Why Equal-Cost Fiscal Actions Can Have Different Effects
A government considering a fiscal action may ask how much output will change for a given budget cost. An increase in government purchases enters planned expenditure directly. A lump-sum tax cut or a transfer increase first changes households’ disposable income; only the portion households spend becomes an initial increase in planned expenditure. This distinction makes the output effects different even when the three actions have the same direct budget cost.
The comparison here uses the simple expenditure model, holding the marginal propensity to consume and the other model conditions fixed. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The Model and the Meaning of Equal Cost
Let \(Y\) denote output, \(C\) consumption, \(I\) planned investment, \(G\) government purchases of goods and services, \(T\) lump-sum taxes, and \(TR\) transfers to households. A lump-sum tax is fixed independently of current output in this model. Let \(a\) be autonomous consumption and \(c\) the marginal propensity to consume (MPC): the fraction of an additional dollar of disposable income that households spend. Assume \(0<c<1\).
The model assumes a closed economy with no imports, no proportional income taxes, and no change in planned investment. The price level is treated as fixed, and the analysis concerns the change from an initial expenditure equilibrium. Planned consumption and equilibrium output are:
The first equation says that consumption depends on autonomous consumption and disposable income. Disposable income is \(Y-T+TR\). The second equation says output equals planned expenditure in equilibrium. When comparing policy changes, a positive \(\Delta T\) is a tax increase; a tax cut of size \(B\) is therefore \(\Delta T=-B\).
This definition compares the direct budget effect of each policy change. The model abstracts from any additional tax revenue that might result when output changes. It also does not treat purchases and transfers as the same thing: purchases are part of planned expenditure directly, while transfers affect expenditure through household disposable income.
Deriving the Three Output Effects
Start from equilibrium before the policy change and compare it with equilibrium after the change. Since \(a\) and \(I\) are unchanged, subtracting the initial equilibrium equation from the new one gives:
The \(\Delta Y\) inside the consumption change matters: higher output generates higher income, which induces further consumption. Collecting the output-change terms gives:
Because \(0<c<1\), \(1-c\) is positive, so dividing by it is valid. This equation provides a compact comparison: a purchase change enters with coefficient 1, while a tax or transfer change enters with coefficient \(c\).
For equal direct costs \(B\), set \(\Delta G=B\), \(\Delta T=-B\), and \(\Delta TR=B\) in turn. The purchase increase raises output by \(B/(1-c)\); either the tax cut or the transfer increase raises output by \(cB/(1-c)\). Since \(0<c<1\), \(cB/(1-c)\) is smaller than \(B/(1-c)\) for any positive \(B\). The tax-cut and transfer effects are equal in this model because each increases disposable income by \(B\), and households initially spend the same fraction \(c\) of that increase.
Worked Comparisons
Worked Example: Equal $50 Million Costs with an MPC of 0.80
A hypothetical economy has an MPC of \(0.80\). Compare a $50 million increase in government purchases, a $50 million lump-sum tax cut, and a $50 million transfer increase. Assume all the conditions of the model above and express output changes in millions of dollars.
Method. Calculate each policy’s first-round effect on planned expenditure, then divide by the fraction of an output change that does not return as induced consumption, \(1-c\). Here \(1-c=1-0.80=0.20\).
Purchases. The direct increase in planned expenditure is $50 million. Thus:
Tax cut. A $50 million tax cut means \(\Delta T=-50\). The first-round increase in consumption is \(0.80\times 50=40\) million dollars. Therefore:
Transfer increase. The transfer raises disposable income by $50 million, so the first-round consumption increase is also \(0.80\times 50=40\) million dollars:
The calculation is checked by comparing the multipliers: purchases have multiplier \(1/0.20=5\), while a tax cut or transfer increase has multiplier \(0.80/0.20=4\). Multiplying each by the $50 million policy amount gives $250 million and $200 million, respectively. The common mistake avoided is applying the purchases multiplier to a tax cut or transfer.
Worked Example: Tracking the Spending Rounds for Equal $140 Million Costs
A hypothetical economy has an MPC of \(0.65\). Each of three alternative policies has a direct budget cost of $140 million: increased purchases, a lump-sum tax cut, or increased transfers. Find the output change for each and show how the first rounds differ.
Purchases. The purchases increase adds $140 million directly to planned expenditure. The next consumption round is \(0.65\times 140=91\) million dollars. The following round is \(0.65\times 91=59.15\) million dollars, and the next is \(0.65\times 59.15=38.4475\) million dollars. The complete series is:
Tax cut. The $140 million tax cut initially raises disposable income by $140 million, but the first-round consumption increase is \(0.65\times 140=91\) million dollars. The later rounds are \(0.65\times 91=59.15\) million dollars, then \(0.65\times 59.15=38.4475\) million dollars, and so on. Thus:
Transfer increase. It produces the same first-round consumption increase, \(0.65\times 140=91\) million dollars, and the same subsequent rounds. Its total output change is also \(91/0.35=260\) million dollars.
