Calculus :

Mj inP C

j


1

=

c (z)

p(z)cj (z)dz

0

s.t.




1

cj (z)

θ−1 θ

0

dz

θ  θ−1

= Cj

(λ multiplier)

FOC: on c(z) θ 
1  θ−1 −1 θ−1 θ j −1 θ j p(z) = λ c (z) θ dz c (z) θ θ−1 0 θ−1 1 
1  θ−1 θ−1 −1 p(z) = λ cj (z) θ dz cj (z) θ

0

using




1

cj (z)

θ−1 θ

dz

0

θ  θ−1

= C j , we get

p(z) = λcj (z)

−1 θ

equation (1) for good z p(z  ) = λcj (z  )

 j  θ1 C

(1)



(2)

−1 θ

so that equations (1) and (2) yield

Cj

 1θ

−1

1

p(z) cj (z) θ cj (z  ) θ = = −1 1 p(z  ) cj (z) θ cj (z  ) θ θ  p(z) j  j c (z ) = c (z) p(z  ) FOC: on C j P =λ Using (3), (1) then implies 1 −1  p(z) = P cj (z) θ C j θ  j  θ1 p(z) C = j P c (z)   p(z) −θ j c(z) = C P

Demand for good z (static problem for the consumer) depends on • overall consumption • relative price avec elasticity of substitution θ 1

(3)