Inflation Persistence in New Keynesian Phillip Curve with Wage Staggering Model

Please follow previous discussion (here) for the history of thoughts on Inflation and how and why New Keynesian Phillip Curve (NKPC) appear in the macroeconomic models and how they are different/similar to Keynesian and/or Rational Expectationist School of thought. Derivation can be found (here). In this this blog we will how to solve the data fitting problem of NKPC via Modified NKPC using Wage Staggering model then see the model of Menu cost (here) to theoretically model "why price in the first place is sticky?"

Another approach to model wage/price stickiness is staggered contracts models of Stanley Fischer (1977) and John Taylor (1979, 1980). and Fuhrer and Moore (1995). However, Fischer and Taylor approach don't exhibit inflation persistence while simple modifications proposed by Fuhrer and Moore incorporates the inflation persistence. 

Real wage is defined as nominal wage adjusted for price and when we consider their logarithmic transformation, we can define log of real wage ${{\varphi }_{t}}={{x}_{t}}-{{p}_{t}}$, where ${{x}_{t}}$ is log of the nominal value of the wage contract and ${{p}_{t}}$ is log of price level. Usually, firms and workers negotiate labor contracts once a year which specifies fixed nominal wages for two periods. Then the average real wage is ${{V}_{t}}=\tfrac{1}{2}\left( {{\varphi }_{t}}+{{\varphi }_{t-1}} \right)$. 

Taylor (1980) assumed that contract wages ${{\varphi }_{t}}$ are set as the average of the lagged and expected future $\tfrac{1}{2}\left( {{V}_{t}}+{{E}_{t}}{{V}_{t+1}} \right)$ adjusted for excess demand ${{y}_{t}}$. Then, Taylor's contracted wage is ${{\varphi }_{t}}=\tfrac{1}{2}\left( {{V}_{t}}+{{E}_{t}}{{V}_{t+1}} \right)+k{{y}_{t}}$, substituting values of ${{V}_{t}}$ as $\tfrac{1}{2}\left( {{\varphi }_{t}}+{{\varphi }_{t-1}} \right)$and ${{E}_{t}}{{V}_{t+1}}$ as ${{E}_{t}}\left[ \tfrac{1}{2}\left( {{\varphi }_{t+1}}+{{\varphi }_{t}} \right) \right]$ and simplifying \[{{\varphi }_{t}}=\frac{1}{2}\left( \left[ \frac{1}{2}\left( {{\varphi }_{t}}+{{\varphi }_{t-1}} \right) \right]+{{E}_{t}}\left[ \frac{1}{2}\left( {{\varphi }_{t+1}}+{{\varphi }_{t}} \right) \right] \right)+k{{y}_{t}}\] \[{{\varphi }_{t}}=\frac{1}{2}\left( \frac{1}{2}{{\varphi }_{t}}+\frac{1}{2}{{\varphi }_{t-1}}+\frac{1}{2}{{E}_{t}}{{\varphi }_{t+1}}+\frac{1}{2}{{E}_{t}}{{\varphi }_{t}} \right)+k{{y}_{t}}\] \[{{\varphi }_{t}}=\frac{1}{2}\left( \frac{1}{2}{{\varphi }_{t}}+\frac{1}{2}{{\varphi }_{t-1}}+\frac{1}{2}{{E}_{t}}{{\varphi }_{t+1}}+\frac{1}{2}{{\varphi }_{t}} \right)+k{{y}_{t}}\] \[{{\varphi }_{t}}=\frac{1}{2}\left( \frac{1}{2}{{\varphi }_{t-1}}+\frac{1}{2}{{E}_{t}}{{\varphi }_{t+1}}+{{\varphi }_{t}} \right)+k{{y}_{t}}\] \[{{\varphi }_{t}}=\left( \frac{1}{2}{{\varphi }_{t-1}}+\frac{1}{2}{{E}_{t}}{{\varphi }_{t+1}} \right)+2k{{y}_{t}}\] This is known as Taylor's contracting equation. Substituting ${{\varphi }_{t}}={{x}_{t}}-{{p}_{t}}$ relation in Taylor's contracting equation we can derive: \[{{\varphi }_{t}}=\left( \frac{1}{2}{{\varphi }_{t-1}}+\frac{1}{2}{{E}_{t}}{{\varphi }_{t+1}} \right)+2k{{y}_{t}}\] \[{{x}_{t}}-{{p}_{t}}=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+{{E}_{t}}{{x}_{t+1}}-{{E}_{t}}{{p}_{t+1}} \right)+2k{{y}_{t}}\] Since ${{p}_{t}}=\tfrac{1}{2}({{x}_{t}}+{{x}_{t-1}})$, ${{p}_{t+1}}=\tfrac{1}{2}({{x}_{t+1}}+{{x}_{t}})$ and ${{E}_{t}}{{p}_{t+1}}=\tfrac{1}{2}({{E}_{t}}{{x}_{t+1}}+{{x}_{t}})$, then, substituting these values we can get: \[{{x}_{t}}-{{p}_{t}}=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+{{E}_{t}}{{x}_{t+1}}-{{E}_{t}}{{p}_{t+1}} \right)+2k{{y}_{t}}\] \[{{x}_{t}}-\left[ 0.5({{x}_{t}}+{{x}_{t-1}}) \right]=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+{{E}_{t}}{{x}_{t+1}}-{{E}_{t}}\left[ 0.5({{E}_{t}}{{x}_{t+1}}+{{x}_{t}}) \right] \right)+2k{{y}_{t}}\] \[{{x}_{t}}-0.5{{x}_{t}}-0.5{{x}_{t-1}}=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+{{E}_{t}}{{x}_{t+1}}-0.5{{E}_{t}}{{x}_{t+1}}-0.5{{x}_{t}} \right)+2k{{y}_{t}}\] \[{{x}_{t}}-0.5{{x}_{t}}-0.5{{x}_{t-1}}=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+0.5{{E}_{t}}{{x}_{t+1}}-0.5{{x}_{t}} \right)+2k{{y}_{t}}\] As, ${{E}_{t}}{{x}_{t+1}}=2{{E}_{t}}{{p}_{t+1}}-{{x}_{t}}$ \[{{x}_{t}}-0.5{{x}_{t}}-0.5{{x}_{t-1}}=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+0.5(2{{E}_{t}}{{p}_{t+1}}-{{x}_{t}})-0.5{{x}_{t}} \right)+2k{{y}_{t}}\] \[{{x}_{t}}-0.5{{x}_{t}}-0.5{{x}_{t-1}}=\frac{1}{2}\left( {{x}_{t-1}}-{{p}_{t-1}}+{{E}_{t}}{{p}_{t+1}}-{{x}_{t}} \right)+2k{{y}_{t}}\] 

