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ExpenditFun

# ExpenditFun - Two Properties of Expenditure functions Proof...

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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 ...
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