Question

Please explain and show work.

Joseph’s utility function is given by UJ = xA + 2xB, where xA denotes his consumption of apples and xB his consumption of bananas. Clara’s utility function is given by UC = 3xA + 2xB. Joseph and Clara shop at the same grocery store.

a. When we observe that Joseph leaves the store with some bananas, then we can deduce that Clara also buys some bananas.

b. There exist prices of apples and bananas such that both consumers buy strictly positive amounts of both goods.

c. When we observe that Joseph leaves the store with some apples and some bananas, then we can deduce that Clara also buys some apples and some bananas.

d. When we observe that Joseph leaves the store with some apples, then we can deduce that Clara also buys some apples.

e. Apples and bananas are perfect complements for Joseph.

Answer 1

(a)

when we observe that Joseph leaves with some bananas then it is clear that Clara also buys some bananas.

Because it is a case of perfect substitutes between the two goods or the more is better strategy is preferred so both the consumers will buy some quantity of both goods rather than buying one of them which is a case when consumer just increase the consumption of one good and more or less of the banana doesnot affect Clara's level of satisfaction which means its marginal utility is zero which is not the case here so Clara will also buy bananas from the grocery store.

(b)

there exists prices of both goods pA for apples and pB for bananas which is given to the two consumers then one can find the equilibrium level of consumption for both consumers which is given by

and the budget constraint is as

also depends on the money income of both consumers.

(c) the MRS of apples for bananas for the two consumers is as

For Joseph:

For Clara:

Joseph has to give up 1unit of apple in order to gain 2units of bananas. Similarly, Clara has to give up 3units of apples in order to get 2units of bananas. So, Clara has to give up more units of apples than Joseph but both will buy some units to the two goods so that they lie on a higher indiiference curve rather than choosing a corner solution and go with one good which again depends on their market prices.

(d)

the explanation is similar to part (a) that when both buy some units of bananas so both will also but some units of apples because more is better and not worse is better in this case because both marginal utilities are positive for either goods.

(e)

Apples and bananas are perfect substitutes than complements which can be observed from the utility function itself where a definate combination of the goods is being consumed by the two consumers. Though, it is necessary to have both goods in order to be better off doesnot imply perfect complements bacause in this case one good substitutes for the other rather consumed along with the other good and in case price of one good increase with respect to the other, consumer will shift towards consumption of the cheaper good and lower the consumption of the expensive one.