Question

Suppose you can spend your monthly income on two specific goods X and Y. For each of the following cases a) to c), answer the following questions:

i) Sketch some indifference curves in the appropriate graph. Tip: Start by connecting bundles that yield the same level of utility.

ii) Derive the utility function if possible. iii) What kind of preferences are the indifference curves derived from? Are they rational? Why, why not?

iv) State a unique example of what good X and Y could be in this case.

a) You like to consume 3 units of X together with 1 unit of Y. More of both goods gives you additional utility but only at this ratio. A basket containing 3 units of X and 2 units of Y does not yield any more utility than a basked of 3 units of X and 1 unit of Y. Neither good gives you any utility alone.

b) You like good X, but you do not like good Y. So for every additional Y you consume, you have to be compensated with 2 additional units of X to be as well off as before. You are indifferent between consuming bundle (X,Y) and bundle (X+2,Y+1).

c) You like both good X and Y, but good X is always more important. You always prefer baskets containing more X to baskets containing fewer X, regardless of how much Y they contain. But if two baskets contain the same number of X, you prefer the basket with the most Y

Answer 1

a)i)

ii) utlility function of such type of goods would be

U= min {3X,Y}

ie with 3 units of X we consume 1unit of Y .so this means we take minimum of both in proprtion ie if we have 3  units of X and 2 units of Y it does not yield any more utility than a basket of 3 units of X and 1 unit of Y

iii) the preferences that are complete and transitive . that is if x is preferred to y and y to z then x is oreferred to z. And the preferences sould be complete. so if preferences are both complete and transitive , then they are rational too.

iv) These types of goods which are consumed in fixed proportion are called complimentary goods. no extra utility would be derived if one of goods is consumed out of the given fixed proportion. eg: shoes.(both left and right would be worn together . one extra left show wont give more utility without the right shoe.

b)i)

ii)U(x, y) = −ax + by is the utility function ie for every additional unit of good y we are compensated with 2 units of good x

iii)the preferences that are complete and transitive . that is if x is preferred to y and y to z then x is preferred to z. And the preferences sould be complete. so if preferences are both complete and transitive , then they are rational too.

iv) such type of goods are called bad goods ie they reduce the utility so extra units of the other good are consumed to increase the utility.

c)i) such type of functions cant be represented on an indifference curve because any continuous preference ordering can be represented by a utility function, but such prefernces preferences are obviously not continuous. With lexicographic preferences, indi§erence curves are single points.

ii) it has no utility representation (x1; x2) >~ (y1; y2)<=> x1 > y1 or x1 = y1

and x2 >= y2

iii) lexicographic preferences are complete and transitive yet they dont have indifference curve.

iv) such type of goods are called lexicographic goods where the most of one good is preferred the most no matter how much the other good is.