Advanced Macroeconomics
60
The Consumption Function
Advanced Macroeconomics
61
The Consumption Function Thus, permanent income from the consumer’s present value includes human capital. The present value of future labor income is included in the above equation.
Friedman, along with Ando-Modigliani, assumes that the consumer wants to smooth his actual income stream into a more or less flat consumption pattern. This gives a level of permanent consumption
ip
C that is proportional to Ypi
i i i
p p
C =K Y (2.22)
The individual rate of permanent consumption to permanent income is k’, and presumably depends on the interest rate. The return on saving-individual tastes shapes the indifference curves and the variability of expected income.
If there is no reason to expect these factors to be associated with the level of income, we assume that the average Ki for all income classes will be the same: equal to the population averageK−. We can classify a sample of the population by income as
C−pi =K Y P− − (2.23)
If strata are included in the cross section budget studies, we would expect that the average permanent consumption to each income class i would be K− times its average permanent income for all income classes i.
We observed that the total income in a given period is made up permanent income Ypi which the individual has imputed to himself; plus a random transitory income component Ytiwhich can be positive, negative or zero. Here, subscript t refers to “transitory” not time. It gives us thus the measured income as the sum of the permanent and transitory component, which can be defined in an equation as
i i i
p t
Y =Y Y+ (2.24)
Similarly, total consumption in any period is permanent consumption. Cpiis the random transitory consumption component andCti represents positive, negative or zero deviation from the “normal” or permanent and transitory consumption.
i i i
p t
C C= +C (2.25)
Advanced Macroeconomics
62
The Consumption Function
Friedman’s theory of the cross sectional results in MPC < APC:
1. Friedman assumes that there is no correlation between transitory and permanent income. In other words, Yt is just randomly fluctuating around, so that the co-variation of Ypiand Yti across individuals is zero.
Implications
Suppose we take a sample of families from a roughly normal income distribution and then sort them out by income classes. Since Ypiand Ytiare not related, the income class that centers on the population average income will have an average transitory income component Y−t =0 and for that income classY Y− = −p.
2. Friedman assumes that there is no relationship between permanent and transitory consumption so that Ct is just a random verification around Cp.
3. Finally, Friedman assumes that there is no relationship between transitory consumption income and transitory income.
Friedman assumes that the co-variance of Ct and Yt is also zero. The last two assumptions that transitory consumption is not co-related with either permanent consumption or transitory income means that when we sample the population and classify the sample by income levels for each income class, the transitory variation in consumption will cancel out so that for each income class the equation becomes
ti 0 C− =
C C− =−Pi (2.26)
For each income class i, we can now bring this series of assumptions together into an explanation of the cross section result that MPC < APC even when the basic hypothesis of the theory is that the ratio of permanent consumption to permanent income and a constantK−.
Consider a randomly selected sample of the population classified by income levels. There is a group I with average observed income of Y−i. The above average population income will have a positive average transitory income component Y−ti >0for this above average group. The observed average income will be greater than the average permanent income, that is,
i pi
Y− >Y−
Advanced Macroeconomics
63
The Consumption Function
All income groups will have an average permanent consumption, given byC−pi =K Y− −pi. But since C−tiis not related to either Cpi or Yti, all groups including the above average income group will have a zero average transitory consumption component, so thatC−i =CPi. Linking these two consumption conditions gives us
pi
i Pi
C− =C =K Y− − (2.27)
Thus, the above average income group will have an average measured consumption equal to permanent consumption but the average measured income is greater than permanent income so that its measured
/ i
C Y−i − ratio will be less than K−. Similarly, a below average income group j will have a measured C Y−j/ −j
ratio >K−.
In the figure,K− represents the relationship between permanent consumption and income. The point Y− is the population average measured income if the sample is taken in a normal year when measured.
Average income is a trend. Average transitory income will be zero so that Y Y− = −P. The point C−pis the population average measured and the permanent consumption.
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