Finding Equilibrium Output from Expenditure Schedules

Why Equilibrium Output Is an Equality

In the Keynesian-cross framework, planned expenditure summarizes intended spending on current output. Equilibrium output is the level of production at which that planned spending equals output. The practical task is therefore to compare the two quantities on the same basis and find where they match—not simply to select the row with the largest expenditure or the smallest apparent gap.

This tutorial uses the short-run expenditure model: output and planned expenditure are measured for the same period and in the same monetary units; the price level is treated as fixed; and the expenditure schedule describes planned, rather than realized, spending. When the schedule is linear, its slope is less than one, so a unique equilibrium can be found by solving an equation. The model’s equilibrium condition is the focus here; the response of inventories and the derivation of fiscal multipliers are separate topics.

Definition: Equilibrium output, denoted $Y^*$, is the level of output $Y$ at which planned expenditure $E^P$ equals output: $$ Y^*=E^P(Y^*) $$ A table has an exact equilibrium at a listed output only when its planned-expenditure entry equals that output. If a table gives no exact match, additional information about the schedule is needed to determine whether interpolation or another method is appropriate.

Read a Table Before Solving an Equation

For a table of output and planned expenditure, compare the two entries at each output level. If $E^P>Y$, planned expenditure exceeds output at that row; if $E^P<Y$, it is below output. Neither comparison alone identifies an equilibrium. An exact match does. If the schedule is explicitly linear between adjacent entries, a change from $E^P>Y$ to $E^P<Y$ indicates that the equality lies between those output levels, and linear interpolation can locate it.

Worked Example: Interpolating Between Two Table Rows

A hypothetical economy reports output and planned expenditure, both in billions of dollars. Its expenditure schedule is linear over the range shown.

Output $Y$	Planned expenditure $E^P$	$E^P-Y$
200	260	60
300	320	20
400	380	-20
500	440	-60

Method. First check for an exact match. None of the listed rows satisfies $E^P=Y$. Since the schedule is stated to be linear, use the two adjacent rows where the difference changes sign: at $Y=300$, the difference is $320-300=20$; at $Y=400$, it is $380-400=-20$.

The slope of planned expenditure between these rows is $ (380-320)/(400-300)=60/100=0.6 $. Using the point $ (Y,E^P)=(300,320) $, the linear schedule between the rows is $E^P=320+0.6(Y-300)=140+0.6Y$.

Set output equal to planned expenditure: $Y=140+0.6Y$. Subtract $0.6Y$ from both sides to obtain $0.4Y=140$, so $Y^*=350$ billion dollars.

Check. At $Y=350$, planned expenditure is $140+0.6(350)=140+210=350$, equal to output. The two table rows at 300 and 400 are each 20 billion away from equality, but in opposite directions. Neither row is an equilibrium; the equality occurs between them.

Solving a Linear Expenditure Schedule

A linear planned-expenditure schedule can be written $E^P=a+bY$, where $a$ is the expenditure intercept and $b$ is the change in planned expenditure for a one-unit change in output. In a basic model with consumption responding to income and other spending components held fixed, the induced part of expenditure is captured by $bY$. The equilibrium calculation is an equation-solving step: set $Y$ equal to the entire planned-expenditure expression, including all of its components.

Result: If planned expenditure is $E^P=a+bY$, equilibrium satisfies $Y^*=a+bY^*$. For $b\ne1$, the algebraic solution is $$ Y^*=\frac{a}{1-b}. $$ For the standard stable linear Keynesian-cross schedule, $0\le b<1$. The formula gives a meaningful equilibrium only when the specified schedule and its assumptions apply; it is not a rule for an arbitrary nonlinear schedule or for a table with no stated interpolation rule.

The result follows directly from the definition, rather than from an additional behavioral claim. At equilibrium, substitute the schedule into $Y^*=E^P(Y^*)$, giving $Y^*=a+bY^*$. Move the term involving output to the left: $Y^*-bY^*=a$. Factor out $Y^*$ to get $(1-b)Y^*=a$. If $1-b\ne0$, divide both sides by $1-b$. Under $0\le b<1$, the denominator is positive, and the schedule has one algebraic intersection with the 45-degree equality condition.

Write planned expenditure.
Include every component specified by the problem, with its signs and units.

Apply the equilibrium condition.
Set $Y=E^P(Y)$; do not set output equal to one component such as consumption alone.

Solve and verify.
Find $Y^*$, substitute it into the original expenditure expression, and confirm that planned expenditure equals output.

Worked Example: Solving from Consumption, Investment, and Government Purchases

In a hypothetical economy, all amounts are in billions of dollars. Consumption is $C=60+0.8(Y-T)$, where lump-sum taxes are $T=25$. Planned investment is $I=45$, and government purchases are $G=40$. Assume these components define the planned-expenditure schedule and there are no other expenditure components. Find equilibrium output.

