Minimization Expenditure

Question 1

we look at the problem of
trying to minimize the expenditure at prices p necessary to attain at least

Answer 1

a level of utility u.

Question 2

If the utility function is____ then that lowest line will give us a unique expenditure minimizing bundle x∗.

Answer 2

strictly quasiconcave

Question 3

We call the consumption plan (x∗1 , x∗2 ) that solves the expenditure minimization problem (1) ______ at prices p and utility u and denote it by ______.

Answer 3

the Hicksian demands, h(p, u) = (h1(p, u), h2(p, u))

Question 4

h(p, v (p, w )) = x(p, w ).

Answer 4

keep the prices the same and set the
level of utility u in the expenditure minimisation problem equal to the level
we attained in the utility maximisation

Question 5

expenditure necessary to obtain that level of utility will be
exactly the ?

Answer 5

wealth w that we started with. That is, e(p, v (p, w )) = w

Question 6

Shephard’s Lemma

Answer 6

states that if u is very well be-haved (or even as well behaved as u is in this exercise) and that e is the expenditure function defined for u then the Hicksian demand functions may be found as hℓ(p, u) = ∂e(p, u)/∂pℓ

Question 7

Inverting the indirect utility function gives

Answer 7

the expenditure function and vice versa

Question 8

Consider the expenditure minimisation problem
min
x1,x2
p1x1 + p2x2
subject to: x1^αx2^1−α = u
Write down the Lagrangian for this problem.

Answer 8

L(x1, x2, λ, p1, p1, u) = p1x1 + p2x2 + λ u − x1^αx2^1−α