Economics/Microeconomics - Choices, Menus, and WARP
First things first i really appreciate your help on this forum.
I have been preparing myself on Preferences and Consumer Theory in Microeconomics. After practicing a few question in my text book, I have been particularly stuck on this one.
I have attached the question as a JPG file to this message, and would appreciate your help on this.
Thank you for your kind words and for asking a very technical question. You have mentioned that you are stuck while preparing yourself on preference and consumer theory. The help you have kindly asked of me involves a couple of sufficiency conditions, in set notation, to clinch the point that a choice correspondence satisfies the weak axiom revealed preference (WARP).
May I mention, for your kind prescience, that the problem, by itself, is simple? So far I understand from the tenor of your question, you are an intelligent student committed to learning these rather abstract microeconomic constructs. Make no mistake, even though this requires not much mathematical finesse, without adequate prerequisite, any perceptive student like you is likely to stumble into almost insurmountable difficulty. So, Bella, the stumbling block that you are confronted with does not at all indicate that there is something you are lacking in tackling the problem. Rather, it is the inadequate preview –I daresay, inadequate preview, because otherwise the answer could itself be extracted easily from the two conditions –that prevents a full-dress grasp of what follows in the problem under question.
THE REQUIRED ARSENAL TO SOLVE THIS PROBLEM
To deal with this problem requires a sound knowledge of the set theory together with an understanding of consumer behaviour theory as it has evolved from demand theory based on utility into an attempt to measure underlying utility function with great certainty. You can overcome your hurdle in the way of solving this problem only if you are armed with: (a) the requisite set-theory background; and (b) a good grasp of the preference theory which has culminated in Samuelson’s revealed preference theory, a concrete improvement over demand based on diminishing marginal rate of substitution.
I presume you have grounding in the set theory. It would not be a bad idea to refresh your memory. Get hold of any intermediate-level book, such as Taro Yamane’s “Mathematics for Economists” or A. C. Chiang’s “Fundamental Methods of Mathematical Economics.”
Secondly, I would strongly urge you to get a good grasp of what is meant by “the preference revealed” in contrast to “preference,” which is not observable. We assume something about what we cannot observe and ultimately make predictions about something we can observe, consumer demand behaviour. Consider two consumption bundles, x and y, both affordable at some prices and income level. [Think of the budget constraint in indifference curve analysis, M= pX+PY, where M=money, p=price of x, and P=price of y (take note of smaller-case p and capital P).] If the consumer buys one bundle instead of the other, then the bundle bought (chosen) is considered to be revealed preferred (RP) to the other.
A consumer’s behaviour satisfies WARP if whenever x is preferred to y, y is never revealed to be preferred to x. This fortunately goes to quash the “loop” we may encounter in preference analysis –such as, “A is preferred to B, and B is preferred to C, but C may be preferred to A.”
Note that x is revealed preferred to y means that x is chosen when both x and y are affordable. For y never to be revealed preferred to x, we must have x NOT affordable whenever y is chosen; i.e., the cost of x must be more than the cost of y at all prices y is chosen. Suppose x is revealed preferred to y at prices p, and that y is chosen at some other prices P [note the two prices, p and P]. Then, we formally express WARP as:
py< or= px => Px>PY
[“py is less than or equal to px” implies “Px is greater than Py, where p and P are two prices]
The weak axiom does not say that y will never be chosen under any circumstances. The bundle y may very well be chosen at some price vector P. What the weak axiom indicates is that if y is chosen at some price P, then x will be more expensive than y at prices P.
As I have already mentioned about budget equation, when consumer behaviour satisfies WARP, the consumer chooses x facing prices p while spending all income M. This is the choice function:
As you already know that, if prices and income both change by a constant factor, the real income is unchanged and the behaviour of the consumer is unchanged. Quite the same, here our choice function, x(p, M), must be of degree zero in prices and income for any good of x.
[You can find in the above-referred books, or in any other such textbooks, an explanation of how when two independent variables change by a factor t, the function f in f(a, b) is multiplied by t, with t raised to an exponent k; and in the case of homogeneity of degree zero, k=0. Here, thus, we have x(tp, tM) = (t^k)x(p, M) = x(p, Y), since t raised to k=o yields 1.]
Remember WARP also implies that the substitution effects of own price changes cannot be positive. Draw a simple diagram for the case of two goods, and observe that. It is, in fact, possible to show that the substitution matrix in the general case of n commodities is negative semidefinite [you may find treatment of semidefinite matrix in Chiang, but I think a very simple and detailed treatment is available in Edward T. Dowling’s “Introduction to Mathematical Economics.”]. In any case, if you are not comfortable with matrix algebra, never mind; you have nothing to lose in solving your problem.
Also, if demand behaviour is generated by maximising preferences then it must satisfy WARP. Should observed choices of a consumer violate WARP, that consumer, we know, cannot possibly be a preference-maximising consumer.
THE ANSWER TO YOUR PROBLEM
Now you are in a position to solve your problem without any difficulty. You can get details which will satisfy all such answers related to WARP and choice theory in an excellent book if you can lay your hand on it [it is available from eBay]. This book is “Microeconomic Theory,” by Mas-Colell, Whiston, and Green. The first chapter gives an excellent exposition to choice theory and WARP, intelligible to good students with basic mathematics like you. This is an easy chapter, and I would strongly recommend this chapter to a student with the elementary background as I have already enumerated above. This is a thick book, and don’t worry about the rest of the chapters that are at too advanced level meeting the requirements of master’s and doctoral students.
FIRST TRY NOW BY YOURSELF
It is always better if you solve the problem by yourself. I strongly suggest try now. However, if you still need the solution –and you will certainly not need that if you get the aforesaid book –then please send me an email (firstname.lastname@example.org), and I will send you an attachment [because such mathematical notations can’t be placed on this page, and the image doesn’t admit of lengthy explanation which might be required for your better grasp].
If you have the above book, and have the background I have asked of you, you have your answer. Should you have problem in getting the book, please be in touch with me.
I hope I have been able to make myself clear to you. My best wishes are for your success in the pursuit of advanced economics knowledge.