Using the algebra in a worked example.

Using IS and LM equations to find the Equilibrium National Income and Interest Rate

Using the algebra in a worked example

As section 19.3 in Economics (7th edition) demonstrates, equilibrium in the goods market is achieved at all points along the IS curve, while equilibrium in the money market is achieved at all points along the LM curve. Simultaneous equilibrium in both the goods and money market is thus achieved where the IS curve crosses the LM curve. This is shown in the following diagram, which is similar to Figure 19.13 in the text. (Note, though, that we are using a linear version of the model: i.e. IS and LM are straight lines. This makes the maths simpler.)

[pic 1]

Equilibrium national income and the equilibrium real rate of interest are given by Ye and re respectively. If we know the IS and LM functions, we can use algebra to find Ye and re.

The IS function

For simplicity, let us assume a closed economy (i.e. one where there is no international trade and hence no imports or exports). This will give the following equilibrium national income (Y):

        [pic 2]        (1)

Again, for simplicity, let us assume that consumption depends solely on disposable income, that investment depends solely on the rate of interest and that taxes are of a constant amount, irrespective of the level of national income. This gives the following simplified version of equation (3a) on page 552:

        [pic 3]        (2)

Let us assume that the consumption function^[1] is:

        [pic 4]        (3)

The term Y – T is gross income minus net taxes and thus represents disposable income. This means that the marginal propensity to consume from disposable income (mpc´) is 0.5. Let us also assume that the investment function is:

        [pic 5]        (4)

In other words, investment varies inversely with the real rate of interest. Finally, let us assume that government expenditure and taxes are the following fixed amounts:

        [pic 6]        (5)

        [pic 7]        (6)

Substituting equations (3) to (6) in equation (2), gives the following equation for the IS curve:

        [pic 8]

Simplifying gives:

        [pic 9]        (7)

Subtracting 0.5Y from both sides gives:

        [pic 10]                

giving an IS function of:

        [pic 11]        (8)

Alternatively, this could be expressed as:

        [pic 12]

i.e.        [pic 13]        (9)

The position and slope of the IS curve

From equation (8) we can see that the intercept of the IS curve with the horizontal axis is 4000 (i.e. when r is zero, Y is £4000bn). We can also see from equation (8) that the IS curve has a slope of –400 (i.e. for each 1 percentage point rise in r, Y falls by £400bn).

        We can also see how the slope and position of the IS curve depends on the mpc'. This can be seen from equation (3). The larger the mpc', the flatter and further to the right would be the IS curve. Thus, if the mpc' rose from 0.5 to 0.8, equation (7) would become:

        [pic 14]

Subtracting 0.8Y from both sides and dividing by 0.2 gives an IS function of

        [pic 15]

The IS curve would now have a slope of –1000 (a less steep line) and would cross the horizontal axis at Y = £9400bn (a point further to the right).

        We can also see how the position and slope of the IS curve depend on the interest sensitivity of investment. The greater the sensitivity, the flatter would be the IS curve. For example, assume that equation (4) became:

        [pic 16]   (a greater sensitivity to changes in interest rates)

Equation (8) would become:

        [pic 17]

The horizontal intercept would remain at Y = £4000bn, but the curve would become flatter. For every 1 percentage point rise in real interest rates, national income would fall by £800bn (compared with £400bn in the original situation).

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