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Calvo Aggregate Price Dynamics
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This question actually comes from economics but, AS USUAL, only the name of the variable and idea (price) is from economics there. The technique used is pure math and since economists are mostly sloppy when explaining the math in the background I decided to ask it here to mathematicians. So it goes as follows:
There are a continuum number of firms over the interval $ [0,1] $ indexed by $ i $. Time framework is discrete. In each time period $t$, only $ (1-theta) $ fraction of firms get the chance of setting (optimizing) their prices (this means nothing else but setting price $P_t(i) = P_t(i)^star $ that maximizes its profit). The rest $ theta $ fraction leave there prices $ P_t(i) = P_t-1(i) $. Since each firm $ i $ is identical each firm who gets the chance to set the price will set identical price $ P_t(i) = P_t^star $.
Aggregate Price Level in each period $t$ is defined in the following way:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon $$
Given this framework the book Monetary Policy Inflation and Business Cycle by Gali(2008) chapter 3 page 62 explains Aggregate Price Level Dynamics in the following passage:
Let $ S(t) subset [0,1] $ represent the set of firms not reoptimizing their posted price in period $t$. Using the definition of the aggregate price level and the fact that all firms resetting the prices will choose an identical price $ P_t^* $,
$$ mathbbP_t = bigg[int_S(t) P_t-1(i)^1-epsilon di + (1-theta)(P_t^*)^1-epsilon bigg] ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon$$
where the second equality follows from the fact that the distribution of prices among firms not adjusting in period $t$ corresponds to the distribution of effective prices in period $t-1$, though with total mass reduced to $ theta $.
I have the problem in understanding this second equality where it substitutes integral with $theta (mathbbP_t-1)^1-epsilon$. Can anybody explain it? Does the above explanation that the author provides make sense to you?
integration
edited Jul 22 at 19:27
rubik
6,69622357
asked Jul 22 at 9:50
G.T.
1009
add a comment |Â
up vote
0
down vote
favorite
This question actually comes from economics but, AS USUAL, only the name of the variable and idea (price) is from economics there. The technique used is pure math and since economists are mostly sloppy when explaining the math in the background I decided to ask it here to mathematicians. So it goes as follows:
There are a continuum number of firms over the interval $ [0,1] $ indexed by $ i $. Time framework is discrete. In each time period $t$, only $ (1-theta) $ fraction of firms get the chance of setting (optimizing) their prices (this means nothing else but setting price $P_t(i) = P_t(i)^star $ that maximizes its profit). The rest $ theta $ fraction leave there prices $ P_t(i) = P_t-1(i) $. Since each firm $ i $ is identical each firm who gets the chance to set the price will set identical price $ P_t(i) = P_t^star $.
Aggregate Price Level in each period $t$ is defined in the following way:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon $$
Given this framework the book Monetary Policy Inflation and Business Cycle by Gali(2008) chapter 3 page 62 explains Aggregate Price Level Dynamics in the following passage:
Let $ S(t) subset [0,1] $ represent the set of firms not reoptimizing their posted price in period $t$. Using the definition of the aggregate price level and the fact that all firms resetting the prices will choose an identical price $ P_t^* $,
$$ mathbbP_t = bigg[int_S(t) P_t-1(i)^1-epsilon di + (1-theta)(P_t^*)^1-epsilon bigg] ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon$$
where the second equality follows from the fact that the distribution of prices among firms not adjusting in period $t$ corresponds to the distribution of effective prices in period $t-1$, though with total mass reduced to $ theta $.
I have the problem in understanding this second equality where it substitutes integral with $theta (mathbbP_t-1)^1-epsilon$. Can anybody explain it? Does the above explanation that the author provides make sense to you?
integration
edited Jul 22 at 19:27
rubik
6,69622357
asked Jul 22 at 9:50
G.T.
1009
1
what is epsilon? In page 146 of this book "Optimal Monetary policy under uncertainty", he gives the derivation for aggregate price level very similar to yours, yours is a little bit complicated with $epsilon$ otherwise, it is the same.
