The hard problems: the Delhi School set

Four habits separate these questions from the ones in earlier lessons.

1. The answer is at a kink or a corner. When the utility is a minimum, a maximum, a convex function or a sum with a switch, tangency is either impossible or the wrong answer. List the candidates (corners and kinks), compute the utility at each, and compare.
2. “Must be true” means find the counter-example. Statements such as “the price of x is necessarily equal to the price of y” are false if one case breaks them. Try the cases that could break the statement before the one that confirms it.
3. Check the constraint after the tangency. A grant that can only buy books, a subsidy up to 35 kg, a subsistence quantity: solve ignoring the restriction, then test whether the answer obeys it. If not, the restriction binds and the answer sits on it.
4. Neat numbers are a hint. √200, 1/3 and 14/3 appear because the question was built from a clean setting. If your working produces something ugly, check the setting.

Before solving a problem, say which technique it needs. Eight problems from the set below, each asking only for the choice of method. The six options are:

1. Tangency, then check the constraint
2. Compare the corners and the kink
3. Test the ordering's properties
4. Table of costs (revealed preference)
5. Compensate: Hicks, Slutsky or the expenditure function
6. Two-period budget and the Euler condition

These questions ask about the properties of preferences, not about maximising. The method is mechanical: write out every case a property applies to and test them.

Two quicker ones. U = 10(x² + 4xy + 4y²) + 20 = 10(x + 2y)² + 20 is an increasing function of x + 2y, so the goods are perfect substitutes with MRS ½ (DSE 2008). And “the more you eat, the more you want” means increasing marginal utility of cheese, which u = x² + y has (DSE 2012).

With a minimum, a maximum or a convex exponential, tangency does not help. The method is to list the candidates and compare.

Two more. U = min{x + y, 2y} and she buys (3, 3): the kink is at x = y, and she is there only if x is no dearer than y, so p_x ≤ p_y (DSE 2008). U = e^{x² + y²} has curves bowing away from the origin, so the best is a corner: with Rs 100 at prices 5 and 10 she buys 20 dosas and no pizza (DSE 2009).

These build on Lesson 7: take the tangency, then see whether a restriction binds. DSE 2004 set five questions on one setting, which is a good way to see how the toolkit fits together.

Buy one get one free (DSE 2010, u = xy): the free unit halves the effective price of x, but the Cobb–Douglas demand for x, in total units, is m/p, double what it was. Half of those units are free, so the units she pays for are unchanged, and so is her y. The scheme leaves her purchases alone and doubles her holdings of x. A fixed fee (DSE 2013, u = min{x₁, x₂}): under the single price she gets M/3 of each; with a lower price 2 − θ plus a fee T she gets (M − T)/(3 − θ). Equating the two gives θ = 3T/M.

The Workbench is set to the DSE 2004 problem: Cobb–Douglas with equal weights, income 10, the price of bread 3, and the price of water going from 1 to 2. The Hicks compensating variation it prints, +4.142, is √200 − 10. The decomposition table shows the income effect of Question 15, and the bundles are the ones above.

That answers DSE 2007: she borrows exactly when (1 + δ)w₂ > (1 + r)w₁. Given she borrows, a larger δ (more impatience) raises c₁, so she borrows more. A rise in the interest rate lowers a borrower’s borrowing: the substitution effect (today is dearer relative to tomorrow) and the income effect (she owes more) both point towards less borrowing, so it falls unambiguously. For a saver the two effects conflict.

Labour. U = C^β + (L̄ − L)^β with C = wL gives, from the first-order condition w(wL)^β−1 = (L̄ − L)^β−1, the supply L = L̄w^a/(1 + w^a) with a = β/(1 − β) (DSE 2008). And with logarithmic utility, U = ln c + b ln(1 − l) and c = wl, hours are l = 1/(1 + b) whatever the wage: the income and substitution effects cancel exactly (DSE 2020).

DSE 2019 compares two utilities that differ only in the middle zone, where y/x lies between ½ and 2: u₁ uses max{x, y} there and u₂ uses x + y, which is never smaller. So u₂ ≥ u₁ everywhere, the maximum utility v₂ ≥ v₁, and the demand set of u₁ sits inside that of u₂ (at equal prices u₂ is flat along the whole middle zone and u₁ is best only at its two ends). The cheapest bundles for the same utility, though, are different sets, and neither contains the other, so none of the inclusions holds necessarily.

Two shorter ones. A consumer’s budget share of an inferior good falls as income rises, since its quantity falls while income rises (DSE 2006). For u = s + ln c, the spending on sandwiches is the price of stamps and nothing else (DSE 2019).