## Thursday, October 13, 2016

### Proof of Permanent Income Hypothesis by Milton Friedman

Here is the proof of the Permanent Income Hypothesis that any change in consumption is due to the difference in the permanent component of income only!

Let's assume consumption $(c)$ is the liner function of income $(y)$ given as: $c=\alpha +\beta y$. Then, with usual OLS estimation: $b=\frac { \sum { \left( c-\overline { c } \right) \left( y-\overline { y } \right) } }{ \sum { { \left( y-\overline { y } \right) }^{ 2 } } }$ and $a=\overline { c } -b\overline { y }$, where: $b$ and $a$ are the estimates of $\beta$ (change in consumption w.r.t income) and $\alpha$ (autonomous consumption); and $\overline { c }$ and  $\overline { y }$ are mean consumption and income.

Let's purpose the income $y$ comprises two components: a permanent (anticipated and planned) component ${ y }_{ p }$ and a transitory (windfall gain/unexpected) component ${ y }_{ t }$. With similar logic, the consumption $c$ comprises two components: a permanent (anticipated and planned) component ${ c }_{ p }$ and a transitory (windfall gain/unexpected) component ${ c }_{ t }$. Formally, $y={ y }_{ p }+{ y }_{ t }$ and $c={ c }_{ p }+{ c }_{ t }$.

The ratio of permanent consumption to permanent income $\frac { { c }_{ p } }{ { y }_{ p } }$ depends upon: the rate of interest $i$ that consumer can buy and lend; the relative importance of property and non-property income, symbolized by the ratio of non-human wealth to income $(w)$; and the consumer's tastes and preferences $(u)$. Formally, it can be expressed as: ${ c }_{ p }=k(i,w,u){ y }_{ p }$.

It seems plausible to define that the transitory and permanent components are uncorrelated with one another and with the corresponding transitory component, or, ${ \rho }_{ { y }_{ t }{ y }_{ p } }={ \rho }_{ { c }_{ t }{ c }_{ p } }={ \rho }_{ { y }_{ t }c_{ t } }=0$.

Then we can re-write:

$\sum { \left( c-\overline { c } \right) \left( y-\overline { y } \right) } =\sum { \left( { c }_{ p }+{ c }_{ t }+{ \overline { c } }_{ p }+{ \overline { c } }_{ t } \right) \left( { y }_{ p }+{ y }_{ t }+{ \overline { y } }_{ p }+{ \overline { y } }_{ t } \right) } \\ \\ \qquad \qquad \qquad \quad \quad \quad \quad =\sum { ({ c }_{ p }-{ \overline { c } }_{ p })({ y }_{ p }-{ \overline { y } }_{ p }) } +\sum { ({ c }_{ p }-{ \overline { c } }_{ p })({ y }_{ t }-{ \overline { y } }_{ t }) } +\sum { ({ c }_{ t }-{ \overline { c } }_{ t })({ y }_{ p }-{ \overline { y } }_{ p }) } +\sum { ({ c }_{ t }-{ \overline { c } }_{ t })({ y }_{ t }-{ \overline { y } }_{ t }) } \\ \\ \because { c }_{ p }=k{ y }_{ p }\quad and\quad { y }_{ p }=\frac { 1 }{ k } { c }_{ p }\\ \\ \qquad \qquad \qquad \qquad \quad =\sum { ({ ky }_{ p }-{ k\overline { y } }_{ p })({ y }_{ p }-{ \overline { y } }_{ p }) } +\sum { ({ ky }_{ p }-{ k\overline { y } }_{ p })({ y }_{ t }-{ \overline { y } }_{ t }) } +\sum { ({ c }_{ t }-{ \overline { c } }_{ t })\left( \frac { 1 }{ k } { c }_{ p }-\frac { 1 }{ k } { \overline { c } }_{ p } \right) } +\sum { ({ c }_{ t }-{ \overline { c } }_{ t })({ y }_{ t }-{ \overline { y } }_{ t }) } \\ \\ taking\quad commons\\ \\ \qquad \qquad \qquad \qquad \quad =k\sum { { ({ y }_{ p }-{ \overline { y } }_{ p }) }^{ 2 } } +k\sum { ({ y }_{ p }-{ \overline { y } }_{ p })({ y }_{ t }-{ \overline { y } }_{ t }) } +\frac { 1 }{ k } \sum { ({ c }_{ t }-{ \overline { c } }_{ t })\left( { c }_{ p }-{ \overline { c } }_{ p } \right) } +\sum { ({ c }_{ t }-{ \overline { c } }_{ t })({ y }_{ t }-{ \overline { y } }_{ t }) } \\ \\ \\ \because { \rho }_{ { y }_{ t }{ y }_{ p } }={ \rho }_{ { c }_{ t }{ c }_{ p } }={ \rho }_{ { y }_{ t }c_{ t } }=0\quad i.e\quad \sum { ({ y }_{ p }-{ \overline { y } }_{ p })({ y }_{ t }-{ \overline { y } }_{ t }) } =\frac { 1 }{ k } \sum { ({ c }_{ t }-{ \overline { c } }_{ t })\left( { c }_{ p }-{ \overline { c } }_{ p } \right) } =\sum { ({ c }_{ t }-{ \overline { c } }_{ t })({ y }_{ t }-{ \overline { y } }_{ t }) } =0\\ \\ \\ \\ \\ \therefore \sum { \left( c-\overline { c } \right) \left( y-\overline { y } \right) } \quad =\quad k\sum { { ({ y }_{ p }-{ \overline { y } }_{ p }) }^{ 2 } }$

