function does not involve non-human assets in an explicit manner. However under the life cycle theory, the non-human assets are involved in an explicit manner.

Criticisms

Relies on variables, like the exact anticipation of future income , that are hard to predict.
Direct relationship between consumption and assets is not always valid.
Fails to account for liquidity constraints in capital markets.

Normal Income Hypothesis

Associated with M.J. Farrell, influenced by Friedman, Brumberg, and Modigliani.

Departure from Friedman

Main departure from Friedman's permanent income hypothesis is with respect to the time span taken into consideration.

Maximizing Utility

Annual consumption depends on resources available over a person's lifetime.

Perfect Capital Market

If the individuals expect a constant annual income Y for the remainder of his life span and Y is such that the current value of this income stream is just equal to v, then Y shall be considered as the normal