Monday, October 31, 2016

The Meaning of Bellman Equation of Shapiro-Stiglitz Model on Efficiency Wage

You need to read my previous posts (here), (here) and (here) to understand this write up.

The bellman equation of value of being employed is give as:

${ V }_{ E }(∆t)=\int _{ t=0 }^{ ∆t }{ { e }^{ -bt }{ e }^{ -\rho t }\left( w-\overline { e } \right) dt } +{ e }^{ -\rho ∆t }\left[ { e }^{ -b∆t }{ V }_{ E }(∆t)+\left( 1-{ e }^{ -b∆t } \right) { V }_{ U }(∆t) \right] \\ \underbrace { { V }_{ E }(∆t) }_{ The\quad value\quad of\\ being\quad employed } =\underbrace { \underbrace { \int _{ t=0 }^{ ∆t }{ } }_{ during\quad \\ interval\\ (0,∆t) } \underbrace { { e }^{ -bt } }_{ the\quad prob\quad \\ that\quad the\quad \\ worker\quad is\quad \\ still\quad \\ employed\quad \\ at\quad time\quad t. } \underbrace { { e }^{ -\rho t }\left( w-\overline { e } \right) }_{ the\quad discounted\\ flow\quad utility } dt }_{ The\quad utility\quad during\quad interval\quad (0,∆t)\quad \\ discounted\quad to\quad time\quad 0.\quad } +\underbrace { \underbrace { { e }^{ -b∆t } }_{ discounting\\ beyound\quad \\ ∆t\quad to\quad t=0 } \left[ \underbrace { { e }^{ -b∆t } }_{ the\quad prob\quad \\ that\quad a\quad \\ worker\\ is\quad still\\ employed\quad \\ after\quad ∆t } \underbrace { { V }_{ E }(∆t) }_{ the\quad value\quad \\ of\quad being\\ employed } +\underbrace { \left( 1-{ e }^{ -b∆t } \right) }_{ the\quad prob\quad \\ that\quad a\quad \\ worker\\ is\quad not\\ employed\quad \\ after\quad ∆t } \underbrace { { V }_{ U }(∆t) }_{ the\quad value\quad \\ of\quad not\\ being\quad \\ employed } \right] }_{ The\quad utility\quad after\quad ∆t\quad \\ discounted\quad to\quad time\quad 0.\quad }$

The solution will emerge as: $\rho { V }_{ E }=\left( w-\overline { e } \right) -b\left( { V }_{ E }-{ V }_{ U } \right)$. For detail see (here).

${ n }^{ th }$ Period Backward Iterations and Repeated Substitutions Method to Solve Adaptive Expectation

Read previous related post first (here) then (here)

Let's model following expression.

“Today’s price expectation ${ P }_{ t }^{ e }$ is yesterday’s price expectation ${ P }_{ t-1}^{ e }$and some weight ${ \varphi }$ on the differences of yesterday’s actual to expected price ${ P }_{ t-1}- { P }_{ t }^{ e }$ or the mistake”.

Formally, ${ P }_{ t }^{ e }= {P }_{ t-1}^{ e }+ { \varphi }({ P }_{ t-1}- { P }_{ t -1}^{ e })$, where $0<{ \varphi }<1$, and for ${ \varphi }=0$ represents myopic expectations.

Let's re-arrange above equation:
${ P }_{ t }^{ e }={ P }_{ t-1 }^{ e }+{ \varphi }{ P }_{ t-1 }-\varphi { P }_{ t -1}^{ e }$

Let's take the common ${ P }_{ t-1 }^{ e }$, we will get
${ P }_{ t }^{ e }={ \varphi }{ P }_{ t-1 }+(1-\varphi ){ P }_{ t-1 }^{ e }$     say $eqn 1$

Now, lets perform first backward iteration, we can write,
${ P }_{ t-1 }^{ e }={ \varphi }{ P }_{ t-2 }+(1-\varphi ){ P }_{ t-2 }^{ e }$

