Sunday, November 27, 2016

Emergences of Rational Expectation Thoughts

Reading list before this blog post!
Read the history (here) | Read why adaptive expectation fall apart? (here) | Example of adaptive expectation solution (here) | Repeated Substitutions Method to Solve Adaptive Expectation (here)


Lucas induced Muth (1961) ideas on forward looking expectation into his "Expectation and the Neutrality of Money (1972, 1981) article. He, firstly, faintly praised Keynesian macroeconomics for having engaged in econometric model and empirical testing, and then, damned the Keynesian econometric models as $\theta $ being fixed parameters (estimates being/were independent with any changes in institutional regime) and $F$ being specified in advanced without structural linkage of agents rational behavior, preferences and technological constraints. Lucas' revealed that consumer’s and businesses’ expectations change as government alters its policy thus the effectiveness of government policies nullify. Such strong rejection of the inflation-unemployment tradeoff statement based on the rational expectation theory has profound importance in the economic models. 

Compare to adaptive expectation, the rational expectation mimics reality differently. Some intuitive examples are, we take debts in current time in the expectation of higher gain later in future, firms make their investment decision based on their future expectation but not with their past experiences. The rational agents– who know the structure of the model/economy– can best use information available till today (rather than using past information), and develops the expectations for today based on what may happen in future (mostly with no systematic mistake) and they are vulnerable to unpredictable exogenous shocks only. In the figure, it can be seen that the expectation of inflation (captured by the Livingston survey and survey of professional forecasters (SPF)) are divergent and contaminated with systematic mistakes during high volatile inflation periods (which shows the vulnerability during shocks) but during low volatile periods the expectation of inflation are fairly accurately captured (Mendes, 2014). 

Figure-2: Vulnerability at High Inflation and Correct Expectation in low volatile Period (Justification toward Rational Expectation)
 Source: Mendes (2014).


Tuesday, November 22, 2016

A Quick Thoughts on "Earlier" Theories of Inflation

The classical belief of flexible wages and prices prophesized that the economy, aftershock, rebounds to natural rate of output ${{Y}_{n}}$ and defined aggregated supply as vertical curve. After the great depression of 1929, the prophecy hiccupped, and, Keynes fixed prices doctoring to boost aggregated demand without prices hike re-defined aggregated supply as horizontal curve. 

Then 1958 discovery of Phillip Curve –tradeoff between wage-inflation $\left( {\Delta w}/{w}\; \right)$ and unemployment $\left( U-{{U}_{n}} \right)$ given as: ${\Delta w}/{w}\;=-h(U-{{U}_{n}})$, where $h$ is tradeoff coefficient– delineated an upward sloping supply curve. And, ${{\pi }_{t}}=c+\beta {{g}_{t}}+{{e}_{t}}$ gives their econometric specification, where ${{\pi }_{t}}$ is inflation, ${{g}_{t}}$ is some measure of economic activities and $c$ is constant. 

Post 1960, such tradeoff was no longer apparent. In 1967, Friedman augmented expectation in the Phillip Curve, pointing that, rather than nominal wages, firms and workers are concern on the real wage hence they try to adjust wage with expected prices $\left( {\Delta w}/{w-{{\pi }^{e}}}\; \right)$. Such expectations-augmented Phillip curve is given as: ${\Delta w}/{w-{{\pi }^{e}}=-h(U-{{U}_{n}})}\;$ or ${\Delta w}/{w=-h(U-{{U}_{n}})}\;-{{\pi }^{e}}$ and re-vamped estimation equation is given as: ${{\pi }_{t}}=\pi _{t+1}^{e}+\varphi {{g}_{t}}+{{e}_{t}}$. Friedman's clairvoyant plausibly explained the 1970 stagflation (high inflation with high unemployment). 

