The Fisher equation shows the relationship between the real and nominal interest rates.

\((1+i)=(1+r)(1+\pi )\)

For small values:

\(i\approx r+\pi\)

We have:

\(Y=C(Y-T(Y))+I(r)+G+NX(Y)\)

Where:

\(Y\) is output

\(C\) is consumption

\(T\) is taxes

\(I\) is investment

\(r\) is the real interest rate

\(G\) is government spending

\(NX\) is net exports

The Keynesian cross plots:

\(Y\)

Against:

\(C(Y-T(Y))+I(r)+G+NX(Y)\)

This identifies an equilibrium level of output.

The IS curve plots the equilibrium level of output from the Keynesian cross against the real interest rate.

As the real interest rate rises, investment and therefore output falls.

The slope of the IS curve depends on taxes and net exports.

Money demand is:

\(L=L(i, Y)\)

As income rises, demand for money rises.

As the nominal interest rate rises, the demand for money falls, due to the opportunity cost.

Money supply is:

\(\dfrac{M}{P}\)

In equilibrium money supply and demand match. We have:

\(\dfrac{M}{P}=L(i,Y)\)

We can plot the level of output which corresponds to the nominal interest rate.

This is the LM curve.

The IS curve plots output against the (real) interest rate. As (real) interest rates rise, investment and therefore output falls.

The LM curve plots output against the (nominal) interest rate. As output rises, (nominal) interest rates fall to ensure clearing.

As prices are fixed in the IS-LM model, we can use the real and nominal rates interchangably.

The IS-LM model identifies the intercepts of the two curves and the equilibrium output and interest rate.

This model takes prices, money supply, taxes and government spending to be exogenous.

In the LM model a monetary expansion lowers interest rates.

In the IS-LM model this effect is lessened. The lower interest rates cause higher output, increasing money demand, and raising interest rates.

Is the IS model a fiscal expansion caused a corresponding increase in output.

In the IS-LM model this is lessened because the increase also causes more real money demand, raising interest rates, and lowering output.

For any given price level there is a corresponding IS-LM equilibrium, with an output level.

The Aggregate Demand curve models the relationship between the price level and equilibrium output.

As the price level rises, the real money supply falls. This means nominal interest rates rise to ensure LM equilibrium.

This rise in interest rates causes the IS curve to shift inwards, reducing output.

\(Y_d=Y_d(\dfrac{M}{P},G,T)\)

The slope of the Aggregate Demand curve

In neoclassical models Aggregate Supply does not depend on price.

The Aggregate Supply curve is informed by the Phillips curve.

As prices rise, so too does output.

Firms could increase production as nominal prices rise, as nominal contracts on wages mean that real costs have fallen.