Preferences: x f y
Consumption Set: {x,y,z,}
Complete, Reflexive
Transitive, Continuous
Utility Function
U(x) > U(y) iff x f y
Budget Set
B( p, m) = {x ∈ X | p ⋅ x ≤ m}
Budget Exhaustion
Weak Monotinicity + Strong Convexity
Consumer Maximization Problem
Max x1 , x2 U ( x1 , x2 )
Subject to: p1 x1 + p2 x2 = m
Monotonic
Transformations
Utility Function
(Ordinal)
U ( x) = min p V ( p, m) st x ⋅ p = m
Utility Maximization
V ( p, m) = max x U ( x) x ⋅ p = m
Marshallian Demand Function
∂v( p, m)
∂pi
xi ( p, m) =
∂v( p, m)
∂m
V ( p, m) : Homogenous of Degree 0,
Continuous, non increasing in P
Expenditure Minimization
e( p, u ) = min x ⋅ p st U ( x) > u
Hicksian Demand Function h i ( p, u ) =
∂e( p, u )
∂pi
e( p, u ) : Homogeneous of degree 1 in p
Continuous, nondecreasing in p
h i ( p, u ) :Homogeneous of degree 0 in p
e( p, v( p, m)) = m
v( p, e( p, u )) = u
xi ( p, e( p, u )) = hi ( p, u )
hi ( p, v( p, m)) = xi ( p, m)
Matrix Symmetry
⎡ ∂e( p, u )
⎢ ∂p ∂p
⎢ 1 1
⎢ ∂e( p, u )
⎢ ∂p ∂p
⎣ 2 1
∂e( p, u ) ⎤
∂p1∂p2 ⎥
⎥ is Neg SemiDef :
∂e( p, u ) ⎥
∂p2 ∂p2 ⎥⎦
⎡ ∂e( p, u )
⎢ ∂p ∂p
⎢ 1 1
⎢ ∂e( p, u )
⎢ ∂p ∂p
⎣ 2 1
∂e( p, u ) ⎤ ⎡ ∂h1 ( p, u )
∂p1∂p2 ⎥ ⎢ ∂p1
⎥=⎢
∂e( p, u ) ⎥ ⎢ ∂h1 ( p, u )
∂p2 ∂p2 ⎥⎦ ⎢⎣ ∂p2
∂h1 ( p, u ) ⎤
∂p2 ⎥
⎥ → ∂h1 ( p, u ) ≤ 0, ∂h1 ( p, u ) = ∂h1 ( p, u )
∂h2 ( p, u ) ⎥
∂p1
∂p2
∂p2
⎥
∂p2 ⎦
SlutskyMatrix
∂hi ( p, u ) ∂xi ( p, e( p, m) xi ( p, e( p, u )) ∂e( p, u )
=
+
∂p j
∂p j
∂e( p, u )
∂pi
so
∂xi ( p, m) ∂hi ( p, u ) ∂xi ( p, m)
=
−
xi
∂p j
∂p j
∂m
Looking for Corners
−
px
py
−
px
py
Possible Interior solutions
−
Multiple Price Vectors
at the Kink x=y
px
py
No Corner Solutions
WAPM and GARP
Line : p x = p y = 1, (2,8) Selected
Comparison:
(4,1),(3,6)
BT
WT
WAPM and GARP
BT
WT
Line : p x = p y = 1, (2,8) Selected
Comparison:
(6,4),(3,8)
WAPM and GARP
Line 1: p x = p y = 1 (2,8) Selected
Line 2: p X = 1, p y = 3 : (15,0) Selected
Comparison:
(5,2),(0,2.5)
BT
WT
Uncertainty Axioms
Continuity :
{α ∈ [0,1]: α L + (1 − α ) L ' f L ''} ⊂ [0,1]
%
{α ∈ [0,1]: α L + (1 − α ) L ' p L ''} ⊂ [0,1]
%
are closed
Independence axiom
if L f L ' then α L + (1 − α ) L '' f α L '+ (1 − α ) L ''
%
%
3
Suppose L ~ L'
Since L ~ L' → α L ~ α L' → α L+(1-α )L' ~ α L'+(1-α )L'=L'
Thus Indifference over probabilities must be straight lines
