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Joey Derives IS/LM

ResponsiblyIrresponsible
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6/21/2015 12:31:56 PM
Posted: 1 year ago
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.

Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!

LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.

And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.
~ResponsiblyIrresponsible

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6/21/2015 4:17:18 PM
Posted: 1 year ago
Eh, I can keep going I guess.

AD Curve

From above, IS Curve:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x) (1)

Then we define the MP Curve :

r = r-bar + lambda * pi (2)

where:

r = riskless policy rate (real)
r-bar = autonomous changes to monetary policy
lambda > 0 = slope coefficient denoting how aggressively central banks respond to increases in inflation, and via the Taylor Principle must be greater than 0 such that nominal interest rates rise by more than the increase in inflation
pi = inflation

Then, to get the AD curve, we need only substitute the MP curve into the IS curve, or (2) into (1), and solve for pi.

Therefore:

r-bar + lambda * pi = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)] (3)

pi = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)] / (lambda) (3)

Again... super messy, but we can fix that.

pi = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)] / (lambda)
pi = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / [lambda * (b + c + x)]
pi = [[(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT)]/ [lambda * (b + c + x)]] + [(Y)/ [lambda * (b + c + x)]]

The first term, thus, is the y-intercept, which I'll label D, and 1/(lambda * (b+c+x)) is the slope of the AD curve, which I'll call T.

Therefore:

pi = D - TY

where pi is inflation, and D and T are constants > 0.

*drops mic*
~ResponsiblyIrresponsible

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ResponsiblyIrresponsible
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6/21/2015 4:25:00 PM
Posted: 1 year ago
Eh, somehow I forgot to subtract r-bar. Bleh. The proof would be a little different, but it's a constant so I'd end up tossing it out eventually anyway.
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6/21/2015 4:56:46 PM
Posted: 1 year ago
I don't know how to speak Asian.
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6/21/2015 4:57:21 PM
Posted: 1 year ago
At 6/21/2015 4:56:46 PM, PetersSmith wrote:
I don't know how to speak Asian.

lol
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Zaradi
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6/21/2015 5:00:21 PM
Posted: 1 year ago
At 6/21/2015 4:57:21 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 4:56:46 PM, PetersSmith wrote:
I don't know how to speak Asian.

lol

Rev's Asian right? Go ask him to translate this for me? ^_^
Want to debate? Pick a topic and hit me up! - http://www.debate.org...
ResponsiblyIrresponsible
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6/21/2015 5:00:59 PM
Posted: 1 year ago
At 6/21/2015 5:00:21 PM, Zaradi wrote:
At 6/21/2015 4:57:21 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 4:56:46 PM, PetersSmith wrote:
I don't know how to speak Asian.

lol

Rev's Asian right? Go ask him to translate this for me? ^_^

I'm pretty sure Rev actively taunts me for being an "econ nerd." I doubt he'll understand this, lol
~ResponsiblyIrresponsible

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ResponsiblyIrresponsible
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6/21/2015 5:02:04 PM
Posted: 1 year ago
If anyone wants a graphical derivation - via screenshots from Twiddla or something - I may be willing to accommodate your request.
~ResponsiblyIrresponsible

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RevNge
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6/21/2015 5:04:40 PM
Posted: 1 year ago
At 6/21/2015 5:00:21 PM, Zaradi wrote:
At 6/21/2015 4:57:21 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 4:56:46 PM, PetersSmith wrote:
I don't know how to speak Asian.

lol

Rev's Asian right? Go ask him to translate this for me? ^_^

I'm only half, so only the math part looks somewhat intelligible.
ResponsiblyIrresponsible
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6/21/2015 5:05:18 PM
Posted: 1 year ago
Fack, another mistake:

"C = C-bar + mpc * (Y-bar-T-bar) + cr"

That's..... wrong. I should be subtracting cr from the consumption function.

#oops
~ResponsiblyIrresponsible

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ResponsiblyIrresponsible
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6/21/2015 5:07:30 PM
Posted: 1 year ago
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.
~ResponsiblyIrresponsible

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Zaradi
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6/21/2015 5:43:46 PM
Posted: 1 year ago
At 6/21/2015 5:07:30 PM, ResponsiblyIrresponsible wrote:
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.

