Equilibrium Income Numericals: Solve C = a + bY

You are given C = 90 + 0.75Y and investment of Rs 60 crore. You recognise the consumption function, but the question asks for income. Where do you begin?

Begin with one sentence: at equilibrium, output equals planned spending. In the simple two-sector economy, that gives you Y = C + I.

Substitute the consumption function, collect the terms containing Y, and solve:

Y = 90 + 0.75Y + 60
0.25Y = 150
Y = Rs 600 crore

That is the basic method. The interesting part is learning to recognise the same calculation when the question gives a saving function, hides investment, or asks how much income changes.

What the letters mean

In the consumption function C = a + bY:

Symbol	Meaning	In C = 90 + 0.75Y
C	Consumption expenditure	Depends on Y
a	Autonomous consumption	Rs 90 crore
b	Marginal propensity to consume, or MPC	0.75
Y	Income	The amount to find
I	Planned investment	Given separately

The coefficient b tells you the fraction of additional income that is consumed. With b = 0.75, an extra Rs 100 of income leads to Rs 75 of extra consumption. The remaining Rs 25 is additional saving, so MPS is 1 - b = 0.25.

Here we use a closed economy with households and firms, no government or foreign trade, fixed prices, and investment given independently of current income. In this setting, disposable income equals Y. The usual numerical assumes 0 < b < 1. These are the simplifying assumptions behind the textbook result, rather than a description of every real economy. See NCERT’s chapter on income and employment.

Derive the equilibrium income formula once

Write the equilibrium condition:

Y = C + I

Replace C with a + bY:

Y = a + bY + I

Move the income term to the left:

Y - bY = a + I

Factor out Y:

Y(1 - b) = a + I

Therefore:

Notice the brackets. (a + I)/(1 - b) is different from a + I/(1 - b). In our opening question, the incorrect second expression gives 90 + 60/0.25 = 330, which does not satisfy the equilibrium condition.

You can always return to the equation Y = a + bY + I if you forget the shortcut. A few lines of algebra are safer than a half-remembered formula.

Example 1: Find income, consumption, and saving

Question: In a two-sector economy, C = 90 + 0.75Y and planned investment is Rs 60 crore. Calculate equilibrium income, consumption, and saving.

Step 1: Calculate income

Y = (90 + 60)/(1 - 0.75)
Y = 150/0.25
Y = Rs 600 crore

Step 2: Put that income into the consumption function

C = 90 + 0.75(600)
C = 90 + 450
C = Rs 540 crore

Step 3: Find saving

S = Y - C
S = 600 - 540
S = Rs 60 crore

Step 4: Check both routes

• Consumption plus planned investment: 540 + 60 = 600, which equals Y.
• Saving: 60, which equals planned investment.

Both checks pass. Write a final sentence: the economy is in equilibrium at income of Rs 600 crore, with consumption of Rs 540 crore and saving of Rs 60 crore.

Why the saving method gives the same answer

Since S = Y - C, substitute the consumption function:

S = Y - (a + bY)
S = -a + (1 - b)Y

For our example:

S = -90 + 0.25Y

Set saving equal to planned investment:

-90 + 0.25Y = 60
0.25Y = 150
Y = 600

This is exactly the earlier calculation. You have rearranged the same relationship, rather than discovered a separate formula to memorise.

The minus sign before a matters. At zero income, consumption of 90 corresponds to saving of negative 90. Writing S = 90 + 0.25Y would reverse that starting position.

Example 2: The question gives a saving function

Question: S = -45 + 0.30Y and investment is Rs 75 crore. Find equilibrium income and consumption.

Use the information as given:

S = I
-45 + 0.30Y = 75
0.30Y = 120
Y = Rs 400 crore

At equilibrium, saving is 75. Therefore:

C = Y - S = 400 - 75
C = Rs 325 crore

For a second check, derive C = 45 + 0.70Y. At Y = 400, consumption is 45 + 280 = 325.

The coefficient 0.30 in the saving function is MPS. Do not treat it as MPC and subtract it from 1 again when solving the saving equation.

Read equilibrium from a schedule

Use the original function C = 90 + 0.75Y and investment of 60. All amounts below are in crore rupees.

Income Y	Consumption C	Planned C + I	Saving S
400	390	450	10
500	465	525	35
600	540	600	60
700	615	675	85
800	690	750	110

At income of 600, planned spending equals output. Saving also equals investment of 60.

At income of 500, planned spending is 525, exceeding output by 25. If that spending occurs, inventories fall below the level firms intended. Firms have a reason to increase production.

At income of 700, planned spending is only 675. The unsold 25 adds to inventories unexpectedly, encouraging firms to reduce production.

This inventory adjustment explains why the model moves toward the equilibrium row. Equilibrium need not be full employment. The OpenStax expenditure-output lesson explains both the adjustment mechanism and the possibility of equilibrium below potential output.