As the stagflation seems to rule out Phillip curve, two different thought emerged: rational expectation school and New-Keynesian school each having different philosophies toward the issue of inflation. Rational expectation school of thought led by Lucas and Sargent in 1970's excelled rational expectation modeling based on micro-foundation but rejected Keynesian thoughts. Meanwhile, another school of thought invented by John Taylor and Stanley Fischer known as New Keynesians (defendant of Keynesian thoughts), argued that the prices are sticky thus markets don't clear instantaneously thus an increase in money supply stock can provoke short run output boost. Their approach was also micro-foundational but borrowed rational expectation thoughts when needed.


Tuesday, November 15, 2016

The Detailed Solution of Bellman Equation of Shapiro-Stiglitz Model on Efficiency Wage (Part-III: The Value of Unemployed)

See my previous posts here, herehere, here and here and after this see posts here here and here.

The value of being unemployed is given as:
 ${{V}_{U}}(\Delta t)={{e}^{-\rho \Delta t}}\left[ {{e}^{-a\Delta t}}{{V}_{U}}(\Delta t)+(1-{{e}^{-a\Delta t}}){{V}_{E}}(\Delta t) \right]$
${{V}_{U}}(\Delta t)={{e}^{-\rho \Delta t}}{{e}^{-a\Delta t}}{{V}_{U}}(\Delta t)+{{e}^{-\rho \Delta t}}(1-{{e}^{-a\Delta t}}){{V}_{E}}(\Delta t)$
${{V}_{U}}(\Delta t)={{e}^{-(a+\rho )\Delta t}}{{V}_{U}}(\Delta t)+({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}}){{V}_{E}}(\Delta t)$

Arranging for
${{V}_{U}}(\Delta t)={{e}^{-(a+\rho )\Delta t}}{{V}_{U}}(\Delta t)+({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}}){{V}_{E}}(\Delta t)$
 ${{V}_{U}}(\Delta t)-\left( {{e}^{-(a+\rho )\Delta t}} \right){{V}_{U}}(\Delta t)=({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}}){{V}_{E}}(\Delta t)$ $\left( 1-{{e}^{-(a+\rho )\Delta t}} \right){{V}_{U}}(\Delta t)=({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}}){{V}_{E}}(\Delta t)$
${{V}_{U}}(\Delta t)=\frac{({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}})}{\left( 1-{{e}^{-(a+\rho )\Delta t}} \right)}{{V}_{E}}(\Delta t)$

Taking limit's on both sides
$\underset{\Delta t\to 0}{\mathop{\lim }}\,{{V}_{U}}(\Delta t)=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}})}{\left( 1-{{e}^{-(a+\rho )\Delta t}} \right)}{{V}_{E}}(\Delta t)$
 ${{V}_{U}}={{V}_{E}}\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{({{e}^{-\rho \Delta t}}-{{e}^{-(a+\rho )\Delta t}})}{\left( 1-{{e}^{-(a+\rho )\Delta t}} \right)}$

Using L'hopital's rule which suggets, if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.
${{V}_{U}}={{V}_{E}}\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{-\rho {{e}^{-\rho \Delta t}}+(a+\rho ){{e}^{-(a+\rho )\Delta t}}}{(a+\rho ){{e}^{-(a+\rho )\Delta t}}}$
${{V}_{U}}={{V}_{E}}\frac{-\rho +a+\rho }{a+\rho }$
${{V}_{U}}\left( a+\rho \right)=a{{V}_{E}}$
$\rho {{V}_{U}}=a\left( {{V}_{E}}-{{V}_{U}} \right)$

This gives the solution for value of being unemployed.

We will discuss this diagram next.

Tuesday, November 8, 2016

The Detailed Solution of Bellman Equation of Shapiro-Stiglitz Model on Efficiency Wage (Part-II: The Value of Shirking)

See (here,  herehere and here) for previous posts:

