ANSWER the foll6

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\" nancial 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 20%.. After mentioning that the central topic of cone is oonsmner 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 sens. After all, she knew this all along without having studied ems. Since you rely on her support, her doubts naturally alarm you. 1t'ou desperately try to search your memory for something less trivial to tell her. You vaguely recall the \"compensated law of demand": For any {13,13}, [p', 112'] with of = p' - m{p,w}| we have {PF Pl ' [1351191 slim-ml} :1 E] with strict inequality if IQ], to] % :cfrp', w'}. [With this notation, p E 111i, is a price vector1 where L is the number of commodities in the economy; to E 1R... denotes the consumers wealth; .r[p, to] denotes Walrasian demand at prices p and wealth to. Throughout, we assume that z[p,w} is single-valued in \"L for any is and ILL] a.) In order to discuss the \"compensated law of demand\" with your grandinother, you need a verbal interpretation of it. Give a verbal interpretation of the I'compensated law of deman ". b.) Your grandmother asks in what seme 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 1(3), to] is homoge- neous of degree zero and satises Walras' law. Then :r[p, to] satises the weak ax- iom of revealed preference if and only if it satises the compensated law of demand. We recall the weak axiom of revealed preference: The Walrasian demand function r{p,ur]| satises the weak axiom of revealed preference iffor any {p,w} and [p',w'}, lfp-Illp',wr] E to and zw,wr] it! :r[p,w}, thanpr-I@,wl } tor. Let's see whether you are able to prove the rst direction of the proposition: If r{p, ur] satises the weak axiom of revealed preference then it satises the compen- sated law of demand. I will guide you step-by-step through a proof: 1. Consider the case in which clip\Question 1 continued 2. Consider now the non-trivial case r(p', w') * r(p, w). Rewrite (p'-p)-(x(p', w) - x(p, w)) = p'-(x(p,w) - a(p,w))-p-(x(p', w') - a(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' - r(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 r(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.)Question 2 In class we saw a stronger version of the first fundamental theorem of welfare economics (FFTWE) for exchange economies than for production economies. We also had to work hard to prove the second fundamental theorem (SFTWE) for production economies. In this exercise you obtain: (i) a strong version of the FFTWE for production economies; and (ii) a simple proof of the SFTWE for exchange economies. 1. Consider a production economy (J,d, (u', when, (Yea, (s")(nexal, where each u' is locally-nonsatiated and Y # @. Suppose that there are as many firms as there are individuals, and that shi = 1 for all i. Say that an allocation (x, y) is in the core of this production economy if there do not exist H C J and (&', 9))text such that: (a) for all i e H, y' er'; (b) ZIEH X' = ZiEn Wit LiEn y'; (c) for all i e H, ul(x') > u'(x' ); and (d) for some i E H, u'(x' ) > u'(x' ). Argue the following version of the FFTWE for this economy: If (p, x, y) is a competitive equilibrium, then allocation (x, y) is in the core of the economy. 2. Consider a society ], where the preferences of the individuals are represented by the utility functions (ul : R4 - R)iej. Assume that all these functions are continu- ous, strictly quasiconcave and strictly monotone. Recall that, given these assumptions, our existence theorem says that for any profile (wileg of individual endowments, ex- change economy {], (u', whey) possesses a competitive equilibrium. Now, using this existence result: (a) Argue that if a profile of individual endowments (when is a Pareto efficient al- location, then there exist prices p such that the pair (p. (w')ey) is a competitive equilibrium of the exchange economy {], (u', w' heal. (b) Argue the following version of the SETWE: Fix initial endowments (when. If allocation & = (X' heg is Pareto efficient, then there exists (p, (when) such that [. wi = [, w and (p, &) is a competitive equilibrium for economy (], (u', when). Note that this definition is sensible only under our assumption on the ownership structure of the industry! This simply says that if the initial endowments of the economy constitute an efficient allocation, then the society does not need to trade