高级微观经济学公式总结Calculation methods and examples

2. the inverse function of v( p, m) 1. e( p, v(q, m)) Money metric Indirect utility function u ( p, q, m)
Expenditure function
2.
∂u ( p, q, m) = xi ( p, u ( p, q, m)), i = 1, 2...n ∂pi u ( q , q , m) = m
x=
Utility function
U = x+ y
m ,y=0 px m px m py
px < p y px = p y px > p y
Mashallian demand function
x+ y =
x = 0, y =
px = p x =
Indirect utility function
u=
f ( x)
1. from xi ( p, y ) to y = f ( x)
1. ∑in=1 pi xi ( p, y )
c ( p, y )
1. max py
Profit function
π ( p)
st : y ∈ Y 2. max{ py − c( y )} st : y ≥ 0 1. from max py st : y ∈ Y max{ py − c( y )} st : y ≥ 0
U = min{x, y}
Formula
Mashallian demand function
x= y=
m px + p y
x< y x= y x> y
Indirect utility function
u=
m px + p y
Inverse demand function
m , py = 0 x m px + p y = x m px = 0, p y = y px =
Formula
Utility function
U = (∑ xiρ )1/ ρ −∞ < ρ ≤ 1
ρ
1− ρ
Mashallian demand function
xi =
1 m( )1− ρ pi [∑
n j =1
1
1 ( )1− ρ ] pj
ρ
Indirect utility function
u = m[∑ nj =1 (
4. xi = hi ( p, e( p, u )) 1. min v( p, m)
st : ∑ pi xi = m
Inverse demand function
pi = Di−1 ( x m)
2. pi = m
∂u ( x m) ∂v( x, m) / ∑in=1 xi ∂xi ∂xi
3. The inverse functions of x = D( p, m)
m min{ px , p y }
Inverse demand function
m x+ y m px ≥ p y = y m p y ≥ px = x
x = u, y = 0 x + y = u, x = 0, y = u
x ≠ 0, y ≠ 0 x = 0, y ≠ 0 x ≠ 0, y = 0
+∞ p > w1 + w1 x1 = x2 = any number p = w1 + w1 p < w1 + w1 0
Cost function
c = ( w1 + w2 ) y
Conditional demand function
x1 = y x2 = y
Linear production function Formula Formula w1 w2 x1 = +∞, x2 = 0 p > min{ , } a1 a2 w w p = min{ 1 , 2 } any number a1 a2 w w x1 = 0, x2 = +∞ p < min{ 1 , 2 } a1 a2 y x1 = , x2 = 0 a1 a1 x1 + a1 x2 = y y x1 = 0, x2 = a2
px < p y px = p y px > p y
Expenditure function
Money metric Indirect utility function
e = min{ px , p y }u
Hicksian demand function
u=
min{ px , p y }m min{qx , q y } CES utility function Formula
1 1− ρ ) ] pj
ρ
Inverse demand function
pi =
mxiρ −1 ∑ nj =1 x ρ j
Expenditure function
Biblioteka Baidu
e=
[∑
n j =1
u
1 ( )1− ρ ] pj
ρ
ρ
1− ρ
u(
ρ
Hicksian demand function
xi = [∑ nj =1 (
CES production function Formula Production function c= [∑ nj =1 ( y = (∑ xiρ )1/ ρ −∞ < ρ ≤ 1 y 1 1− ρ ) ] wj
ρ
1− ρ
Formula
Demand function
Cost function
ρ
x= y =u
Expenditure function Money metric Indirect utility function
e = ( px + p y )u
u=
Hicksian demand function
( px + p y )m (qx + q y )
Linear utility function Formula Formula
Conditional demand function
xi = [∑
1 y ( )1− ρ wi
n j =1
1
1 ( )1− ρ ] ρ wj
ρ
1
α w2 1−α y ] (1 − α ) w1 A (1 − α ) w1 α y x2 = [ ] A α w2
x1 = [
Leontief production function Formula Production function y = min{x1 , x2 } Demand function Formula
f ( x)
1. from xi ( p, y ) to y = f ( x)
1. ∑in=1 pi xi ( p, y )
c ( p, y )
1. max py st : y ∈ Y
Profit function
π ( p)
2. max{ py − c( y )} st : y ≥ 0 3. max{ pf ( x1 , x2 ) − w1 x1 − w2 x2 } st : x1 ≥ 0 x2 ≥ 0 1. from max py st : y ∈ Y max{ py − c( y )} st : y ≥ 0
Demand function
yi = yi ( p ) 2. from
1. min{∑ pi xi } Conditional demand function xi = hi ( p y ) st : f ( x) = y 2. xi =
∂c( p, y ) ∂pi
Production calculation methods Notation Calculation methods Production function Cost function
1. max U ( x)
st : ∑ pi xi = m
Mashallian demand function 2. xi = −
xi = Di ( p m)
∂v( p, m) ∂v( p, m) / ∂pi ∂m
3. xi =
m∂v( p, m) ∂v( p, m) / ∑in=1 pi ∂pi ∂pi
1 1− ρ ) pi
1
1 1− ρ ρ ) ] pj
ρ
1
Money metric Indirect utility function
m[∑ u= [∑
n j =1
1 ( )1− ρ ] qj
ρ
1− ρ
ρ
n j =1
1 ( )1− ρ ] pj
1− ρ
ρ
Production calculation methods Notation Calculation methods Production function Cost function
Cobb Douglas production function Formula Production function Cost function y = Ax1α x21−α Demand function
Formula
c=
w1 w y Aα α (1 − α )1−α
α
1−α 2
Conditional demand function
1. max U ( x) st : ∑ pi xi = m Indirect utility function v ( p, m) 2. U ( D( p, m)) 3. the inverse function of e( p, u ) 1. ∑in=1 pi hi ( p, u ) e( p , u )
Demand function
yi = yi ( p )
2. from
3. yi =
∂π ( p ) ∂pi
1. min{∑ pi xi } Conditional demand function xi = xi ( p y ) st : f ( x) = y 2. xi =
∂c( p, y ) ∂pi
Utility calculation methods Notation Calculation methods
1. min v( p, m)
Utility function
U ( x)
st : ∑ pi xi = m
2. v( p p , 1), where i = Di−1 ( x,1) m m
w1 w2 < a1 a2 w1 w2 = a1 a2 w1 w2 > a1 a2
Production function
y = a1 x1 + a1 x2
Demand function
Cost function
c = min{
w1 w2 , }y a1 a2
Conditional demand function
Expenditure function Money metric Indirect utility function
e=
α p y 1−α u ] A (1 − α ) px (1 − α ) px α u y =[ ] A α py
u=
pα p1−α m x y qα q1−α x y
Leontief utility function Formula Utility function
1. min{∑ pi xi } st : U ( x) = u Hicksian demand function xi = hi ( p u )
2. xi = ∂e( p, u ) ∂pi
3. xi = Di ( p, e( p, u )) Cobb Douglas utility function Formula Utility function Indirect utility function U = Axα y1−α Mashallian demand function
Formula
x=
αm
px
,y=
(1 − α )m py
(1 − α )m y
u=
Aα α (1 − α )1−α m pα p1−α x y pα p1−α u x y Aα (1 − α )
α
1−α
Inverse demand function
px =
αm
x
, py =
x =[ Hicksian demand function