${{V}_{S}}(\Delta t)=\int\limits_{t=0}^{\Delta t}{{{e}^{-(b+q)t}}{{e}^{-\rho t}}wdt}+{{e}^{-\rho \Delta t}}\left[ {{e}^{-(b+q)\Delta t}}{{V}_{S}}(\Delta t)+(1-{{e}^{-(b+q)\Delta t}}){{V}_{U}}(\Delta t) \right]$
We can arrange, ${{e}^{-bt}}{{e}^{-\rho t}}={{e}^{-(b+\rho )t}}$, then above equation can be written as: 
${{V}_{S}}(\Delta t)=\int\limits_{t=0}^{\Delta t}{{{e}^{-(b+q+\rho )t}}wdt}+{{e}^{-(b+q+\rho )\Delta t}}{{V}_{S}}(\Delta t)+({{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}){{V}_{U}}(\Delta t)$ 
Taking the integral part and solving it \[\int\limits_{t=0}^{\Delta t}{{{e}^{-(b+q+\rho )t}}wdt}\] \[\frac{-1}{b+q+\rho }w\left. \left( {{e}^{-(b+q+\rho )t}} \right) \right|_{0}^{\Delta t}\] \[\frac{-1}{b+q+\rho }w\left( {{e}^{-(b+q+\rho )\Delta t}}-{{e}^{0}} \right)\] \[\frac{-1}{b+q+\rho }w\left( {{e}^{-(b+q+\rho )\Delta t}}-1 \right)\] \[\frac{w}{b+q+\rho }\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)\] \[{{V}_{S}}(\Delta t)=\frac{w}{b+q+\rho }\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)+{{e}^{-(b+q+\rho )\Delta t}}{{V}_{S}}(\Delta t)+({{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}){{V}_{U}}(\Delta t)\] \[{{V}_{S}}(\Delta t)-{{e}^{-(b+q+\rho )\Delta t}}{{V}_{S}}(\Delta t)=\frac{w}{b+q+\rho }\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)+({{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}){{V}_{U}}(\Delta t)\] \[\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right){{V}_{S}}(\Delta t)=\frac{w}{b+q+\rho }\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)+({{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}){{V}_{U}}(\Delta t)\] 
Now,
\[{{V}_{S}}(\Delta t)=\frac{\frac{w}{b+q+\rho }\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)}{\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)}+\frac{({{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}){{V}_{U}}(\Delta t)}{\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)}\] \[{{V}_{S}}(\Delta t)=\frac{w}{b+q+\rho }+\frac{({{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}})}{\left( 1-{{e}^{-(b+q+\rho )\Delta t}} \right)}{{V}_{U}}(\Delta t)\] Taking limits on both sides, \[\underset{\Delta t\to 0}{\mathop{\lim }}\,{{V}_{S}}(\Delta t)=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{w}{b+q+\rho }+\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}}{1-{{e}^{-(b+q+\rho )\Delta t}}}{{V}_{U}}(\Delta t)\] \[{{V}_{S}}=\frac{w}{b+q+\rho }+{{V}_{U}}\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}}{1-{{e}^{-(b+q+\rho )\Delta t}}}\] Using L'hopital's rule which suggets, if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. \[\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+q+\rho )\Delta t}}}{1-{{e}^{-(b+q+\rho )\Delta t}}}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{m(\Delta t)}{n(\Delta t)}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{m}'(\Delta t)}{{n}'(\Delta t)}\] \[=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{-\rho {{e}^{-\rho \Delta t}}-(b+q+\rho )\left( -{{e}^{-(b+q+\rho )\Delta t}} \right)}{(b+q+\rho )\left( -{{e}^{-(b+q+\rho )\Delta t}} \right)}\] \[=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{-\rho {{e}^{-\rho \Delta t}}+(b+q+\rho )\left( {{e}^{-(b+q+\rho )\Delta t}} \right)}{(b+q+\rho )\left( -{{e}^{-(b+q+\rho )\Delta t}} \right)}\] \[=\frac{-\rho +b+q+\rho }{b+q+\rho }\] \[=\frac{b+q}{b+q+\rho }\] Interestingly, now we have very neat equation. Let's do further simplification. \[{{V}_{S}}=\frac{w}{b+q+\rho }+{{V}_{U}}\left( \underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}} \right)\] \[{{V}_{S}}=\frac{w}{b+q+\rho }+{{V}_{U}}\left( \frac{b+q}{b+q+\rho } \right)\] \[{{V}_{S}}\left( b+q+\rho \right)=w+\left( b+q \right){{V}_{U}}\] \[\rho {{V}_{S}}=w-\left( b+q \right)\left( {{V}_{S}}-{{V}_{U}} \right)\]

We will talk about this diagram in later post.

