05 Jun Question 1. Consider the following utility functio
Question
1. Consider the following utility function U(xM,xB)=xM×xB (Cobb-Douglas utility function) where xM and xB are the amount of magazines and books, respectively, that the consumer consumes. The consumer has an income of $24. The price of a book is $6. The price of a magazine is $4
a) Draw a set of indifference curves showing the utility levels U=6 and U=12.
b) Draw the consumer’s budget line with magazines on the horizontal axis and the books on the vertical axis. Hint: The budget line is a kinked budget line and the kink point is (4 magazines, 8/6 books).
c) Write the consumer’s demand for books and the demand for magazines. (Hint: Ch 5 of the book.)
d) Find the optimal choice of the consumer.
e) Find the utility of the consumer when she consumes her optimal choice.
f) Now, suppose that the consumer gets a discount if she buys a sufficient number of magazines. If the consumer buys more than 4 magazines, then each of the additional magazine costs only $1. That is, if the consumer buys more than 4 magazines, the fifth magazine will cost her $1, the sixth magazine will cost her $1 and so on so forth. Given the optimal bundle in part‘d’ (and the utility that the consumer gets from consuming this bundle as you calculated in part ‘e’), is it possible to have another optimal bundle for the consumer? If yes, find the other optimal bundle. If no, explain the reasoning.
2. Suppose that books and magazines are perfect substitutes for the consumer. As we mentioned in the class, the indifference curves of the consumer will be negatively sloped linear lines. As we also mentioned, in the case of perfect substitutes, the optimal choice of the consumer will be a boundary optimum. The consumer has $24 and the budget constraint of the consumer is a kinked one as described in part f of the previous question.
a) If the utility function of the consumer is U(xM,xB)=xM+4xB, then what will be the optimal bundle for the consumer?
b) If the utility function of the consumer is U(xM,xB)=xM+2xB, then what will be the optimal bundle for the consumer?
c) Write a utility function that represents the consumer’s preferences such that the consumer will be indifferent between spending all of her budget on magazines or books. Remember books and magazines are perfect substitutes.
3. Consider question 1 and the utility function U(xM,xB)=xM×xB (Cobb-Douglas utility function) where xM and xB are the amount of magazines and books, respectively, that the consumer consumes, again. The consumer has an income of $24. The price of a book is $6. The price of a magazine is $4. (The consumer’s optimal choice is (3 magazines, 2 books)).
a) If the price of one magazine falls to $1, what is the optimal choice of the consumer?
b) What is the total change in demand for magazines due to the decrease in magazine price?
c) If, after the price change, the consumer’s income is compensated so that he can afford a bundle that is just indifferent to his initial bundle. With respect to this compensation method, what will be the optimal bundle of the consumer for the compensated budget?
d) With respect to this compensation method, calculate the substitution and income effects.
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