Need a math help for the Cagan's model in macroeconomics
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From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.
To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is
$m_t − p_t = −gamma( p_{t+1} − p_t)$,
where $m_t$ is the log of the quantity of money at time t, $p_t$ is the log of the price level at time t, and $gamma$ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, $m_t − p_t$ is the log of real money balances, and $p_{t+1} − p_t$ is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
Shouldn't $(p_{t+1} - p_t)$ be the log of inflation rate? Why it says just "the inflation rate"?
This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all.
$ln frac{M}{P} = ln (frac{p_{t+1}}{p_t})^{-gamma} rightarrow frac{M}{P} = (frac{p_{t+1}}{p_t})^{-gamma}$
So, if the $(p_{t+1} - p_t)$ is just the log of inflation rate, then $frac{p_{t+1}}{p_t}$ is the inflation rate and,
inflation goes up by 1 percentage point
would mean $frac{p_{t+1}}{p_t}$ is going to get +1, right? But I couldn't possibly think it would result the fall of $frac{M}{P}$ by the $gamma$ point. What am I missing?
And secondly, if the $(p_{t+1} - p_t)$ is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "$−gamma(1 + p_{t+1} − p_t)$" at the right side, right? How could this be the case?
mathematical-economics
edited yesterday
Giskard
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asked yesterday
dolcodolco
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$begingroup$
From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.
To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is
$m_t − p_t = −gamma( p_{t+1} − p_t)$,
where $m_t$ is the log of the quantity of money at time t, $p_t$ is the log of the price level at time t, and $gamma$ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, $m_t − p_t$ is the log of real money balances, and $p_{t+1} − p_t$ is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
Shouldn't $(p_{t+1} - p_t)$ be the log of inflation rate? Why it says just "the inflation rate"?
This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all.
$ln frac{M}{P} = ln (frac{p_{t+1}}{p_t})^{-gamma} rightarrow frac{M}{P} = (frac{p_{t+1}}{p_t})^{-gamma}$
So, if the $(p_{t+1} - p_t)$ is just the log of inflation rate, then $frac{p_{t+1}}{p_t}$ is the inflation rate and,
inflation goes up by 1 percentage point
would mean $frac{p_{t+1}}{p_t}$ is going to get +1, right? But I couldn't possibly think it would result the fall of $frac{M}{P}$ by the $gamma$ point. What am I missing?
And secondly, if the $(p_{t+1} - p_t)$ is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "$−gamma(1 + p_{t+1} − p_t)$" at the right side, right? How could this be the case?
mathematical-economics
edited yesterday
Giskard
13.3k32248
asked yesterday
dolcodolco
1183
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.
To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is
$m_t − p_t = −gamma( p_{t+1} − p_t)$,
where $m_t$ is the log of the quantity of money at time t, $p_t$ is the log of the price level at time t, and $gamma$ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, $m_t − p_t$ is the log of real money balances, and $p_{t+1} − p_t$ is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
Shouldn't $(p_{t+1} - p_t)$ be the log of inflation rate? Why it says just "the inflation rate"?
This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all.
$ln frac{M}{P} = ln (frac{p_{t+1}}{p_t})^{-gamma} rightarrow frac{M}{P} = (frac{p_{t+1}}{p_t})^{-gamma}$
So, if the $(p_{t+1} - p_t)$ is just the log of inflation rate, then $frac{p_{t+1}}{p_t}$ is the inflation rate and,
inflation goes up by 1 percentage point
would mean $frac{p_{t+1}}{p_t}$ is going to get +1, right? But I couldn't possibly think it would result the fall of $frac{M}{P}$ by the $gamma$ point. What am I missing?
And secondly, if the $(p_{t+1} - p_t)$ is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "$−gamma(1 + p_{t+1} − p_t)$" at the right side, right? How could this be the case?
mathematical-economics
edited yesterday
Giskard
13.3k32248
asked yesterday
dolcodolco
1183
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
From the appendix after the chapter 4 in Macroeconomics 7th edition by Gregory Mankiw.
