The IS curve plots the relationship between the interest rate and the level of national income, which arises in the market for goods and services. To understand this curve, we must begin with the Keynesian Cross, which graphs planned expenditure (PE) as a function of income (Y). Moreover, the Keynesian Cross describes economic equilibrium as the point where planned expenditure equals actual expenditure ($\mathrm{PE} \= Y$).

The equation for planned expenditure is $\mathrm{PE} \= C \+ I \+ G$, where $C \= C\left(Y-T\right)$ is the consumption as a function of $Y-T$, disposable income ($T$ is taxes), $I \= I\left(r\right)$ is the investment as a function of $r$, the real investment rate, and G is the amount of government expenditures.

The investment function, $I\left(r\right)$, depicts a negative relationship between the interest rate and the level of planned investment. Since higher interest rates lead to higher interest charges on loans, businesses are deterred from making large-scale investments, such as buying new equipment or constructing new buildings.

The IS curve shows a negative relationship between r and Y. As the interest rate rises, the level of planned investment expenditure falls, causing the total planned expenditure and national income to decrease. An increase in r causes the PE function to shift downward, shifting the equilibrium point in the Keynesian Cross leftward to a lower level of Y. This relationship is summarized as the movement of a point along the IS curve.

The Effects of Fiscal Policy

The IS curve shows the level of national income which brings the goods market into equilibrium for any given interest rate. But, as we can see from the Keynesian Cross, the level of income also depends on the fiscal policy: the amount of government expenditures (G) and the level of taxes (T). We draw the IS curve for given levels of G and T, meaning that the fiscal policy is fixed as we draw it. As a result, when fiscal policy changes, the IS curve must shift in response. When the government is enacting a fiscal expansion, either by increasing G or cutting T, total planned expenditure rises, thereby raising equilibrium income for every interest rate in the goods market. This forces the entire IS curve to shift outward (to the right). When the government is enacting a fiscal contraction, either by decreasing G or raising T, total planned expenditure falls, thereby reducing equilibrium income for every interest rate. This causes the entire IS curve to shift inward (to the left).

The LM curve plots the relationship between the interest rate and the level of national income which arises in the market for real money balances. To understand this curve, we must first look at the market for real money balances, which plots the money supply curve ${\left(\frac{M}{P}\right)}^s$ and the money demand curve ${\left(\frac{M}{P}\right)}^d$, and describes an equilibrium as: the point at which the quantity of real money balances supplied is equal to the quantity of real money balances demanded.

The quantity of real money balances supplied is an exogenous value equal to ${\left(\frac{M}{P}\right)}^s\= \frac{\stackrel{\mathit{\&conjugate0;}}{M}}{\stackrel{\mathit{\&conjugate0;}}{P}}$ where $\stackrel{\mathit{\&conjugate0;}}{M}$ is the nominal money supply (as determined by the central bank's monetary policy) and $\stackrel{\mathit{\&conjugate0;}}{P}$ is the short-term fixed price level.

The quantity of real money balances demanded is equal to ${\left(\frac{M}{P}\right)}^d \= L\left(r\, Y\right)$ where $L\left(r\, Y\right)$ is a function of the interest rate (r) and national income (Y). According to the Theory of Liquidity Preference, the interest rate is a key determinant of how much money people want to hold because it is the opportunity cost of holding money; it is what you forgo by holding your wealth as cash money rather than an interest-bearing asset. So, when the interest rate rises, people want to hold less of their wealth in the form of money, and therefore the quantity of real money balances demanded falls, giving the money demand curve a negative slope. At the same time, national income also affects the demand for real money balances because as income rises, so too does total expenditure, and thus more money is needed to make these purchases. Higher income increases the quantity of real money balances demanded at every interest rate, meaning that the entire $L\left(r\, Y\right)$ shifts to the right. So, money demand is negatively related to r and positively related to Y.

The LM curve is upward sloping because of the positive relationship between Y and r. As the level of income rises, the total demand for real money balances increases while the total money supply remains unchanged, thus forcing the interest rate to rise.

An increase in Y causes the $L\left(r\, Y\right)$ curve to shift outward (to the right), which moves the equilibrium point upwards to a higher level of r. This relationship is summarized by the movement of a point along the LM curve.

The Effects of Monetary Policy

The LM curve shows us the interest rate which brings the market for real money balances into balance, for any level of income. Yet, we know that equilibrium in this market also depends on monetary policy, as decided through the nominal supply of money $\left(\stackrel{\mathit{\&conjugate0;}}{M}\right)$. The LM curve is drawn for a given level of $\stackrel{\mathit{\&conjugate0;}}{M}$, meaning that monetary policy is held fixed when we draw it, and so when monetary policy changes, the LM curve must shift in response. When the government is enacting expansionary monetary policy by increasing $\stackrel{\mathit{\&conjugate0;}}{M}$, the equilibrium in the money market shifts rightward and lowers the interest rate for every level of income. This forces the entire LM curve to shift outward (to the right). When the government is enacting contractionary monetary policy by decreasing $\stackrel{\mathit{\&conjugate0;}}{M}$, the equilibrium in the money market shifts leftward and raises the interest rate for every level of income. This causes the entire LM curve to shift inward (to the left).

Using the IS-LM Model of aggregate demand, equilibrium is said to occur at the point where the IS and LM curves intersect.

The equation of the IS curve is $Y \= C\left(Y-T\right) \+ I\left(r\right) \+ G$ and the equation of the LM curve is $\frac{\stackrel{\mathit{\&conjugate0;}}{M}}{\stackrel{\mathit{\&conjugate0;}}{P}} \= L\left(r\,Y\right)$, where fiscal policy (G and T), monetary policy ($\stackrel{\mathit{\&conjugate0;}}{M}\)$ and the price level ($\stackrel{\mathit{\&conjugate0;}}{P}\)$ are taken to be exogenous variables.

Since the IS curve provides the combinations of Y and r which satisfy equilibrium in the market for goods and services, and the LM curve provides the combinations of Y and r which satisfy equilibrium in the market for real money balances, the point at which they cross will provide a combination of Y and r, which satisfies the equilibrium conditions of both markets.

So, at this equilibrium point, planned expenditure equals actual expenditure and the demand for real money balances equals the supply of real money balances.