Questions: Quasi-Linear Preferences and Their Properties

Question 1

An auction designer increases every bidder's cash endowment by $1000. Under quasi-linear preferences, what happens to each bidder's valuation for the auctioned object?

• A. Valuations increase because higher income raises willingness to pay for most goods
• B. Valuations decrease because bidders now care less about winning relative to keeping their cash
• C. Valuations are unchanged because under quasi-linear preferences the MRS for good x depends only on x, not on income
• D. Valuations change unpredictably, depending on the shape of each bidder's v(x) function

Answer: (C)

This is the defining property of quasi-linear preferences. With u(x, y) = v(x) + y, the marginal rate of substitution between x and the numeraire is v'(x), which depends only on how much x the consumer has — not on their income or wealth. Adding $1000 to everyone's endowment shifts the y-component of utility upward uniformly, but leaves the allocation of x completely undisturbed. This is precisely why mechanism designers love quasi-linear preferences: cash transfers are a 'clean' instrument that redistributes surplus without distorting who should get what. Option A describes the standard income effect present in general preferences but absent here.

Question 2

What geometric property of the indifference curves under u(x, y) = v(x) + y reflects the absence of income effects for good x?

• A. Indifference curves are straight lines because v(x) is linear
• B. Indifference curves are vertical translates of each other — all have the same shape, just shifted up or down
• C. Indifference curves become flatter as income increases, reflecting declining marginal value of x
• D. Indifference curves are L-shaped, indicating that x and y are perfect complements

Answer: (B)

Because y enters linearly, every indifference curve u(x, y) = c has the form y = c − v(x). For different constant levels c, these curves are identical in shape — they are vertical translations of one another. The MRS at any point (x, y) is v'(x), which does not depend on y (or equivalently, on income). This means at any given quantity of x, the consumer's willingness to trade x for money is the same regardless of how much money they have. The 'parallel' indifference curves are the visual signature of zero income effects.

Question 3

Under quasi-linear preferences u(x, y) = v(x) + y, a consumer's demand for good x is independent of their income level.

• T. True
• F. False

Answer: True

This follows directly from the Marshallian demand for x. Maximizing v(x) + y subject to px + y = I gives the first-order condition v'(x) = p, which determines optimal x as a function of price alone. Income I drops out entirely — it only determines how much y is consumed. This is the income effect equal to zero for x. Graphically, an expansion of the budget set shifts the optimal bundle upward (more y) but leaves the x-coordinate unchanged.

Question 4

Quasi-linear preferences are a poor choice for mechanism design contexts because monetary transfers distort the allocation of the good being sold.

• T. True
• F. False

Answer: False

This is the opposite of the truth. Quasi-linear preferences are the workhorse assumption in mechanism design *precisely because* monetary transfers do NOT distort the allocation of good x. Under quasi-linearity, a bidder's valuation v(x) for the object is a fixed number independent of how much money changes hands. The designer can optimize the allocation (who gets x) to maximize total surplus, then use transfers to satisfy incentive compatibility and individual rationality constraints — without worrying that the transfers will shift bidders' valuations. This separability between allocation and transfers is what makes clean mechanism design results (like the Vickrey-Clarke-Groves mechanism) possible.

Question 5

Why does the linear entry of numeraire y in u(x, y) = v(x) + y eliminate the income effect for good x, and why does this property matter for mechanism design?

Model answer: The income effect arises when a change in wealth shifts a consumer's marginal valuation for a good. In u(x, y) = v(x) + y, the marginal utility of y is always 1 — an extra dollar is worth exactly one util regardless of how much money you have. This means the consumer's willingness to pay for good x (their MRS between x and y) is determined entirely by v'(x), the marginal value of x itself, not by their wealth. Income changes only how much y they consume, leaving x demand unchanged. In mechanism design, this property means the designer can adjust monetary transfers freely without distorting the allocation of x. The allocation problem (maximize total v(x)) and the transfer problem (satisfy constraints) can be solved independently — a separation that enables clean characterizations of efficient mechanisms like VCG.

The linearity in y is a strong but analytically indispensable restriction. It fails when wealth constraints bind (you can't pay more than you have) and when income effects for x are substantial (e.g., housing). In those settings, quasi-linearity is a poor approximation and the mechanism design problem becomes significantly harder. But for many auction and regulatory settings where stakes are small relative to wealth, it is both realistic and tremendously useful.