Wow the Econ nerd can't even do Econ....

One of us! One of us! One of us!
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ResponsiblyIrresponsible
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6/21/2015 5:50:09 PM
Posted: 1 year ago
At 6/21/2015 5:43:46 PM, Zaradi wrote:
At 6/21/2015 5:07:30 PM, ResponsiblyIrresponsible wrote:
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.

Wow the Econ nerd can't even do Econ....

Just for that, I'm going to redo all of this sh1t and post it to your wall - in tiny, miniature chunks - just to increase your suffering.

One of us! One of us! One of us!

Oh, it's on.
~ResponsiblyIrresponsible

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Zaradi
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6/21/2015 6:09:28 PM
Posted: 1 year ago
At 6/21/2015 5:50:09 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 5:43:46 PM, Zaradi wrote:
At 6/21/2015 5:07:30 PM, ResponsiblyIrresponsible wrote:
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.

Wow the Econ nerd can't even do Econ....

Just for that, I'm going to redo all of this sh1t and post it to your wall - in tiny, miniature chunks - just to increase your suffering.


One of us! One of us! One of us!

Oh, it's on.

;)
Want to debate? Pick a topic and hit me up! - http://www.debate.org...
ResponsiblyIrresponsible
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6/21/2015 6:19:08 PM
Posted: 1 year ago
At 6/21/2015 6:09:28 PM, Zaradi wrote:
At 6/21/2015 5:50:09 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 5:43:46 PM, Zaradi wrote:
At 6/21/2015 5:07:30 PM, ResponsiblyIrresponsible wrote:
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.

Wow the Econ nerd can't even do Econ....

Just for that, I'm going to redo all of this sh1t and post it to your wall - in tiny, miniature chunks - just to increase your suffering.


One of us! One of us! One of us!

Oh, it's on.

;)

Make your comments public, and expect some notification spam within the next hour, lmfao
~ResponsiblyIrresponsible

DDO's Economics Messiah
ResponsiblyIrresponsible
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6/21/2015 6:37:02 PM
Posted: 1 year ago
IS Curve

Y = Yp = Ip + C + I + G + NX

Ip = I-bar - b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) - cr
G = G-bar
NX = NX-bar - xr

Y = I-bar - b (r + f) + C-bar + mpc * (Y-bar-T-bar) - cr + G-bar + NX-bar - xr
Y = I-bar - br - bf + C-bar + mpc * Y-bar - mpc * T-bar - cr + G-bar + NX-bar - xr
Y = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - r (b + c + x)
r (b + c + x) = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y
r = [(I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y) / (b + c + x)]

^^ That's simplified enough, lol.
~ResponsiblyIrresponsible

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Zaradi
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6/21/2015 6:46:21 PM
Posted: 1 year ago
At 6/21/2015 6:19:08 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 6:09:28 PM, Zaradi wrote:
At 6/21/2015 5:50:09 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 5:43:46 PM, Zaradi wrote:
At 6/21/2015 5:07:30 PM, ResponsiblyIrresponsible wrote:
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.

Wow the Econ nerd can't even do Econ....

Just for that, I'm going to redo all of this sh1t and post it to your wall - in tiny, miniature chunks - just to increase your suffering.


One of us! One of us! One of us!

Oh, it's on.

;)

Make your comments public, and expect some notification spam within the next hour, lmfao

Make my comments public? But I thought you knew what you were doing! ;)
Want to debate? Pick a topic and hit me up! - http://www.debate.org...
ResponsiblyIrresponsible
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6/21/2015 6:47:17 PM
Posted: 1 year ago
At 6/21/2015 6:46:21 PM, Zaradi wrote:
At 6/21/2015 6:19:08 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 6:09:28 PM, Zaradi wrote:
At 6/21/2015 5:50:09 PM, ResponsiblyIrresponsible wrote:
At 6/21/2015 5:43:46 PM, Zaradi wrote:
At 6/21/2015 5:07:30 PM, ResponsiblyIrresponsible wrote:
Yuck, that... mucks up a lot.

