Alternative Approaches to Consumption Smoothing: Fisher, Modigliani, Friedman and Hall

Term Paper, 2004

25 Pages, Grade: A

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Alternative approaches to Consumption Smoothing: Fisher, Modigliani, Friedman and Hall

By Selim Dedekarginoglu

Fundamental to studying an economy in terms of it’s current and future welfare and therefore of great importance, is the level of current consumption

As consumption makes up about two – thirds of the GDP, it is of great importance to appreciate it’s determinants in order to understand the long run growth path of GDP and it’s short run fluctuations.

Consumption is also of great significance in terms of the influences government policies (taxes and transfer payments etc) have on an aggregate economy, as an expansionary government policy may lead to an increase in disposible income, thus to an increase in the transaction demand for money and therefore may influence the interest rate and the level of investment, which determines the future capital stock of an economy and so it’s future capacity of production and therefore it’s future level of welfare.

In this paper, I will try to provide different theoretical approaches to the empirically proven fact that consumption is less fluctuating than income,

whereas I will concentrate on the most recent work done by Hall

The first approach to analyzing individual consumption (on a microeconomic basis) was developed by Irving Fisher already in the 1930s.

Fisher suggested that consumption was a function of the present value of an income stream and not only of current income.

Model of Intertemporal Consumption Smoothing (Irving Fisher,1930s)

We assume a rational, individual within a two – period model, who tries to maximize his overall utility, by choosing the level of consumption in period t and the level of consumption in t+1, discounted with the individual rate of time preference . Thus his utility function has the form

 = .

Furthermore we want to assume that the underlying, intertemporal budget constraint takes on the usual form

 = .

With , : level of consumption in period I

 : level of income in period I

 r : interest rate

 The budget constraint implies that the present value of the income stream has to equal the present value of the consumption levels chosen for t and t+1.


 = ,

 such that =

The above constrained maximization problem can be solved by following Lagrangian:

 :  + )

Optimality implies:

 , thus :

 , thus :

 yields : .

Thus, we can conclude that the individual maximizes his utility and solves his intertemporal problem of consumption by consuming that amount in each period, where, given the interest rate and given the rate of time preference, the marginal utilities in periods one and two equal each other.

 If we assume that the interest rate equals the rate of time preference, we get


which means that the marginal utility of consumption in period 1 has to equal the marginal utility in period 2 .

And furthermore, if we assume:

  • the usual concavity of the utility function,
  • a zero interest rate and a zero rate of time preference
  • a symmetric and homothetic utility function

we can conclude that:


which would be the special case of perfect consumption smoothing, and the proof that a stable lifestyle maximizes utility.

In order to stay in the chronological order, we now want to turn to Modigliani/Brumberg’s approach to consumption smoothing.

In order to do so, we have to derive the consumption function Modigliani/Brumberg suggested.

Life Cycle Hypothesis ( Modigliani/Brumberg, 1954)

The first step in their derivation of a consumption function was Modigliani/Brumberg’s assumption that current consumption of an individual is proportionally related to the present value of his income stream.


whereas k is defined as a proportionality factor depending on the individuals age, income level and preferences.

Assuming these three determinants to be constant, Modigliani/Brumberg derived a stable aggregate function of the form:


The problem they were confronted with in this context was that there was no data available on the present value of income.

So, they divided income  into labor income  and property income .

Thus, the present value of an income stream was defined to be:


whereas, r : interest rate

 T : remaining future periods of life.

But as this representation of the present value of an income stream still was not operational Modigliani/Brumberg furthermore assumed that the present value of the property-income stream, under the assumption of perfect capital markets, equaled it’s market value, denoted by .

Therefore: , or equivalently:


Now, the first and the last terms on the right hand side were observables, but not the middle term, which represents the present value of the expected future labor income.

They went on by defining


as the average expected future labor income in time period zero.

Or equivalently,


and therefore: .

But still they were confronted with the problem that there was no data available on the magnitude of .

Modilgliani/Brumberg’s crucial assumption in this context was to assume that the average expected future labor income had to be proportionally related to the level of current labor income:


And therefore: ,

whereas  is the sensitivity of the average expected future labor income on current labor income.

