ECON8010 Masterlist

kms

x is at least as good as y

x≿y

x is strictly preferred to y

x≻y

x is indifferent to y

x∼y

Compensated Law of Demand

(p′ − p) · (x(p′, p′ · x(p, w)) − x(p, w)) ≤ 0

Slutsky Matrix (x(p,w))

S(p,w) = D_px(p,w) + D_wx(p,w) * x(p,w)^T

Slutsky Compensation

w^c = p_a¹a⁰ + p_b¹b⁰

Slutsky Substitution + Income Effects

Slutsky substitution effect = x^c − x⁰
Slutsky income effect = x¹ − x^c

The Slutsky Equation (Hicksian)

D_ph(p,ū) = S(p,w)

These are equal to the substitution effect

Negative Semidefinite

2×2
Check the principal minors.
| a b | a ≤ 0
| c d | ad-bc ≥ 0

3×3
Check if its skew-symmetric:
diagonals = 0, off-diagonals are negatives of each other

Walras’ Law (Demand)

p ⋅ x(p,w) = ∑​p_i​x_i​(p,w) = w

the consumer always spends all their wealth

Setup to check WARP

Set up choices:
At prices p,w, bundle x is chosen.
At prices p′,w′ bundle x′ is chosen.

Check affordability:

Is x′ affordable at (p,w)? i.e. p ⋅ x′ ≤ w

Is x affordable at (p′,w′)? i.e. p′ ⋅ x ≤ w′

Check violation:

If x chosen at (p,w) and x′ affordable there, then we need x′ not chosen at (p′,w′) when x is affordable.

If both cross-affordability conditions hold (each is affordable when the other is chosen), then WARP is violated.

homogeneous of degree zero (HD0)

x(p,w) = x(αp,αw) for all α > 0

Lagrangian for the UMP with 2 goods

L(x₁​,x₂​,λ) = u(x1​,x2​) + λ(w−p1​x1​−p2​x2​)

(λ = marginal utility of income)

How to find the MRS

MU₁/MU₂ = ∂u(x)/∂x₁ / ∂u(x)/∂x₂ = λp₁/λp₂ = p1/p2

Walrasian-Hicks Relations

x(p,w) = h(p, v(p,w))
h(p, ū) = x(p, e(p, ū))

v(p, e(p, ū)) = ū
e(p, v(p,w)) = w

Shepard’s Lemma

∇_pe(p, ū) = h(p, ū)

Roy’s Identity

x_i(p, w) = (-D_piv(p, w))
.. . . . . .(^_∂/∂w v(p, w))

Compensating variation CV

e(p′, v(p⁰, w)) − w

∫_p0^p1 h(p,u⁰) dp

Equivalent variation EV

w − e(p⁰, v(p′, w))

∫_p0^p1 h(p,u¹) dp

Envelope Theorem

dc(w,q)/dq = dL(w,q,z,λ)/dq

UMP
EMP
PMP
CMP

x(p,w), v(p,w)
h(p,u), e(p,u)
π(p), y(p)
z(w,q), c(w,q)

Arrow-Pratt coefficient of absolute risk aversion

r_A(x,u) = -u’’(x) / u’(x)

Expected Utility (Discrete)

EU = p₁u(x₁) + p₂u(x₂) + …

Expected Utility (Continuous)

EU = ∫_-∞^∞ u(x)g(x)

Only when G(x) has a density

Certainty Equivalent

u(c) = E[u(w)]

Condition for FOSD

F ≿_FOSD G if G(x) ≥ F(x) for all x

Condition for SOSD

If F & G have the same mean, F ≿_SOSD G if ∫₀^xG(t)dt > ∫₀^xF(t)dt

Maximum Buying Price
Minimum Selling Price

u(without lottery) = u(with lottery (include buying price P))
u(with lottery) = u(sell lottery (include selling price S))

Bayes’ Rule

P(A∣B) = P(B∣A)P(A) / P(B∣A)P(A)+P(B∣A^c)P(A^c)