\[{{x}_{t}}-0.5{{x}_{t}}-0.5{{x}_{t-1}}=0.5{{x}_{t-1}}-0.5{{p}_{t-1}}+0.5{{E}_{t}}{{p}_{t+1}}-0.5{{x}_{t}}+2k{{y}_{t}}\] \[{{x}_{t}}-0.5{{x}_{t-1}}-0.5{{x}_{t-1}}=-0.5{{p}_{t-1}}+0.5{{E}_{t}}{{p}_{t+1}}+2k{{y}_{t}}\] \[{{x}_{t}}-{{x}_{t-1}}=\frac{1}{2}\left( {{E}_{t}}{{p}_{t+1}}-{{p}_{t-1}} \right)+2k{{y}_{t}}\] \[\Delta {{x}_{t}}=\frac{1}{2}\left( {{E}_{t}}{{p}_{t+1}}-{{p}_{t-1}} \right)+2k{{y}_{t}}\] Adding and subtracting ${{P}_{t}}$ in this equation: \[\Delta {{x}_{t}}=\tfrac{1}{2}\left( {{E}_{t}}{{p}_{t+1}}+{{p}_{t}}-{{p}_{t}}-{{p}_{t-1}} \right)+2k{{y}_{t}}\] which is same as: \[\Delta {{x}_{t}}=\tfrac{1}{2}\left( \underbrace{{{E}_{t}}{{\pi }_{t+1}}}_{{{E}_{t}}{{p}_{t+1}}-{{p}_{t}}}+\underbrace{{{\pi }_{t}}}_{{{p}_{t}}-{{p}_{t-1}}} \right)+2k{{y}_{t}}\]. Now, let's define inflation ${{\pi }_{t}}={{p}_{t}}-{{p}_{t-1}}$, substituting ${{p}_{t}}=\tfrac{1}{2}({{x}_{t}}+{{x}_{t-1}})$ and${{p}_{t-1}}=\tfrac{1}{2}({{x}_{t-1}}+{{x}_{t-2}})$in it we get: ${{\pi }_{t}}=0.5{{x}_{t}}-0.5{{x}_{t-1}}+0.5{{x}_{t-1}}-0.5{{x}_{t-2}}=0.5(\Delta {{x}_{t}})+0.5(\Delta {{x}_{t-1}})$. Now substituting the values of $\Delta {{x}_{t}}$ and $\Delta {{x}_{t-1}}$we get: \[{{\pi }_{t}}=0.5(\Delta {{x}_{t}})+0.5(\Delta {{x}_{t-1}})\Delta {{x}_{t}}\] \[{{\pi }_{t}}=0.5\left[ \tfrac{1}{2}\left( {{E}_{t}}{{\pi }_{t+1}}+{{\pi }_{t}} \right)+2k{{y}_{t}} \right]+0.5\left[ \tfrac{1}{2}\left( {{E}_{t-1}}{{\pi }_{t}}+{{\pi }_{t-1}} \right)+2k{{y}_{t-1}} \right]\] \[{{\pi }_{t}}=\tfrac{1}{4}{{E}_{t}}{{\pi }_{t+1}}+\tfrac{1}{4}{{\pi }_{t}}+k{{y}_{t}}+\tfrac{1}{4}{{E}_{t-1}}{{\pi }_{t}}+\tfrac{1}{4}{{\pi }_{t-1}}+k{{y}_{t-1}}\] \[{{\pi }_{t}}-\tfrac{1}{4}{{\pi }_{t}}=\tfrac{1}{4}{{E}_{t}}{{\pi }_{t+1}}+k{{y}_{t}}+\tfrac{1}{4}{{E}_{t-1}}{{\pi }_{t}}+\tfrac{1}{4}{{\pi }_{t-1}}+k{{y}_{t-1}}\] \[\tfrac{3}{4}{{\pi }_{t}}=\tfrac{1}{4}{{E}_{t}}{{\pi }_{t+1}}+k{{y}_{t}}+\tfrac{1}{4}{{E}_{t-1}}{{\pi }_{t}}+\tfrac{1}{4}{{\pi }_{t-1}}+k{{y}_{t-1}}\] \[{{\pi }_{t}}=\tfrac{1}{3}{{E}_{t}}{{\pi }_{t+1}}+\tfrac{1}{3}k{{y}_{t}}+\tfrac{1}{3}{{E}_{t-1}}{{\pi }_{t}}+\tfrac{1}{3}{{\pi }_{t-1}}+\tfrac{1}{3}k{{y}_{t-1}}\] \[{{\pi }_{t}}=\tfrac{1}{3}{{E}_{t}}{{\pi }_{t+1}}+\tfrac{1}{3}{{E}_{t-1}}{{\pi }_{t}}+\tfrac{1}{3}{{\pi }_{t-1}}+\tfrac{1}{3}\left( k{{y}_{t}}+k{{y}_{t-1}} \right)\] This results is very profound especially for two reasons: one it complements the NKPC literature to explain the inflation persistence and another it explains the structural relationship rather than the reduced form (accelerationist approach).

Please follow previous discussion (here) for the history of thoughts on Inflation and how and why New Keynesian Phillip Curve (NKPC) appear in the macroeconomic models and how they are different/similar to Keynesian and/or Rational Expectationist School of thought. Derivation can be found (here). In this this blog we will how to solve the data fitting problem of NKPC via Modified NKPC using Wage Staggering model then see the model of Menu cost (here) to theoretically model "why price in the first place is sticky?"

Derivation of New Keynesian Phillip Curve Using Calvo Pricing and Forward Looking Approach

Please follow previous discussion (here) for the history of thoughts on Inflation and how and why New Keynesian Phillip Curve (NKPC) appear in the macroeconomic models and how they are different/similar to Keynesian and/or Rational Expectationist School of thought. After this blog see (here) for how to solve the data fitting problem of NKPC via Modified NKPC using Wage Staggering model then see the model of Menu cost (here) to theoretically model "why price in the first place is sticky?"

To model the price stickiness, let's consider the calvo-pricing. Suppose, there is a chance $q$ that firms change their prices and remaining firms keep their price unchanged. Hence, the expected loss of the firms changing the price is given by "expected loss function" as \[{{\left( {{p}_{it}}-p_{t}^{*} \right)}^{2}}+\left( 1-q \right)\beta {{E}_{t}}{{\left( {{p}_{it}}-p_{t+1}^{*} \right)}^{2}}+{{\left( 1-q \right)}^{2}}{{\beta }^{2}}{{E}_{t}}{{\left( {{p}_{it}}-p_{t+1}^{*} \right)}^{2}}+\cdots \] alternatively, it can be re-expressed as \[\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}{{\left( {{p}_{it}}-p_{t+j}^{*} \right)}^{2}}}\]which represents totals of discounted expected relative loss of profit for firms who changed the price and $0<\beta <1$ implies the firm places less weight on the future losses than today's loss. Therefore, each price changing ${{i}^{th}}$ firm tries to minimize the this loss function by choosing the price ${{p}_{it}}$ . The above expression can be re-state as: \[\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}{{\left( {{p}_{it}}-p_{t+j}^{*} \right)}^{2}}}\] \[\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}\left[ {{E}_{t}}{{\left( {{p}_{it}} \right)}^{2}}-2{{p}_{it}}{{E}_{t}}\left( p_{t+j}^{*} \right)+{{E}_{t}}{{\left( p_{t+j}^{*} \right)}^{2}} \right]}\] \[\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}{{\left( {{p}_{it}} \right)}^{2}}}-\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}2{{p}_{it}}{{E}_{t}}\left( p_{t+j}^{*} \right)+}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}{{\left( p_{t+j}^{*} \right)}^{2}}}\] \[{{\left( {{p}_{it}} \right)}^{2}}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}}-2{{p}_{it}}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)+}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}{{\left( p_{t+j}^{*} \right)}^{2}}}\] \[\because {{E}_{t}}{{\left( {{p}_{it}} \right)}^{2}}={{\left( {{p}_{it}} \right)}^{2}}\] The FOC (taking FOD w.r.t ${{p}_{it}}$ and set to zero) is: \[\frac{{{\left( {{p}_{it}} \right)}^{2}}\partial \sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}}}{\partial {{p}_{it}}}-\frac{2{{p}_{it}}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}}{\partial {{p}_{it}}}+\frac{\partial \sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}{{\left( p_{t+j}^{*} \right)}^{2}}}}{\partial {{p}_{it}}}=0\] \[2{{p}_{it}}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}}-2\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}+0=0\] \[{{p}_{it}}\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}}=\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] \[{{p}_{it}}\frac{1}{\left( 1-\left( 1-q \right)\beta \right)}=\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] \[{{p}_{it}}=\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] Let's assume ${{p}_{it}}={{x}_{t}}$ for sake of simplicity. \[{{p}_{it}}={{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=0}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] say equation (1) Instead of indexing $j$ from 0, let's index $j$ from 1 such that we can re-express ${{x}_{t}}$ as: \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right){{\left( 1-q \right)}^{0}}{{\beta }^{0}}{{E}_{t}}\left( p_{t+0}^{*} \right)+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] \[\because {{E}_{t}}\left( p_{t+0}^{*} \right)={{E}_{t}}\left( p_{t}^{*} \right)=p_{t}^{*}\And {{z}^{0}}=1\], we get \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)p_{t}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\], say equation (2). Now, let's again re-express the second expression of this equation indexing $j$ from 2 we get: \[\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}=\left( 1-\left( 1-q \right)\beta \right){{\left( 1-q \right)}^{1}}{{\beta }^{1}}{{E}_{t}}\left( p_{t+1}^{*} \right)+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=2}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\]\[\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}=\left( 1-\left( 1-q \right)\beta \right)\left( 1-q \right)\beta {{E}_{t}}\left( p_{t+1}^{*} \right)+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=2}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] Let's forward ${{x}_{t}}$ of equation (2) by one period and take the expectation. \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)p_{t}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] \[{{E}_{t}}({{x}_{t+1}})=\left( 1-\left( 1-q \right)\beta \right){{E}_{t}}p_{t+1}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j+1}^{*} \right)}\] Now, let's multiply both sides by $(1-q)\beta $, we get \[(1-q)\beta {{E}_{t}}({{x}_{t+1}})=\left( 1-\left( 1-q \right)\beta \right)(1-q)\beta {{E}_{t}}p_{t+1}^{*}+\left( 1-\left( 1-q \right)\beta \right)(1-q)\beta \sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j+1}^{*} \right)}\] \[(1-q)\beta {{E}_{t}}({{x}_{t+1}})=\left( 1-\left( 1-q \right)\beta \right)(1-q)\beta {{E}_{t}}p_{t+1}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j+1}}{{\beta }^{j+1}}{{E}_{t}}\left( p_{t+j+1}^{*} \right)}\] We can now re-index. \[(1-q)\beta {{E}_{t}}({{x}_{t+1}})=\left( 1-\left( 1-q \right)\beta \right)(1-q)\beta {{E}_{t}}p_{t+1}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=2}^{\infty +1}{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] \[(1-q)\beta {{E}_{t}}({{x}_{t+1}})=\left( 1-\left( 1-q \right)\beta \right)(1-q)\beta {{E}_{t}}p_{t+1}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=2}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] Note that this expression $(1-q)\beta {{E}_{t}}({{x}_{t+1}})$ is same as the re-expressed second expression of ${{x}_{t}}$. So, now we can write: \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)p_{t}^{*}+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}\] \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)p_{t}^{*}+\underbrace{\underbrace{\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=1}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}}_{\left( 1-\left( 1-q \right)\beta \right)\left( 1-q \right)\beta {{E}_{t}}\left( p_{t+1}^{*} \right)+\left( 1-\left( 1-q \right)\beta \right)\sum\limits_{j=2}^{\infty }{{{\left( 1-q \right)}^{j}}{{\beta }^{j}}{{E}_{t}}\left( p_{t+j}^{*} \right)}}}_{(1-q)\beta {{E}_{t}}({{x}_{t+1}})}\] \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)p_{t}^{*}+(1-q)\beta {{E}_{t}}({{x}_{t+1}})\] Let's say above as equation (3).