Sunday, November 6, 2016

A Quick Thoughts on "Earlier" Theories of Consumptions

Although, saving accumulates wealth for the nation, but if, investments don’t absorb savings, or if demand deficiency exists– depression is inevitable. By nature, investments are volatile whereas consumption seems predictable aggregate-demand component. Therefore, Keynes's consumption function seemed to explain 1930s depression and availability of national accounts raw-data reinforced empirical studies later. (Bold letter topics are explained in footnotes)

Keynes Absolute Income Hypothesis (AIH) centers on “psychological law”–ceterius paribus, rise/change in consumption $\left( \Delta C \right)$ is smaller than rise/change in income $\left( \Delta Y \right)$i.e,$0<\frac{\Delta C}{\Delta Y}<1$. This is known as multipliers. AIH was plausible for short-run and micro-level but, it short-lived because, it didn’t reconcile for long-run (Kuznet Paradox) and also failed to explain Engel Curve. To answer, Dorothy Brady and Rose Friedman hypothesized Relative Income Hypothesis (RIH) and Samuelson (1943) proposed a “ratchet” model . Later, Duesenberry (1949) empirical work verified both RIH and ratchet model. 

However, Modigliani and Friedman criticized these myopic consumption function and introduced forward looking assumption in consumption function. Modigliani proposed a Lifecycle Consumption Hypothesis (LCH) and stated that consumers save to smooth consumption over a lifetime or for retirement. Further, Friedman proved his Permanent Income Hypothesis (PIH) that consumption only depends upon the shock in the permanent part of income but not in transitory income. Proof (here). Further other post PIH theories, proved that consumption are unpredictable and follows random walk known as random walk hypothesis (RWH). And later the literature evolved/revolved to test whether consumption validates RWH or not. See (here) for RWH. 

Source:http://www.pfsweb.com/blog/infosheet-the-evolution-of-commerce-through-consumer-behavior/

Consumption Function: A consumption function is list of the variables that influence consumption together with the direction and magnitude of their effects and whatever relationship are summarized in a consumption function can be summarize equally well in saving function.

Availability of National Accounts Raw-Data: In 1931, at Mitchell’s behest, Kuznets took charge of the NBER’s work on U.S. national income accounts. In 1934, an assessment of the national income of the United States for the period 1929–1932 was given; further, it was extended to 1919–1938, and then, until 1869. Although Kuznets was not the first economist to try this, his work was so comprehensive and meticulous that it set the standard in the field (https://en.wikipedia.org/wiki/Simon_Kuznets)

Multiplier: Say, ∆C/∆Y=b which is known as marginal propensity to consume (mpc) which is the foundation of Keynesian thought that – to close any ∆Y output gap for any economy, ∆Y(1-b) policy intervention is required, which is known as multiplier. See (here) for more.

Kuznet ParadoxKuznet and later by Goldsmit during long run (1869-1929 period), the (mpc) remained relatively constant while income had quadrupled. This is known as Kuznet Paradox.

Engel Curve: When households are classified into income brackets and the average consumption for a bracket is plotted against income, the scatterplot traces out a upward bending path known as Engle curve.

RIH : RIH states that consumption may depend on the relative income– “upon the percentile position in the total income distribution.

Ratchet Model: Samuelson tried to reconcile the short-run, or cyclical fluctuation of consumption to its long-run consumption. He stated that in long-run consumption proportionately grows with income; but during cyclical interruptions of long-run growth, consumers defend living standards already attained, and consequently consumption follows a flatter (lower rape) Keynesian path.

Saturday, November 5, 2016

The Detailed Solution of Bellman Equation of Shapiro-Stiglitz Model on Efficiency Wage (Part-I: The Value of Being Employed)

The value of being employed:


See (here, here, here) for previous posts and  here  and here for more on value of shirking and unemployed.

