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
y
x
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: 12 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
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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
y
WARP
satisfied violated
b
a
x
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
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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.
x2
c
c
c
P .X = M
d
X
c
X
a
a
a
b
P .X = M
X
a
X
b
b
b
x
P .X = M
1
Consistency of preferences requires SARP “strong axiom
of revealed preference”: for any chain a, b, c, . . . j, k,
If xa RP xb , xb RP xc , . . . xj RP xk , then xk RP xa false
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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
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