Now, we can substitute value of ${ P }_{ t-1 }^{ e }$ in $eqn 1$, we get,
${ P }_{ t }^{ e }={ \varphi }{ P }_{ t-1 }+(1-\varphi )\underbrace { \left[ { \varphi }{ P }_{ t-2 }-(1-\varphi ){ { P }_{ t-2 }^{ e } } \right] }_{ { P }_{ t-1 }^{ e } }$

which is equivalent to,
${ P }_{ t }^{ e }={ \varphi }{ P }_{ t-1 }+{ \varphi }(1-\varphi ){ P }_{ t-2 }+{ (1-\varphi ) }^{ 2 }{ { P }_{ t-2 }^{ e } }$     say $eqn 2$

Now, lets perform second backward iteration
${ P }_{ t-2 }^{ e }={ \varphi }{ P }_{ t-3 }+(1-\varphi ){ P }_{ t-3 }^{ e }$

let's substitute ${ P }_{ t-2 }^{ e }$ in $eqn 2$ we get,
${ P }_{ t }^{ e }={ \varphi }{ P }_{ t-1 }+{ \varphi }(1-\varphi ){ P }_{ t-2 }+{ (1-\varphi ) }^{ 2 }{ \underbrace { \left[ { \varphi }{ P }_{ t-3 }+(1-\varphi ){ P }_{ t-3 }^{ e } \right] }_{ { P }_{ t-2 }^{ e } } }$     $eqn 3$

which is equivalent to,
${ P }_{ t }^{ e }={ \varphi }{ P }_{ t-1 }+{ \varphi }(1-\varphi ){ P }_{ t-2 }+{ \varphi }{ (1-\varphi ) }^{ 2 }{ P }_{ t-3 }+{ (1-\varphi ) }^{ 3 }{ P }_{ t-3 }^{ e }$

In general form we can write this second period backward iteration and repeated substitution as:
${ P }_{ t }^{ e }=\underbrace { { \varphi }{ P }_{ t-1 }+{ \varphi }(1-\varphi ){ P }_{ t-2 }+{ \varphi }{ (1-\varphi ) }^{ 2 }{ P }_{ t-3 } }_{ \sum _{ i=0 }^{ 3-1 }{ { \varphi }{ (1-\varphi ) }^{ i }{ P }_{ t-1-i } } } \underbrace { + }_{ + } \underbrace { { (1-\varphi ) }^{ 3 }{ P }_{ t-3 }^{ e } }_{ { (1-\varphi ) }^{ 3 }{ P }_{ t-3 } }$

Which is:
${ P }_{ t }^{ e }=\sum _{ i=0 }^{ 3-1 }{ { \varphi }{ (1-\varphi ) }^{ i }{ P }_{ t-1-i } } +{ (1-\varphi ) }^{ 3 }{ P }_{ t-3 }^{ e }$

Then for the ${ n }^{ th }$ Period Backward Iterations and Repeated Substitutions, we can generalize as:
${ P }_{ t }^{ e }=\sum _{ i=0 }^{ n-1 }{ { \varphi }{ (1-\varphi ) }^{ i }{ P }_{ t-1-i } } +{ (1-\varphi ) }^{ n }{ P }_{ t-n }^{ e }$

or,
${ P }_{ t }^{ e }={ (1-\varphi ) }^{ n }{ P }_{ t-1 }^{ e }+\sum _{ i=0 }^{ n-1 }{ \varphi } { (1-\varphi ) }^{i }{ P }_{ t-1-i }$

and, $\lim _{ n\rightarrow \infty }{ { (1-\varphi ) }^{ n } } =0$ because we have assumed $0<{ \varphi }<1$.

Then, the solution is:
${ P }_{ t }^{ e }=\sum _{ i=0 }^{ n-1 }{ \varphi } { (1-\varphi ) }^{ n }{ P }_{ t-1-i }$ .

The possible solution on various values of  ${ \varphi }$
For $-1<{ \varphi }<0$ the $\lim _{ n\rightarrow \infty }{ { (1-\varphi ) }^{ n } }$ would be oscillatory but will converge to the stable solution.
For ${ \varphi }>1$ the $\lim _{ n\rightarrow \infty }{ { (1-\varphi ) }^{ n } }$ would be explosive thus we won't have the stable solution.
For ${ \varphi }< -1$ the $\lim _{ n\rightarrow \infty }{ { (1-\varphi ) }^{ n } }$ would be explosive oscillatory and won't stable solution.