This led them to their consumption function of the form:


It is interesting to note that in the case of an increase in transitory income,

 will take on a value close to zero, or zero.

The rational behind this is that an increase in only  will not make the people believe that this will have any/ a large impact on the present value of their expected future labor income. Thus, the sensitivity parameter  will be zero, or close to zero.

The individual’s saving in this case is:


We can see that, the more an income change is perceived as transitory ( is zero or close to zero), ceteris paribus the higher saving will be.

Again, the consumer smoothes consumption through saving and borrowing.

In order to stay in the chronological order of the theories, and in order to stress the similarities and parallels, it now seems to be appropriate to deal with a simplified version of Friedman’s model and to see which implications it has regarding consumption smoothing.

The Permanent Income Hypothesis (Friedman, 1957)

Similar to Modigliani/Brumberg’s approach, Friedman starts out by assuming that consumption for an individual i, has to be a function of the present value of it’s income stream ,

whereas i represents a specific individual.

Then he defines  as “permanent income” (similar to the concept of expected average lifetime income).

He goes on by defining ,

which means that the level of permanent consumption of the individual i ,

is a function of a parameter k (which, according to Friedman depends on “individual rate of returns”, or skills and is defined to be between zero and one), and the level of permanent income.

By taking averages over all individuals within a specific income class he gets an aggregate “permanent consumption function” of the form:


whereas i now refers to income classes and  is assumed to be constant over all income classes (which intuitively makes sense that every income class consumes the same proportion of it’s permanent income level).

Then, he defines: ,

wheras  denotes the observed income level of income class i,

 and  the level of transitory (random) income.

And furthermore: ,

whereas  denotes the level of observed consumption and

  the level of transitory (random) consumption.

Then he makes following crucial assumptions:

1. . Meaning that  is a random fluctuation around . That means that regardless of the level of permanent income, each income class can experience same level of transitory income.

Assuming a normal distribution for income classes we get:

As can be seen in the graph, this implies that the stream of positive transitory incomes of income classes below the income class i in the graph is bigger than the sum of negative transitory incomes, income classes above i experience.

Therefore it is possible to conclude that income classes above/below the population average, experience positive/negative transitory income on average, and thus experience an average observed income level which is higher/lower than their average permanent income level.

For income classes below population average: >0 thus >

For income classes above population average: <0 thus < .

2. , meaning that transitory consumption is a random variation around permanent consumption

3. , meaning that the transitory components do not relate to each other.

Furthermore, he assumes that for each income class i the average level of transitory (random) consumption, , is zero.

This leads him to the formulation:


which means that average observed consumption for income class i, is proportional to the level of average permanent income for income class i.


Explanation of graph : The dotted line plots observed average income against observed average consumption. Thus, observed points like a,b and d (in d, average permanent income equals average observed income, as we are at the population average, where average transitory income equals zero), yield a consumption function (dotted, sloped line), similar to the Keynesian one.

The more an income class is above population average, the bigger it’s average transitory income will be, and the more an income class is below population average, the more negative it’s average transitory income will be.

That explains the divergence between the levels of observed and permanent income for classes i and j.

Note that consumption is determined by the level of permanent income along the k-bar curve.

An interseting implication of this graph is that the bigger the level of average permanent income is, the bigger the level of average transitory income will be.

And the bigger the level of average transitory income is, the bigger the level of observed average income will be.

But as consumers consume according to the level of their permanent income, the APC (=C/Y) will decline which means that the average propensity to save will increase (APS=S/Y).

Thus, again we get the result that a higher level of transitory income results in a higher level of saving, which again implies that consumers use saving and borrowing to smooth their consumption.

Consumption under Certainty: The Life-Cycle / Permanent-Income Hypothesis

 Before dealing with the case of uncertainty, it seems to be appropriate to summarize Friedman’s and Modigliani’s finding, that the path of consumption is smoothed by saving and borrowing decisions of an individual, in one model.

We assume an individual who lives for T periods and maximizes lifetime utility. Furthermore, we assume a utility function which satisfies the usual assumptions ( positive but diminishing marginal utility).