${{V}_{E}}(\Delta t)=\int\limits_{t=0}^{\Delta t}{{{e}^{-bt}}{{e}^{-\rho t}}(w-\bar{e})dt}+{{e}^{-\rho \Delta t}}\left[ {{e}^{-b\Delta t}}{{V}_{E}}(\Delta t)+(1-{{e}^{-b\Delta t}}){{V}_{U}}(\Delta t) \right]$ 
We can arrange, ${{e}^{-bt}}{{e}^{-\rho t}}={{e}^{-(b+\rho )t}}$, then above equation can be written as: ${{V}_{E}}(\Delta t)=\int\limits_{t=0}^{\Delta t}{{{e}^{-(b+\rho )t}}(w-\bar{e})dt}+{{e}^{-\rho \Delta t}}\left[ {{e}^{-b\Delta t}}{{V}_{E}}(\Delta t)+(1-{{e}^{-b\Delta t}}){{V}_{U}}(\Delta t) \right]$ 
This equation has two parts $\int\limits_{t=0}^{\Delta t}{{{e}^{-(b+\rho )t}}(w-\bar{e})dt}$and ${{e}^{-\rho \Delta t}}\left[ {{e}^{-b\Delta t}}{{V}_{E}}(\Delta t)+(1-{{e}^{-b\Delta t}}){{V}_{U}}(\Delta t) \right]$. Say them part A and B. Let's solve integral for part A, we get:
 \[\begin{align} & \int\limits_{t=0}^{\Delta t}{{{e}^{-(b+\rho )t}}(w-\bar{e})dt} \\ & =\frac{-1}{b+\rho }(w-\bar{e})\left. \left( {{e}^{-(b+\rho )t}} \right) \right|_{0}^{\Delta t} \\ & =\frac{-1}{b+\rho }(w-\bar{e})\left( {{e}^{-(b+\rho )\Delta t}}-{{e}^{0}} \right) \\ & =\frac{-1}{b+\rho }(w-\bar{e})\left( {{e}^{-(b+\rho )\Delta t}}-1 \right) \\ & =\frac{w-\bar{e}}{b+\rho }\left( 1-{{e}^{-(b+\rho )\Delta t}} \right) \\ \end{align}\] 
And, if we simplify part B, we get: \[\begin{align} & {{e}^{-\rho \Delta t}}\left[ {{e}^{-b\Delta t}}{{V}_{E}}(\Delta t)+(1-{{e}^{-b\Delta t}}){{V}_{U}}(\Delta t) \right] \\ & ={{e}^{-\rho \Delta t}}{{e}^{-b\Delta t}}{{V}_{E}}(\Delta t)+{{e}^{-\rho \Delta t}}(1-{{e}^{-b\Delta t}}){{V}_{U}}(\Delta t) \\ & ={{e}^{-(b+\rho )\Delta t}}{{V}_{E}}(\Delta t)+\left( {{e}^{-\rho \Delta t}}-{{e}^{-\rho \Delta t}}{{e}^{-b\Delta t}} \right){{V}_{U}}(\Delta t) \\ \end{align}\] \[={{e}^{-(b+\rho )\Delta t}}{{V}_{E}}(\Delta t)+\left( {{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}} \right){{V}_{U}}(\Delta t)\] 
And, then we can write: \[\begin{align} & {{V}_{E}}(\Delta t)=\frac{w-\bar{e}}{b+\rho }\left( 1-{{e}^{-(b+\rho )\Delta t}} \right)+{{e}^{-(b+\rho )\Delta t}}{{V}_{E}}(\Delta t)+\left( {{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}} \right){{V}_{U}}(\Delta t) \\ & {{V}_{E}}(\Delta t)-{{e}^{-(b+\rho )\Delta t}}{{V}_{E}}(\Delta t)=\frac{w-\bar{e}}{b+\rho }\left( 1-{{e}^{-(b+\rho )\Delta t}} \right)+\left( {{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}} \right){{V}_{U}}(\Delta t) \\ \end{align}\] \[\left( 1-{{e}^{-(b+\rho )\Delta t}} \right){{V}_{E}}(\Delta t)=\frac{w-\bar{e}}{b+\rho }\left( 1-{{e}^{-(b+\rho )\Delta t}} \right)+\left( {{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}} \right){{V}_{U}}(\Delta t)\] \[{{V}_{E}}(\Delta t)=\frac{\frac{w-\bar{e}}{b+\rho }\left( 1-{{e}^{-(b+\rho )\Delta t}} \right)}{\left( 1-{{e}^{-(b+\rho )\Delta t}} \right)}+\frac{\left( {{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}} \right)}{\left( 1-{{e}^{-(b+\rho )\Delta t}} \right)}{{V}_{U}}(\Delta t)\] \[{{V}_{E}}(\Delta t)=\frac{w-\bar{e}}{b+\rho }+\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}}{{V}_{U}}(\Delta t)\] 
taking limits on both sides \[\underset{\Delta t\to 0}{\mathop{\lim }}\,{{V}_{E}}(\Delta t)=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{w-\bar{e}}{b+\rho }+\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}}{{V}_{U}}(\Delta t)\] \[{{V}_{E}}=\frac{w-\bar{e}}{b+\rho }+{{V}_{U}}\left( \underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}} \right)\] 
Using L'hopital's rule which suggets, if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. \[\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{m(\Delta t)}{n(\Delta t)}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{m}'(\Delta t)}{{n}'(\Delta t)}\] \[=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{-\rho {{e}^{-\rho \Delta t}}-(b+\rho )\left( -{{e}^{-(b+\rho )\Delta t}} \right)}{(b+\rho )\left( -{{e}^{-(b+\rho )\Delta t}} \right)}\] \[=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{-\rho {{e}^{-\rho \Delta t}}+(b+\rho )\left( {{e}^{-(b+\rho )\Delta t}} \right)}{(b+\rho )\left( {{e}^{-(b+\rho )\Delta t}} \right)}\] \[=\frac{-\rho +b+\rho }{b+\rho }\] \[=\frac{b}{b+\rho }\] 
Interestingly, now we have very neat equation. Let's do further simplification. \[{{V}_{E}}=\frac{w-\bar{e}}{b+\rho }+{{V}_{U}}\left( \underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}} \right)\] \[{{V}_{E}}=\frac{w-\bar{e}}{b+\rho }+{{V}_{U}}\left( \frac{b}{b+\rho } \right)\] \[{{V}_{E}}\left( b+\rho \right)=\left( w-\bar{e} \right)+b{{V}_{U}}\] \[\rho {{V}_{E}}=\left( w-\bar{e} \right)-b\left( {{V}_{E}}-{{V}_{U}} \right)\]