For optimum $p_{t}^{*}$, we assume: $p_{t}^{*}={{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}}$ say equation (4) 

Re-expressing for the $p_{t}^{*}$ we get: \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)\overbrace{\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]}^{p_{t}^{*}}+(1-q)\beta {{E}_{t}}({{x}_{t+1}})\], say equation (5) While some firms (${{q}^{th}}$ portion of total) changed the price and remaining didn't change the price so the actual price ${{p}_{t}}$ is the weighted sum of prices responses of all the firms. It can be expressed as: \[{{p}_{t}}=q{{x}_{t}}+(1-q){{p}_{t-1}}\] say equation (6) Let's forward by one period and take expectation, we get: \[{{E}_{t}}{{p}_{t+t}}=q{{E}_{t}}{{x}_{t+t}}+(1-q){{E}_{t}}{{p}_{t}}\] \[{{E}_{t}}{{p}_{t+t}}=q{{E}_{t}}{{x}_{t+t}}+(1-q){{p}_{t}}\] Arranging for $q{{E}_{t}}{{x}_{t+t}}$ we get \[q{{E}_{t}}{{x}_{t+t}}={{E}_{t}}{{p}_{t+t}}-(1-q){{p}_{t}}\] Now, taking equation (6) and Substituting value of ${{x}_{t}}$, we get: \[{{p}_{t}}=q{{x}_{t}}+(1-q){{p}_{t-1}}\] \[{{x}_{t}}=\left( 1-\left( 1-q \right)\beta \right)\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]+(1-q)\beta {{E}_{t}}({{x}_{t+1}})\] \[{{p}_{t}}=q\left[ \left( 1-\left( 1-q \right)\beta \right)\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]+(1-q)\beta {{E}_{t}}({{x}_{t+1}}) \right]+(1-q){{p}_{t-1}}\] \[{{p}_{t}}=q\left( 1-\left( 1-q \right)\beta \right)\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]+q(1-q)\beta {{E}_{t}}({{x}_{t+1}})+(1-q){{p}_{t-1}}\] Substituting value of $q{{E}_{t}}{{x}_{t+t}}$ we get: \[{{p}_{t}}=q\left( 1-\left( 1-q \right)\beta \right)\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]+(1-q)\beta \left[ {{E}_{t}}{{\pi }_{t+1}}+q{{p}_{t}} \right]+(1-q){{p}_{t-1}}\] say equation (7) Simplifying equation 7 \[{{p}_{t}}=q\left( 1-\left( 1-q \right)\beta \right)\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]+(1-q)\beta \left[ {{E}_{t}}{{\lambda }_{t+1}}+q{{p}_{t}} \right]+(1-q){{p}_{t-1}}\] \[or,{{p}_{t}}=\left( q-q\theta \beta \right)\left[ {{p}_{t}}+\gamma {{y}_{t}}+{{\omega }_{t}} \right]+\theta \beta \left[ {{E}_{t}}{{\lambda }_{t+1}}+q{{p}_{t}} \right]+\theta {{p}_{t-1}}\] \[or,{{p}_{t}}=q{{p}_{t}}+q\gamma {{y}_{t}}+q{{\omega }_{t}}-q\theta \beta {{p}_{t}}-q\theta \beta \gamma {{y}_{t}}-q\theta \beta {{\omega }_{t}}+\theta \beta {{E}_{t}}{{\lambda }_{t+1}}+q\theta \beta {{p}_{t}}+\theta {{p}_{t-1}}\] \[or,{{p}_{t}}-q{{p}_{t}}-\theta {{p}_{t-1}}=q\gamma {{y}_{t}}+q{{\omega }_{t}}-q\theta \beta \gamma {{y}_{t}}-q\theta \beta {{\omega }_{t}}+\theta \beta {{E}_{t}}{{\lambda }_{t+1}}\] \[or,(1-q){{p}_{t}}-(1-q){{p}_{t-1}}=q(\gamma {{y}_{t}}+{{\omega }_{t}})-q\theta \beta (\gamma {{y}_{t}}+{{\omega }_{t}})+\theta \beta {{E}_{t}}{{\lambda }_{t+1}}\] \[or,(1-q)({{p}_{t}}-{{p}_{t-1}})=(q-q\theta \beta )(\gamma {{y}_{t}}+{{\omega }_{t}})+\theta \beta {{E}_{t}}{{\lambda }_{t+1}}\] \[or,({{p}_{t}}-{{p}_{t-1}})=\frac{\theta \beta {{E}_{t}}{{\lambda }_{t+1}}}{(1-q)}+\frac{(q-q(1-q)\beta )(\gamma {{y}_{t}}+{{\omega }_{t}})}{(1-q)}\] \[\therefore {{\pi }_{t}}=\beta {{E}_{t}}{{\lambda }_{t+1}}+\frac{q(1-(1-q)\beta )}{(1-q)}(\gamma {{y}_{t}}+{{\omega }_{t}})\because (1-q)=\theta \] say equation (8) This equation gives the New-Keynesian Phillip Curve (NKPC),which states inflation as the function of two factors: next period expected inflation ${{E}_{t}}{{\pi }_{t+1}}$ if rises inflation rises and if gap between frictionless optimal price and current prices (firm's re-setting the price) increases inflation increases. 

In the empirical quest, the gap between frictionless optimal price and current prices cannot be measured thus researchers have used output gap as proxy. Output gap can be estimated via various techniques like: Hedrick Prescott filter, Kalman Filter, Baxter and King Filter etc. For estimation purpose above equation 8 is written as: ${{\pi }_{t}}=\beta {{E}_{t}}{{\pi }_{t+1}}+{\gamma }'{{y}_{t}}+{{{\omega }'}_{t}}$ and is the econometric specification of NKPC. However, in the context of econometric approach rather than NKPC ${{\pi }_{t}}=\beta {{E}_{t}}{{\pi }_{t+1}}+{\gamma }'{{y}_{t}}+{{{\omega }'}_{t}}$ approach to fit ${{\pi }_{t+1}}$ to ${{\pi }_{t}}$, fitting lagged value ${{\pi }_{t}}={{\alpha }_{0}}+{{\alpha }_{1}}{{\pi }_{t-1}}+{{\alpha }_{2}}{{y}_{t}}+{{\varepsilon }_{t}}$ (accelerationist approach) performs statistically better - due to stronger correlation between ${{\pi }_{t-1}}$ to ${{\pi }_{t}}$ and possibly have more policy relevancies. 

