The IS LM model is a model used in macroeconomics to help explain the possible relationships between the interest rate and real GDP. While not very accurate for real world analysis, it gives an interesting look at possible outcomes of various policy tools for a classroom setting.

The IS (Investment and savings equilibrium) equation:

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

Where

Y = national income or real GDP

C(Y-T(Y)) = consumption or consumer spending which is a
function of disposable income

(Y-T(Y)) = disposable income which is equal to national
income minus tax (which is a function of income)

I(r) = Investment which is a function of the real interest
rate

G = Government spending/expenditures

NX(Y) = Net exports, where imports depend on income (Y)

The IS/LM model showing possible shifts in the IS curve |

The IS curve is downward sloping in the graph because a
lower real interest rate motivates higher investment (look at the above
equation) which translates to a higher Y.
Note that Y is endogenous, and responds to changes in r, and the IS
curve shows this relationship. Remember
that the IS curve shows every possible point with respect to the interest rate
and real GDP where investment is equal to savings. For example, if the interest rate falls
savings must fall (because people don’t like the lower return) which means that
real GDP must rise so that people can save more. The increase in Y will be caused by an
increase in investment which must be matched by an increase in savings either
from the private, public (government), or foreign sector.

The LM (Liquidity preference and money supply equilibrium) equation:

M/P = L(i,Y)

Where

M/P = the real money supple where M is the actual amount of
money in the economy and P is the overall price level

L(i,Y) = the real demand for money which is a function of
the interest rate (i) and national income (Y)

The same IS/LM graph showing possible shifts in the IS curve |

The LM curve is upward sloping because it shows the possible
interest rate and real GDP values that allow for equilibrium to exist in the
money market. You can see in the LM
equation that i and Y both enter on the right hand side in the real demand for
money function. Since people demand less
money when the interest rate is high, L has a negative relationship with i. But people demand more money when their
income rises so L has a positive relationship with Y. This means that if i goes up, Y go up as well
so that L doesn’t change. And if i goes
down, Y must fall as well so L doesn’t change.
This means that they have a positive relationship hence the upward
slope.

Equilibrium level of national income in the IS-LM model is
considered to be aggregate demand. And
any change in the price level (P) will result in a shift in the LM curve and
its new equilibrium with respect to IS will show the new point on the AD
(aggregate demand curve).

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