Our individual is assumed to have an initial wealth of  and a sequence of labor incomes ( we refer to labor incomes, in reference to the Life Cycle Theory) .

To stress, that we are in a non-stochastic world, we assume the income stream to be given, thus to be non-stochastic.

For simplicity the interest rate equals the time preference rate equals zero.

Furthermore the individual faces a budget consraint of the following form:


and as marginal utility was assumed to be positive (consumption is always chosen at a level below the bliss point), we can conclude that the individual will satisfy the budget constraint with equality. Therefore:


which means that the sum of all consumption levels throughout the individuals life has to equal the sum of his initial wealth, plus the sum of his future income stream.

We want to solve this maximization problem by setting up following Lagrangian: L: ,

The first-order condition for consumption in period t implies,


Since this implies that the marginal utility stays constant in each period, and since our utility function is strictly concave in consumption, we can conclude that


or in other words, that consumption is perfectly smooth.

If we take this into account in our wealth constraint described above, we get following consumption function:


This consumption function implies that consumption in a certain period is determined, not only by the income level in that period, but by the initial level of wealth, and the future income stream of the individual (usually referred to as “total lifetime resources”).

In Friedman’s terminology, the right hand side of our consumption function can be qualified as permanent income.

An interesting point to note is that although the income pattern seems not to be critical for the level of consumption, it seems to be critical for the level of saving. Considering the macroeconomic identity that income is divided up into saving an consumption



We can clearly see that whenever current income is high, ceteris paribus saving in that period will be high too. This makes us conclude that a big part of a positive transitory income is saved, whereas it only has little impact on that periods actual consumption.

This leads us again to our crucial result that the individual uses saving and borrowing to smooth consumption, just as suggested by Modigliani/Brumberg (1954) and Friedman (1957).

Having dealt with the case of certainty, we now want to relax our assumption af a nonstochastic world, and allow for uncertainty.

 Before going deeper into the issue of consumption under uncertainty, it seems appropriate to give a definition of what the “uncertainty” refers to.

In the following context, uncertainy will refer to the level of income and consumption.

This means that both, consumption and income are stochastic.

An economic model (R.Hall, 1978)

We assume a rational individual ( homo oeconomicus), who maximizes following function of expected utility


so that following intertemporal budget constraint holds true

 ) + = + -

which can be rewritten to

 ) = +[ -(1+r) ]

 whereas : expectations, conditional on the information available at time t

  : rate of time preference

 : standard utility function (concave in consumption)

  : consumption at time t

 t : time

 : income in t

 r : interest rate

 : wealth accumulated in time t


The individual is supposed to choose a consumption plan over time and thus maximize his discounted utility, whereas the discount factor is his rate of time preference.

The underlying, intertemporal constraint is the standard macroeconomic identity that income has to equal consumption and saving, thus the term -(1+r) can be interpreted as saving in period t.

Plugging in the budget constraint ) into the expected utility function yields, by an Euler approach, following solution for the optimal choice of consumption at each time t : = [(1+ ) / (1+r)]

This result has following implications:

  1. No information available at time t is going to affect the choice of the level of , apart from  itself. In other words, current information about the level of income and/or the level of wealth, does not help predicting future consumption.
  2. Marginal utility can be written in the form of following regression relation:

 = ,

 whereas =[(1+ ) / (1+r)], and  is , meaning that  is equal to zero.

  1. If we assume a quadratic utility function of the following form

 = -1/2 ,

 whereas  is the bliss point in consumption ( where  equals

 zero), then consumption will exactly obey

 = ,

 with = [(r- ) / (1+r)]. Again we can see that neither income nor

 wealth in period t ,included into this regression, will have a non-zero


Taking expectations at time t and assuming that:

  • The interest rate equals the rate of time preference ( r = ),

we get

  • The utility function has it’s usual concave form, yields following crucial result

 , and and as the expectations based earlier always “win”,


which means that consumption follows a Martingale, which means that the expected value of next periods consumption, conditional on all informations available at time t, is just consumption itself at time t.

