ExpenditFun - Two Properties of Expenditure functions Proof...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Two Properties of Expenditure functions Proof that e(p, u) is a concave function of p. Proof: We want to show that for any u and any two price vectors p and p , and for any λ between 0 and 1, λe(p, u) + (1 − λ)e(p , u) ≤ e(λp + (1 − λ)p , u). Let h = h(p, u) and h = h(p , u), and let hλ = h(λp + (1 − λ)p , u). We note that u(h) = u(h ) = u(hλ ) since h(p, u) is the cheapest consumption vector that yields utility u at price vector p. Then e(p, u) = ph(p, u) ≤ phλ (because u(hλ ) = u) Similarly, e(p , u) = p h(p , u) ≤ p hλ . It follows from these two inequalities that λe(p, u) + (1 − λ)e(p , u) ≤ (λp + (1 − λ)p )hλ = e(λp + (1 − λp , u). Notes on Proving Shepherd’s lemma. ∂e(p, u) = xi (p, u). ∂pi Proof: e(p, u) = j pj xh (p, u). Differentiate this to find that j ∂e(p, u) = xi (p, u) + ∂pi pj j ∂xj (p, u) . ∂pi Note also that u(xh (p, u)) = u for all p. Differentiate this uj (xh (p, u)) j ∂xh (p, u) = 0. ∂pi But uj (xh (p, u)) = λpj . Now finish proof by substituting uj /λ for pj in the above equation and noticing that all the complicated stuff disappears. 1 ...
View Full Document

This note was uploaded on 12/25/2011 for the course ECON 210A taught by Professor Bergstrom during the Fall '09 term at UCSB.

Ask a homework question - tutors are online