The economist John Hicks wrote out Keynes' prose as an economic model that came to be known as the IS-LM model. I already derived this model before in a way that followed the way it is introduced in macroeconomics classes (as an IS and LM market). This derivation will acheive the same result, but approached fundamentally as an information transfer market system. The basis for the IS-LM model is that there are two markets: the real economy (IS) and the money market (LM) that couple to each other through the interest rate. As an information transfer system, we'll take a more direct route. We will posit that aggregate investment (demand) is a source of information that sends signals into the market that are detected by interest rate changes. The aggregate investment supply (the money supply) receives this information. See this post for a more detailed description of how information moves around the economy.

Let's start with the market $r : I \rightarrow M$, with interest rate $r$, nominal investment $I$ and money supply $M$ so that

$$

\text{(1) } r = \frac{dI}{dM} = \frac{1}{\kappa} \; \frac{I}{M}

$$

from the basic information transfer model. Looking at constant information source $I = I_{0}$, we have

$$

r = \frac{1}{\kappa} \; \frac{I_{0}}{\langle M\rangle}

$$

where $\langle M \rangle$ is the expected level of the money supply. Solving the differential equation (1), we obtain

$$

\Delta I \equiv I-I_{ref} = \frac{I_{0}}{\kappa}\log \frac{\langle M\rangle}{M_{ref}}

$$

where $ref$ refers to reference values of the variables $I$ and $M$. We can combine the previous two equations into a single function that defines the IS curve:

$$

\text{(2) }\log r = \log \frac{I_{0}}{\kappa M_{ref}} - \kappa \frac{\Delta I}{I_{0}}

$$

We can also look at constant $M = M_{0}$ so that we have, solving the differential equation (1) again:

$$

r = \frac{1}{\kappa} \; \frac{\langle I\rangle}{M_{0}}

$$

$$

\Delta M \equiv M-M_{ref} = \kappa M_{0} \log \frac{\langle I\rangle}{I_{ref}}

$$

where we can eliminate $\langle I\rangle$ to produce (after some re-arranging)

$$

\text{(3) } \log r = - \log \frac{\kappa M_{0}}{I_{ref}} + \frac{\Delta M}{\kappa M_{0}}

$$

Equations (2) and (3) represent how the market adjusts the interest rate to changes in the money supply given fixed investment demand and to changes in the investment demand given fixed money supply, respectively.

One more piece -- we need to relate output $Y$ to investment $I$ and money $M$. I'll add a simple market $p : N \rightarrow I$ where $N$ is NGDP (aggregate demand, sending information to aggregate investment) so that

$$

\frac{dN}{dI} = \frac{1}{\eta} \; \frac{N}{I}

$$

which we solve without holding either constant, resulting in:

$$

N \sim I^{1/\eta}

$$

It turns out empirically (see below), $\eta \simeq 1$ so that $\log N$ is proportional to $\log I$. If we hold the price level constant (we are looking at short run effects [1]), then we can say that $Y = N/P$ is proportional to $I$. This is the first part. To get the relationship with $M$, we'll match first order shifts of the IS and LM curves. If we have a small change in $r = r_{0} + \delta r$, then we have for the IS curve:

$$

\log r_{0} + \frac{\delta r}{r_{0}} + \cdots = \log \frac{I_{0}}{\kappa M_{ref}} - \kappa \frac{\Delta I}{I_{0}}

$$

So that

$$

\frac{\delta r}{r_{0}} \simeq - \kappa\frac{\Delta I}{I_{0}}

$$

Similarly for the LM curve

$$

\frac{\delta r}{r_{0}} \simeq \frac{\Delta M}{\kappa M_{0}}

$$

So that we have (after taking the absolute value as a postitive shift of the LM curve causes the interest rate to fall, but a positive shift in the IS curve causes the interest rate to rise)

$$

\Delta I \simeq \frac{I_{0}}{\kappa^{2} M_{0}} \Delta M

$$

Which means $\Delta Y \sim \Delta I \sim \Delta M$ are all proportional to each other and we can scale the IS and LM curves such that they become functions of $Y$. This allows us to plot them on the same graph, like the ISLM model. On the left, we have the relationship between $I$ and $Y$ and on the right, we have the IS-LM model graph (IS curve in blue, LM curve in red):

