Chapter 3: Basics of Consumer Demand

Question 1

Consumer’s Problem

Answer 1

Maximize u(x) subject to p\cdot x\leq y and x\geq 0

Question 2

If u is quasi-concave, the set of solutions to the CP…

Answer 2

for any p and y are convex. if Strictly quasi concave, then there is a unique solution for each p and y.

Question 3

If preferences are locally insatiable, and x is a solution to the CP

Answer 3

p\cdotx=y.

Question 4

Marshallian demand

Answer 4

Fixing u, let D(p,y) represent the set of solutions given the values p and y.

Question 5

Indirect utility function

Answer 5

the map between p and y to the utility at optimality.

Question 6

Berge’s Theorem for the CP (3)

Answer 6

Assume u is continuous:

1. For all prices and positive income, demand does not depend on numeraire.
2. Marshallian demand is upper semi-continuous and for some it is a continuous function.
3. The indicrect utility is continuous.

Question 7

Kuhn Tucker Conditions

Answer 7

If x* is a solution tot his problem, then the optimality conditions hold for x.
If u is concave and x satisfies the optimality conditions, then x* solves the CP.

optimality conditions are necessary for a CP solution and sufficient if concave.

Question 8

Optimality Conditions (3)

Answer 8

1. p\cdotx\leq y
2. for some lambda\geq 0,
   MU_j(@x) \leq \lambdap_j. Equality if x_j > 0.
3. p\cdot x* < y then lambda = 0.