However, New-Keynesian critique such accelerationist reduced form and advocates NKPC being as a structural form and concludes a strong statement that low inflation can be achieved immediately by the central bank announcing (and the public believing) that it is committing itself to eliminating positive output gaps in the future.

Please follow previous discussion (here) for the history of thoughts on Inflation and how and why New Keynesian Phillip Curve (NKPC) appear in the macroeconomic models and how they are different/similar to Keynesian and/or Rational Expectationist School of thought. After this blog see (here) for how to solve the data fitting problem of NKPC via Modified NKPC using Wage Staggering model then see the model of Menu cost (here) to theoretically model "why price in the first place is sticky?"

Please follow previous discussion (here) for the history of thoughts on Inflation and how and why New Keynesian Phillip Curve (NKPC) appear in the macroeconomic models and how they are different/similar to Keynesian and/or Rational Expectationist School of thought. After this blog see (here) for how to solve the data fitting problem of NKPC via Modified NKPC using Wage Staggering model then see the model of Menu cost (here) to theoretically model "why price in the first place is sticky?"

Tests of Excess Sensitivity of Consumption

This is a continuation of previous blogs (here), (here), (here), (here), (here) and (here)

The Empirical Evidences Hall stated that the theory can be directly tested with first order condition of the model i.e., Euler equation. He used slight modified versions as: \[{{c}_{t+1}}={{\varphi }_{0}}+{{\varphi }_{1}}{{c}_{t}}+\underset{i=1}{\overset{k}{\mathop \sum }}\,{{\varphi }_{i}}{{y}_{t-i}}+{{\epsilon }_{t+1}}\] Halls (1978) considered aggregated quarterly data on non-durable real consumption per capita $\left( {{c}_{t}} \right)$ and real disposable income per capital $\left( {{y}_{t}} \right)$. For the theory to be correct, under the null the ${{H}_{0}}:{{\tilde{\varphi }}_{1}}=0$ . Under various lag control on income ${{y}_{t-i}}$, for $i=1,4 \And 12 $, Hall failed to reject the null hypothesis thus confirmed the RWH (see Table-1). 

Instead using aggregated data to test CEQ as Hall (1978), Campbell and Mankiw (1989), specified an alternative approach where they assumed a fraction $\left( \lambda \right)$ of consumer are 'hand-to-mouth' with $c_{t}^{h}=y_{t}^{h}$ while remaining consumer follows CEQ with. Then, aggregate consumption change is then the sum of the consumption change of the hand-to-mount consumer and of the CEQ, i.e., \[\Delta {{c}_{t+1}}=\lambda \Delta {{y}_{t+1}}+\left( 1-\lambda \right){{\epsilon }_{t+1}}\] To access such theoretical model, they estimated the log-linearized CEQ as: \[\Delta {{c}_{t}}=\mu +\lambda \Delta {{y}_{t}}+\theta {{r}_{t}}\] where they defined $\left( {{c}_{t}} \right)$ as log transformed consumption as real purchases of consumer non durable and services per person and income $\left( {{y}_{t}} \right)$ as real disposable income per person with the control variable $\left( {{r}_{t}} \right)$ 3-month T-bill rate over the quarter minus the rate of change in the PCE deflator. Then performed the OLS and various 2SLS to suppress the endogeneity between the error term and income with various lagged value of change in consumption and controls. They found $\left( \lambda \right)$ to be similar/roughly as 0.5 and significant with most of above econometric specification (see Table-2 and 3) which signifies significant departure from RWH because consumption appears to increase by fifty cents for one dollar anticipated change in income, hence provided strong support to PIH. Other researchers like Shae (1995), Parker (1999), Souleles (1999), Hsieh (2003), also converge to Campbell and Mankiw (1989) like conclusion while they consider more disaggregated household dataset. 

The Policy Implications:

Previously, AIH has a short run policy implication which illuminated the effect of multipliers. While RIH opened the floor to differentiate the level of consumption on various income brackets and 'Ratchet model' helped to model behaviors of how/why people don't like to retrench consumption even thou economy is down. RIH and Ratchet model jointly solve the Kuznet paradox and Engel Paradox. However, LCH and PIH came with forward looking approach rather than previous myopic models. LCH's interesting policy implications helped to draw a distinction between how the consumption occurs through the lifetime of individuals. While, PIH almost tried to close the policy debate/implication on differentiating the components of consumption as the predictable and unpredictable parts. While RWH mostly illuminated the unpredictable part while PIH answered the predictable component of consumption. Most of the empirical evidences suggested that consumption roughly grows by fifty cents for a dollar increment on the anticipated increase. 

Table-1

Hall (1978)

Table-2

Campbell and Mankiw (1989) 

Table-3

Campbell and Mankiw (1989) 

Why the Consumption is Unpredictable?

This is a continuation of previous blogs (here), (here) (here) (here) and (here).

Till now, we have derived that consumption is unpredictable, now, we seek the answer–"Why?". Let's difference the consumption in two consecutive period $\Delta {{c}_{t}}={{c}_{t+1}}-{{c}_{t}}$, we get: $\Delta {{c}_{t}}=\left( 1-\beta \right)\left( {{a}_{t+1}}-{{a}_{t}}+{{h}_{t+1}}-{{h}_{t}} \right)$ which is same as: $\Delta {{c}_{t}}=\left( 1-\beta \right)\left( {{h}_{t}}-{{E}_{t+1}}{{h}_{t}} \right)$ then we get $\Delta {{c}_{t}}=\left( 1-\beta \right)\underset{t=0}{\overset{\infty }{\mathop \sum }}\,\left[ {{E}_{t+1}}{{y}_{t}}-{{E}_{t}}{{y}_{t}} \right]$ (see Proof below). 

The results is remarkable that expression $\left[ {{E}_{t+1}}{{y}_{t}}-{{E}_{t}}{{y}_{t}} \right]$ represents the revision of expectation about the future income and such revision is unpredictable as of time period ${t}$. Hence we can conclude the change in consumption is related to the news that the household receives about the future income. Previous theories suggested when output declines, consumption decline but will recover, while, Hall's random walk theory remarked consumption is unpredictable, further, Hall's extension to PIH showed when output declines, consumption decline only by amount of the permanent income and not expected to recover. These contrary thought provoked a rigorous empiric to proof/disproof these hypothesis or to analyze whether predictable changes in income feedback the predictable changes in consumption or not and test of hypothesis is known as "excess sensitivity of consumption" (Flavin, 1981). We will see them (here).

Proof: \[\Delta {{c}_{t}}=\left( 1-\beta \right)\left( {{h}_{t}}-{{E}_{t+1}}{{h}_{t}} \right)=\left( 1-\beta \right)\underset{t=0}{\overset{\infty }{\mathop \sum }}\,\left[ {{E}_{t+1}}{{y}_{t}}-{{E}_{t}}{{y}_{t}} \right]\] The${{c}_{t}}$ and ${{c}_{t+1}}$ are given as: \[{{c}_{t}}=\left( 1-\beta \right)\left( {{a}_{t}}+{{E}_{t}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t}}{{y}_{t}} \right] \right)\] \[{{c}_{t+1}}=\left( 1-\beta \right)\left( {{a}_{t+1}}+{{E}_{t+1}}\left[ \underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t-1}}{{y}_{t+1}} \right] \right)\] Let's difference and substitute${{a}_{t+1}}=\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)R$, and we get: \[{{c}_{t+1}}-{{c}_{t}}=\left( 1-\beta \right)\left( {{a}_{t+1}}-{{a}_{t}}+{{E}_{t+1}}\left[ \underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t-1}}{{y}_{t+1}} \right]-{{E}_{t}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t}}{{y}_{t}} \right] \right)\] \[\Delta {{c}_{t}}=\left( 1-\beta \right)\left( R\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)-{{a}_{t}}+\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t-1}}{{E}_{t+1}}{{y}_{t+1}}-\underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t}}{{E}_{t}}{{y}_{t}} \right)\] \[\Delta {{c}_{t}}=\left( 1-\beta \right)\left( \left( R-1 \right){{a}_{t}}+R{{y}_{t}}-R{{c}_{t}}+R\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{t}}{{E}_{t+1}}{{y}_{t+1}}-\underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{t}}{{E}_{t}}{{y}_{t}} \right)\] \[\Delta {{c}_{t}}=\left( 1-\beta \right)\left( R\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{t}}{{E}_{t+1}}{{y}_{t+1}}-R\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{t}}{{E}_{t}}{{y}_{t}} \right)\] \[\Delta {{c}_{t}}=\left( R-\beta R \right)\left( \underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{t}}{{E}_{t+1}}{{y}_{t+1}}-\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{t}}{{E}_{t}}{{y}_{t}} \right)\] \[\Delta {{c}_{t}}=\left( R-1 \right)\left( {{R}^{-1}}\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{E}_{t+1}}{{y}_{t+1}}-{{R}^{-1}}\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{E}_{t}}{{y}_{t}} \right)\] \[\Delta {{c}_{t}}=\left( R{{R}^{-1}}-{{R}^{-1}} \right)\left( \underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{E}_{t+1}}{{y}_{t+1}}-\underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{E}_{t}}{{y}_{t}} \right)\] \[\Delta {{c}_{t}}=\left( 1-\beta \right)\left( \underset{t=1}{\overset{\infty }{\mathop \sum }}\,{{E}_{t+1}}{{y}_{t+1}}-\underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{E}_{t}}{{y}_{t}} \right)=\frac{r}{1+r}\underset{t=1}{\overset{\infty }{\mathop \sum }}\,\left( {{E}_{t+1}}{{y}_{t+1}}-{{E}_{t}}{{y}_{t}} \right)\]

