Using the algebra in a worked example.

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The LM function

Assume that real money supply (Ms/P) is fixed by the central bank at 1000 and that the real demand for money function is as given on page 000, namely:

[pic 18]

Assume that the actual real demand for money function is:

[pic 19] (10)

Thus, given that in equilibrium real money supply (= 1000) equal real money demand, then:

[pic 20]

Expressing this in terms of r gives:

[pic 21] (11)

This is the LM function.

The position and slope of the LM curve

From equation (10), the intercept of the LM curve with the vertical axis is –1.25% (i.e. the value of r where Y is zero). We can also see that the slope of the LM curve is 0.4/800 = 0.0005 (i.e. for each £1bn rise in Y, r rises by 0.0005 of a percentage point).

We can also see how the slope and position of the LM curve depends on the responsiveness of the demand for money (L) to changes in national income (Y). This can be seen from equation (10). The greater the responsiveness, the steeper will be the LM curve. Thus if equation (10) became:

[pic 22] (a greater sensitivity to changes in Y)

the LM function (equation (11)) would become:

[pic 23]

This represents a steeper LM curve, since for every £1bn rise in Y, r rises by 0.6/800 = 0.00075 (a bigger rise in r than with the original LM curve). Note, however, that the vertical intercept remains at r = –1.25%.

How does the responsiveness of the demand for money (L) to changes in the real interest rate affect the LM curve? This can again be seen from equation (10). The greater the responsiveness, the steeper will be the LM curve. Thus if equation (10) became:

[pic 24] (a greater sensitivity to changes in r)

the LM function (equation (11)) would become:

[pic 25]

This represents a flatter LM curve, since for every £1bn rise in Y, r rises by 0.4/1000 = 0.0004 (a smaller rise in r than with the original LM curve). The vertical intercept also moves upwards to r = –1%.

Equilibrium

Setting equation (8) (the IS function) equal to equation (11) (the LM function) and solving for Y, gives:

[pic 26]

Multiplying both sides by 800 gives:

[pic 27]

Substituting Y = 3750 into either equation (9) or (11) gives r = 0.625. Thus, in the ISLM model illustrated in the diagram above, the values of Ye and re are 3750 and 0.625 respectively.

Equilibrium values of C and I can also be found. This involves substituting Y = 3750 and r = 0.625 in equations (3) and (4) respectively. Thus:

[pic 28]

and

[pic 29]

These values can be verified from equation (1)

[pic 30]

Question

Given the above IS and LM equations, solve for equilibrium Y and r if government expenditure increases to £600bn. Confirm that the result is consistent with a rightward shift in the IS curve in the above diagram.

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