Yes, Joon, you do have a different edition of the text. You have the "first edition" of the Jehle/Reny text. You may be confused because there was a prior edition of the text by Jehle, alone, published by Prentice Hall.

The new edition, published by Addison Wesley this past summer, is the only one that says "2nd edition" in its title.

Sorry for the confusion.



Your first question concerns what is now Exercise 1.25

Here's my hint: First, let's restrict attention to the strictly positive orthant.
If preferences are monotonic there, then one thing we know is that utility can not go down as consumption of either x_1 or x_2 increases, holding the other constant. Differentiate u(x_1,x_2) with respect to x_1, and do the same with respect
to x_2, and see what this tells you must be true of the value of alpha.

That's really only part of the picture, though. We know that convex monotonic preferences are represented by increasing and quasiconcave utility functions. Hence, we need to check if quasiconcavity requires anything additional of alpha. One way to do that is to apply the test for quasiconcavity given in Exercise A2.18 of the text. (Unfortunately, Joon, in the edition of the text you have, there is an error in the description of that test that is given in exercise 2.18. The error is corrected in the new edition, so I have attached a pdf file containing the corrected version to this reply for you to consult. It is called "Arrow_Enthoven") When you apply that test, however, you'll see nothing more is needed than what you will have deduced above.




Your second question concerns what is now Exercise 1.27

The hint in the text suggests you sketch the indifference map. When you do, you will find that the indifference curves are "V-shaped", and convex away from the origin. The Vee's all kink along the line x_2=x_1: above that line all indifference curve segments have a slope of -1/a, while below the kink their slope is -a. (Note that because 0 < a < 1, the former is absolutely steeper than the latter, forming the Vee.)

Now suppose any level of income, and ask yourself the following questions: what happens when p1/p2 < a? How about when
a< p1/p2 < 1/a ? How about when 1/a < p1/p2? Answer all these, and resolve what happens when equality holds in the ranges I've indicated, and you will have described Marshallian demand at all relative prices.

Does that help?

Arrow_Enthoven.pdf
A test for quesiconcavity (52,251 bytes)