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Economics 301
Intermediate Macroeconomics

IS/LM

Winter 2000

Last updated: February 14, 2000

Note: These notes are preliminary and incomplete and they are not guaranteed to be free of errors. Please let me know if you find typos or other errors. 

Derivation of the LM curve

The LM curve ( "L" denotes Liquidity and "M" denotes money) is a graph of combinations of real income, Y, and the real interest rate, r, such that the money market is in equilibrium (i.e. real money supply = real money demand). The graphical derivation of the LM curve is illustrated below.

lm.gif (7753 bytes)

The left-hand side of the graph illustrates money market equilibrium for a given level of Y. For example, when Y = Y0 the equilibrium real interest rate is 5%. The right-hand-side of the graph gives the LM curve. The LM curve is plotted with the real interest rate on the vertical axis and real income (GDP) on the horizontal axis. Each point on the LM curve represents a money market equilibrium for a particular real interest rate and income pair (r, Y). For example, the money market equilibrium at (r = 5%, Y = Y0) is given by the black (middle) dot on the LM curve.

At a higher level of income, Y1 > Y0, the money demand curve shifts up and right and a new equilibrium occurs at r = 7%. This equilibrium is represented by the blue (upper) dot on the LM curve. Similarly, at a lower level of income Y2 < Y0 the money demand curve shifts down and left and a new equilibrium occurs at r = 3%. This equilibrium is given the by the red (lower) dot on the LM curve.

Factors that Shift the LM Curve

The above analysis shows that the LM curve is an upward sloping curve in the graph with r on the vertical axis and Y on the horizontal axis. Every point on the LM curve represents an intersection between the real money supply MS and real money demand MD. The LM curve will shift whenever the variables we hold fixed, other than Y, in the money-supply/money-demand diagram change. These variable are M/P and pe. In particular, if M/P increases holding expected inflation fixed then r falls in the money market and so the LM curve shifts down and right. Similarly, if expected inflation increases real money demand falls, lowering the interest rate, and the LM curve shifts down and to the right.

Derivation of the IS curve

The IS curve ("I" denotes investment and "S" denotes saving) represents all combinations of income (Y) and the real interest rate (r) such that the market for goods and services is in equilibrium. That is, every point on the IS curve is an income/real interest rate pair (Y,r) such that the demand for goods is equal to the supply of goods (where it is implicitly assumed that whatever is demanded is supplied) or, equivalently, desired national saving is equal to desired investment. The graphical derivation of the IS curve is given below.

is.gif (8890 bytes)

Consider an initial equilibrium in the goods market where r = 5% and income is equal to Y0. This equilibrium is illustrated in the graph on the right with r on the vertical axis and Y on the horizontal axis as the big black dot (middle dot). Now suppose Y increases to Y1 (say supply increases). This increase in Y shifts the desired savings curve down and right lowering the equilibrium real interest rate to 3%. The new equilibrium in the goods market with higher income and a lower real interest rate is illustrated in the graph on the right as the big blue dot (bottom dot). Similarly, if Y decreases from Y0 to Y2 then the savings curve shifts up and left and the equilibrium real interest rises. The new equilibrium in the goods market with lower income and a higher real interest rate is illustrated in the graph on the right as the big red dot (top dot). Notice that as income increases (decreases) the real interest must fall (rise) in order to maintain equilibrium in the goods market. This is the relationship that is represented in the downward sloping IS curve.

Factors that Shift the IS Curve

Every point on the IS curve represents an intersection between desired national saving and desired investment for some income/interest rate pair (Y,r). As such the IS curve is derived holding the determinants of saving and investment, other than Y and r, fixed. When these factors change the IS curve will shift. Since points on the IS curve represent points where aggregate demand is equal to aggregate supply any factor that increases the demand for goods and services will shift the IS curve up and to the right and any factor that decreases the demand for goods and services will shift the IS curve down and to the left. From the savings/investment diagram it follows that any shift of the savings or investment curve that increases the real interest rate, holding Y fixed, will shift up the IS curve.