Why Interest-Sensitive Investment Matters
A rise in government purchases adds directly to planned expenditure. In the simplest expenditure model, that initial increase is followed by further rounds of consumption, so the total change in output exceeds the initial spending change. But that calculation holds other components of expenditure constant apart from consumption induced by income. If investment falls when interest rates rise, some of the fiscal expansion is offset: higher output raises money demand, the interest rate increases in the model considered here, and firms undertake less investment.
This reduction in private investment caused by expansionary fiscal policy is called crowding out. It weakens the expenditure-multiplier effect; it does not necessarily eliminate it. The size of the effect depends on how strongly investment responds to the interest rate and how the money market responds to higher output.
The interest rate is denoted by \(r\), measured in percentage points, and is the rate relevant to investment decisions in the model. This differs from the actuarial notation \(i\) for an effective annual rate. Output, government purchases, consumption, and investment are measured in billions of dollars in the examples.
The Model and Its Assumptions
Consider a short-run, closed-economy model with a fixed price level. Consumption depends on disposable income, taxes are lump-sum and fixed during the spending change, and investment falls linearly as the interest rate rises. The central bank holds the nominal money supply fixed; with the price level fixed, real money balances are therefore also fixed. The fiscal expansion is an increase in government purchases \(G\), with taxes and the other autonomous components held constant.
Here \(Y\) is equilibrium output; \(C_0\) is autonomous consumption; \(c\), with \(0<c<1\), is the marginal propensity to consume; \(T\) is lump-sum taxes; \(I_0\) is autonomous investment; and \(b>0\) measures how much investment falls for a one-percentage-point rise in \(r\). The money-market equilibrium condition is
In this equation, \(m\) is fixed real money supply, \(k>0\) measures how money demand responds to output, and \(h>0\) measures how money demand responds to the interest rate. A rise in output raises money demand; with money supply fixed, the interest rate must rise to restore money-market equilibrium.
These equations are a simplified comparative-statics model, not a claim that every fiscal expansion must raise rates. The result depends on the stated assumptions: fixed real money supply, a responsive money market, and investment that responds to the rate. If monetary policy accommodates the expansion by increasing money supply, or if investment is insensitive to rates, the crowding-out effect can be smaller or absent. The equations also abstract from changes in prices, foreign trade, and other possible responses.
Deriving the Crowding-Out Multiplier
The expenditure multiplier without an interest-rate response is \(1/(1-c)\). With interest-sensitive investment, the multiplier must account for the investment decline as well as the induced consumption spending. The following derivation uses changes from the initial equilibrium, so constant autonomous terms cancel.
Because \(m\) is fixed, a change in money-market equilibrium requires \(0=k\Delta Y-h\Delta r\). Thus \(\Delta r=(k/h)\Delta Y\): higher equilibrium output raises the interest rate.
Since \(I=I_0-br\), the change in investment is \(\Delta I=-b\Delta r=-(bk/h)\Delta Y\). The minus sign records the decline in investment when the rate rises.
With taxes fixed, the change in consumption is \(c\Delta Y\). Therefore \(\Delta Y=c\Delta Y+\Delta I+\Delta G\). Substituting the investment response gives \((1-c+bk/h)\Delta Y=\Delta G\).
The extra term \(bk/h\) in the denominator captures the interest-rate feedback. A larger \(b\) means investment responds more strongly to rates; a larger \(k\) means output raises money demand more strongly; and a smaller \(h\) means a given money-demand increase requires a larger rate change. Each condition increases the offset to fiscal spending.
For positive \(b\), \(k\), and \(h\), the denominator exceeds \(1-c\), so the multiplier is smaller than \(1/(1-c)\). This is the mathematical statement that crowding out weakens, but does not automatically reverse, the fiscal expansion in this model. With \(\Delta G>0\), the multiplier remains positive, so output still rises.
