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B. Monetary policy and the classical aggregate supply curve
2.2 The Ando-Modigliani Approach: The life cycle hypothesis
Advanced Macroeconomics
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The Consumption Function At point B, the individual’s consumption pattern is C0,C1. The position of the budget line is determined by two variables, namely: income period and interest rates. The relationship between the present value of the income stream and current consumption from the above figure gives us the first general formulation of the consumption function
Ct = f (PV1) and f > 0 (2.7)
where PV1 is the present value of current and future income at time t is (1 )t t
Y
∑ r +
Thus, it can be simply stated that an individual’s consumption in time t is an increasing function of the present value of his income in time t.
Both Ando-Modigliani and Friedman began their analysis of the consumption function with the general form of the function given in equation (2.8).
Advanced Macroeconomics
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The Consumption Function We assume that in the absence of any particular reason to favor consumption in any period over any other, for representative consumer i, if present value (PVi) rises, all of his Cif rises more or less proportionately.
In other words, for consumer i,
( )...0 1
i i i
t t
C =K PV < <k (2.8)
Here Ki is the fraction of the consumer’s PV that he wants to consume in period t. It would depend on the shape of his indifference curve. The above equation explains that if there is an increase in any income, present or expected, raising the consumer’s estimate of PV, he will consume the fraction of Ki of the increase in the current period. If the population distribution by age and income is relatively constant and the tastes between present and future consumption are stable through time, we can add up all the individual consumption functions (2.8) to a stable aggregate function as
Ct = k (PVt) (2.9)
The theory involves consumption as a function of expected income which, of course, cannot be measured.
Ando-Modigliani began to make the present value term operational by noting that income can be divided into income from labor YL and income from assets or property Yp. Thus, permanent income is presented as follows:
0 0 (1 ) 0 (1 )
L P
t T
t t
t
Y Y
PV = r + r
+ +
∑ ∑
(2. 10)where time 0 is the current period and t ranges from 0 to the remaining years of life T, we assume that the PV of the income from an asset is equal to the value of the asset itself measured at the beginning of the current period, i.e.,
0 (1 ) 0
t t
t
Y a
r =
∑
+ (2.11)where α is the real household net worth at the beginning of the period. We can separate out known current labor income from the unknown or expected future labor income. Thus given, PV0 can be deduced as
0 0 1 0
1 (1 )
T L L
t
PV Y Y q
= + r +
∑
+ (2.12)The next step in this sequence is to determine how the expected labor income in time 0 evolves, such that
0 1
1
1
1 (1 )
T L i
t
Y Y
T r
= −
∑
+ (2.13)Advanced Macroeconomics
57
The Consumption Function where T-1 is the average remaining life expectancy of the population and the term 1
T−1 the average PV of future labor income over T-1 year. Thus, expected labor income (2.12) can be written as
1 0
( 1) (1 )
T L t e
t
Y T Y
r = −
∑
+ (2.14)This gives us an expression for the PV of the income stream as
0 0L ( 1) 0e 0
PV Y= + T− Y +a (2.15)
There is only one remaining variable that is not yet measurable: the average expected labor income Ye. The simplest assumption would be that average expected labor income is just a multiple of present labor income.
0e 0L
Y =
β
Y and β > 0The assumption is that if current income rises, people adjust their expectation of future income upwards so that Ye rises by the fraction β of the increase in YL.
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Advanced Macroeconomics
58
The Consumption Function Alternatively, we could assume that Ye is related to both the present labor income and employment on the theory that as employment t goes up, people will expect their chances for future employment and thus income, to rise, too.
This assumption can be formulated as
0e 0L ( ).W 0L
Y Y f Y
β L
= = and S’ > 0
where N = employment
L = size of the labor force
Ando-Modigliani tried a number of similar assumptions and found that the simplest assumption is Ye = βYL
Substituting
β
Y0L for Y0ein equation (2.15) for PV, we obtain0 [1 ( 1)] 0L 0
PV = +
β
T− Y +a (2.16)As an operational expression for PV in equation (2.16), both YL and a can be measured statistically.
Substituting o in this equation into equation (2.9) from the consumption yield, the expression becomes
0 [1 ( 1)] 0L 0
C =K +
β
T− Y +Ka (2.17)The above equation is a statistically measurable form of the Ando-Modigliani consumption function.
The co-efficient of YL and a in equation (2.11) were estimated statistically by Ando-Modigliani using annual U.S.A. data.
A typical result of their procedure is explained as follows:
0 0.7 tL 0.06 t
C = Y + a (2.18)
This says that an increase of $1 billion in real labor income will raise the real consumption by $0.7 billion. The marginal propensity to consume (MPC) out of labor income is 0.7. Similarly, the MPC out of assets is 0.06.
The value of β from equation (2.17) that is implicit in the estimate of the YL co-efficient in equation (2.18) 0.7 = k [1+β (T-1)]
= 0.06 (1+44β)
Advanced Macroeconomics
59
The Consumption Function So that β is about 0.25 percent. This suggests that when the current labor income goes up by $100 in the aggregation, estimates of average expected labor income rise by $1.25.
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6ORSH .GW
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Figure 2.4 Consumption and labor income
The Ando-Modigliani consumption function of equation (2.11) is shown in the figure. It shows consumption against labor income. The intercept of the consumption income function is set by the level of asset a. Figure 2.4 shows a constraint consumption income ratio trend as the economy grows.
Thus, constancy of the trends c/y ratio can be derived from the Ando-Modigliani function as follows.
We can divide all the terms in equation (2.18) by total real income to be obtained as 0.7 L 0.06
t t t
t t t
C Y a
Y = Y + Y (2.19)
The Ando-Modigliani model of consumption behavior explains all three of the observed consumption phenomena, namely:
1. Cross sectional budget studies show that s/y increases as income (y) rises, so that in cross sections of the population, MPC < APC.
2. Business cycle or short run data shows that the c/y ratio is smaller than average during boom periods; greater than average during boom periods; greater than average during stumps so that in the short run, income fluctuates and MPS < APS.
3. Long run data trend as MPC = APC.
It explains the MPC < APC, the result of the cross sectional budget studies by the life cycle hypothesis. It provides an explanation for the cyclical behavior of consumption with the consumption income ratio inversely related to income along a short term function.
Advanced Macroeconomics
60
The Consumption Function