– Satish Ramanathan
Jul 22 at 11:10
1
books.google.co.in/…
– Satish Ramanathan
Jul 22 at 11:11
$epsilon$ is elasticity of substitute in economic terms but mathematically it is just positive real number.
– G.T.
Jul 22 at 11:23
You can substitute $P_t(i)^1-epsilon$ and $P_t^*1-epsilon$ to simple $p_t$ and $p^*_t$ in the derivation of the book and you will see the derivation in the book makes sense.
– Satish Ramanathan
Jul 22 at 11:27
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This question actually comes from economics but, AS USUAL, only the name of the variable and idea (price) is from economics there. The technique used is pure math and since economists are mostly sloppy when explaining the math in the background I decided to ask it here to mathematicians. So it goes as follows:
There are a continuum number of firms over the interval $ [0,1] $ indexed by $ i $. Time framework is discrete. In each time period $t$, only $ (1-theta) $ fraction of firms get the chance of setting (optimizing) their prices (this means nothing else but setting price $P_t(i) = P_t(i)^star $ that maximizes its profit). The rest $ theta $ fraction leave there prices $ P_t(i) = P_t-1(i) $. Since each firm $ i $ is identical each firm who gets the chance to set the price will set identical price $ P_t(i) = P_t^star $.
Aggregate Price Level in each period $t$ is defined in the following way:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon $$
Given this framework the book Monetary Policy Inflation and Business Cycle by Gali(2008) chapter 3 page 62 explains Aggregate Price Level Dynamics in the following passage:
Let $ S(t) subset [0,1] $ represent the set of firms not reoptimizing their posted price in period $t$. Using the definition of the aggregate price level and the fact that all firms resetting the prices will choose an identical price $ P_t^* $,
$$ mathbbP_t = bigg[int_S(t) P_t-1(i)^1-epsilon di + (1-theta)(P_t^*)^1-epsilon bigg] ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon$$
where the second equality follows from the fact that the distribution of prices among firms not adjusting in period $t$ corresponds to the distribution of effective prices in period $t-1$, though with total mass reduced to $ theta $.
I have the problem in understanding this second equality where it substitutes integral with $theta (mathbbP_t-1)^1-epsilon$. Can anybody explain it? Does the above explanation that the author provides make sense to you?
integration
edited Jul 22 at 19:27
rubik
6,69622357
asked Jul 22 at 9:50
G.T.
1009
This question actually comes from economics but, AS USUAL, only the name of the variable and idea (price) is from economics there. The technique used is pure math and since economists are mostly sloppy when explaining the math in the background I decided to ask it here to mathematicians. So it goes as follows:
There are a continuum number of firms over the interval $ [0,1] $ indexed by $ i $. Time framework is discrete. In each time period $t$, only $ (1-theta) $ fraction of firms get the chance of setting (optimizing) their prices (this means nothing else but setting price $P_t(i) = P_t(i)^star $ that maximizes its profit). The rest $ theta $ fraction leave there prices $ P_t(i) = P_t-1(i) $. Since each firm $ i $ is identical each firm who gets the chance to set the price will set identical price $ P_t(i) = P_t^star $.
Aggregate Price Level in each period $t$ is defined in the following way:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon $$
Given this framework the book Monetary Policy Inflation and Business Cycle by Gali(2008) chapter 3 page 62 explains Aggregate Price Level Dynamics in the following passage:
Let $ S(t) subset [0,1] $ represent the set of firms not reoptimizing their posted price in period $t$. Using the definition of the aggregate price level and the fact that all firms resetting the prices will choose an identical price $ P_t^* $,
$$ mathbbP_t = bigg[int_S(t) P_t-1(i)^1-epsilon di + (1-theta)(P_t^*)^1-epsilon bigg] ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon$$
where the second equality follows from the fact that the distribution of prices among firms not adjusting in period $t$ corresponds to the distribution of effective prices in period $t-1$, though with total mass reduced to $ theta $.