Wednesday, November 2, 2016

A Quick Thoughts on Theories of Unemployment

The persistence average unemployment and the cyclical unemployment are two major discussions on unemployment. For former, two extreme theories exist, unemployment is: first, as illusory–friction in the process of matching up workers to job, and second as non-Walrasian–that it represents waste of resources. For later, also two extreme theories exist, cyclical unemployment, first, as Walrasian feature of labor market–elastic labor supply (only weak empirical evidences were found) and moderate pro-cyclical real wage, and second again as non-Walrasian. Hence, unemployment theories mostly focus on non-Walrasian possibilities. 

In a Walrasian labor market, unemployed identical workers would bid lower wage until supply clears demand, but, in-reality, this mechanism fails to operate. Because, for such lower bid, firms have at-least four possible responses: first, firm may don’t want to reduce wages (efficiency-wage-theories here) or possibly would pay high wages to rise productivity; second, even thou firm wishes to cut wages, but explicit and implicit contracts may prevent them (contracting models here); third, firm can accept the offer due to heterogeneity among workers (search and matching models here); and lastly, firm can accept the offer as the possibility of Walrasian labor market.

Picture Source:http://www.insideronline.org/2016/10/the-us-labor-market-questions-and-challenges-for-public-policy-2/

Reference
Advanced Macroeconomics (The Mcgraw-Hill Series in Economics) 4th Edition by David Romer