# Simplification of the IS - LM model

As I've previously posted, I'm currently working through Macroeconomics: a European Perspective. I've just gone through their construction of the IS-LM model, which is all straightforward, apart from one step, which I find mystifying.

We have the two equilibrium conditions in the goods market and financial market given by:

$$Y = C(Y-T) +I(Y,i)+G$$ $$M/P = Y\times L(i)$$

Where: $$Y$$ is output/income; $$C(.)$$ is the consumption function of disposable income; $$T$$ is the tax level; $$I(Y,i)$$ is the level of investment as a function of income and $$i$$, the interest rate;G is government spending not including transfers; $$M$$ is the money supply; $$P$$ is the price level; and $$L(i)$$ is a decreasing function of the interest rate which gives the level of money demand for a fixed level of income $$Y$$.

Now what we are interested in is where both of these markets are in equilibrium at the same time. I believe that the standard approach here would be to derive the IS and LM curves, and then look at where these two curves intersect, which gives us the equilibrium.

The book does derive the IS curve, which we do by looking at the relationship between the interest rate and the equilibrium level of output in the goods market for given values of $$G$$ and $$T$$.

At this point the text takes a bit of an odd turn. I expected to do the same thing to derive the LM curve. That is, look at the relationship between the interest rate and the equilibrium point in the financial market. This would then give the LM curve, we plot them on the same chart and we are away.

However, instead, the book simply argues that in reality central banks no aim to make choices about the money supply. Instead they explicitly target a given level of interest rate, $$\bar{i}$$ adjusting the money supply to meet this. Therefore, we can take a simply define out LM curve as: $$i=\bar{i}$$

This would give us an IS-LM chart that looks a bit like this: I don't really follow this last step of just assuming that the LM curve is flat.

I understand that central banks essentially undertake an inflation targeting exercise now. Indeed, we can think of the aim of a central bank, in simple terms (all I understand!), as looking to minimise a loss function of $$L = (y_t - y_e)^2+\beta (\pi_t - \pi^T)^2$$ which is balancing its interests in keeping inflation on target and output near equilibrium.

Two questions:

1. Can anyone explain to me this simplifying step of assuming that the LM curve is constant with respect to the interest rate?
2. Is this a standard treatment of the LM curve? As I say, it is not what I was expecting.

1. Can anyone explain to me this simplifying step of assuming that the LM curve is constant with respect to the interest rate?

This is direct consequence of assuming that central bank targets $$i$$. If central bank targets $$i$$, and $$i$$ only, then this is the logical consequence.

As you mention in your question money market equilibrium can be described as:

$$M/P=L(Y,i) \quad \text{or} \quad M/P=YL(i)$$

Let's make it even more concrete (so you are better able to understand) and lets say its given by:

$$M/P = Y(1 - bi/Y) \implies M/P = Y-bi$$

Now lets solve this equation for $$i$$ which gives us:

$$-M/(Pb) +Y/b=i \quad \text{or} \quad -M/(Pb) +Y/b=0.5$$

Now if central bank wants to always keep $$i$$ constant at 5% what central bank has to do is to manipulate $$M$$ in a way that no matter what $$P$$, $$Y$$ or $$b$$ happens to be you always get $$0.5$$ out of that equation.

For example if in original equilibrium $$b$$ is 1000, $$Y$$ is 5000 and $$P=1$$ we would have:

$$-4500/(1 \cdot 1000) +5000/1000=0.5$$

Now if $$Y$$ would change to $$10000$$, immediately at the same time central bank would have to change $$M$$ to $$9500$$ to keep $$i=0.5$$.

This would happen for any change in $$Y$$, $$P$$ or $$b$$. They would always be immediately canceled by central bank's intervention in quantity of money.

$$LM$$ curve is curve that plots relationship between $$Y$$ and $$i$$. If $$i$$ is always $$0.5$$ regardless of $$Y$$ and if the relationship never changes because changes to other parameters such as $$P$$ or $$b$$ are also immediately offset, what you will be left with is a constant horizontal curve, since such central bank behavior simply means that $$i(Y,P,b) = 0.5$$.

Is this a standard treatment of the LM curve? As I say, it is not what I was expecting.

Yes it is standard textbook treatment. In real life CBs do not religiously stick to interest rate targets so its not 100% realistic but it is important mental exercise (to see what consequences of fanatically sticking to such target would be) and it is useful didactic tool to get comfortable with workings of more complex macro models.

You would probably find exercise like that in almost any modern macro textbook. If not mentioned directly in text then at least as one of the practice exercise at end of the chapter. Some books might omit it as its not that important as other parts of IS-LM model, but it is not anything that would raise eyebrows (at least definitely not in academic circles).

It seems to me that there is another issue that perplexes you.

You wrote:

the book simply argues that in reality central banks no aim to make choices about the money supply. Instead ,they explicitly target a given level of interest rate […]

I understand that central banks essentially undertake an inflation targeting exercise now. Indeed, we can think of the aim of a central bank, in simple terms (all I understand!), as looking to minimize a loss function of $$L = (y_t - y_e)^2+\beta (\pi_t - \pi^T)^2$$ which is balancing its interests in keeping inflation on target and output near equilibrium.

That is, you are puzzled by the idea, advanced by Blanchard, that the goal of the central bank is to keep the rate of interest at a fixed level, while its goal should be, according to the loss function you mentioned, inflation and level of income.

The issue is that, in economic policy, a distinction is made between intermediate targets and final targets. The rate of inflation and the level of income fall into the category of final targets, while the rate of interest and money supply are intermediate targets.

Intermediate targets are not in themselves the ultimate goal of economic policy, but serve to achieve the final targets.

This theoretical scheme is linked to a conception of a two-stage monetary policy: monetary authorities control some tools$$^1$$ through which they manage to determine, more or less correctly, the intermediate targets (traditionally, rates of interest and quantity of money) and the latter can determine the final targets of economic policy.

Therefore, the Blanchard model refers to an intermediate target, the rate of interest, while your loss function refers to final targets, inflation and level of income.

$$^1$$ Traditional tools of monetary policy can be, for instance, open market operations or bank reserve requirements.