The multiplier calculation checks the round-by-round result: \(140/0.35=400\) for purchases and \((0.65\times 140)/0.35=91/0.35=260\) for either income-support policy. The mistake to avoid is beginning the tax-cut or transfer series with $140 million; that is the change in disposable income, not the first-round change in consumption.
Worked Example: Finding the Cost of a Target Output Increase
A hypothetical economy has an MPC of \(0.75\). Policymakers want output to rise by $180 million. Under the model assumptions, find the cost of achieving this target through (a) increased government purchases and (b) either a lump-sum tax cut or a transfer increase. Then compare the three policies if the available budget cost is $60 million.
Here \(1-c=1-0.75=0.25\). The purchases multiplier is \(1/0.25=4\). To obtain a $180 million output increase using purchases, divide the target by 4:
Check: a $45 million purchases increase gives \(45\times 4=180\) million dollars of additional output.
The tax-cut and transfer multipliers are both \(0.75/0.25=3\). The budget cost needed to reach the same target with either policy is:
Check: a $60 million tax cut or transfer increase gives \(60\times 3=180\) million dollars of additional output. If the budget cost is instead fixed at $60 million for all three options, purchases raise output by \(60\times4=240\) million dollars, whereas either income-support option raises it by \(60\times3=180\) million dollars. The comparison is consistent: purchases need $15 million less to reach the target, and the equal-cost purchase option produces $60 million more output.
This inverse calculation avoids a related error: using the output multiplier as though it were the cost required to reach a target. To find required cost, divide the target output change by the applicable multiplier.
Reading the Comparison Correctly
| Policy change of direct cost \(B\) | First-round increase in planned expenditure | Output change |
|---|---|---|
| Government purchases increase by \(B\) | \(B\) | \(B/(1-c)\) |
| Lump-sum tax cut of \(B\) | \(cB\) | \(cB/(1-c)\) |
| Transfer increase of \(B\) | \(cB\) | \(cB/(1-c)\) |
The table separates two stages that are easy to conflate. First, the policy changes planned expenditure directly or indirectly. Then, the resulting output and income changes lead to further consumption rounds. An increase in purchases contributes its full amount in the first stage. A tax cut or transfer contributes only \(cB\) in that stage because households spend fraction \(c\) of the added disposable income.
The equality between the tax-cut and transfer results depends on the model treating each as an equal lump-sum increase in disposable income and applying the same MPC. It is not a claim that every real-world tax change and transfer program has identical effects. For example, a policy whose amount depends on income, or one affecting households with different spending responses, would not meet the comparison’s assumptions.
Common Mistake: Using the Purchases Multiplier for Every Policy
Suppose the MPC is \(0.80\) and a tax cut costs $50 million. A mistaken calculation might apply the purchases multiplier \(1/(1-0.80)=5\), giving \(50\times5=250\) million dollars of additional output. The correction is to calculate the tax cut’s first-round consumption effect: \(0.80\times50=40\) million dollars. Its output effect is \(40/(1-0.80)=40/0.20=200\) million dollars. The purchases multiplier applies to a direct purchases increase; the tax-cut multiplier applies to the tax change.
These comparisons rely on the simple expenditure model’s fixed conditions. If its assumptions change, the calculations may change too; the formulas above should not be used without checking that the model applies.
Check Your Understanding
For each problem, use the simple expenditure model and state output changes in the units specified.
- A hypothetical economy has \(c=0.70\). Compare the output changes from a $30 million purchase increase and an equal-cost $30 million lump-sum tax cut. (Answer: purchases, $100 million; tax cut, $70 million.)
- With \(c=0.60\), a government considers a $45 million transfer increase. Find the first-round consumption change and the total output change. (Answer: first-round consumption, $27 million; output, $67.5 million.)
- A hypothetical economy has \(c=0.80\), and policymakers seek a $320 million output increase. Find the direct cost using purchases and the direct cost using either a tax cut or transfers. (Answer: purchases, $64 million; tax cut or transfers, $80 million.)
- For \(c=0.75\), compare a $24 million tax cut with a $24 million transfer increase. Calculate the first-round planned-expenditure increase and total output change for each, and explain why they match. (Answer: each has a $18 million first-round increase and a $72 million output increase; each raises disposable income by $24 million, of which households initially spend 75%.)
- A policy analyst claims that, with \(c=0.50\), an equal-cost $90 million purchases increase and tax cut each raise output by $180 million. Identify the error and give the correct output change for each policy. (Answer: the tax cut was assigned the purchases multiplier; purchases raise output by $180 million, while the tax cut raises it by $90 million.)