A. Then $\beta$ can be re-expressed as:

$b=k\frac { \sum { { \left( { y }_{ p }-{ \overline { y } }_{ p } \right) }^{ 2 } } }{ \sum { { \left( y-\overline { y } \right) }^{ 2 } } } =k{ P }_{ y }$

The $b$ estimates the change in consumption w.r.t income. The ${ P }_{ y }$ is the fraction of the total variance of income contributed by the permanent component of income. The $b$ doesn't comprise of any transitory income or consumption in above equation, then, we can define, any change in consumption  is due to the difference in the permanent component. This is the important concept of the Permanent Income Hypothesis (PIH)

Source: https://en.wikipedia.org/wiki/Milton_Friedman

B. the Further we can re-write $\alpha$ as:

$a=\overline { c } -b\overline { y } \\ a=\left( { \overline { c } }_{ p }+{ \overline { c } }_{ t } \right) -b\left( { \overline { y } }_{ p }+{ \overline { y } }_{ t } \right) \\ a={ k\overline { y } }_{ p }+{ \overline { c } }_{ t }-k{ P }_{ y }\left( { \overline { y } }_{ p }+{ \overline { y } }_{ t } \right) \quad \because { \overline { c } }_{ p }={ k\overline { y } }_{ p }\quad and\quad b=k{ P }_{ y }\\ a={ \overline { c } }_{ t }-k{ P }_{ y }{ \overline { y } }_{ t }+k(1-{ P }_{ y }){ \overline { y } }_{ p }$.

C. The elasticity of consumption w.r.t income at the point $(c,y)$ is:

${ \eta }_{ cy }=\frac { dc }{ dy } \frac { y }{ c } =b\frac { y }{ c } =k{ P }_{ y }\frac { y }{ c }$

D. A especial case:
And say the mean of transitory component of both the income and consumption are zero such that
$\frac { \overline { y } }{ \overline { c } } =\frac { 1 }{ k }$ and their elasticity will be ${ \eta }_{ cy }={ P }_{ y }$.

References:

Friedman, M. (1957). The Permanent Income Hypothesis. In A Theory of the Consumption Function (pp. 20–37). Princeton University Press. Retrieved from http://www.nber.org/chapters/c4405

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