I have the problem in understanding this second equality where it substitutes integral with $theta (mathbbP_t-1)^1-epsilon$. Can anybody explain it? Does the above explanation that the author provides make sense to you?
integration
edited Jul 22 at 19:27
rubik
6,69622357
asked Jul 22 at 9:50
G.T.
1009
edited Jul 22 at 19:27
rubik
6,69622357
edited Jul 22 at 19:27
rubik
6,69622357
edited Jul 22 at 19:27
rubik
6,69622357
6,69622357
asked Jul 22 at 9:50
G.T.
1009
asked Jul 22 at 9:50
G.T.
1009
asked Jul 22 at 9:50
G.T.
1009
1009
1
what is epsilon? In page 146 of this book "Optimal Monetary policy under uncertainty", he gives the derivation for aggregate price level very similar to yours, yours is a little bit complicated with $epsilon$ otherwise, it is the same.
– Satish Ramanathan
Jul 22 at 11:10
1
books.google.co.in/…
– Satish Ramanathan
Jul 22 at 11:11
$epsilon$ is elasticity of substitute in economic terms but mathematically it is just positive real number.
– G.T.
Jul 22 at 11:23
You can substitute $P_t(i)^1-epsilon$ and $P_t^*1-epsilon$ to simple $p_t$ and $p^*_t$ in the derivation of the book and you will see the derivation in the book makes sense.
– Satish Ramanathan
Jul 22 at 11:27
add a comment |Â
1
what is epsilon? In page 146 of this book "Optimal Monetary policy under uncertainty", he gives the derivation for aggregate price level very similar to yours, yours is a little bit complicated with $epsilon$ otherwise, it is the same.
– Satish Ramanathan
Jul 22 at 11:10
1
books.google.co.in/…
– Satish Ramanathan
Jul 22 at 11:11
$epsilon$ is elasticity of substitute in economic terms but mathematically it is just positive real number.
– G.T.
Jul 22 at 11:23
You can substitute $P_t(i)^1-epsilon$ and $P_t^*1-epsilon$ to simple $p_t$ and $p^*_t$ in the derivation of the book and you will see the derivation in the book makes sense.
– Satish Ramanathan
Jul 22 at 11:27
1
1
what is epsilon? In page 146 of this book "Optimal Monetary policy under uncertainty", he gives the derivation for aggregate price level very similar to yours, yours is a little bit complicated with $epsilon$ otherwise, it is the same.
– Satish Ramanathan
Jul 22 at 11:10
what is epsilon? In page 146 of this book "Optimal Monetary policy under uncertainty", he gives the derivation for aggregate price level very similar to yours, yours is a little bit complicated with $epsilon$ otherwise, it is the same.
– Satish Ramanathan
Jul 22 at 11:10
1
1
books.google.co.in/…
– Satish Ramanathan
Jul 22 at 11:11
books.google.co.in/…
– Satish Ramanathan
Jul 22 at 11:11
$epsilon$ is elasticity of substitute in economic terms but mathematically it is just positive real number.
– G.T.
Jul 22 at 11:23
$epsilon$ is elasticity of substitute in economic terms but mathematically it is just positive real number.
– G.T.
Jul 22 at 11:23
You can substitute $P_t(i)^1-epsilon$ and $P_t^*1-epsilon$ to simple $p_t$ and $p^*_t$ in the derivation of the book and you will see the derivation in the book makes sense.
– Satish Ramanathan
Jul 22 at 11:27
You can substitute $P_t(i)^1-epsilon$ and $P_t^*1-epsilon$ to simple $p_t$ and $p^*_t$ in the derivation of the book and you will see the derivation in the book makes sense.
– Satish Ramanathan
Jul 22 at 11:27
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
After several hours of research I finally found out (thanks to commented book by Satish Ramanathan in the comment section of the question) that the author skips several steps to derive the Aggregate Price Level in period $ t $. For those who love deep understanding and will question the same step in the future or those who blindly use that equality here, I will provide those steps and detailed derivation of how to get to the following equality:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon $$
Before that let's remember that:
For those firms who set their price in period $t$, $P_t(i) = P_t^*$ and the length of the interval over which $P_t(i) = P_t^*$ will be $ 1-theta $.