Certainty Equivalent (CEQ) theory of Consumption

This is a continuation of previous blogs (here), (here) (here) and (here).

This concept is not yet a decision rule for consumption. For decision rule, let's forward iterate dynamic budget constraint ${{a}_{t+1}}=\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)R$ to derive an inter temporal budget constraint as: \[{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \frac{1}{1+r} \right)}^{t}}{{c}_{t}} \right]={{a}_{0}}+{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \frac{1}{1+r} \right)}^{t}}{{y}_{t}} \right]\] or, \[{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \beta \right)}^{t}}{{c}_{t}} \right]={{a}_{0}}+{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \beta \right)}^{t}}{{y}_{t}} \right]\] \[{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t}}{{c}_{t}} \right]={{a}_{0}}+{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( R \right)}^{-t}}{{y}_{t}} \right]\] This can be interpreted as the present value sum of current and future household consumption is current stock of capital or financial wealth (a_0) and present value sum of current and future incomes or human wealth say, ${{h}_{0}}={{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \beta \right)}^{t}}{{y}_{t}} \right]$ With Euler equation then simplification of geometric progression for infinite horizon, we re-express \[{{E}_{0}}\left[ \underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \frac{1}{1+r} \right)}^{t}}{{c}_{t}} \right]={{c}_{0}}\underset{t=0}{\overset{\infty }{\mathop \sum }}\,{{\left( \frac{1}{1+r} \right)}^{t}}={{c}_{0}}\frac{\left( 1+r \right)}{r}\] Hence, we get, \[{{c}_{0}}=\frac{r}{1+r}\left( {{a}_{0}}+{{h}_{0}} \right)\] and for any $t$ time period we get, \[{{c}_{t}}=\frac{r}{1+r}\left( {{a}_{t}}+{{h}_{t}} \right)=\left( 1-\frac{1}{1+r} \right)\left( {{a}_{t}}+{{h}_{t}} \right)=\left( 1-\beta \right)\left( {{a}_{t}}+{{h}_{t}} \right)\] This is a closed form solution. Different books, paper uses these equation equivalently. This equation converge to tell consumption follows the PIH in expectation. The term in right-hand is the expected total wealth where the expectation is over future labor income. And consumption decisions are made as if future income is certain and equal to expected value. Hence this model is called as "Certainty Equivalent" (CEQ) and it also derived the household is smoothing consumption. Above result only hold true for Hall's quadratic utility specification which is symmetric and has linear marginal utility, however, may not hold for more general utility function. Till now, we have derived that consumption is unpredictable, now, we seek the answer–"Why?". (here)

Consumption Follows a Random Walk (Random Walk Hypothesis Theory of Consumption)

Let’s consider household maximizing the preferences over consumption sequences as: $\begin{matrix} Max & U={{E}_{0}}\left[ \sum\limits_{t=0}^{\infty }{{{\beta }^{t}}u\left( {{c}_{t}} \right)}|{{\omega }_{0}} \right] \\ \end{matrix}$ under the budget constraints as: ${{a}_{t+1}}=\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)\left( 1+r \right)$ where, ${{\omega }_{0}}$ is information available till current time and $E\left[ \left. \centerdot \right|{{\omega }_{0}} \right]$ captures conditional expectation on information available till current time . I will skip writing this to lighten notations. 

Further, ${u}'\left( \centerdot \right)>0$ and ${u}''\left( \centerdot \right)<0$and $\beta$ is discount factor and $t$ is time till infinite horizon. ${{y}_{t}}-{{c}_{t}}$ captures income residual after consumption, and ${{a}_{t}}$ is net-assets and ${{a}_{0}}$ is given. 

There exist an implicit assumption that one can borrow and lend freely with certainty interest rate of $r$ and $\left( 1+r \right)=R$, However, household will not be allowed to run Ponzi-scheme i.e., $\underset{T\to \infty }{\mathop{\lim }}\,{{\beta }^{T}}{{a}_{T}}\ge 0$ and $T=t+n$and $n$ is sufficiently large time-index. 

Since, ${{a}_{t}}$ assets are the sole endogenous state variables; we can write the value function that maximizes the utility of the agents as a function of state variable ${{a}_{t}}$ as: \[V\left( {{a}_{0}} \right)=\underbrace{max}_{\left\{ {{c}_{t}} \right\}}{{E}_{0}}\left[ \sum\limits_{t=0}^{\infty }{{{\beta }^{t}}u\left( {{c}_{t}} \right)} \right]\]. 

Given the recursive structure of this equation, we can re-express as Bellman equation: \[V\left( {{a}_{t}};{{y}_{t}} \right)=\underbrace{max}_{{{c}_{t}}}u\left( {{c}_{t}} \right)+\beta {{E}_{t}}\left[ V\left( {{a}_{t+1}} \right) \right]\] subject to \[{{a}_{t+1}}=\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)\left( 1+r \right)\] The first order condition is given as: \[\frac{dV\left( {{a}_{t}};{{y}_{t}} \right)}{d{{c}_{t}}}={u}'\left( {{c}_{t}} \right)+\beta {{E}_{t}}\left[ \frac{dV\left( {{a}_{t+1}};{{y}_{t}} \right)}{d\left( {{a}_{t+1}} \right)}\frac{d\left( {{a}_{t+1}} \right)}{d{{c}_{t}}} \right]=0\] \[{u}'\left( {{c}_{t}} \right)=-\beta {{E}_{t}}\left[ \frac{dV\left( {{a}_{t+1}};{{y}_{t}} \right)}{d\left( {{a}_{t+1}} \right)}\frac{d\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)R}{d{{c}_{t}}} \right]\] \[\therefore {u}'\left( {{c}_{t}} \right)=\beta {{E}_{t}}\left[ \frac{dV\left( {{a}_{t+1}};{{y}_{t}} \right)}{d\left( {{a}_{t+1}} \right)}R \right]=\beta R{{E}_{t}}\left[ {V}'\left( {{a}_{t+1}} \right) \right]\to eqn(1)\] Let’s take total derivative of Bellman equation to analyze the effect of change in ${{a}_{t}}$ on the value function $V\left( {{a}_{t}};{{y}_{t}} \right)$, for notational simplicity, I will write $V\left( {{a}_{t}};{{y}_{t}} \right)$ as $V\left( {{a}_{t}} \right)$. And we get: \[{V}'\left( {{a}_{t}} \right)da={u}'\left( {{c}_{t}} \right)dc+\beta {{E}_{t}}\left[ {V}'\left( {{a}_{t+1}} \right)Rda-{V}'\left( {{a}_{t+1}} \right)Rdc \right]\] \[{V}'\left( {{a}_{t}} \right)da={u}'\left( {{c}_{t}} \right)dc+\beta {{E}_{t}}{V}'\left( {{a}_{t+1}} \right)Rda-\beta {{E}_{t}}{V}'\left( {{a}_{t+1}} \right)Rdc\] \[{V}'\left( {{a}_{t}} \right)da={u}'\left( {{c}_{t}} \right)dc-\beta {{E}_{t}}{V}'\left( {{a}_{t+1}} \right)Rdc+\beta {{E}_{t}}{V}'\left( {{a}_{t+1}} \right)Rda\] taking commons we get: \[{V}'\left( {{a}_{t}} \right)da=\left[ {u}'\left( {{c}_{t}} \right)-\beta {{E}_{t}}\left[ {V}'\left( {{a}_{t+1}} \right)R \right] \right]dc+\beta {{E}_{t}}\left[ {V}'\left( {{a}_{t+1}} \right)R \right]da\] From equation 1, ${u}'\left( {{c}_{t}} \right)=\beta R{{E}_{t}}\left[ {V}'\left( {{a}_{t+1}} \right) \right]$. So, we can write:${V}'({{a}_{t}})=\beta R{{E}_{t}}[{V}'({{a}_{(t+1)}})],$ this is an implication of envelop theorem. Combining the results of FOC and envelop theorem, we get, \[{u}'\left( {{c}_{t}} \right)={V}'\left( {{a}_{t}} \right)=\beta R{{E}_{t}}\left[ {V}'\left( {{a}_{t+1}} \right) \right]=\beta R{{E}_{t}}\left[ {u}'\left( {{c}_{t+1}} \right) \right]=\beta \left( 1+r \right){{E}_{t}}\left[ {u}'\left( {{c}_{t+1}} \right) \right]\] or , \[{u}'\left( {{c}_{t}} \right)=\beta \left( 1+r \right){{E}_{t}}\left[ {u}'\left( {{c}_{t+1}} \right) \right]\] which is a simple Euler equation.