I also should be subtracting the "r" term from the investment and NX functions, lol.

Wow the Econ nerd can't even do Econ....

Just for that, I'm going to redo all of this sh1t and post it to your wall - in tiny, miniature chunks - just to increase your suffering.


One of us! One of us! One of us!

Oh, it's on.

;)

Make your comments public, and expect some notification spam within the next hour, lmfao

Make my comments public? But I thought you knew what you were doing! ;)

Yes, which is why I wanted my math to be publicly available, lollol.

But it's above.

I'm too lazy to sub an MP curve into that and watch it and the IS curve a baby in the form of the AD curve, so you'll just have to believe me that the math checks out.
~ResponsiblyIrresponsible

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16kadams
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6/22/2015 11:45:29 AM
Posted: 1 year ago
At 6/21/2015 12:31:56 PM, ResponsiblyIrresponsible wrote:
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.


Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!


LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.


And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.

give me your lunch money, nerd
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ResponsiblyIrresponsible
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6/22/2015 11:46:10 AM
Posted: 1 year ago
At 6/22/2015 11:45:29 AM, 16kadams wrote:
At 6/21/2015 12:31:56 PM, ResponsiblyIrresponsible wrote:
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.


Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!


LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.


And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.

give me your lunch money, nerd

Not until you pry it from my cold, dead hands
~ResponsiblyIrresponsible

DDO's Economics Messiah
16kadams
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6/22/2015 11:52:19 AM
Posted: 1 year ago
At 6/22/2015 11:46:10 AM, ResponsiblyIrresponsible wrote:
At 6/22/2015 11:45:29 AM, 16kadams wrote:
At 6/21/2015 12:31:56 PM, ResponsiblyIrresponsible wrote:
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.


Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!


LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.


And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.

give me your lunch money, nerd

Not until you pry it from my cold, dead hands

dat demand doe
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https://rekonomics.wordpress.com...
"A trend is a trend, but the question is, will it bend? Will it alter its course through some unforeseen force and come to a premature end?" -- Alec Cairncross
ResponsiblyIrresponsible
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6/22/2015 12:00:59 PM
Posted: 1 year ago
At 6/22/2015 11:52:19 AM, 16kadams wrote:
At 6/22/2015 11:46:10 AM, ResponsiblyIrresponsible wrote:
At 6/22/2015 11:45:29 AM, 16kadams wrote:
At 6/21/2015 12:31:56 PM, ResponsiblyIrresponsible wrote:
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.


Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!


LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.


And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.

give me your lunch money, nerd

Not until you pry it from my cold, dead hands

dat demand doe

Lol

Yup.. your demand for my lunch money far exceeds it's supply, and it's a relatively illiquid market with a highly elastic AS curve, so that magnifies the price movement. Hence, even the "cold dead hands" standard may be too lenient.
~ResponsiblyIrresponsible

DDO's Economics Messiah
lannan13
Posts: 23,017
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6/22/2015 12:34:20 PM
Posted: 1 year ago
At 6/21/2015 6:37:02 PM, ResponsiblyIrresponsible wrote:
IS Curve

Y = Yp = Ip + C + I + G + NX

Ip = I-bar - b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) - cr
G = G-bar
NX = NX-bar - xr

Y = I-bar - b (r + f) + C-bar + mpc * (Y-bar-T-bar) - cr + G-bar + NX-bar - xr
Y = I-bar - br - bf + C-bar + mpc * Y-bar - mpc * T-bar - cr + G-bar + NX-bar - xr
Y = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - r (b + c + x)
r (b + c + x) = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y
r = [(I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y) / (b + c + x)]


^^ That's simplified enough, lol.

In English?
-~-~-~-~-~-~-~-Lannan13'S SIGNATURE-~-~-~-~-~-~-~-

If the sky's the limit then why do we have footprints on the Moon? I'm shooting my aspirations for the stars.