To keep the math as simple as possible, we posit a money demand function that is linear in the natural logarithms of all the variables. The money demand function is
$m_t − p_t = −gamma( p_{t+1} − p_t)$,
where $m_t$ is the log of the quantity of money at time t, $p_t$ is the log of the price level at time t, and $gamma$ is a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, $m_t − p_t$ is the log of real money balances, and $p_{t+1} − p_t$ is the inflation rate between period t and period t+1. This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
Shouldn't $(p_{t+1} - p_t)$ be the log of inflation rate? Why it says just "the inflation rate"?
This equation states that if inflation goes up by 1 percentage point, real money balances fall by $gamma$ percent.
My math level is like that of a high school. Would anyone be so nice and explain this for me? To me, it doesn't make sense at all.
$ln frac{M}{P} = ln (frac{p_{t+1}}{p_t})^{-gamma} rightarrow frac{M}{P} = (frac{p_{t+1}}{p_t})^{-gamma}$
So, if the $(p_{t+1} - p_t)$ is just the log of inflation rate, then $frac{p_{t+1}}{p_t}$ is the inflation rate and,
inflation goes up by 1 percentage point
would mean $frac{p_{t+1}}{p_t}$ is going to get +1, right? But I couldn't possibly think it would result the fall of $frac{M}{P}$ by the $gamma$ point. What am I missing?
And secondly, if the $(p_{t+1} - p_t)$ is just the inflation rate,(not the log of any) then it bugs me more than the former. So, +1 change to the inflation rate is like nothing but that we would get "$−gamma(1 + p_{t+1} − p_t)$" at the right side, right? How could this be the case?
mathematical-economics
mathematical-economics
edited yesterday
Giskard
13.3k32248
asked yesterday
dolcodolco
1183
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited yesterday
Giskard
13.3k32248
asked yesterday
dolcodolco
1183
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
Giskard
13.3k32248
edited yesterday
Giskard
13.3k32248
edited yesterday
Giskard
13.3k32248
13.3k32248
asked yesterday
dolcodolco
1183
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
dolcodolco
1183
asked yesterday
dolcodolco
1183
1183
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
dolco is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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1 Answer
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The answer to both your questions is that for small $x$ values
$$
ln(1+x) approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)
So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.
This should also answer your second question, as the approximation
$$
gamma x approx ln(1+ gamma x),
$$
works as well.
It may also be worthwhile to look into elasticity.
answered yesterday
GiskardGiskard
13.3k32248
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
The answer to both your questions is that for small $x$ values
$$
ln(1+x) approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)
So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.
This should also answer your second question, as the approximation
$$
gamma x approx ln(1+ gamma x),
$$
works as well.
It may also be worthwhile to look into elasticity.
answered yesterday
GiskardGiskard
13.3k32248
$endgroup$
add a comment |
$begingroup$
The answer to both your questions is that for small $x$ values
$$
ln(1+x) approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)
So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.
This should also answer your second question, as the approximation
$$
gamma x approx ln(1+ gamma x),
$$
works as well.
It may also be worthwhile to look into elasticity.
answered yesterday
GiskardGiskard
13.3k32248
$endgroup$
add a comment |
$begingroup$
The answer to both your questions is that for small $x$ values
$$
ln(1+x) approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)
So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.
This should also answer your second question, as the approximation
$$
gamma x approx ln(1+ gamma x),
$$
works as well.
It may also be worthwhile to look into elasticity.
answered yesterday
GiskardGiskard
13.3k32248
$endgroup$
The answer to both your questions is that for small $x$ values
$$
ln(1+x) approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)
So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.
This should also answer your second question, as the approximation
$$
gamma x approx ln(1+ gamma x),
$$
works as well.
It may also be worthwhile to look into elasticity.
answered yesterday
GiskardGiskard
13.3k32248
answered yesterday
GiskardGiskard
13.3k32248
answered yesterday
GiskardGiskard
13.3k32248
answered yesterday
GiskardGiskard
13.3k32248
13.3k32248
add a comment |
add a comment |
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