Why marshallian demand curves for normal goods slope down

Question

Problem 1: An individual consumes two goods, food q1 and clothing q2. Their total budget is y, the prices of food and clothing are P1 and P2 and their utility is v. Suppose their preferences are represented by expenditure function

c(v, p₁, p₂) = √(p₁p₂)e^p1^v/p2

Assume that prices and utilities take values such that this function has all the required properties of an expenditure function. including concavity in prices. and that consumers demand positive quantities of food and clothing. Note that v is allowed to be positive or negative.

(a) Use Shephard's Lemma to show that compensated budget shares of food and clothing are 1/2 + vp₁/p₂ and 1/2 - vp₁/p₂

Suppose that prices in a base period are p₁ = p₂ = 1. In a later period they have changed to p₁ = 2 p₂ = 1/2. You are asked to consider whether the overall cost of living should be regarded as having risen or fallen.

(b) Explain what a true or Kona& cost of living index is and show that a Koniis cost of living index evaluated at initial utility vu for these preferences and these prices has the form k(va ) = e^3v0

(c) Explain what a Laspeyres cost of living index is and show that a Laspeyres cost of living index for these preferences and these prices has the form

L(vo) = 5/4 + 3/2 v0

(d) Discuss the facts that

i. according to either index. whether the cost of living has risen of fallen depends on the initial utility v0 at which it is evaluated

ii. the cost of living has risen according to L(v0) but fallen according to K(v0) if initial utility is such that 0 > v0 > -1/6.

Problem 2:

Marie has preferences over two goods, cake q1 and bread q₂. She chooses quantities to consume so as to best satisfy these preferences subject to the budget constraint p₁q₁ +p₂q₂ ≤ y where p1 and p₂ are prices and y is total budget.

Suppose that Marie's preferences are represented by utility function u(q₁, q₂) = q₁ + ln (bq₁ + q₂) where b ≥ 0 is a preference parameter.

Assume that p₁/bp₂ ≥ (P₁ - bp₂) ≥ 1.

(a) Show that Marie's indifference curves are downward sloping and that her weakly preferred sets are convex for all possible values of b.

(b) Show that her Marshallian demand for cake q1 is

f1(y, p₁, p₂) = y/p₁ - bp₂ - 1

and find her Marshallian demand for bread, f₂ (y,p₁,p₂). Discuss the shape of Engel curves for the two goods.

(c) Explain why Marshallian demand curves for normal goods slope down. Are there any values of b for which either cake or bread could be a Giffen good for Marie? Discuss.

(d) Find the form of the indirect utility function and expenditure function and hence establish expressions for the Hicksian demands for the two goods.