The Delhi School economies

Four things recur, and the 74 questions sorted into this lesson’s practice sets fall into four groups: is this allocation efficient, which prices clear the markets, what happens when tastes are lexicographic or an equilibrium fails to exist, and transfers, production and fairness. Almost every question gives the same kind of data: two agents, two goods, endowments, and a taste for each agent that is not Cobb–Douglas.

A four-step habit solves most of them.

1. Normalise. Let the prices sum to one (the DSE often asks for them that way) or set one to 1.
2. Write each agent’s demand from the table below. Ask what she does when the price ratio is below, equal to, and above the one that matters, and what she does when a price is zero.
3. Find the scarce good. Compare totals. If the economy has more of x than anyone can use (complements), x is free.
4. Check the market for one good, then use Walras’ law for the other; and check whether demand jumps over the price you need.

Two consequences explain most of the surprises. First, only perfect complements can sit at an equilibrium where a price is zero, because only they stop wanting more of a free good. So when the economy has a surplus of one good and everyone wants goods in fixed proportions, the surplus good is free. Second, with lexicographic tastes demand jumps, which is how equilibria vanish.

Choose tastes and endowments for two agents, or load one of the past-paper economies from the menu. The solver finds every price (as the share of the price sum, p_x out of 1) at which the markets clear, with open ends marked (“0 < p_x/p_y ≤ 1”), and gives the allocation. If there is no equilibrium it says what goes wrong. The graph shows the excess demand for x as its price rises from 0 to 1: a flat stretch where an agent is indifferent, a jump where demand switches, a red tick where someone wants a free good without limit.

The test is always the same: can anyone gain with no one losing? Look for a swap. Four tricks cover most questions.

• Perfect complements. Waste is the signal: if someone holds more of a good than her partner good lets her use, the spare can be moved to the other person, unless that person is also tied. With two Leontief agents and totals (X, Y), each unit of utility costs one x and one y, so utilities are limited to u₁ + u₂ ≤ min(X, Y), and an allocation is efficient exactly when its utilities add up to that (DSE 2010, 2017).
• Perfect substitutes. Efficient means each person holds only the good she values relatively more, unless the MRSs are equal (DSE 2009, 2011).
• Non-convex or max tastes. Interior points are not efficient: shift to the extremes (DSE 2006, 2012).
• Utility depends on others, or on location. Draw the possible pairs of utilities and look for the frontier (DSE 2012 Antony and Cleopatra, 2013 power plant, 2014 consumption externality).

One more of the set-logic kind (DSE 2014): if a is Pareto optimal in the large set S, it is optimal in any subset containing it (nothing in the subset beats it). The converse fails: a may be optimal in a subset only because the alternative that beats it was left out. Of the four statements about S = {a, b, c, d, e} and T = {a, b, c, d}, exactly two are true.

Normalise the prices to sum to one and check each case in the table. Most of these questions have a surplus good whose price must be zero, or an agent with perfect substitutes whose indifference pins the price ratio down to a range.

Sequential choice (DSE 2007). A variant where there is no market at all: agent 1 chooses first, then agent 2 takes the best of what is left. Each faces x ≤ 3, y ≤ 3 and x² + y² ≤ 12. Agent 1 maximises x²y: on the circle x² + y² = 12 the tangency is at x = √8, y = 2. Agent 2 takes the remainder (3 − √8, 1). If agent 2 chooses first with u = 2x + y, the circle’s tangency would put x above 3, which is not allowed, so she takes the corner x = 3, y = √3, leaving agent 1 (0, 3 − √3). With a = 1 and the bound on consumption, the first of these allocations is efficient (Q28).

A lexicographic agent compares bundles the way a dictionary orders words: she looks at the first good first, and only if the amounts tie does she look at the second. At positive prices she therefore spends everything on her first good. Her demand is a corner, and as prices move across a single value nothing smooth happens: she either has the money to buy a bit more of her first good, or she does not. The upshot is that excess demand can jump from positive to negative without being zero at any price. When that happens there is no equilibrium.

A lexicographic agent also never accepts a zero price: if her first good is free she wants an unlimited amount, and if only the other good is free she wants an unlimited amount of it as a tie-break. So prices must be strictly positive, and the question becomes whether they can be.

Four conceptual ones. No-envy (DSE 2013). With u_A = αx₁ + x₂ and u_B = y₁y₂, and A holding 4 of good 1 and B holding (1, 5): A envies B if α + 5 > 4α, so there is no envy if α ≥ 5/3. The same allocation is Pareto optimal only if α ≥ 5: B’s MRS at (1, 5) is 5, and A holds only good 1, so A would sell it for any price above α, and B would pay up to 5. Rawls (DSE 2011). With voters 1 and 2 ranking A ≻ B ≻ C and voter 3 ranking C ≻ B ≻ A, nobody ranks B last, while A is last for voter 3 and C for voters 1 and 2: a planner who maximises the worst rank chooses B. Arrow (DSE 2018). If society’s ranking is whatever a “holy book” says, whatever the voters think, the Pareto principle fails: the book can rank A above B when everyone prefers B to A. The other axioms hold. The first theorem (DSE 2021). The assumption not required is convex preferences. Property rights, no externalities and monotonic preferences are needed.

74 questions from the DSE entrance on exchange economies, efficiency and welfare, in four sets; within each, moderate questions come before hard ones. Foundation and standard questions from other exams are in the lessons.