Question

Charlie likes two goods, pears and kiwis, and consumes no other goods. We write his consumption bundle as (pears,kiwis), so that pears are the x variable and kiwis the y variable. Last month, Charlieconsumed 20 pears and 5 kiwis. The indifference curve through this bundle is described by y=100/x.Furthermore, you are told that the set of bundles for which he is indifferent between them and a newbundle at(10,15)is the set of bundles with y=150/x.

(a) Plot the two indifference curves. Note the slope at the two points given.

(b) Are the following true or false? (No explanation needed here, but it’s all or nothing for credit.)

i.(30,5)∼(10,15)

ii.(10,15)(20,5)

iii.(20,5)(10,10)

(c) Is the set of bundles Charlie weakly prefers to (20,5) convex?

(d) Recall the indifference curve through the point at (20,5) is defined by y=100/x. This has slope for any given x of −100/x2. What is the marginal rate of substitution at

i.(10,10) ii.(5,20) iii.(20,5)

(e) Do these preferences represent diminishing marginal rate of substitution?

Answer 1

a) The utility function of Charlie is a Cobb-douglas consumption function. So the shape of the indifference curves will be that of a rectangular hyperbola. The figure below plots the two ICs for two different utility levels.

b) *The symbols mentioned are not clear*

c) Since Charlie's preferences reveal that the indifference curves will be rectangular hyperbola. The set of bundles which Charlie weakly prefers to (20,5) is the shaded area below IC1 in figure 1. So, according to the definition of a convex set which goes as:

A set S in a vector space over R is called a convex set if the line segment joining any pair of points of S lies entirely in S.

We can say that the set of bundles that Charlie weakly prefers to (20,5) is not a convex set.

d) Since the utility function is U = xy then MRS = y/x. So the marginal rate of substitution for different consumption bundles will be (i) 1 ;(ii) 4 ;(iii) 1/4

e) Yes, Charlie's preferences represent diminishing marginal utility.This means that as consumption of commodity x increases, the amount of another commodity that the consumer would give up to aquire an additional unit of x decreases.