Question 1 Assume that your grandmother pays for your studies. You probably agree that this assumption is more realistic than most assumptions we make in economics given the cost of living and our "generous" financial support at UC Davis. Imagine the following story: You pay a visit to your grandmother to thank her for her truly generous support. She asks what path-breaking insights you have learned in 200A. After mentioning that the central topic of 200A is consumer theory you proudly tell her that you learned that when prices go up then demand may go up or down. She looks at you with some consternation and starts to wonder whether supporting your studies makes any sense. After all, she knew this all along without having studied 200A. Since you rely on her support, her doubts naturally alarm you. You desperately try to search your memory for something less trivial to tell her. You vaguely recall the "compensated law of demand": For any (p. w), (p', w') with w = p . r(p, w) we have (p' - P) . (x(p, w) - r(p,w)) so with strict inequality if r(p, w) # a(p, w'). (With this notation, pe Ry, is a price vector, where L is the number of commodities in the economy; w E R+ denotes the consumer's wealth; r(p, w) denotes Walrasian demand at prices p and wealth w. Throughout, we assume that r(p, w) is single-valued in R' for any p and w.) a.) In order to discuss the "compensated law of demand" with your grandmother, you need a verbal interpretation of it. Give a verbal interpretation of the "compensated law of demand". b.) Your grandmother asks in what sense the "compensated law of demand" is a law. You recall the following proposition that we proved in class: Proposition: Suppose that the Walrasian demand function r(p, w) is homoge- neous of degree zero and satisfies Walras' law. Then a(p, w) satisfies the weak ax- iom of revealed preference if and only if it satisfies the compensated law of demand. We recall the weak axiom of revealed preference: The Walrasian demand function r(p, w) satisfies the weak axiom of revealed preference if for any (p, w) and (p', w), if p . x(p, w) w. Let's see whether you are able to prove the first direction of the proposition: If r(p, w) satisfies the weak axiom of revealed preference then it satisfies the compen- sated law of demand. I will guide you step-by-step through a proof: 1. Consider the case in which a(p, w) = x(p, w). This should be easy.2. Consider now the non-trivial case (p', w') * x(p, w). Rewrite (p'-p).(x(p. w) - a(p, w)) = p.(x(p,w') - a(p,w))-p.(x(p, w') - r(p, w)). Consider separately each term of the right-hand side. Derive the signs of each of the two terms using (some) assumptions of the proposition and the fact that we assumed w' = p' .x(p, w) for the compensated law of demand. To derive the sign of the second term of the right-hand side you will need to make use of the weak axiom of revealed preference. 3. Put everything together to conclude the proof of the first direction. c.) Given the characterization of the compensated law of demand by the weak axiom of revealed preference, you need to explain to your grandmother the meaning of the weak axiom of revealed preference. Provide a verbal interpretation of the weak axiom of revealed preference. d.) The weak axiom of revealed preference may be viewed as a condition on the "con- sistency of consumer choice". We may be inclined to label a consumer violating this condition as being "irrational" (unless a more complex setting is considered). So the above proposition characterizes the compensated law of demand in terms of consistency of consumer choice. But is it really a characterization in terms of rational consumer choice in the sense of having complete and transitive preferences over consumption bundles? That is, is the following conjecture true? Conjecture: Suppose that the Walrasian demand function (p, w) is homoge- neous of degree zero and satisfies Walras' law. Then a(p, w) satisfies the com- pensated law of demand if and only if the consumer has complete and transitive preferences over consumption bundles. Consider both directions of the conjecture separately. Moreover, consider the spe- cial case of L = 2 separately. (You don't have to present a detailed proof. You can make use of results learned in class. The line of arguments should be clear.)