This Euler equation is in the general form thus not very tractable. Hall (1978) imposed few assumptions to utility function and derived stunning results. He assumed a particular quadratic preference function as: $u\left( {{c}_{t}} \right)=g{{c}_{t}}-\frac{bc_{t}^{2}}{2}$ where $g>0;b>0$ and, ${u}'(.)=g-b{{c}_{t}}>0$ and${u}''\left( . \right)=-b<0$ . He also assumed a perfect capital market i.e,$\begin{matrix} \beta R=1 & or & \beta ={{R}^{-1}}=\frac{1}{\left( 1+r \right)} \\ \end{matrix}$ Then, the simple Euler equation turns to: ${u}'({{c}_{t}})={{E}_{t}}[{u}'({{c}_{t+1}})]$ or, $g-b{{c}_{t}}={{E}_{t}}\left[ g-b{{c}_{t+1}} \right]$ which gives:${{c}_{t}}={{E}_{t}}\left[ {{c}_{t+1}} \right]$. 

Using law of iterated expectation on the Euler equation we get:${{E}_{0}}\left[ {{E}_{t}}\left[ {{c}_{t+1}} \right] \right]={{E}_{0}}\left[ {{c}_{t+1}} \right]={{E}_{0}}\left[ {{c}_{t}} \right]$ . More generally, ${{c}_{t}}={{E}_{t}}\left[ {{c}_{k}} \right]\forall k>0$

This surprising result states that consumption follows a Random Walk (known as Random Walk Hypothesis (RWH)) i.e., – consumption is entirely unpredictable. And at the best, we can do a naïve prediction that, “tomorrow’s consumption would be same as today’s consumption”. Such represents consumption smoothing. 

This concept is not yet a decision rule for consumption. For more see (here) and (here)

Inflation is Based on Rational Expectation of Future Monetary Policy

Cagan supposed that the demand for real money balances is negative function of expected rate of price rise or inflation ${{e}^{-\gamma {{E}_{t}}{{\pi }_{t+1}}}}$, where $\gamma >0$ is elasticity of money demand w.r.t expected inflation. If it's expected no inflation, the money demand is a unit which represents the public desires to hold a quantity of nominal money balances $M$ equal to price level. If they expect inflation to rise/fall, they desire to hold less/more money because real value of money deteriorates/appreciates.

The inflation expectation is growth of the price i.e. \[{{E}_{t}}{{\pi }_{t+t}}=\ln \left( \tfrac{{{E}_{t}}{{P}_{t+1}}}{{{P}_{t}}} \right)=\ln {{E}_{t}}{{P}_{t+1}}-\ln {{P}_{t}}\] When real money demand equates to real money supply, the money market equilibrates i.e. \[\tfrac{{{M}_{t}}}{{{P}_{t}}}={{e}^{-\gamma {{E}_{t}}{{\pi }_{t+1}}}}\] taking log we get: \[\ln {{M}_{t}}-\ln {{P}_{t}}=-\gamma {{E}_{t}}{{\pi }_{t+1}}\] and substituting value of ${{E}_{t}}{{\pi }_{t+t}}$, we get: \[\ln {{M}_{t}}-\ln {{P}_{t}}=-\gamma \left( \ln {{E}_{t}}{{P}_{t+1}}-\ln {{P}_{t}} \right)\] and arranging for ${{P}_{t}}$, we get: \[\ln {{P}_{t}}=\tfrac{1}{1+\gamma }\ln {{M}_{t}}+\tfrac{\gamma }{1+\gamma }\ln {{E}_{t}}{{P}_{t+1}}\] For notational simplicity let's express as: \[{{p}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\tfrac{\gamma }{1+\gamma }{{E}_{t}}{{p}_{t+1}}\] This represents that current price is weighted average of current money supply $\left( {{m}_{t}} \right)$ and public's expectation on tomorrow's inflation ${{E}_{t}}{{p}_{t+1}}$.

Let's forward for one period, take expectation and with law of iterated expectation we get, \[{{E}_{t}}{{p}_{t+1}}=\tfrac{1}{1+\gamma }{{E}_{t}}{{m}_{t+1}}+\tfrac{\gamma }{1+\gamma }{{E}_{t}}{{p}_{t+2}}\] and substitute this value we have: \[{{p}_{t}}=\tfrac{1}{1+\gamma }\left( {{m}_{t}}+\tfrac{1}{1+\gamma }{{E}_{t}}{{m}_{t+1}} \right)+{{\left( \tfrac{\gamma }{1+\gamma } \right)}^{2}}{{E}_{t}}{{p}_{t+2}}\] and with ${{n}^{th}}$ period forward iteration and repetitive substitution we get the solution as: (click (here) for long-hand solution) \[{{p}_{t}}=\tfrac{1}{1+\gamma }\sum\limits_{i=0}^{n}{{{\left( \tfrac{1}{1+\gamma } \right)}^{i}}{{E}_{t}}{{m}_{t+i}}}+{{\left( \tfrac{\gamma }{1+\gamma } \right)}^{i+1}}{{E}_{t}}{{p}_{t+i+1}}\] and $\underset{i\to \infty }{\mathop{\lim }}\,{{\left( \tfrac{\gamma }{1+\gamma } \right)}^{i+1}}{{E}_{t}}{{p}_{t+i+1}}=0\because \gamma >0\And \tfrac{\gamma }{1+\gamma }<1$, or this represents the expectation of all the future fundaments converges to zero. Therefore the solution emerges as: \[{{p}_{t}}=\tfrac{1}{1+\gamma }\sum\limits_{i=0}^{\infty }{{{\left( \tfrac{1}{1+\gamma } \right)}^{i}}{{E}_{t}}{{m}_{t+i}}}\] and this expression represents price today is the weighted average of current and future expected nominal money stocks and more weight is placed on near future than far future money stocks. This solution defines a equilibrium price and inflation is based on rational expectation of future monetary policy.

How Money Supply Affects Prices– A Rational Expectation Solution Based on Methods of Undetermined Co-efficients

Let's define the money supply as: \[{{m}_{t}}={{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}t+\lambda {{m}_{t-1}}\] where ${{m}_{t}}$ is the $Ln{{M}_{t}}$.

Now we want to define what is price as per the rational expectation. For this we will use the method of undetermined coefficients.