L
1
L’
2
Preferences:
Complete
Continuous
Transitive
Utility
Uncertainty
Certainty
Independence
Continuity
Expected
Utility Function
N
Utility Function
(Ordinal)
U(L)= ∑ pi u ( xi )
i=1
Gambles Create Cardinality
Unique up to Affine
Transformations
Risk Aversion
Let α be the probability of outcome x1
E[U ( L)] = α u ( x1 ) + 1 − α u ( x2 )
U ( E ( x)) = u (α x1 + (1 − α ) x2 )
→ E[U ( x)] < U ( E ( x)) when u(x) is concave
(Jensen's Inequality)
U(x)
U (α x1 + (1 − α ) x2 )
α u ( x1 ) + (1 − α )u ( x2 )
α x1 + (1 − α ) x2
x
Some Useful Definitions
Certainty Equivalent:
The certainty equivalent of any simple gamble g
over wealth levels is an amount of wealth CE,
offered with certainty, such that u(g)=u(CE).
Risk Premium:
The amount of wealth, P, such that u(g) ≡ u(E(g)-P)
Note that P ≡ E(g)-CE
Example:
u(w) = ln(w)
Consider a 50/50 bet of winning and losing wealth h:
1
1
1
u(g)= ln( w0 + h) + ln( w0 − h) = ln( w0 2 − h 2 ) 2
2
2
Thus :
1
2 2
CE ≡ ( w0 − h )
2
1
2 2
P ≡ w0 − ( w0 − h )
2
Risk Aversion
U (.)
U ( E ( g ))
u( g )
P
CE
E(g )
x
Production
Technology
f(x)
Profit Maximization
Cost minimization
Profit Function π ( p, w) = max x pf ( x) − wx
Cost Function c( w, y ) = min x wx
Input Demand Function x ( p, w)
SubjectTo : f ( x) = y
Conditional Demand Function x ( w, y )
Supply Function y ( p, w)
⎧ Nondecreasing in p
⎪ Nondecreasing in p
⎪
π ( p, w) : ⎨
⎪ Homogeneous of Degree 1
⎪⎩Continuous and Convex in p
⎧ Homogeneous of degree 1
⎪
c ( w, y ) : ⎨ Nondecreasing in ω
⎪Concave in w
⎩
Solve max y py − c( w, y )
max x1 , x2 p min(2 x1 , x1 + x2 ) − w1 x1 − w2 x2 )
1) Draw an Isoquant:
2) Think about production function:
f(tx1 ,tx 2 ) = min(2tx,tx1 + tx2 ) = tf(x1 ,x 2 )
Homogeneous of degree 1 → CRTS → c(w1 , w2 , y ) = c( w1 , w2 ) y
max x1 , x2 p min(2 x1 , x1 + x2 ) − w1 x1 − w2 x2 )
3) Think about corners
3) Write out possible outcomes:
a) Use all x1 : Second part binds so f(x1 , x2 ) = x1
b) Use both: Both will bind so 2x1 = x1 + x2 → x1 = x2 f(x1 , x2 ) = 2 x1
max x1 , x2 p min(2 x1 , x1 + x2 ) − w1 x1 − w2 x2 )
4) Solve the easier problems:
When w1 < w2 : x1 = y
y
When w1 > w2 : x1 = x2 =
2
w1 y
w1 < w2
⎧
⎪
c( w1 , w2 , y ) = ⎨ ( w1 + w2 ) y
w1 ≥ w2
⎪⎩
2
min( w1 , w2 ) + w1
or c(w1 , w2 , y ) =
y
2