Question 2

QUESTION 2 : DERIVATION OF THE SLUTSKY EQUATION FOR A CHANGE IN THE CONSUMPTION OF COMMODITY X2 AS A RESULT OF A CHANGE IN THE PRICE OF COMMODITY X1.

with budget constraint Suppose a consumers utility function is U ( X 1 , X 2 , X 3 )

, I = P1 X 1 + P2 X 2 + P3 X 3

then he can maximize his utility as shown below.

L = u ( x1 , x 2 , x3 ) + λ ( I − p1x1 − p 2x 2 − p 3x 3) ∂L ∂u = − λ p1 = 0 ∂x1 dx1 u1 − λ p1 = 0 .....................(1) ∂L ∂u = − λ p2 = 0 dx2 dx2 u2 − λ p2 = 0 ...................(2) ∂L ∂u = − λ p3 = 0 ∂x3 ∂x3 u3 − λ p3 = 0 .....................(3) ∂L = I − p1 x1 − p 2 x 2 − p3 x3 = 0 ∂λ I − p1 x1 − p2 x2 − p3 x3 = 0 ...............(4)

Where

is the Lagrange function and

is the Lagrange multiplier which represents the marginal

λ

L

utility of money and

>0. Taking the total differentials of (1),(2),(3), and (4), we have;

λ

u11dx1 + u12 dx2 + u13 dx3 − λ dp1 − p1 d λ = 0 u21dx2 + u22 dx2 + u23 dx3 − λ dp2 − p2 d λ = 0 u31dx3 + u32 dx2 + u33 dx3 − λ dp3 − p3 d λ = 0 − p1dx1 − p2 dx2 − p3 dx3 − x1 dp1 − x2 dp2 − x3 dp3 + dI = 0

In matrix form, we have;  u11   u21  u31   − p1

u12

u13

u22

u 23

u32

u 33

− p1   dx1   λ dp1  λ      − p 2   dx2   λ dp 2  = 0 =  0 − p 3   dx3   λ dp3       0   d λ   x1dp1 + x2dp 2 + x3dp 3 − dI   x1

− p2 − p3

let u11

u12

u13

∆ = u21

u22

u23

u31

u32

u33

Now, the total change in x2 as a result of a change in p1 , we have

λ u13 0 u23 0 u3 3 x1 − p3 ∆

u11 u21 u31 ∂x2 − p1 = ∂p

− p1 − p2 − p3 0

u21

u23

− p2

u11

u13

− p1

u31

u33

− p3

u21

u23

− p2

u31

u33 ∆

− p3

− p1 ∂x2 =λ ∂p1

− p3 ∆

0

+ x1

u21

u 23

− p2

u 11

u 13

− p1

let ∆12 = u31

u 33

− p3

and ∆ 42 = u 21

u 23

− p2

u 31

u 33

− p3

− p1

− p3

therefore, ∂x2 ∆ ∆ = λ 12 + x1 42 ∂p1 ∆ ∆

0

0 0 0   dp1    λ 0 0   dp2  0 λ 0   dp3    x2 x3 − 1  dI 

Now, we consider a change in x2 with respect to a change in income,

u11

0

u13

− p1

u21

0

u23

− p2

u31

0

u33

− p3

∂x2 − p1 − 1 − p3 = ∂I ∆ u11

u13

− p1

u21

u23

− p2

∂x2 u31 = ∂I

u33 ∆

− p3

∂x2 ∆ = − 42 ∂I ∆

0

................(5)

Also, we try to investig1ate the effect of a change in p1 holding p2, p3 and I (income) constant. We then have;

3

∴ ∑ pi dxi = 0 i =1

3

The budget line is also given as I= ∑ pi xi 1=1

3

3

i =1

i =1

dI = ∑ pi dxi + ∑ xi dpi but 3

∑ p dx i =1

i

i

=0 3

∴ dI = ∑ xi dpi i =1

with p2 and p3 constant, then dpi = 0, i = 2, 3 ∴ dI = x1 p1

u = u ( x1 , x2 , x3 ) du = u1dx1 + u 2 dx2 + u3dx3 = 0 3

∴ ∑ ui dxi = 0 i =1

but, from (1) u i = λ pi so,

3

λ ∑ pi dxi = 0 i =1

but λ ≠ 0, (since the marginal utility of money is positive).

Hence;

 u11 u12   u21 u22  u31 u32   − p1 − p2

u13 u23 u33 − p3

u11 u21 u31 − p1 ∂x2 − = ∂p1 u

− p1  dx1   λ   dp1    − p2  dx2   0   0  = − p3  dx3   0   0       0  d λ   0  0 

λ u13 − p1 0 u 23 − p 2 0 u33 − p3 0 − p3 0 ∆

but u22

u23

− p2

∆12 = λ u 32

u 33

− p3

− p2

∴

− p3

0

∂x2 ∆12 .......................... (6) − =λ ∂p1 u ∆

, Now, from (5) and (6), we have that

∂x2 ∂x2 ∂x 2 = − − x1 ∂p1 ∂p1 u ∂I

of a This represents the Slutsky equation of a change in the quantity of commodity x2 as a result change in the price of commodity X.