Statements and Inferences from Hall’s results:

 - Surprise changes in Permanent Income ( ) are unpredictable

 - Therefore, changes in consumption (  ) are unpredictable

 as well.

 - The individual adjusts his current consumption to the point where

 consumption, on average, does not change over time, as this will

 maximize his expected lifetime utility.

This result is also referred to as the expected consumption smoothing plan.

This implies that people expect to live a stable lifestyle.

Taking expectations on both sides of the budget constraint will yield:

 , and as

 , we get


We can see that the consumption function under uncertainty, is very similar to the one we got in the case of certainty.

The implication of this result is that if an individual expects his income to increase in future, he’ll change his consumption today

In order to see the implications of this result, we again state that:

 , equivalently, ,which means that the expected change in consumption, conditional on information available in t is zero.

If we now assume an infinitely long living individual ( in this case, transitory changes in income would have no effect on consumption, as ), we can argue that changes in consumption can only occur due to surprise changes in permanent income, .

Thus we can say that , where  is a random variable with mean zero.

And as , we can say that:


 which means that consumption obeys a Random Walk and is smilar to Friedman’s finding that consumption only changes in the case of changes in permanent income, as  is defined in that way.

Having worked out the most crucial results Hall developed, it seems appropriate to stress one more time that all three theories, Modigliani – Brumberg’s Life Cycle Hypothesis, Friedman’s Permanent Income Hypothesis and Hall’s Random Walk Theory of Consumption imply that predictable changes in income should not cause consumers to revise their spending plans when the income change occurs.

The reasoning behind this is that every theory suggests, within it’s own framework, that consumers would adjust their current consumption to expected changes in their future income level or in other words: smooth consumption.

At this point, we want to introduce and define the concepts of Excess Sensitivity and Excess Smoothness .

Excess Sensitivity means that consumption does respond to predictable income changes at the time the income change has occurred.

In other words Excess Sensitivity implies that observed consumption (data) responds more to predictable income changes than predicted by the Permanent Income - / Life Cycle Hypothesis and the Random Walk Theory of Consumption. Therefore, in order to verify the Permanent Income Hypothesis and the Random Walk Theory, it has to be empirically proven that there is no Excess Sensitivity, in other words, that individuals do not revise their consumption plans due to expected changes in their permanent income, as the levels of current and future consumption already take these expected future changes into account.

Excess Smoothness means that consumption responds less to surprise changes in income, transitory income, than predicted by Permanent Income - / Life Cycle Hypothesis and the Random Walk Theory of consumption.

Thus the concept of Excess Smoothness could be a very useful tool to falsify the implication of the Permanent Income / Random Walk Theory, that changes in consumption are only due to surprise changes in permanent income.

In order to see the implications of Excess Sensitivity and Excess Smoothness,

we want to introduce a simplified version of the:

Test for Excess Sensitivity ( Campbell & Mankiw, 1989)

We start out by assuming two types of consumers.

The first type of consumer acts according to the Life Cycle / Permanent Income Hypothesis & the Random Walk Theory of Conumpion.

As we have stated earlier, the Random Walk Theory as well as the Permanent Income Hypothesis, imply that changes in consumption only appear due to a surprise change in permanent income, which was described as  in our earlier discussion of Hall’s Random Walk Theory.

Thus for this type of consumer it mathematically seems to be plausible to assume that changes in consumption are due to the variable  , described above

The behaviour of the other type of consumer is described by the following total differential of a Keynesian consumption function


with: dC: total change in consumption

 c: MPC

 : total change in disposable income

 : Income minus taxes

In other words we assume that the first type of consumer smoothes consumption, whereas the second type of consumer does not and just spends a proportional amount of his current disposable income on consumption goods.

Now, we want to define

as the fraction of Keynesian consumers who’s changes in consumption are due to a change in their disposable income, and

as the fraction of consumers who’s change in consumption is due to a surprise change in their permanent income.

Mankiw and Campbell’s main purpose in this context, as mentioned earlier, was to figure out if changes in consumption were really only due to surprise changes in permanent income, it seemed to be plausible to run the following regression  or equivalently


Here, we are confronted with an econometric problem, because  and  will surely be correlated (as it is quite plausible that surprising, “good news“ about future income mostly appear in times where the economy is expanding).

