n

L u( x1 ,..., xn ) m pi xi
i 1 diagram for n=2

L x2
ui pi 0 for each i 1,..., n
xi
m p2
n
L
m pi xi 0
i 1
x̂2

n 1 equations, solve for x1 ,..., xn and

xi xˆi (p1 ,.., pn , m)

(Marshallian) demand functions 0 x̂1 m p1
x1
ˆ (p1 ,.., pn , m)
1
 xˆi (p1 ,.., pn , m) uˆ u(xˆ1 ,.., xˆn )

uˆ v(p1 ,.., pn , m) the indirect utility function

û xˆ1 xˆ
vm u1 ... un n
m m m
but at the optimum:

ui ˆ pi vm ˆ p1 xˆ1 ... pn xˆn

m m

xˆ1 xˆ dm
p1 xˆ1 ... pn xˆn m p1 ... pn n 1
m m dm
differentiate totally w.r.t. m

vm ˆ
2
 xˆi (p1 ,.., pn , m) uˆ u(xˆ1 ,.., xˆn )

uˆ v(p1 ,.., pn , m) the indirect utility function

xˆ1 xˆn
Partial derivative w.r.t pj: vj u1 ... un
pj pj

ui ˆ pi vj ˆ p1 xˆ1 ... pn xˆn

pj pj

xˆ1 xˆ
p1 xˆ1 ... pn xˆn m xˆ j p1 ... pn n 0 dm 0
pj pj
differentiate totally w.r.t. pj
xˆ1 xˆ
p1 ... pn n xˆ j
pj pj
vj ˆ xˆ
j
3
 xˆi (p1 ,.., pn , m) uˆ u(xˆ1 ,.., xˆn )

uˆ v(p1 ,.., pn , m) the indirect utility function

vm ˆ

vj ˆ xˆ
j

vj
xˆ j Roy’s Identity
vm

René François Joseph Roy

(1894-1977)

4
 vm ˆ v1 ˆ xˆ1
x2 x2

m p2 m p2

x̂2 x̂2

u v(p1 , p2 , m) u v(p1 , p2 , m)

0 x̂1 m p1 0 x̂1 m p1
x1 x1

5
 minimise expenditure subject to u(x1 ,..., xn ) u
n
L pi xi u u(x1 ,..., xn )
i 1
x2
L
pi ui 0 for each i 1,..., n
xi

L
x2 u u(x1 ,..., xn ) 0

n 1 equations, solve for x1 ,..., xn and

u u

xi xi (p1 ,.., pn ,u )
0 x1 Hicksian demand functions
x1
(p1 ,.., pn ,u )
6
 xi (p1 ,.., pn ,u ) p1 x1 ... pn xn e(p1 ,.., pn ,u ) the expenditure function

x1 x
Partial derivative w.r.t pj: ej xj p1 ... pn n
pj pj

x1 xn
pi ui xj u1 ... un
pj pj

x1 x
u(x1 ,..., xn ) u u1 ... un n 0 du 0
pj pj
differentiate totally w.r.t. pj

ej xj Shephard’s Lemma

7
For any given utility function u(x1 ,..., xn )

the indirect utility function the expenditure function

v(p1 ,.., pn , m) e(p1 ,.., pn ,u)

are inverses of each other.

Given p1 ,...pn : u v(p1 ,.., pn , m) if and only if m e(p1 ,.., pn ,u)

vj
in which case: xˆ j x j ej
vm

Shephard’s Lemma

Roy’s Identity
8