Government, taxes and trade in the multiplier

With a government, households spend out of disposable income: income less taxes plus transfers. With a lump-sum tax T and transfers TR, C = C₀ + c(Y − T + TR), and output is Y = C + I + G. Solving:

Read off the three multipliers:

Why is a tax cut weaker than the same rise in G? A rupee of G is spent in full in the first round. A rupee of tax cut is income, and households save 1 − c of it before any spending starts. The tax multiplier is the purchases multiplier with its first round missing.

Raise G and lump-sum taxes by the same amount. Output rises by 1/(1 − c) − c/(1 − c) = (1 − c)/(1 − c) = 1 times the change. The spending adds its full value in the first round; the tax takes away exactly the later rounds. So a balanced increase in the budget still expands output, rupee for rupee.

Hyderabad 2010 used this as a policy puzzle: with MPC 0.8, how do you offset a fall in income of 1,000? Raising both spending and taxes by 1,000 does it exactly.

Real taxes rise with income. With T = T₀ + tY, a rupee of extra income becomes only 1 − t of disposable income, and households spend c(1 − t) of it. That is the marginal propensity to consume out of income, and it replaces c in the multiplier.

With c = 0.8 and t = 0.25, c(1 − t) = 0.6 and k = 2.5, down from 5: introducing the tax halves the multiplier (Hyderabad 2021). JNU likes the intermediate step: with a marginal tax rate of 1/5 and an MPC out of disposable income of 3/5, the MPC out of income is 3/5 × 4/5 = 12/25.

A smaller multiplier is a steadier economy. When investment collapses, falling income cuts tax payments, which cushions disposable income. That is why a proportional income tax is an automatic stabiliser, and why “taxes make the multiplier larger” was the wrong statement in CUET 2025.

The Keynesian cross again, now with taxes, transfers and government purchases. Each preset is a past paper. Try raising G and then raising T₀ by the same amount, and read the “since the preset” box: output ends up higher by exactly the rise in G.

Imports rise with income: M = mY, where m is the marginal propensity to import. Spending on imports is income for foreign firms, so it leaks out of each round just as saving and taxes do. Exports depend on foreign income, not ours, so they are autonomous: a rise in foreign income raises our exports and our output (GATE 2021).

JNU 2012: MPS 0.15 and MPM 0.05 give a multiplier of 1/0.2 = 5. JNU 2010: c = 0.8, t = 0.2, m = 0.1 give 1/(1 − 0.64 + 0.1) = 50/23. JNU 2021 wrote imports as M = −6 + 0.3Y alongside C = 10 + 0.8Y: the multiplier is 1/(0.2 + 0.3) = 2.

Lesson 1’s identity, S + T + M = I + G + X, rearranges into the three sector balances that must add up to zero:

JNU 2015 set a trap with it: with a fiscal deficit of 3% of GDP and a current account deficit of 3%, must aggregate saving equal aggregate investment? Private saving equals investment (3 − 3 = 0), but national saving, which includes the government’s −3%, falls short of investment by the 3% borrowed from abroad. So the statement is false. GATE 2025 used the same identity: with a trade surplus, lower private saving or higher investment reduces it.

The balanced budget multiplier of 1 assumes the tax falls on people with the average propensity to spend. JNU 2012 dropped that assumption: workers earn three-quarters of output and spend all or most of it, while non-workers spend a fixed amount whatever their income. Taxing non-workers then cuts no spending at all, and taxing workers cuts a lot.

The four answers: taxing non-workers with workers spending everything, 4; with workers spending 0.8, 2.5; taxing workers who spend everything, 0 (“none of the above”); taxing workers who spend 0.8, 0.5. The Delhi School built a family of questions on the same idea, with two groups of households and different propensities to consume; they are in Lesson 9.