Why Taxes and Imports Reduce the Spending Multiplier
A government-purchases increase adds directly to planned expenditure, but its full effect on output is usually larger than the initial increase. Households spend part of each additional dollar of disposable income, creating further rounds of expenditure. Proportional income taxes reduce the disposable income households receive, while income-dependent imports direct part of additional spending toward goods produced abroad rather than domestic output.
The spending multiplier with both leakages measures the change in equilibrium domestic output caused by a change in government purchases, holding the model’s other conditions fixed. It is smaller than the multiplier when either proportional taxes or income-dependent imports are absent, provided the relevant marginal propensities are nonnegative.
The model assumes that \(c\), \(t\), and \(m\) are fixed over the change being analyzed; investment and exports are autonomous; and the price level, interest rate, and other determinants of expenditure do not respond to the spending change. It also treats the economy as having enough capacity to increase output in response to higher planned expenditure. These are short-run expenditure-model assumptions, not claims that all actual economies behave this way.
Planned expenditure is consumption plus investment, government purchases, and exports, less imports. Equilibrium output occurs where output equals planned expenditure. Substituting the consumption, tax, and import relationships gives:
Deriving the Spending Multiplier
The key is to track how a change in output affects the expenditure components that depend on income. If output rises by \(\Delta Y\), proportional taxes rise by \(t\Delta Y\). Disposable income therefore rises by \((1-t)\Delta Y\), and consumption rises by \(c(1-t)\Delta Y\). Imports rise by \(m\Delta Y\). The increase in imports is subtracted from domestic planned expenditure.
For a change in government purchases, \(\Delta Y=\Delta G+\Delta C-\Delta M\), because investment, exports, autonomous consumption, and autonomous imports are held fixed.
Use \(\Delta C=c(1-t)\Delta Y\) and \(\Delta M=m\Delta Y\), so \(\Delta Y=\Delta G+c(1-t)\Delta Y-m\Delta Y\).
Rearranging gives \([1-c(1-t)+m]\Delta Y=\Delta G\).
When the denominator is nonzero, divide both sides by it to find the output change caused by the purchases change.
This derivation explains the signs in the denominator. The induced domestic consumption propensity is \(c(1-t)\), which supports further expenditure and reduces the denominator. The import propensity \(m\) is a leakage from domestic expenditure, so it increases the denominator. The tax rate does not subtract from the multiplier directly: it reduces the fraction of additional income that becomes disposable income, and therefore reduces induced consumption.
Worked Calculations
Worked Example: Finding Equilibrium Before and After a Purchases Increase
Suppose a hypothetical economy has \(C_0=100\), \(c=0.8\), a proportional tax rate \(t=0.25\), \(I_0=80\), \(G=120\), \(X_0=30\), and \(M=M_0+mY\) with \(M_0=20\) and \(m=0.1\). Find equilibrium output, then calculate the change in output if government purchases rise by 25.
Method. First substitute the income-dependent terms into planned expenditure to find the equilibrium level. Then use the spending multiplier for the purchases shock.
Disposable income is \(Y-T=Y-0.25Y=0.75Y\). Consumption is therefore \(C=100+0.8(0.75Y)=100+0.6Y\). Imports are \(M=20+0.1Y\). Planned expenditure is:
At equilibrium, \(Y=AE\), so \(Y=310+0.5Y\). Subtract \(0.5Y\) from both sides: \(0.5Y=310\). Thus the initial equilibrium output is \(Y=310/0.5=620\).
The net induced expenditure coefficient is \(q=c(1-t)-m=0.8(0.75)-0.1=0.6-0.1=0.5\). The multiplier is \(1/(1-q)=1/(1-0.5)=2\). With \(\Delta G=25\), the output change is \(\Delta Y=2(25)=50\), so new equilibrium output is \(620+50=670\).
Check. Raising \(G\) from 120 to 145 changes the expenditure intercept from 310 to 335. The new equilibrium satisfies \(Y=335+0.5Y\), hence \(0.5Y=335\) and \(Y=670\), agreeing with the multiplier calculation. This method avoids confusing the change in output with the initial change in purchases.
Worked Example: Reading the Spending Rounds
In another hypothetical economy, \(c=0.75\), \(t=0.20\), and \(m=0.15\). Autonomous imports and all other autonomous expenditure are fixed. Government purchases increase by 55. Find the total equilibrium output change and show how the first four rounds contribute.
Method. Compute the induced domestic expenditure coefficient, then apply the geometric spending-round series. The initial 55 is the first round; later rounds are based on the previous round.