Method. First express consumption in terms of output. Disposable income is $Y-T=Y-25$, so $C=60+0.8(Y-25)=60+0.8Y-20=40+0.8Y$.

Add all planned expenditure components: $E^P=C+I+G=(40+0.8Y)+45+40=125+0.8Y$. The equilibrium condition is $Y=125+0.8Y$. Subtract $0.8Y$ from both sides: $0.2Y=125$. Therefore $Y^*=125/0.2=625$ billion dollars.

Check. At $Y=625$, disposable income is $625-25=600$. Consumption is $60+0.8(600)=60+480=540$. Total planned expenditure is $540+45+40=625$, equal to output.

Common mistake avoided. Using $60+0.8Y$ as consumption would omit the tax deduction inside disposable income and overstate consumption by $0.8(25)=20$. Keeping $Y-T$ intact until substitution prevents that error.

When a Table Gives an Exact Match

A table may already contain the equilibrium, but this should be established by checking the entries rather than by visual estimation. A direct match is especially useful when the table is not said to be linear: it identifies an equilibrium at that listed value without requiring any assumption about values between rows. If the table has no match, do not silently assume that the schedule is linear.

Worked Example: Checking a Component Table for Equality

A hypothetical economy’s table lists planned consumption $C$, planned investment $I$, and government purchases $G$, all in millions of dollars. No other planned expenditure is included.

Output $Y$	$C$	$I$	$G$	$E^P=C+I+G$
100	100	45	35	180
200	160	45	35	240
300	220	45	35	300
400	280	45	35	360

Method. Calculate total planned expenditure in each row. At output 100, it is $100+45+35=180$; at 200, it is $160+45+35=240$; at 300, it is $220+45+35=300$; and at 400, it is $280+45+35=360$. Compare each total with the output in that same row.

Only the row $Y=300$ has $E^P=Y$, so the table identifies equilibrium output of 300 million dollars. There is no need to interpolate to report this listed equilibrium.

Check. At the matching row, the three components sum to $220+45+35=300$, exactly the output value. The other rows have differences $80$, $40$, and $-40$, respectively, so they are not additional equilibria.

Common Errors and Their Corrections

Choosing the nearest row as if it were an equilibrium. A small gap is still a gap. In the first example, $Y=300$ has $E^P-Y=20$, and $Y=400$ has $E^P-Y=-20$. They are tied for the smallest absolute difference among those rows, since both gaps have magnitude 20. Neither is an equilibrium; under the stated linearity assumption, the equilibrium is $Y=350$.
Omitting an expenditure component. If the problem defines $E^P=C+I+G$, use the sum of all three components. Equating output to consumption alone does not apply the equilibrium condition.
Using the wrong sign or base for taxes. If consumption depends on disposable income $Y-T$, substitute that expression before simplifying. Do not replace $Y-T$ with $Y+T$, or treat a fixed tax as though it were part of output.
Interpolating without justification. A table gives values at listed output levels. Interpolation is justified only when linearity over the relevant interval is specified or otherwise supplied by the model.
Mixing units. If output and expenditure are in billions, the resulting equilibrium is in billions. Component values must use the same scale before they are added or compared.

Key takeaway: Equilibrium is an equality, not a close approximation. For a table, add the listed planned components and compare their total with output. For a linear schedule, set $Y=E^P(Y)$, solve, and substitute the result back into the original schedule as a check.

Check Your Understanding

For each problem, state the equilibrium output in the units provided. Use only the assumptions stated in that problem.

A linear planned-expenditure schedule is $E^P=110+0.6Y$, with output and expenditure in millions of dollars. Find equilibrium output. (Answer: 275 million dollars)
Consumption is $C=50+0.75Y$, planned investment is 40, and government purchases are 10, all in billions of dollars. There are no other expenditure components. Find equilibrium output. (Answer: 400 billion dollars)
A schedule is linear between the two listed rows: at $Y=240$, $E^P=260$; at $Y=340$, $E^P=320$. Both are in millions of dollars. Find the equilibrium between the rows. (Answer: 290 million dollars)
Consumption is $C=75+0.6(Y-T)$, with fixed taxes $T=50$. Planned investment is 55 and government purchases are 25, all in millions of dollars. Find equilibrium output. (Answer: 312.5 million dollars)
In billions of dollars, a linear schedule has $E^P=220+0.7Y$. Find equilibrium output and verify that planned expenditure equals it. (Answer: $Y^*=733\frac{1}{3}$ billion dollars; $220+0.7(733\frac{1}{3})=733\frac{1}{3}$)

What You'll Learn