Taxes, subsidies, transfers and non-standard budgets

Her tastes are Cobb–Douglas, with prices x Rs 2 and y Rs 1 and income Rs 60. Open each tab, move the slider and read the three boxes. The comparison in every tab is between the policy and its alternative at the same cost to the state.

A subsidy of s per unit on x lowers its price to p_x − s. She picks the best bundle S on the new, pivoted line, and the state pays s times the amount of x she buys: call that G. Now give her the same G in cash instead. Her budget line shifts out in parallel, with the old prices, and by construction S is still on it: S costs (p_x − s)x + p_yy = m, so at the old prices it costs m + s·x = m + G. So she can afford S and perhaps do better.

The keyed answer to the opening question is (B), and it is the best of the four. Strictly, “always” is a little too strong: for perfect complements she buys the same bundle either way and is indifferent. But (A) is plain wrong, (D) cannot happen, and (C) holds only in special cases. The sharper version is the one to remember: cash is never worse, and usually better.

Tax x at t per unit and she picks a bundle B on the pivoted-in line. The tax raises R = t·x_B. Now take R from her as a lump sum instead: her line shifts in in parallel, at the old prices, and passes through B. She can still choose B and, in general, does better. So an excise tax puts her on a lower indifference curve than a lump-sum tax raising the same revenue, because it distorts her choice by changing the relative price. That is the reason for the keyed answer to Hyderabad 2016 and 2020: an excise tax on one good places the consumer on a lower indifference curve.

The other options in that question are tests of detail. The tax does not rotate the line outward, which would make her better off. It need not lower her spending on x including the tax (for Cobb–Douglas it leaves it unchanged). And it does not reduce both goods equally (with Cobb–Douglas, y stays the same). Only the lower indifference curve always holds.

A voucher for V units of x that cannot be resold changes the budget in a different way. Every bundle she can reach contains at least V units of x, and beyond V she pays the ordinary price. If, with the cash value of the voucher, she would have bought at least V anyway, the voucher and the cash lead to the same bundle: the grant does not bind. But if she would have chosen less than V, the voucher forces her to take more x than she wants, she ends up at the corner x = V, and she is worse off than with cash.

In the lab, move V up from 0. For a small voucher the two bundles coincide; once V is larger than the x she would buy with the equivalent cash, the box turns red.

Three more shapes, all from past papers.

• A quantity limit. If she may buy at most X̄ of x, the budget is cut off at X̄. If her free choice is below X̄ nothing changes; if it is above, she buys exactly X̄ and puts the rest into y. For GATE 2021, u = x + 4y with prices 2 and 3, income 120 and caps x ≤ 60, y ≤ 30: utility per rupee is ½ on x and 4/3 on y, so she buys the cap of 30 on y (Rs 90), and spends the remaining Rs 30 on x: 15 units. The optimum is (15, 30).
• Discrete goods. If only whole units exist, the budget set is a set of points, not a triangle. Prices Rs 4 and Rs 3 and income Rs 10 allow exactly eight bundles: (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (2,0). If both prices fall by 10 paise and income rises by 10 paise, the same eight are affordable, so her best bundle does not change (JNU 2010). With u = √(xy), prices 1 and 1, income 3, a bundle of whole units is best at (1, 2) or (2, 1) (JNU 2012); if x is limited to 0 or at most 1, it is (1, 2).
• A refund scheme. LPG cylinders cost Rs 50 and an empty returned earns Rs 20; she cannot borrow. The first costs 50, and each later one costs a net Rs 30 once she returns the previous empty. After k cylinders she has spent 50k − 20(k − 1) = 30k + 20, so the largest k with 30k + 20 ≤ 4000 is 132 (JAM 2021).

The two-constraint question of Lesson 4, rotis and dal at Rs 13 and 5,500 calories, belongs to this family too: when two limits both bind there is one bundle and no choice (JNU 2013).