At first let's propose a guess as: \[{{p}_{t}}={{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}\] where ${{p}_{t}}$ is the $Ln{{P}_{t}}$. Forwarding one period and taking expectation, \[{{E}_{t}}{{p}_{t+1}}={{A}_{0}}+{{A}_{1}}(t+1)+{{A}_{2}}{{E}_{t}}{{m}_{t+1}}\] substituting value of ${{E}_{t}}{{m}_{t+1}}={{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}(t+1)+\lambda {{m}_{t}}$ we get \[{{E}_{t}}{{m}_{t+1}}={{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}(t+1)+\lambda {{m}_{t}}\] We know that ${{p}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\tfrac{\gamma }{1+\gamma }{{E}_{t}}{{p}_{t+1}}$, see previous post (here) so, substituting proposal and value of ${{E}_{t}}{{p}_{t+1}}$ we get: \[{{p}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\tfrac{\gamma }{1+\gamma }{{E}_{t}}{{p}_{t+1}}\] \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\tfrac{\gamma }{1+\gamma }{{E}_{t}}{{p}_{t+1}}\] \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\tfrac{\gamma }{1+\gamma }\left[ {{A}_{0}}+{{A}_{1}}+{{A}_{1}}t+{{A}_{2}}\left[ {{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}(t+1)+\lambda {{m}_{t}} \right] \right]\] \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\left[ \tfrac{\gamma }{1+\gamma }{{A}_{0}}+\tfrac{\gamma }{1+\gamma }{{A}_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{1}}t+\tfrac{\gamma }{1+\gamma }\left[ {{A}_{2}}{{\theta }_{0}}+{{A}_{2}}(1-\lambda )({{\theta }_{1}}t+{{\theta }_{1}})+{{A}_{2}}\lambda {{m}_{t}} \right] \right]\] \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\tfrac{1}{1+\gamma }{{m}_{t}}+\left[ \tfrac{\gamma }{1+\gamma }{{A}_{0}}+\tfrac{\gamma }{1+\gamma }{{A}_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{1}}t+\tfrac{\gamma }{1+\gamma }\left[ {{A}_{2}}{{\theta }_{0}}+{{A}_{2}}(1-\lambda ){{\theta }_{1}}t+{{A}_{2}}(1-\lambda ){{\theta }_{1}}+{{A}_{2}}\lambda {{m}_{t}} \right] \right]\] \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\tfrac{\gamma }{1+\gamma }{{A}_{0}}+\tfrac{\gamma }{1+\gamma }{{A}_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{2}}{{\theta }_{0}}+\tfrac{\gamma }{1+\gamma }{{A}_{2}}(1-\lambda ){{\theta }_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{1}}t+\tfrac{\gamma }{1+\gamma }{{A}_{2}}(1-\lambda ){{\theta }_{1}}t+\tfrac{1}{1+\gamma }{{m}_{t}}+\tfrac{\gamma }{1+\gamma }{{A}_{2}}\lambda {{m}_{t}}\] \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\tfrac{\gamma }{1+\gamma }\left[ {{A}_{0}}+{{A}_{1}}+{{A}_{2}}{{\theta }_{0}}+{{A}_{2}}(1-\lambda ){{\theta }_{1}} \right]+\left[ \tfrac{\gamma }{1+\gamma }{{A}_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{2}}(1-\lambda ){{\theta }_{1}} \right]t+\left[ \tfrac{1}{1+\gamma }+\tfrac{\gamma }{1+\gamma }{{A}_{2}}\lambda \right]{{m}_{t}}\] Matching coefficient \[{{A}_{0}}+{{A}_{1}}t+{{A}_{2}}{{m}_{t}}=\underbrace{\tfrac{\gamma }{1+\gamma }\left[ {{A}_{0}}+{{A}_{1}}+{{A}_{2}}{{\theta }_{0}}+{{A}_{2}}(1-\lambda ){{\theta }_{1}} \right]}_{{{A}_{0}}}+\underbrace{\left[ \tfrac{\gamma }{1+\gamma }{{A}_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{2}}(1-\lambda ){{\theta }_{1}} \right]t}_{{{A}_{1}}t}+\underbrace{\left[ \tfrac{1}{1+\gamma }+\tfrac{\gamma }{1+\gamma }{{A}_{2}}\lambda \right]{{m}_{t}}}_{{{A}_{2}}{{m}_{t}}}\] Solving for ${{A}_{2}},{{A}_{1}}\And {{A}_{0}}$ Solving for ${{A}_{2}}$: \[{{A}_{2}}=\tfrac{1}{1+\gamma }+\tfrac{\gamma }{1+\gamma }{{A}_{2}}\lambda\] \[{{A}_{2}}-\tfrac{\gamma }{1+\gamma }{{A}_{2}}\lambda =\tfrac{1}{1+\gamma }\] \[{{A}_{2}}\left( 1-\tfrac{\gamma \lambda }{1+\gamma } \right)=\tfrac{1}{1+\gamma }\] \[{{A}_{2}}\left( \tfrac{1+\gamma -\gamma \lambda }{1+\gamma } \right)=\tfrac{1}{1+\gamma }\] \[{{A}_{2}}=\frac{1}{1+\gamma -\gamma \lambda }\] Solving for ${{A}_{1}}$: \[{{A}_{1}}=\tfrac{\gamma }{1+\gamma }{{A}_{1}}+\tfrac{\gamma }{1+\gamma }{{A}_{2}}(1-\lambda ){{\theta }_{1}}\] \[{{A}_{1}}\left( 1-\tfrac{\gamma }{1+\gamma } \right)=\tfrac{\gamma }{1+\gamma }{{A}_{2}}(1-\lambda ){{\theta }_{1}}\] \[{{A}_{1}}\left( \tfrac{1}{1+\gamma } \right)=\tfrac{\gamma }{1+\gamma }\frac{1}{1+\gamma -\gamma \lambda }(1-\lambda ){{\theta }_{1}}\] \[{{A}_{1}}=\gamma \frac{1}{1+\gamma -\gamma \lambda }(1-\lambda ){{\theta }_{1}}\] \[{{A}_{1}}=\frac{\gamma {{\theta }_{1}}(1-\lambda )}{1+\gamma -\gamma \lambda }\] Solving for ${{A}_{0}}$: \[{{A}_{0}}=\tfrac{\gamma }{1+\gamma }\left[ {{A}_{0}}+{{A}_{1}}+{{A}_{2}}{{\theta }_{0}}+{{A}_{2}}(1-\lambda ){{\theta }_{1}} \right]\] \[{{A}_{0}}-\tfrac{\gamma }{1+\gamma }{{A}_{0}}={{A}_{1}}+{{A}_{2}}\left[ {{\theta }_{0}}+(1-\lambda ){{\theta }_{1}} \right]\] \[{{A}_{0}}\left( 1-\tfrac{\gamma }{1+\gamma } \right)=\frac{\gamma {{\theta }_{1}}(1-\lambda )}{1+\gamma -\gamma \lambda }+\frac{1}{1+\gamma -\gamma \lambda }\left[ {{\theta }_{0}}+(1-\lambda ){{\theta }_{1}} \right]\] \[{{A}_{0}}\left( \tfrac{1}{1+\gamma } \right)=\frac{{{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}+\gamma {{\theta }_{1}}(1-\lambda )}{1+\gamma -\gamma \lambda }\] \[{{A}_{0}}=\frac{\left( 1+\gamma \right)\left[ {{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}(1+\gamma ) \right]}{1+\gamma -\gamma \lambda }\] This equation gives the rational expectation model of how the money supply influence the price. \[{{p}_{t}}=\frac{\left( 1+\gamma \right)\left[ {{\theta }_{0}}+(1-\lambda ){{\theta }_{1}}(1+\gamma ) \right]}{1+\gamma -\gamma \lambda }+\frac{\gamma {{\theta }_{1}}(1-\lambda )}{1+\gamma -\gamma \lambda }t+\frac{1}{1+\gamma -\gamma \lambda }{{m}_{t}}\]

Rational Expectation Solution for Inter-temporal Budget Constraint Using Repetitive Substitution Tecnhique

