## Formula for the Spending Multiplier

Some simple algebra permits us to derive a formula for the size of the multiplier effect that arises when an increase in government purchases induces increases in multiplier effect the additional shift» w «qg'vgat* demand tb»t result when expansionary fiscal policy tncroar.es income and thereby irvcreases consumer spending ?.. . b.ii th« nut (ri«f ef«e« ct»n amptify tint shin n aRff cjjalc dunvnd.

AfilfeCoV? demand. a£>, consumer spending. An important number in this formula is the marginal propensity to consume (MPC>—the fraction of extra income that a household consumes rather than saves. Fur example, suppose that the marginal propensity to consume is 'A. This means that for every extra dollar that a household earns, the household spends SO.75 ('/«of the dollar) and saves SO.25. With an MPC of 'A, when the workers and owners of Boeing earn \$20 billion from the government contract, they increase their consumer spending by Vi x \$20 billion, or \$15 billion.

To gauge the impact on aggregate demand of a change in government purchases, we follow the effects step by step. The process begins when the government spends\$20 billion, which implies thai national income (earningsand profits) also rises by this amount. This increase in income in turn raises consumer spending by MFC x \$20 billion, which in turn raises the income for the workers and owners of the firms that produce the consumption goods. This second increase in income again raises consumer spending, this time by MPC X <A1 PC X \$20 billion). These feedback effects go on and on.

To find the total impact on the demand for goods and services, we add tip all these effects:

Change in government purchases ■» \$20 billion

First change in consumption = .MPC x S2U billion

Second change in consumption « MPC! x £20 billion

Third change in oxisumplKui - MPC' X \$20 billion

Toial change in demand

Here .represents an infinite number of similar terms. Thus, we can write the multiplier as follows;

This multiplier tells us the demand for goods and services that each dollar of government purchases generates.

To simplify tins equation for the multiplier, recall from math class that this expression is an infinite geometric series. For x between-1 and +1,

1 +x + x' + x' » ... - 1 /(1-i). In our case, x - MPC. Thus,

For example, if MPC » '/», the multiplier is 1 / {I - which b4. In this case, the \$20 billion of government spending generates \$80 billion of demand for goods and services.

This formula for the multiplier shows an important conclusion: The size of the multiplier depends on the marginal propensity to consume While an MPC of V« leads to a multiplier of 4. an MPC of 14 leads to a multiplier of only 2. Thus, a larger AIPC means a larger multiplier. To see why litis is true, remember that the multiplier arises because higher income induces greater spending on consumption. With a larger MPC, consumption responds more to a change in income, ami so the multiplier is larger.

### Other Applications of the Multiplier Effect

Because of the multiplier effect, a dollar of government purchases can generate more than a dollar of aggregate demand. The logic of the multiplier effect, however, is not restricted to changes in government purchases Instead, it applies to any event that alters spending on any component of GDP—consumption, investment, government purchases, or net exports.

For example, suppose that a recession overseas reduces the demand for U-S. net exports by \$10 billion. This rcduccd spending on US goods and services depresses Ui5. national income, which reduces spending by US. consumers. If the marginal propensity to consume is V* and the multiplier is 4, then the \$10 billion fall in net exports mean» a \$<10 billion contraction in aggregate demand.

As another example, suppose that a stock-market boom increases households' wealth and stimulates their spending on goods and services by \$20 billion. This extra consumer spending increases national income, which in turn generates even more consumer spending. If the marginal propensity to consume is V* and the multiplier is 4, then the initial impulse of \$20 billion in consumer spending translates into an \$80 billion increase in aggregate demand.

The multiplier is an important concept in macroeconomics because it shows how the economy can amplify the impact of changes in spending. A small initial change in consumption, investment, government purchases, or net exports can end up having a large effect on aggregate demand and, therefore, the economy's production of goods and services. 