For those firms who set their price in period $t-1$ and then maintained the same price in period $ t $, $P_t(i) = P_t-1^*$ and the length of the interval over which $ P_t(i) = P_t-1^* $ will be $theta(1-theta) $.
For those firms who set their price in period $t-2$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-2^* $ and the length of the interval over which $ P_t(i) = P_t-2^* $ will be $theta^2(1-theta) $
For those firms who set their price in period $t-3$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-3^* $ and the length of the interval over which $ P_t(i) = P_t-3^* $ will be $theta^3(1-theta) $
We can generalize this and for those firms who set their price in period $t-j$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-j^* $ and the length of the interval over which $ P_t(i) = P_t-j^* $ will be $theta^j(1-theta) $
Since $ sum_j=o^infty theta^j (1-theta) = 1$, we can then write
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon =bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon $$
Using the same transformation
$$ mathbbP_t-1 = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg)^frac11-epsilon $$
this implies that
$$ mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg) $$
now multiply both sides by $ theta$
$$ theta mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^j+1P_t-1-j^*^1-epsilon bigg) $$
Notice that we can rewrite $mathbbP_t $ using $ mathbbP_t-1 $:
$$ mathbbP_t = bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon = bigg ( (1-theta) P_t^* + sum_j=0^infty (1-theta)theta^j+1P_t-j-1^*^1-epsilon bigg)^frac11-epsilon =bigg( (1-theta) P_t^*^1-epsilon + theta mathbbP_t-1^1-epsilon bigg)^frac11-epsilon $$
For those expert in math please edit if you see the lack of rigor.
1
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
After several hours of research I finally found out (thanks to commented book by Satish Ramanathan in the comment section of the question) that the author skips several steps to derive the Aggregate Price Level in period $ t $. For those who love deep understanding and will question the same step in the future or those who blindly use that equality here, I will provide those steps and detailed derivation of how to get to the following equality:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon $$
Before that let's remember that:
For those firms who set their price in period $t$, $P_t(i) = P_t^*$ and the length of the interval over which $P_t(i) = P_t^*$ will be $ 1-theta $.
For those firms who set their price in period $t-1$ and then maintained the same price in period $ t $, $P_t(i) = P_t-1^*$ and the length of the interval over which $ P_t(i) = P_t-1^* $ will be $theta(1-theta) $.
For those firms who set their price in period $t-2$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-2^* $ and the length of the interval over which $ P_t(i) = P_t-2^* $ will be $theta^2(1-theta) $
For those firms who set their price in period $t-3$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-3^* $ and the length of the interval over which $ P_t(i) = P_t-3^* $ will be $theta^3(1-theta) $
We can generalize this and for those firms who set their price in period $t-j$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-j^* $ and the length of the interval over which $ P_t(i) = P_t-j^* $ will be $theta^j(1-theta) $
Since $ sum_j=o^infty theta^j (1-theta) = 1$, we can then write
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon =bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon $$
Using the same transformation
$$ mathbbP_t-1 = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg)^frac11-epsilon $$
this implies that
$$ mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg) $$
now multiply both sides by $ theta$
$$ theta mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^j+1P_t-1-j^*^1-epsilon bigg) $$
Notice that we can rewrite $mathbbP_t $ using $ mathbbP_t-1 $:
$$ mathbbP_t = bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon = bigg ( (1-theta) P_t^* + sum_j=0^infty (1-theta)theta^j+1P_t-j-1^*^1-epsilon bigg)^frac11-epsilon =bigg( (1-theta) P_t^*^1-epsilon + theta mathbbP_t-1^1-epsilon bigg)^frac11-epsilon $$
For those expert in math please edit if you see the lack of rigor.