The induced consumption response to an extra dollar of output is \(c(1-t)=0.75(0.80)=0.60\). After the import leakage, the net induced domestic expenditure coefficient is \(q=0.60-0.15=0.45\). Since \(|0.45|<1\), the rounds converge. The multiplier is \(1/(1-0.45)=1/0.55=20/11\).
Thus \(\Delta Y=(20/11)(55)=100\). The rounds begin with the purchases increase, followed by 45% of each preceding round:
The first four rounds sum to \(55+24.75+11.1375+5.011875=95.899375\). The remaining rounds sum to \(100-95.899375=4.100625\), so all rounds together give the same 100 as the multiplier. The common error avoided here is to stop after one or two rounds and report a partial effect as the total equilibrium change.
Worked Example: Separating Tax and Import Effects
A hypothetical economy has \(c=0.60\), \(t=0.10\), and \(m=0.20\). A government-purchases increase of 66 occurs, while the other autonomous expenditure terms stay fixed. Calculate the output change and verify the result by tracing the induced changes in taxes, disposable income, consumption, and imports.
Method. Apply the multiplier first, then use the resulting income change to check the induced flows. This check distinguishes the tax effect on disposable income from the import leakage.
The induced consumption coefficient is \(c(1-t)=0.60(0.90)=0.54\). The net induced domestic expenditure coefficient is \(q=0.54-0.20=0.34\). The denominator is \(1-q=0.66\), giving a multiplier of \(1/0.66=50/33\). Therefore \(\Delta Y=(50/33)(66)=100\).
For an income increase of 100, taxes rise by \(t\Delta Y=0.10(100)=10\), so disposable income rises by \(100-10=90\). Consumption rises by \(0.60(90)=54\), and imports rise by \(0.20(100)=20\). The induced net domestic expenditure is \(54-20=34\). Adding the original purchases increase gives \(66+34=100\), which checks the equilibrium change.
A tempting but incorrect calculation would apply the MPC to the full 100 increase in income, giving consumption growth of 60. The correction is to apply the MPC to the 90 increase in disposable income after the 10 increase in taxes.
Interpreting the Formula and Avoiding Traps
The multiplier can also be written as \(1/[1-c(1-t)+m]\), which makes the separate roles of taxes and imports visible. Holding the other parameters fixed, a higher \(t\) reduces \(c(1-t)\); a higher \(m\) adds to the denominator. Both changes reduce the multiplier when the denominator is positive. With \(m=0\), the formula becomes the spending multiplier with proportional income taxes. With \(t=0\), it becomes the multiplier with income-dependent imports but no proportional income taxes.
Autonomous imports \(M_0\) affect the level of equilibrium output because they are subtracted from planned expenditure. However, if \(M_0\) is fixed, they do not affect the change in output per unit change in government purchases. The multiplier depends on the marginal response of imports to income, \(m\), not on the initial amount of imports.
The formula is not a forecast that can be applied without checking the model. It assumes the tax rate, MPC, and import propensity remain constant over the relevant range, and that changes in prices, interest rates, investment, and policy responses can be ignored. If any of these conditions are not reasonable, the simple multiplier may not describe the actual output response.
A reliable calculation sequence is to identify the MPC, tax rate, and marginal propensity to import; compute \(c(1-t)\); subtract \(m\) to get \(q\); check the relevant convergence condition; and use \(1/(1-q)\) for the multiplier. Keep changes and levels distinct: the multiplier maps \(\Delta G\) to \(\Delta Y\), while the equilibrium level also depends on all autonomous expenditure terms.
Check Your Understanding
For each problem, treat the stated parameters as fixed and use the short-run expenditure model described above.
- An economy has \(c=0.70\), \(t=0.20\), and \(m=0.10\). Find the spending multiplier. (Answer: \(50/27\), or approximately 1.8519.)
- In a hypothetical economy, \(c=0.80\), \(t=0.25\), and \(m=0.20\). If government purchases rise by 36, what is the equilibrium output change? (Answer: Multiplier \(=5/3\); \(\Delta Y=60\).)
- A purchases increase of 61 causes output to rise by 100. If \(c=0.65\), \(t=0.20\), find the marginal propensity to import \(m\). (Answer: \(m=0.13\).)
- An economy has \(c=0.75\), \(t=0.20\), and a spending multiplier of 2. Find \(m\). (Answer: \(m=0.10\).)
- Let \(c=0.90\), \(t=0.10\), and \(m=0.08\). Calculate the spending multiplier and the output effect of a 27 increase in government purchases. Then calculate the multiplier if imports did not respond to income. (Answer: With imports, multiplier \(=1/0.27\approx3.7037\) and \(\Delta Y=100\); with \(m=0\), multiplier \(=1/0.19\approx5.2632\).)