Empirically, it works best if we take $r \rightarrow r^{c}$ where $c \simeq 1/3$ [2] ($M$ is taken to be the monetary base):

And as promised, here is that plot of investment vs NGDP:

[1] This is effectively where economists' complaint about the IS-LM model not incorporating inflation (by not differentiating real and nominal interest rates) comes in.

[2] This is a fudge that I have yet to figure out. The empirical results for the 10-year and 3-month interest rates are a pretty good motivation. It fits the data. Additionally, it is basically a re-labeling of the interest rates. There is no a priori reason that the "information price" $p_{i}$ that goes into Equation (1) and the real world price represented by the interest rate $r$ need to be related by $r = p_{i}$; any bijective relationship is possible. We encounter this all the time e.g. decibels give us a more intuitive linear feeling of loudness than power. The market treats the cube root of the interest rate as a linear measure of information.

__UPDATE 5/22/2015__
Fixed the fudge factor. See here. Basically introduced another information equilibrium relationship between $r$ and the price of money $p$ so that the market at the top of the page becomes:

(r \rightarrow p) : I \rightarrow M

$$

And therefore $r \sim p^{1/c}$

Jason, is there a way to turn this kind of thing:

ReplyDelete$$

r = \frac{1}{\kappa} \; \frac{\langle I\rangle}{M_{0}}

$$

into human readable format? Is that in TeX or something? I never learned that... do you just read that like it was a regular formula? Lol. Is my browser not doing something it should be doing? Thanks.

Jason, never mind: as soon as I pasted my comment with the above jumble of $s, backslashes, and curly braces into my comment, suddenly all that gibberish turned into human readable formulas. Amazing. :D

DeleteAfter pasting the escape sequence code here and then suddenly seeing it appear as the formula you intended (and then the whole page turn into the formulas you intended) I tried pasting it in the comments to another blogspot blog to see if it showed up as a formula there too... it didn't. What's the difference? When I go to your blog does my browser somehow get clued in to interpret those sequences, but not when I go to other blogspot blogs? How does that work?

DeleteIt's a java applet called mathjax ... You have to put a little snipet of HTML at the top of a post. It will then work for the comments on the same post but not on pages that don't include the snippet. Additionally, Java is sometimes disabled or fails to load properly and you end up with just LaTeX. I haven't figured out how to get the equations to show up in an RSS feed yet (there probably isn't a way). I will find the reference link I use for the snippet of code ...

DeleteHere's the reference link I use to grab the snippet ..

Deletehttp://holdenweb.blogspot.com/2011/11/blogging-mathematics.html

for some reason it doesn't seem to work (or I didn't do it right) if you try to put it in the header of the page where you set the blogger style sheets. I just end up writing the post, then switching to html view in the editor and dropping the snippet at in at the top. Some formatting will interfere with it ... e.g. justified text or indentations.

Great! Thanks for the info... I'll try that out now... I already copied the globe you have in the side bar. I'm not an HTML guy... just monkey see monkey do for me. I found I was able to cut and paste your formulas (not the LaTeX) in a blogger blog w/o the snipped, but then I get a big mess of extra HTML in that post. Plus the little context menu doesn't appear (like it does here), but I can drag the formulas around and position them how I like. I have a garbage blog I just use to point things out to people rather than try to make complicated comments. Also, I could past the formulas themselves (w/o the escape sequences) into comments and they appeared in a mangled but partly formulaic looking set of characters (no horizontal divisor bar for example, but some of the rest was preserved).

DeleteWorked great! Thanks gain.

DeleteCheer, Tom.

DeleteFYI, here are some LaTeX guides:

http://latex-project.org/guides/