1
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
add a comment |Â
up vote
2
down vote
After several hours of research I finally found out (thanks to commented book by Satish Ramanathan in the comment section of the question) that the author skips several steps to derive the Aggregate Price Level in period $ t $. For those who love deep understanding and will question the same step in the future or those who blindly use that equality here, I will provide those steps and detailed derivation of how to get to the following equality:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon $$
Before that let's remember that:
For those firms who set their price in period $t$, $P_t(i) = P_t^*$ and the length of the interval over which $P_t(i) = P_t^*$ will be $ 1-theta $.
For those firms who set their price in period $t-1$ and then maintained the same price in period $ t $, $P_t(i) = P_t-1^*$ and the length of the interval over which $ P_t(i) = P_t-1^* $ will be $theta(1-theta) $.
For those firms who set their price in period $t-2$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-2^* $ and the length of the interval over which $ P_t(i) = P_t-2^* $ will be $theta^2(1-theta) $
For those firms who set their price in period $t-3$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-3^* $ and the length of the interval over which $ P_t(i) = P_t-3^* $ will be $theta^3(1-theta) $
We can generalize this and for those firms who set their price in period $t-j$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-j^* $ and the length of the interval over which $ P_t(i) = P_t-j^* $ will be $theta^j(1-theta) $
Since $ sum_j=o^infty theta^j (1-theta) = 1$, we can then write
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon =bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon $$
Using the same transformation
$$ mathbbP_t-1 = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg)^frac11-epsilon $$
this implies that
$$ mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg) $$
now multiply both sides by $ theta$
$$ theta mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^j+1P_t-1-j^*^1-epsilon bigg) $$
Notice that we can rewrite $mathbbP_t $ using $ mathbbP_t-1 $:
$$ mathbbP_t = bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon = bigg ( (1-theta) P_t^* + sum_j=0^infty (1-theta)theta^j+1P_t-j-1^*^1-epsilon bigg)^frac11-epsilon =bigg( (1-theta) P_t^*^1-epsilon + theta mathbbP_t-1^1-epsilon bigg)^frac11-epsilon $$
For those expert in math please edit if you see the lack of rigor.
1
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
add a comment |Â
up vote
2
down vote
up vote
2
down vote
After several hours of research I finally found out (thanks to commented book by Satish Ramanathan in the comment section of the question) that the author skips several steps to derive the Aggregate Price Level in period $ t $. For those who love deep understanding and will question the same step in the future or those who blindly use that equality here, I will provide those steps and detailed derivation of how to get to the following equality:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon $$
Before that let's remember that:
For those firms who set their price in period $t$, $P_t(i) = P_t^*$ and the length of the interval over which $P_t(i) = P_t^*$ will be $ 1-theta $.
For those firms who set their price in period $t-1$ and then maintained the same price in period $ t $, $P_t(i) = P_t-1^*$ and the length of the interval over which $ P_t(i) = P_t-1^* $ will be $theta(1-theta) $.
For those firms who set their price in period $t-2$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-2^* $ and the length of the interval over which $ P_t(i) = P_t-2^* $ will be $theta^2(1-theta) $
For those firms who set their price in period $t-3$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-3^* $ and the length of the interval over which $ P_t(i) = P_t-3^* $ will be $theta^3(1-theta) $
We can generalize this and for those firms who set their price in period $t-j$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-j^* $ and the length of the interval over which $ P_t(i) = P_t-j^* $ will be $theta^j(1-theta) $
Since $ sum_j=o^infty theta^j (1-theta) = 1$, we can then write
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon =bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon $$
Using the same transformation
$$ mathbbP_t-1 = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg)^frac11-epsilon $$
this implies that
$$ mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg) $$
now multiply both sides by $ theta$
$$ theta mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^j+1P_t-1-j^*^1-epsilon bigg) $$
Notice that we can rewrite $mathbbP_t $ using $ mathbbP_t-1 $:
$$ mathbbP_t = bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon = bigg ( (1-theta) P_t^* + sum_j=0^infty (1-theta)theta^j+1P_t-j-1^*^1-epsilon bigg)^frac11-epsilon =bigg( (1-theta) P_t^*^1-epsilon + theta mathbbP_t-1^1-epsilon bigg)^frac11-epsilon $$
For those expert in math please edit if you see the lack of rigor.