Planned investment versus actual investment

At Y = 700 in our schedule, saving is 85 while planned investment is 60. The unexpected inventory addition is 25, so actual investment becomes 60 + 25 = 85.

Actual saving and actual investment therefore still match in the accounts. What fails away from equilibrium is the equality between saving and planned investment. This distinction is developed in NCERT’s discussion of ex ante and ex post values.

Example 3: Find investment for a target income

Question: Consumption is C = 120 + 0.80Y. What planned investment supports equilibrium income of Rs 1,000 crore?

When Y is already given, calculate consumption first:

C = 120 + 0.80(1,000)
C = 920

Then use I = Y - C:

I = 1,000 - 920
Required investment = Rs 80 crore

You can also rearrange the formula:

I = Y(1 - b) - a
I = 1,000(0.20) - 120 = 80

Check: (120 + 80)/0.20 = 1,000.

The question asks for the level of investment. If current investment were 50 and the question asked for the increase needed, the answer would be 80 - 50 = 30, not 80.

Example 4: Find MPC from equilibrium information

Question: Equilibrium income is Rs 900 crore, autonomous consumption is Rs 90 crore, and investment is Rs 180 crore. Find MPC and MPS.

Start with:

900 = 90 + b(900) + 180
900b = 900 - 270 = 630
b = 630/900

MPC = 0.70
MPS = 0.30

Check consumption: 90 + 0.70(900) = 720.

Check saving: 900 - 720 = 180, equal to investment.

Here APC is 720/900 = 0.80, while MPC is 0.70. If you divide total consumption by total income and call the result MPC, you get the wrong answer. The autonomous consumption term makes the two ratios different.

What if autonomous consumption is missing instead?

Suppose equilibrium income is 800, MPC is 0.75, and investment is 140, all monetary amounts being in crore rupees.

Use a = Y(1 - b) - I:

a = 800(0.25) - 140 = Rs 60 crore

Verify: C = 60 + 0.75(800) = 660, and 660 + 140 = 800.

Example 5: Investment rises, so how much does income rise?

Return to C = 90 + 0.75Y. Suppose investment increases from 60 to 85 crore, with the consumption function unchanged.

Initial equilibrium income:

Y = (90 + 60)/0.25 = 600

New equilibrium income:

Y = (90 + 85)/0.25 = 700

Therefore:

Increase in investment = Rs 25 crore
Increase in income = Rs 100 crore

The ratio is 100/25 = 4. Algebraically, with a and b unchanged:

Change in Y = Change in I/(1 - b)

Here the multiplier is 1/0.25 = 4, so 4 × 25 = 100.

A useful check is the change in consumption. It rises from 540 to 90 + 0.75(700) = 615, an increase of 75. The extra 100 of income is matched by extra consumption of 75 and extra planned investment of 25.

Do not call 100 the new equilibrium income. It is the change. The new level is 600 + 100 = 700.

If MPC changes, calculate both equilibria separately

Suppose investment stays at 60 and autonomous consumption at 90, but MPC rises from 0.75 to 0.80.

Old income = 150/0.25 = 600
New income = 150/0.20 = 750
Increase = Rs 150 crore

Investment has not changed. The earlier change-in-investment shortcut cannot explain this result on its own because its assumption of an unchanged MPC no longer holds.

Break-even income is a different calculation

Break-even income is the income where consumption equals income and saving is zero. For C = 90 + 0.75Y:

Y = 90 + 0.75Y
0.25Y = 90
Break-even income = Rs 360 crore

With investment of 60, equilibrium income was 600. At that equilibrium, saving is 60, not zero.

The difference follows directly from the equations:

Question	Condition	Result
Where is saving zero?	Y = C	Y = a/(1 - b)
Where does planned spending equal output?	Y = C + I	Y = (a + I)/(1 - b)

In this simple model, the two income levels coincide when planned investment is zero.

For the opening example, the distance between them is 600 - 360 = 240. That also equals I/(1 - b) = 60/0.25. This is a handy check when a question asks for both levels.

How to draw the equilibrium diagram from your answer

For C = 90 + 0.75Y and I = 60, the planned expenditure equation is:

AE = C + I = 150 + 0.75Y

Use these points to build your diagram:

1. Put income or output Y on the horizontal axis and planned expenditure AE on the vertical axis.
2. Use equal scales to draw the 45-degree line AE = Y from the origin.
3. Mark the expenditure intercept at 150 on the vertical axis.
4. Plot another expenditure point, such as (400, 450), and draw the straight expenditure line through the points.
5. Mark its intersection with the 45-degree line at (600, 600).
6. Drop a perpendicular to the horizontal axis and label equilibrium income 600.

These are the standard expenditure-output axes described by OpenStax. The vertical axis here measures spending, not the price level.

