Equilibrium Income Calculator

Equilibrium income is the level of national income where aggregate demand exactly equals output: firms produce what buyers plan to spend, inventories stay where firms want them, and income has no built-in reason to rise or fall. This calculator finds that level with the Keynesian cross model. Use the three-sector mode, with government spending and lump-sum taxes, for fiscal policy questions, or the two-sector mode for the classic households-plus-firms case. Either way, the tool returns equilibrium income, the spending multiplier, a step-by-step substitution with your own numbers, and a savings-identity check that proves the answer really is an equilibrium rather than just a number from a formula.

Be clear about what the model is. It is a closed economy with fixed prices, lump-sum taxes only, and no net exports, transfer payments, or proportional tax rate. That simplicity is the point for teaching and scenario work, and the sections below say plainly where the model stops. If your question is about how a change in planned spending shifts total demand before income adjusts, our change in aggregate demand calculator covers that neighbouring step.

Quick Answer

With the pre-filled three-sector example (autonomous consumption $200, investment $300, government spending $400, taxes $100, MPC 0.8), equilibrium income is $4,100 and the spending multiplier is 5. Switch to the no-government tab with consumption $100, investment $200 and MPC 0.75, and equilibrium income is $1,200 with a multiplier of 4. Change any input and press Calculate equilibrium: the formula, the substitution and the savings check all update with your numbers.

Worked example: three-sector economy

Enter autonomous consumption $200, investment $300, government spending $400, taxes $100 and MPC 0.8. The numerator is 200 + 300 + 400 minus 0.8 times 100, which is 820. The denominator is 1 minus 0.8, which is 0.2. So Y = 820 / 0.2 = $4,100. The multiplier is k = 1 / 0.2 = 5, so every extra dollar of autonomous spending would ultimately add five dollars of income in this economy.

The check is what separates an answer from a guess. Consumption at Y is 200 + 0.8 times (4,100 minus 100) = $3,400. Savings are then 4,100 minus 3,400 minus 100 = $600, and investment plus government spending minus taxes is 300 + 400 minus 100 = $600. The two sides match exactly, which is what equilibrium means in this model.

Worked example: two-sector economy

Switch to No government and enter C = $100, I = $200 and MPC = 0.75. Now there are no taxes, so disposable income is national income itself. Y = (100 + 200) / (1 minus 0.75) = 300 / 0.25 = $1,200, with a multiplier of 4. Consumption at equilibrium is 100 + 0.75 times 1,200 = $1,000, and savings of $200 exactly equal investment of $200, as the two-sector identity requires.

The equilibrium income formula, derived

Equilibrium is defined by Y = C + I + G, where consumption depends on disposable income: C = C₀ + MPC times (Y minus T). Substituting the consumption function into the equilibrium condition and solving for Y gives Y = (C₀ + I + G minus MPC times T) / (1 minus MPC). Two details in that formula repay attention. Taxes enter multiplied by MPC, not at full value, because a dollar of tax would only partly have been spent: the rest would have been saved anyway. And the denominator, 1 minus MPC, is the marginal propensity to save. Dividing autonomous spending by the saving rate is what scales a modest injection of spending up into total income, because only the saved fraction leaks out of each spending round.

Without government and taxes, the same algebra collapses to Y = (C₀ + I) / (1 minus MPC). Nothing else changes: it is the same model with G and T set to zero, which is why comparing the two modes is such a clean way to see what fiscal policy adds. For a broader view of how individual incomes add up before any equilibrium condition is imposed, see our aggregate income calculator.

The multiplier, and why MPC drives it

The spending multiplier k = 1 / (1 minus MPC) measures how much total income rises when autonomous spending rises by one dollar. The mechanism is a chain: the government spends a dollar, the recipient treats it as income and spends MPC of it, the next recipient spends MPC of that, and so on. The rounds shrink by the factor MPC each time, and the infinite sum is exactly 1 / (1 minus MPC). With MPC = 0.8, k = 5: a $100 rise in government spending ultimately raises income by $500. This is a different multiplier from the jobs version: if your question is how direct employment ripples into further jobs, use the employment multiplier calculator instead.

The table shows why small differences in MPC matter so much. Moving from 0.75 to 0.9 more than doubles the multiplier, because what matters is the shrinking saving leakage in the denominator, not the MPC itself. It also shows why the tool rejects MPC = 1: the denominator would be zero, every dollar would be respent in full forever, and income would be infinite. Real estimated MPC values sit comfortably below 1, and most textbook exercises use 0.75 to 0.9.

Why the savings check matters

At equilibrium, leakages equal injections: savings = investment + government spending minus taxes. The calculator verifies this numerically with your inputs, in the three-sector mode, and verifies savings = investment in the two-sector mode. If the two sides did not match, the computed Y would not actually be an equilibrium: households would be planning to save more or less than firms and government plan to inject, and income would move. Textbooks state the identity; this tool proves it for your numbers, on every run. Note one labelling quirk carried over from the original tool: inside the check line, the C in "S = Y minus C minus T" refers to consumption at equilibrium, the figure printed just before it, not to the autonomous consumption you entered.

What this model leaves out, honestly

The Keynesian cross in this form assumes fixed prices, a closed economy, lump-sum taxes, and investment that does not respond to income or interest rates. It has no imports or exports, no transfer payments such as benefits, no proportional income tax, and no supply side: output adjusts to demand without limit. Richer textbook treatments, including the OpenStax expenditure-output chapter listed in the references, add a tax rate, imports and a potential-GDP comparison, which change the multiplier formula. Use this calculator for the core mechanism and for standard coursework in the two-sector and lump-sum three-sector cases, and treat real-world fiscal forecasts as a job for far larger models.

Two input behaviours are worth knowing. Leaving autonomous consumption, investment, government spending or taxes blank treats that item as zero; only MPC is required, and the tool says so if it is missing. And the tool does not reject negative values for the spending inputs, because a negative entry can be a deliberate way to model a fall in spending, but a negative equilibrium result is a modelling signal to recheck, not a forecast.

Frequently asked questions

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References

1. OpenStax, Principles of Economics 3e, appendix on the expenditure-output (Keynesian cross) model: equilibrium where aggregate expenditure crosses the 45-degree line.
2. Lumen Learning, Using an Algebraic Approach to the Expenditure-Output Model, worked algebra for solving equilibrium income from the consumption function.
3. Keynes, J. M., The General Theory of Employment, Interest, and Money, the original statement of the consumption function and the multiplier.