"If you are going through hell, keep going." "Sir Winston Churchill

"No one can make you feel inferior without your consent." "Eleanor Roosevelt

Topics I want to debate. (http://tinyurl.com...)
-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~
ResponsiblyIrresponsible
Posts: 12,398
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6/22/2015 12:35:15 PM
Posted: 1 year ago
At 6/22/2015 12:34:20 PM, lannan13 wrote:
At 6/21/2015 6:37:02 PM, ResponsiblyIrresponsible wrote:
IS Curve

Y = Yp = Ip + C + I + G + NX

Ip = I-bar - b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) - cr
G = G-bar
NX = NX-bar - xr

Y = I-bar - b (r + f) + C-bar + mpc * (Y-bar-T-bar) - cr + G-bar + NX-bar - xr
Y = I-bar - br - bf + C-bar + mpc * Y-bar - mpc * T-bar - cr + G-bar + NX-bar - xr
Y = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - r (b + c + x)
r (b + c + x) = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y
r = [(I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y) / (b + c + x)]


^^ That's simplified enough, lol.

In English?

lol... I coulllddddd.
~ResponsiblyIrresponsible

DDO's Economics Messiah
lannan13
Posts: 23,017
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6/22/2015 12:38:14 PM
Posted: 1 year ago
At 6/22/2015 12:35:15 PM, ResponsiblyIrresponsible wrote:
At 6/22/2015 12:34:20 PM, lannan13 wrote:
At 6/21/2015 6:37:02 PM, ResponsiblyIrresponsible wrote:
IS Curve

Y = Yp = Ip + C + I + G + NX

Ip = I-bar - b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) - cr
G = G-bar
NX = NX-bar - xr

Y = I-bar - b (r + f) + C-bar + mpc * (Y-bar-T-bar) - cr + G-bar + NX-bar - xr
Y = I-bar - br - bf + C-bar + mpc * Y-bar - mpc * T-bar - cr + G-bar + NX-bar - xr
Y = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - r (b + c + x)
r (b + c + x) = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y
r = [(I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y) / (b + c + x)]


^^ That's simplified enough, lol.

In English?

lol... I coulllddddd.

Explain in Layman's terms so it's understandable for the rest of us to catch up. I haven't started my Economics Major yet.
-~-~-~-~-~-~-~-Lannan13'S SIGNATURE-~-~-~-~-~-~-~-

If the sky's the limit then why do we have footprints on the Moon? I'm shooting my aspirations for the stars.

"If you are going through hell, keep going." "Sir Winston Churchill

"No one can make you feel inferior without your consent." "Eleanor Roosevelt

Topics I want to debate. (http://tinyurl.com...)
-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~-~
ResponsiblyIrresponsible
Posts: 12,398
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6/22/2015 12:40:52 PM
Posted: 1 year ago
At 6/22/2015 12:38:14 PM, lannan13 wrote:
At 6/22/2015 12:35:15 PM, ResponsiblyIrresponsible wrote:
At 6/22/2015 12:34:20 PM, lannan13 wrote:
At 6/21/2015 6:37:02 PM, ResponsiblyIrresponsible wrote:
IS Curve

Y = Yp = Ip + C + I + G + NX

Ip = I-bar - b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) - cr
G = G-bar
NX = NX-bar - xr

Y = I-bar - b (r + f) + C-bar + mpc * (Y-bar-T-bar) - cr + G-bar + NX-bar - xr
Y = I-bar - br - bf + C-bar + mpc * Y-bar - mpc * T-bar - cr + G-bar + NX-bar - xr
Y = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - r (b + c + x)
r (b + c + x) = I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y
r = [(I-bar + C-bar + G-bar + Nx-bar + mpc * Y-bar - mpc * T-bar - bf - Y) / (b + c + x)]


^^ That's simplified enough, lol.

In English?

lol... I coulllddddd.

Explain in Layman's terms so it's understandable for the rest of us to catch up. I haven't started my Economics Major yet.

All the math does is relate the riskless interest rate set by the Fed to the level of actual output needed to equate planned and actually spending when the goods market is in equilibrium. All the curve does is give us combinations of Y (output) and r (real interest rate). Everything else is mumbo jumbo.
~ResponsiblyIrresponsible

DDO's Economics Messiah
RevNge
Posts: 13,835
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6/22/2015 6:01:16 PM
Posted: 1 year ago
At 6/21/2015 12:31:56 PM, ResponsiblyIrresponsible wrote:
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.


Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!


LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.


And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.

I'm going to dissect this and roast you on anything that you missed.
ResponsiblyIrresponsible
Posts: 12,398
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6/22/2015 6:01:42 PM
Posted: 1 year ago
At 6/22/2015 6:01:16 PM, RevNge wrote:
At 6/21/2015 12:31:56 PM, ResponsiblyIrresponsible wrote:
So, this is a function (a) of me being bored and (b) wanting to see if I actually remember how to derive IS/LM without cheating.

Disclaimer: I hate this model, and you all should as well.

Without further ado...


IS Curve


Y = Yp = Ip + C + I + G + NX

Ip = I-bar + b (r + f)
C = C-bar + mpc * (Y-bar-T-bar) + cr
G = G-bar
NX = NX-bar + xr

whereby b, c, and x are parameters and the notation "bar" indicates that the variable is autonomous or in the case of Y, T, and G, exogenous.

Expanding this:

Y = Yp = I-bar + b (r + f) + C-bar + mpc * (Y-T) + cr + G-bar + NX-bar + xr

I should end up with some y-intercept - which in this case I'll just call B - minus some constant (slope coefficient of the IS curve) multiplied by the level of output.


Y = I-bar + br + bf + C-bar + mpcY - mpcT + cr + G-bar + Nx-bar + xr
Y = I-bar + C-bar + G-bar + NX-bar + r(b+c+x) + bf + mpcY - mpcT
-r(b+c+x) = I-bar +C-bar + G-bar + NX-bar + bf + mpcY - mpcT - Y
-r = (I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)
r = - [(I-bar + C-bar + G-bar + NX-bar + bf + mpcY - mpcT - y)/ (b + c + x)]

Then distribute the negative sign....

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

Realistically I could set this all equal to Y, but I'm too lazy.

Since I'm not using numbers I'm going to use a cute trick. My only things actually being varied within the framework of a model of some economy is r and Y - i.e., per some given level of C, G, NX, and I, the IS curve gives us some combination of Y and r that equates planned spending with actual spending, with the difference being unplanned inventory investment. Thus, I can set everything else equal to some arbitrary constant, which will obviously shift the function up and down, but for the sake of argument we'll assume is fixed.

So let's rewrite this mess of an equation as such:

r = (-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT + Y) / (b + c + x)

is the same as:

r = [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] + Y/(b + c + x)

Therefore, [(-I-bar - C-bar - G-bar - NX-bar - bf - mpcY + mpcT) / (b + c + x)] is our y-intercept, which I'll call B, and [1/(b+ c + x)] is the slope of the IS curve, which I'll call M.

Therefore,

r = B - MY

We have our IS curve!


LM Curve

So we start with an equation of money demand predicated on liquidity preference, whereby income is a parameter which shifts the curve up and down and changes in the riskless interest rate move us along the curve.

In other words, the demand for real money balances can be written as follows:

(M/P) = Y - br

where, again, b is a parameter > 0, and the slope of the Md curve.

Since the LM curve gives us combinations of income and the riskless interest rate where financial markets are in equilibrium, the next step is obviously to equate Md with Ms (i.e., money demand with money supply).

The real money supply, we know, is equal to M/P, whereas real money demand is Y - br.

Therefore,

Y - br = M/P

Thus,

Y = M/P + br

And let's write this, much like our IS curve, in terms of Y:

br = M/P - Y
r = [(M/P - Y) / (b)]

And then go through the same steps as above:

r = [(M/P)/(b)] - [(y)/(b)]

[(M/P)/(b)] is thus our y-intercept whereas (-1/b) is our slope coefficient.


And...... done!

I probably screwed up somewhere, but that's IS/LM, lol.

I'm going to dissect this and roast you on anything that you missed.

Then scroll up for the correction, lol.
~ResponsiblyIrresponsible

DDO's Economics Messiah