Worked Calculations
Worked Example: Calculate the Output and Investment Effects
A hypothetical economy has \(c=0.80\), \(b=40\) billion dollars per percentage point, \(k=0.50\), and \(h=20\). Real money supply and the price level are fixed. Government purchases rise by \(\Delta G=30\) billion dollars. Find the change in output, the change in the interest rate, and the change in investment. Compare output with the result when investment is interest-insensitive.
Method. Use the crowding-out multiplier, then use money-market equilibrium and the investment function to calculate the associated rate and investment changes.
First calculate the interest-rate feedback term:
The denominator of the multiplier is \(1-c+bk/h=1-0.80+1=1.20\). Therefore the multiplier is \(1/1.20=0.8333\), and output changes by
From the money-market equation, \(\Delta r=(k/h)\Delta Y=(0.50/20)(25)=0.025(25)=0.625\) percentage points. Investment then changes by \(\Delta I=-b\Delta r=-40(0.625)=-25\) billion dollars.
Without an interest-rate response, the multiplier would be \(1/(1-0.80)=5\), giving an output increase of \(5(30)=150\) billion dollars. The model with crowding out instead gives a 25-billion-dollar increase. Check the equilibrium change: induced consumption is \(0.80(25)=20\) billion dollars, so \(\Delta G+\Delta C+\Delta I=30+20-25=25\) billion dollars, equal to \(\Delta Y\). The calculation avoids the mistake of treating the increase in government purchases as the only change in expenditure.
Worked Example: Keep the Rate Units Consistent
In a different hypothetical economy, \(c=0.75\), \(b=10\) billion dollars per percentage point, \(k=0.40\), and \(h=8\). With fixed real money supply, government purchases increase by 24 billion dollars. Calculate the output increase, interest-rate increase, and investment decline, and verify the goods-market change.
Method. Compute the multiplier first, keeping \(b\) defined per percentage point throughout. The common mistake is to treat a rate change in percentage points as a decimal rate change without also changing the units of \(b\).
The feedback term is \(bk/h=10(0.40)/8=4/8=0.50\). The multiplier is \(1/(1-0.75+0.50)=1/0.75=4/3\). Thus \(\Delta Y=(4/3)(24)=32\) billion dollars.
The money-market equation gives \(\Delta r=(0.40/8)(32)=0.05(32)=1.6\) percentage points. At 10 billion dollars of investment per percentage point, the investment change is \(\Delta I=-10(1.6)=-16\) billion dollars. The induced consumption change is \(c\Delta Y=0.75(32)=24\) billion dollars.
The goods-market check is \(\Delta G+\Delta C+\Delta I=24+24-16=32\) billion dollars, matching the output change. If investment were held fixed, the multiplier would be \(1/(1-0.75)=4\), implying an output increase of \(4(24)=96\) billion dollars. The lower result is consistent with the 16-billion-dollar investment reduction.
Worked Example: Find the Spending Increase Needed for an Output Target
A hypothetical economy has \(c=0.60\), \(b=20\) billion dollars per percentage point, \(k=0.30\), and \(h=6\), with real money supply fixed. What increase in government purchases is required to raise output by 40 billion dollars? Also calculate the resulting rate and investment changes.
Method. Rearrange the change equation to find the fiscal change that produces the specified output change. This avoids using the no-crowding-out multiplier for a target calculation.
The feedback term is \(bk/h=20(0.30)/6=6/6=1\). Hence the multiplier is \(1/(1-0.60+1)=1/1.40=5/7\). Since \(\Delta Y=(5/7)\Delta G\), the required spending change is
The interest-rate increase is \(\Delta r=(0.30/6)(40)=0.05(40)=2\) percentage points. The investment decline is \(\Delta I=-20(2)=-40\) billion dollars. Induced consumption rises by \(0.60(40)=24\) billion dollars.