If you add the consumption line, its intercept is 90. The planned expenditure line sits 60 above it at every income. Both have slope 0.75.

The consumption line crosses the 45-degree line at 360. The expenditure line crosses it at 600. Labelling both intersections makes the difference between break-even and equilibrium income visible.

Six mistakes worth checking before you finish

• Dividing by MPC: With MPC of 0.75, the denominator is 0.25. Derive Y - 0.75Y = 0.25Y if you are unsure.
• Omitting autonomous consumption: Investment alone is not the numerator when calculating the income level.
• Adding a percentage as a whole number: An MPC of 80% means 0.80, not 80.
• Treating saving as zero at equilibrium: Saving equals planned investment; it need not be zero.
• Mixing rupees, lakh, and crore: Convert to one unit before substituting.
• Confusing an income change with an income level: Label the initial value, final value, and difference separately.

A further warning concerns the scope of the question. If government, taxes, or foreign trade are included, write the full spending equation first. For example, with lump-sum taxes T and government spending G, but no foreign trade, C = a + b(Y - T) gives Y = (a - bT + I + G)/(1 - b). The original two-sector formula cannot simply absorb taxes as an extra item of spending. The broader model is covered in OpenStax’s algebraic treatment.

Practice questions with worked answers

Try the questions before reading the solutions. All monetary amounts are in crore rupees.

1. Given C = 70 + 0.80Y and I = 50, find Y, C, and S

Y = (70 + 50)/0.20 = 600.
C = 70 + 0.80(600) = 550.
S = 600 - 550 = 50.

Check: 550 + 50 = 600.

2. Given S = -36 + 0.25Y and I = 64, find equilibrium income

-36 + 0.25Y = 64
0.25Y = 100
Y = 400.

Consumption is 400 - 64 = 336.

3. Given C = 50 + 0.60Y, find investment for equilibrium income of 500

C = 50 + 0.60(500) = 350.
I = 500 - 350 = 150.

4. Equilibrium income is 1,200, autonomous consumption is 120, and investment is 240. Find MPC

1,200 = 120 + 1,200b + 240
1,200b = 840
MPC = 0.70.

MPS is 0.30. Check: 360/0.30 = 1,200.

5. MPC is 0.80 and investment rises by 18. Find the increase in equilibrium income

With autonomous consumption and MPC unchanged:

Change in Y = 18/0.20 = 90.

The initial income is not given, so you cannot report a numerical final income level.

6. For C = 48 + 0.60Y and I = 32, find break-even and equilibrium income

Break-even income = 48/0.40 = 120.
Equilibrium income = (48 + 32)/0.40 = 200.

At Y = 120, saving is zero. At Y = 200, consumption is 168 and saving is 32, matching investment.

For the wider meaning of the income measure, read our guide to national income. To connect saving and investment with the flow of spending, see leakages and injections in circular flow.

Frequently asked questions

What is the formula for equilibrium income when C = a + bY?

In the two-sector model with autonomous planned investment, Y = (a + I)/(1 - b). It follows by substituting the consumption function into Y = C + I and collecting the Y terms.

Why do we divide by 1 minus MPC?

Because Y - bY = Y(1 - b). The denominator is MPS. With MPC of 0.80, for instance, Y - 0.80Y = 0.20Y.

Can I solve the question using S = I instead?

Yes. Derive S = -a + (1 - b)Y and set it equal to planned investment. It gives the same equilibrium income as the spending method.

Is equilibrium income the same as consumption?

Not generally. In the first example, equilibrium income is 600, consumption is 540, and investment is 60. Consumption is only one component of planned spending.

Is saving always zero at equilibrium?

No. In these questions, saving equals planned investment. Zero saving identifies break-even income. With positive planned investment, the two income levels differ.

What should I do if the question gives MPS?

Use it directly in Y = (a + I)/MPS. If MPS is 0.20, divide by 0.20. Only calculate MPC as 1 - MPS if you need it to form the consumption function.

What happens if MPC is 1?

The denominator becomes zero, so the usual formula cannot give a finite equilibrium. Going back to the original equation gives Y = a + Y + I, or 0 = a + I. If autonomous spending is positive, there is no solution. In the special case a + I = 0, the equation alone does not select a unique Y.

Does a higher MPC increase equilibrium income?

With positive autonomous spending and the other inputs unchanged, yes. In our example, raising MPC from 0.75 to 0.80 increases income from 600 to 750 because the denominator falls from 0.25 to 0.20.

Can I use the investment multiplier to find the new income directly?

The multiplier times the change in investment gives the change in income when the consumption function is unchanged. Add that change to the original equilibrium income to find the new level.

What is the quickest way to check my answer?

Substitute your Y into the consumption function, then subtract consumption from income. If the result equals the given planned investment and the units are consistent, your answer passes the saving check.