This will lead to biased OLS estimates.

The solution to this problem is provided by a more complicated econometric procedure known as Instrumental Variables. The main idea behind this is to take an other variable instead of the change in income, which is related to the change in consumption, but not to .

In the rational expectations framework this means that we have to use a variable at a specific point in time, where the value of  is not in the information set of the consumer, but which is nevertheless correlated to .

It seems quite useful to take the own lagged value of , .

Thus we get following regression equation:


Simply speaking, Mankiw and Campbell regressed changes in consumption on two variables, the changes in disposable income and the surprise change in permanent income. Intuitively it makes sense that if the Permanent Income / Random Walk Theory really holds true,  has to equal zero, as this would imply that changes in consumption are really only due to surprises changes in permanent income, and thus the Keynesian view that consumption is only related to current income, really is wrong.

But surprisingly,  was significantly positive and approximately equaled 0.5.

This result verified the existence of Excess Sensitivity and Excess Smoothness, and furthermore implied the existence of two different types of consumers, the Keynesian consumers and the Friedmanian consumers.

But the Question now was the rational behind this result and it turned out that this result could be due to three possibilities:

1. Liquidity Constraints

This has to do with the fact that the usual assumption in economics of perfect capital markets does not hold.

As (perfect) consumption smothing requires a capital market in which every individual is allowed to borrow or lend, according to his preferneces, it turns out that, especially in the case of a borrowing household, the existence of borrowing constraints of the form (the consumer is only allowed to consume in period i what he can afford in period i) force the level of consumption below it’s permanent level ( assuming a bell-shaped income curve over time). This implies Excess Sensitivity. Therefore, consumption will be more related to current disposable income.

The following graph shows what we have described intuitively.

If the household is a borrower, and the economy is liquidity constrained as described above, an increase in current income, even if it is expected, will lead to a high increase ( in the most extreme case, MPC = 1) in consumption.

 2. Consumers Myopia

These results could be due to the fact that Hall’s rational expectations framework simply is wrong.

It could be possible that consumers are not rational as always assumed in the neoclassical economic theory, but myopic.

In this case, people would simply consider their current income, when deciding how much to consume.


3. Precautionary Saving

Optimality in Hall’s model implied that an individual has to work out a “expected consumption smoothing plan”, where,

 = , by an Euler equation.

Taking expectations,at time t and assuming that the time preference equals the interest rate yields:


which means that the expected value of the marginal utility out of consumption in t+1, conditional on information available at time t, has to equal the marginal utility out of current consumption.

Hall’s result that consumption obeys a Martingale, , is crucially based on the quadratic form assumption of his utility function which has the form:

 = -1/2

This utility function assumes a risk-neutral individual as it also intuitively makes sense that no risk-avert individual would have a future consumption pattern, obeying the relationship , and thus ,  etc., or generally:


This result implies an individual who determines the level of all future consumptions , once having detemined the level of current consumption.

Determining the level of consumption equivalently means determining the level of all future saving.

It is interesting to note that this individual does not seem to care that the level of his future income might change.

Therefore, a risk-avert individual would make “precautionary saving”, to take into account the risk that his income might decrease in the future.

I have provided five different theoretical apporaches to the empirically proven fact that consumption is smoother than income.

The significance of these theories increases as they all approach this issue in a different model framework.

Interesting to note and therefore important to point out is that most of the theories presented, stress the assumption that consumption is perfectly smooth, only in the case where the interest rate equals the time preference rate, regardless of if individuals are in a stochastic or a nonstochastic world.

Due to the significance of consumption as the main determinant of GDP,

the research done on the area of why and how consumption fluctuates, and the reasons for it’s smoothness, will obviously never come to an end.



-Romer David “Advanced Macroeconomics”

-Hall Robert “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence”, JPE, v.86

-Paper notes Dr.Levin (WSU-Detroit)

-Paper notes Dr.Braid (WSU-Detroit)

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Alternative Approaches to Consumption Smoothing: Fisher, Modigliani, Friedman and Hall
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