Alternatively, for $|1-\varphi |<0$ ensures the stable solution.

Saturday, October 29, 2016

Meaning of Adaptive Expectation, Example, Solutions and it's ill-fate (Rational Expectation-II)

Cooley and Prescott (1973) proposed alternative adaptive regression method in which they treated the parameter vector ${ \theta }$ as random variables with random walk as: ${ \theta }_{ t+1 }={ \theta }_{ t }+\vartheta _{ t+1 }$,where $ϑ\_ (t)$ is sequence of $iid$ random variables. Their approach resembled the exponential smoothing on the observation with distance past receiving small weights. This approach has better short term forecasting properties compare to fixed $θ$ regression (here). This adaptive expectation plausibly models rational agents' simple heuristic rule that “a rational agent gathers information about the pasts, takes their mistakes into accounts and introduce past trends to their current decision”.

A simple example can be “today’s price expectation ${ P }_{ t }^{ e }$ is yesterday’s price expectation ${ P }_{ t-1}^{ e }$and some weight ${ \varphi }$ on the differences of yesterday’s actual to expected price ${ P }_{ t-1}- { P }_{ t-1 }^{ e }$ or the mistake”.

Formally, ${ P }_{ t }^{ e }= {P }_{ t-1}^{ e }+ { \varphi }({ P }_{ t-1}- { P }_{ t -1}^{ e })$, where $0<{ \varphi }<1$, and for ${ \varphi }=0$ represents myopic expectations. Via ${ n }^{ th }$ period backward iterations and repeated substitutions method (for Proof (here)), the solution emerges as:

${ P }_{ t }^{ e }={ (1-\varphi ) }^{ n }{ P }_{ t-1 }^{ e }+\sum _{ i=0 }^{ n-1 }{ \varphi } { (1-\varphi ) }^{ n }{ P }_{ t-1-i }$

and, $\lim _{ n\rightarrow \infty }{ { (1-\varphi ) }^{ n } } =0$

then, ${ P }_{ t }^{ e }=\sum _{ i=0 }^{ n-1 }{ \varphi } { (1-\varphi ) }^{ i }{ P }_{ t-1-i }$ .

The major importance of such adaptive expectation in economic models is that it can capture the subjective essence of expectation via objective mechanism. In the world of high uncertainty; clinging to some past-remark seems the rational response (yet, it may provoke systematic mistake). However, the systematic mistakes in forecasting 1982 US recession is important remark of the ill-fated fixed parameter and adaptive forecasting/policy modeling (see Figure-1).

Figure-1: Systematic Mistakes in Forecasting 1982 Recession

Source: Mendes, V.

Monday, October 24, 2016

Fall of Fixed $θ$ regression (Rational Expectation-I)

Keynes doctrine of pressing interest rate low and inflation-unemployment tradeoff were quintessential to answer –“How to recover the demand deficient economy and unemployment?” Hicks (1937) graphical and simultaneous equation transcription of Keynes verbal ideas; Klein and Goldberg (1955) econometric model and later Brookings (1960) model with ~ 400 equations made Keynes linchpin and IS-LM framework as workhorse. These data-based models had dual objectives: predicting economic activities; and simulating the effects of policy change (Vroey and Malgrange, 2011). And, maybe, economist's roles revolve around these objectives.

As per Lucas (1976), these types of models tried to describe the economy at time $t+1$ by vector $y_t$ of state variables, a vector $x_t$ of exogenous forcing variables and a vector $ϵ_t$ of independent identically distributed $(iid)$ random shocks and defined the motion of economy as: $y_(t+1)=f(y_t,x_t,ϵ_t)$. The function $f$ is taken to be fixed but not directly known; the task is then to estimate $f$. For practical purposes, $f(y_t,x_t,ϵ_t )≡F(y_t,x_t,ϵ_t )$ where $θ$ is fixed parameters and $F$ being specified in advanced. The pasts of $x_t$ are observed thus cause no difficulties in estimation of $θ$. Hence, with the knowledge of function $F$ and $θ$, policy evaluation can be viewed as a specification of present and future values of some of $x_t$. However beyond 1960s, the theory developed by these fixed $θ$ regression, didn’t seem to be occurring in reality. Then expectations (adaptive and later rational expectations) in economic models came into light.

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