After several hours of research I finally found out (thanks to commented book by Satish Ramanathan in the comment section of the question) that the author skips several steps to derive the Aggregate Price Level in period $ t $. For those who love deep understanding and will question the same step in the future or those who blindly use that equality here, I will provide those steps and detailed derivation of how to get to the following equality:
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon = bigg[ theta (mathbbP_t-1)^1-epsilon + (1- theta ) (P_t^*)^1-epsilon bigg]^frac11-epsilon $$
Before that let's remember that:
For those firms who set their price in period $t$, $P_t(i) = P_t^*$ and the length of the interval over which $P_t(i) = P_t^*$ will be $ 1-theta $.
For those firms who set their price in period $t-1$ and then maintained the same price in period $ t $, $P_t(i) = P_t-1^*$ and the length of the interval over which $ P_t(i) = P_t-1^* $ will be $theta(1-theta) $.
For those firms who set their price in period $t-2$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-2^* $ and the length of the interval over which $ P_t(i) = P_t-2^* $ will be $theta^2(1-theta) $
For those firms who set their price in period $t-3$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-3^* $ and the length of the interval over which $ P_t(i) = P_t-3^* $ will be $theta^3(1-theta) $
We can generalize this and for those firms who set their price in period $t-j$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_t-j^* $ and the length of the interval over which $ P_t(i) = P_t-j^* $ will be $theta^j(1-theta) $
Since $ sum_j=o^infty theta^j (1-theta) = 1$, we can then write
$$ mathbbP_t = bigg(int_0^1 P_t(i)^1-epsilon di bigg) ^frac11-epsilon =bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon $$
Using the same transformation
$$ mathbbP_t-1 = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg)^frac11-epsilon $$
this implies that
$$ mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^jP_t-1-j^*^1-epsilon bigg) $$
now multiply both sides by $ theta$
$$ theta mathbbP_t-1^1-epsilon = bigg(int_0^1 P_t-1(i)^1-epsilon di bigg) ^frac11-epsilon = bigg( sum_j=0^infty (1-theta)theta^j+1P_t-1-j^*^1-epsilon bigg) $$
Notice that we can rewrite $mathbbP_t $ using $ mathbbP_t-1 $:
$$ mathbbP_t = bigg ( sum_j=0^infty (1-theta)theta^jP_t-j^*^1-epsilon bigg)^frac11-epsilon = bigg ( (1-theta) P_t^* + sum_j=0^infty (1-theta)theta^j+1P_t-j-1^*^1-epsilon bigg)^frac11-epsilon =bigg( (1-theta) P_t^*^1-epsilon + theta mathbbP_t-1^1-epsilon bigg)^frac11-epsilon $$
For those expert in math please edit if you see the lack of rigor.
answered Jul 22 at 18:03
community wiki
G.T.
1
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
add a comment |Â
1
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
1
1
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
I really commend you for the love of understanding and the perseverance.
– Satish Ramanathan
Jul 22 at 23:09
add a comment |Â
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what is epsilon? In page 146 of this book "Optimal Monetary policy under uncertainty", he gives the derivation for aggregate price level very similar to yours, yours is a little bit complicated with $epsilon$ otherwise, it is the same.
– Satish Ramanathan
Jul 22 at 11:10
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books.google.co.in/…
– Satish Ramanathan
Jul 22 at 11:11
$epsilon$ is elasticity of substitute in economic terms but mathematically it is just positive real number.
– G.T.
Jul 22 at 11:23
You can substitute $P_t(i)^1-epsilon$ and $P_t^*1-epsilon$ to simple $p_t$ and $p^*_t$ in the derivation of the book and you will see the derivation in the book makes sense.
– Satish Ramanathan
Jul 22 at 11:27