QUASILINEAR PREFERENCES Utility additive, y and linear in y : U ( - - PDF document

ECO 305 — FALL 2003 — October 7

QUASILINEAR PREFERENCES Utility additive, and linear in y: U(x, y) = F(x) + y, Example: F(x) = x1/2

• Indiff. curves vertically parallel

y = u − F(x) for any constant u Measure prices relative to (in units of) y: Budget constraint: p x + y = M

x y

Substitute from budget constraint: max M + F(x) − p x FONC: F 0(x) = p, SOSC F 00(x) < 0 Invert FONC to get demand function: x = D(p) Example: 1

2 x−1/2 = p, x = 1/(4 p2)

y = M − p D(p), non-negative if M ≥ p D(p) If M < p D(p), then y = 0, x = M/p Example: If M > 1/(4p), y = M − 1/(4p) ”Isolates out” industry x — useful in Ind Org Hicksian and Marshallian demands coincide so conventional consumer surplus analysis valid But no pure income effect on x — unrealistic 1

REVEALED PREFERENCE Inferring indifference map from observed demands Vector notation: Budget constraint P · x ≤ M Demand function x = D(P, M). If these satisfy [1] Adding up: P · D(P, M) ≡ M [2] Homogeneity: D(k P, k M) = D(P, M) [3] Internal Consistency: axiom SARP derived below then there are underlying preferences being maximized DEFINITION: Suppose xa = D(Pa, M a). Call xa revealed preferred to xb if xb is on or within the budget constraint that led to the choice of xa. More formally: xa RP xb if Pa · xb ≤ Pa · xa

a b WARP satisfied violated x y

WARP “Weak axiom of revealed preference”: If xa RP xb is true, then xb RP xa should be false, that is, If Pa · xb ≤ Pa · xa, then Pb · xa > Pb · xb 2

Can construct chain of revealed preferences. Revealed indifference curve through xa is traced out by envelope of the budget lines as chain gets finer and finer.

x

1

x2

P .X

c c

= M

c

P .X

a a

= M

a

P .X

b b

= M

b

X

d

X

c

X

b

X

a

Consistency of preferences requires SARP “strong axiom

• f revealed preference”: for any chain a, b, c, . . . j, k,

If xa RP xb, xb RP xc, . . . xj RP xk, then xk RP xafalse 3

ANOMALIES Psychologists and experimental researchers find behavior inconsistent with rational choice Framing and endowment effects: Preference depends on how posed and status quo, not just on actual final consumption Coffee-mug experiments Lost-ticket vs. lost-money findings Recent experiments show endowment effect decreases as trading experience increases Taxi drivers’ daily target income behavior refuted by Prof. Farber Time inconsistency: trade-off between day 2 and day 3 looks different on day 2 than it did on day 1 Example — When you are 20, you plan to save a lot of your income in your 30’s, but when the 30’s come along . . . Other anomalies later: (1) choices under uncertainty, game interactions e.g. prisoner’s dilemma General lesson — use standard theory as your starting point, but may need to supplement/modify 4