Check the goods market: \(\Delta G+\Delta C+\Delta I=56+24-40=40\) billion dollars, the target output increase. Without the investment response, achieving the same 40-billion-dollar output increase would require only \(40(1-0.60)=16\) billion dollars of purchases. That comparison shows why the rate channel matters to the size of the required fiscal change.
Interpreting Crowding Out and Avoiding Traps
The multiplier formula is a compact description of two linked markets, not an independent rule to apply without checking assumptions. In the goods market, higher output generates additional consumption. At the same time, higher output raises money demand; under the fixed-money-supply condition, the interest rate rises and reduces investment. The output change must satisfy both effects together.
| Model feature | Effect on the multiplier |
|---|---|
| Higher investment sensitivity \(b\) | More investment is lost for a given rate increase; multiplier falls. |
| Stronger output response of money demand \(k\) | A given output increase puts more upward pressure on the rate; multiplier falls. |
| More rate-sensitive money demand \(h\) | A smaller rate increase clears the money market; crowding out weakens. |
| Investment insensitive to the rate, \(b=0\) | The multiplier becomes \(1/(1-c)\), the simple expenditure multiplier. |
A frequent exam trap is to calculate \(\Delta Y=\Delta G/(1-c)\) and stop, even though the problem specifies that investment falls as rates rise. For example, in the first worked example that incorrect calculation gives \(150\) billion dollars. It ignores the money-market rate increase and the \(25\)-billion-dollar investment decline. The corrected multiplier is \(1/(1-c+bk/h)=1/1.20\), giving \(\Delta Y=25\) billion dollars. A second trap is a unit error: if \(b\) is given per percentage point, a rate increase of \(0.625\) percentage points must be multiplied by \(b\) as stated, not converted to \(0.00625\) without converting \(b\)'s units too.
Crowding out is therefore a matter of degree in this model. If investment responds weakly, or if money demand responds strongly to the rate, the offset can be small. If investment responds strongly and money demand responds little to the rate, the offset can be substantial. This framework is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, use the fixed-money-supply model. Rates are measured in percentage points, and monetary amounts are in billions of dollars unless stated otherwise.
- An economy has \(c=0.70\), \(k=0.40\), \(h=10\), and \(b=25\) billion dollars per percentage point. Government purchases rise by 20. Find \(\Delta Y\), \(\Delta r\), and \(\Delta I\). (Answer: \(\Delta Y=15.38\), \(\Delta r=0.6154\) percentage points, \(\Delta I=-15.38\).)
- Let \(c=0.80\), \(k=0.20\), \(h=4\), and \(b=10\) billion dollars per percentage point. If \(\Delta G=18\), calculate the multiplier, output change, rate change, and investment change. (Answer: multiplier \(=1/0.70=1.4286\), \(\Delta Y=25.71\), \(\Delta r=1.2857\) percentage points, \(\Delta I=-12.86\).)
- Suppose \(c=0.65\), \(k=0.30\), \(h=6\), and \(b=14\) billion dollars per percentage point. Find the increase in government purchases required for \(\Delta Y=30\), and calculate the associated rate and investment changes. (Answer: \(\Delta G=31.50\), \(\Delta r=1.50\) percentage points, \(\Delta I=-21.00\).)
- With \(c=0.60\), \(k=0.50\), \(h=10\), and \(b=12\) billion dollars per percentage point, government purchases rise by 20. Calculate output, the rate increase, investment, and induced consumption changes. Verify that the expenditure changes sum to the output change. (Answer: \(\Delta Y=20\), \(\Delta r=1.00\) percentage point, \(\Delta I=-12\), \(\Delta C=12\); \(20-12+12=20\).)
- An economy has \(c=0.75\), \(k=0.40\), and \(h=8\). What value of \(b\), in billions of dollars per percentage point, makes the multiplier equal to 1? For a spending increase of 40, find the resulting output, rate, and investment changes. (Answer: \(b=15\), \(\Delta Y=40\), \(\Delta r=2.00\) percentage points, \(\Delta I=-30\).)