At time ${t}$, the expectation for ${t+1}$ net assets is ${{E}_{t}}\left( {{a}_{t+1}} \right)$, which  is indeed the residuals of  income and consumption plus the net assets at time ${t}$  i.e. $\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)$ and it grows at the fixed rate of $\left( r>0 \right)$. This can be represented by: \[{{E}_{t}}\left( {{a}_{t+1}} \right)=\left( {{y}_{t}}-{{c}_{t}}+{{a}_{t}} \right)\left( 1+r \right)\] Arranging for ${{a}_{t}}$ we get, \[{{E}_{t}}\left( {{a}_{t+1}} \right)={{y}_{t}}\left( 1+r \right)-{{c}_{t}}\left( 1+r \right)+{{a}_{t}}\left( 1+r \right)\] \[{{E}_{t}}\left( {{a}_{t+1}} \right)-{{y}_{t}}\left( 1+r \right)+{{c}_{t}}\left( 1+r \right)={{a}_{t}}\left( 1+r \right)\] \[{{a}_{t}}\left( 1+r \right)={{c}_{t}}\left( 1+r \right)-{{y}_{t}}\left( 1+r \right)+{{E}_{t}}\left( {{a}_{t+1}} \right)\] \[{{a}_{t}}={{c}_{t}}\frac{\left( 1+r \right)}{\left( 1+r \right)}-{{y}_{t}}\frac{\left( 1+r \right)}{\left( 1+r \right)}+\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)\] \[{{a}_{t}}={{c}_{t}}-{{y}_{t}}+\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)\] Let's suppose: ${{z}_{t}}={{c}_{t}}-{{y}_{t}}\And \rho =\frac{1}{\left( 1+r \right)}$ so that we can re-express this equation as: \[{{a}_{t}}={{z}_{t}}+\rho {{E}_{t}}\left( {{a}_{t+1}} \right)\to eqn\left( 1 \right)\] This equation tries to explain that today's net-assets depends upon the expectation for tomorrow's net assets. Thus this is a forward looking equation. We can solve this equation with forward looking approach and with law of iterated expectation. Let's forward one-period we get, \[{{E}_{t}}{{a}_{t+1}}={{E}_{t}}{{z}_{t+1}}+\rho {{E}_{t}}\left( {{E}_{t+1}}\left( {{a}_{t+2}} \right) \right)\] With the law of iterated expectation we can write: \[{{E}_{t}}\left( {{E}_{t+1}}\left( {{a}_{t+2}} \right) \right)={{E}_{t}}{{a}_{t+2}}\] The law of iterated expectation simply states that, today's expectation for tomorrow expectation on day after tomorrow net-asset is same as today's expectation for day after tomorrow's expectation for net-assets. Then we get: \[{{E}_{t}}{{a}_{t+1}}={{E}_{t}}{{z}_{t+1}}+\rho {{E}_{t}}{{a}_{t+2}}\to eqn\left( 2 \right)\] Substituting $eqn\left( 2 \right)$ in $eqn\left( 1 \right)$, we get: \[{{a}_{t}}={{z}_{t}}+\rho \left[ {{E}_{t}}{{z}_{t+1}}+\rho {{E}_{t}}{{a}_{t+2}} \right]\] \[{{a}_{t}}={{z}_{t}}+\rho {{E}_{t}}{{z}_{t+1}}+{{\rho }^{2}}{{E}_{t}}{{a}_{t+2}}\to eqn\left( 3 \right)\] again, consider $eqn\left( 1 \right)$ and forward by two-period we get: \[{{E}_{t}}{{a}_{t+2}}={{E}_{t}}{{z}_{t+2}}+\rho {{E}_{t}}\left( {{E}_{t+2}}\left( {{a}_{t+3}} \right) \right)\] again, we the law of iterated expectation we can get, \[{{E}_{t}}{{a}_{t+2}}={{E}_{t}}{{z}_{t+2}}+\rho {{E}_{t}}{{a}_{t+3}}\] Substituting value of this ${{E}_{t}}{{a}_{t+2}}$ in \[{{a}_{t}}={{z}_{t}}+\rho {{E}_{t}}{{z}_{t+1}}+{{\rho }^{2}}\left[ {{E}_{t}}{{z}_{t+2}}+\rho {{E}_{t}}{{a}_{t+3}} \right]\] \[{{a}_{t}}={{z}_{t}}+\rho {{E}_{t}}{{z}_{t+1}}+{{\rho }^{2}}{{E}_{t}}{{z}_{t+2}}+{{\rho }^{3}}{{E}_{t}}{{a}_{t+3}}\to eqn\left( 4 \right)\] We can re-express $eqn\left( 4 \right)$ as: \[{{a}_{t}}={{\rho }^{0}}{{E}_{t}}{{z}_{t}}+{{\rho }^{1}}{{E}_{t}}{{z}_{t+1}}+{{\rho }^{2}}{{E}_{t}}{{z}_{t+2}}+{{\rho }^{3}}{{E}_{t}}{{a}_{t+3}}\] \[\because {{E}_{t}}{{z}_{t}}={{z}_{t}};{{\rho }^{0}}=1\And {{\rho }^{1}}=\rho\] then \[{{a}_{t}}=\sum\limits_{i=0}^{2}{{{\rho }^{i}}{{E}_{t}}{{z}_{t+i}}}+{{\rho }^{2+1}}{{E}_{t}}{{a}_{t+2+1}}\] Which is a solution for two-period forward iteration. With this logic we can derive a solution for an $n-th$ period forward iteration. \[{{a}_{t}}=\sum\limits_{i=0}^{n}{{{\rho }^{i}}{{E}_{t}}{{z}_{t+i}}}+{{\rho }^{n+1}}{{E}_{t}}{{a}_{t+n+1}}\] For $n\to \infty$, taking limits, we get, \[{{\rho }^{n+1}}{{E}_{t}}{{a}_{t+n+1}}=0\] $iff$ $|\rho |<1$. In our case, $r>0$ ie. $0<\frac{1}{1+r}<1$ or $0<\rho <1$ So, our solution for $n\to \infty$is: \[{{a}_{t}}=\sum\limits_{i=0}^{\infty }{{{\rho }^{i}}{{E}_{t}}{{z}_{t+i}}}\] Further, \[{{a}_{t}}=\sum\limits_{i=0}^{\infty }{{{\left( \frac{1}{1+r} \right)}^{i}}{{E}_{t}}\left[ {{c}_{t+i}}-{{y}_{t+i}} \right]}\]$\because {{z}_{t}}={{c}_{t}}-{{y}_{t}}\And \rho =\frac{1}{\left( 1+r \right)}$ \[{{a}_{t}}=\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{t}}{{c}_{t+i}}}-\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{t}}{{y}_{t+i}}}\] \[\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{t}}{{c}_{t+i}}}={{a}_{t}}+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{t}}{{y}_{t+i}}}\] This expression is known as inter-temporal budget constraint. This can be interpreted as the present value sum of current and futu household consumption is current stock of capital and present value sum of current and future incomes.

The above expression can be re-expressed by indexing time to $t$ to $t=0$. Substituting the value of ${{a}_{t}}={{c}_{t}}-{{y}_{t}}+\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)$ \[\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{t}}{{c}_{t+i}}}={{c}_{t}}-{{y}_{t}}+\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{t}}{{y}_{t+i}}}\] \[{{E}_{t}}{{c}_{t}}+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{c}_{i}}}={{c}_{t}}-{{y}_{t}}+\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)+{{E}_{t}}{{y}_{t}}+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{y}_{i}}}\] \[{{c}_{t}}+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{c}_{i}}}={{c}_{t}}-{{y}_{t}}+\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)+{{y}_{t}}+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{y}_{i}}}\] \[{{E}_{t}}{{c}_{t}}={{c}_{t}}\And {{E}_{t}}{{y}_{t}}={{y}_{t}}\] \[\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{c}_{i}}}=\frac{1}{\left( 1+r \right)}{{E}_{t}}\left( {{a}_{t+1}} \right)+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{y}_{i}}}\] \[\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{c}_{i}}}=\frac{1}{\left( 1+r \right)}{{a}_{t}}\left( 1+r \right)+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{y}_{i}}}\] \[\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{c}_{i}}}={{a}_{t}}+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{y}_{i}}}\] \[\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{c}_{i}}}={{a}_{0}}\left( 1+r \right)+\sum\limits_{i=0}^{\infty }{\frac{1}{{{\left( 1+r \right)}^{i}}}{{E}_{0}}{{y}_{i}}}\to eqn\left( 5 \right)\] This equation gives the foundation to link the inter-temporal budget constraint on the Hall's (1978)  Radom Walk Consumption Hypothesis to derived the Certainty Equivalence Equation (CEQ).

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).

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.

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

See my previous posts here, here, here, 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.

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

See (here,  here, here 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.

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 Paradox: Kuznet 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.

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. \[{{V}_{E}}=\frac{w-\bar{e}}{b+\rho }+{{V}_{U}}\left( \underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{{{e}^{-\rho \Delta t}}-{{e}^{-(b+\rho )\Delta t}}}{1-{{e}^{-(b+\rho )\Delta t}}} \right)\] \[{{V}_{E}}=\frac{w-\bar{e}}{b+\rho }+{{V}_{U}}\left( \frac{b}{b+\rho } \right)\] \[{{V}_{E}}\left( b+\rho \right)=\left( w-\bar{e} \right)+b{{V}_{U}}\] \[\rho {{V}_{E}}=\left( w-\bar{e} \right)-b\left( {{V}_{E}}-{{V}_{U}} \right)\]

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

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.