Suppose that Rose’s tastes for roses (x) and chocolate (y) can be represented by
ID: 1106308 • Letter: S
Question
Suppose that Rose’s tastes for roses (x) and chocolate (y) can be represented by the utility function u( x,y) = 5ln( x +1) + y , where MUx = 5 ( x +1) and MUy = 1. Assume price of roses is px, price of the chocolate is py and Rose’s money income is $M.
c) Derive Rose’s ordinary (Marshallian) demand for roses.
d) Derive Rose’s ordinary (Marshallian) demand for chocolate.
e) Are roses normal or inferior good? Is chocolate normal or inferior good? Explain.
f) Are roses substitutes or complements for chocolate? Is chocolate substitute or complement for roses? Explain.
Assume that Rose’s income is $100, price of the chocolate is $1 and rose also costs 1 dollars each.
g) What is the utility-maximizing choice of the chocolate and roses?
h) What is the level of utility at the optimal basket? Show your result in the optimal choice diagram. Assume that Rose’s income and the price of the chocolate stay at $100 and $1 respectively. Price of the roses, however, increases to 6 dollars each.
i) What is the utility-maximizing choice of the chocolate and roses? (Hint: Is the tangency condition satisfied?)
j) What is the level of utility at the optimal basket? Show your result in the optimal choice diagram.
k) Calculate the compensating and equivalent variation for this price increase.
Explanation / Answer
(c) & (d): Consumption is optimal when MUx / MUy = px / py
[5 / (x + 1)] / 1 = px / py
5 / (x + 1) = px / py
x.px + px = 5py
x.px = 5py - px.......(1)
Substituting in budget line,
M = 5py - px + y.py
y.py = M + px - 5py
y = (M + px - 5py) / py [Demand function for y]
From (1), x.px = 5py - px
x = (5py - px) / px [Demand function for x]
(e) Demand function for Rose (x) is independent of income (M), therefore we cannot conclude whether Roses are normal or inferior good. However, from demand function for chocolates (y), as M increases (decreases), demand for chocolates (y) increases (decreases), therefore Chocolates are normal goods.
(f) From demand function for Roses, as Price of chocolate (py) increases (decreases), demand for x increases (decreases), and from demand function for chocolate, as Price of rose (px) increases (decreases), demand for y increases (decreases). So Rose and chocolate are substitutes.
NOTE: First 4 parts are answered.