Precise solutions

Question 3 Emma is a rst-year Ph.D. student of economics. She is very lucky because she got a place to stay in the Aggie Village, the nicest part of Davis. She lives in a beautiful cottage with a little garden. Her modest wealth from being a TA is to. She spends it on coffee and gardening. Let :1 denote the amount of coffee and 12 the amount of gardening and let p1 and P2 denote the corresponding unit prices. Her budget constraint is given by p1I1+p212 5 w. {1} Coffee is really a private good in the sense that she is the sole beneciary of caHeine in her coffee {unless she calls up in panic her fellovir student in the middle of the night because she cannot solve her ECNEUUA. homework problem]. In contrast, gardening creates a positive externality on others. But so does the gardening of others create a positive externality on her. There is plenty of gardening in the Aggie 1|In'rillage Denote by e the total externality or public good created from gardening in the community. Her utility function u[I1,Ig, e} is concave and continuously dilferentiable with a strictly positive gradient on the interior of its domain. The total externality depends in part on Emma's gardening 12 and on the externality created by the gardening of others1 denoted by e_,-. It is assumed to satisfy e i e_.- + 11:2 {2} for some parameter :1 satisfying or :e U. Emma does not think that she can affect the level of externalities provided by others. For instance, Professor Schipper, who also lives in the Aggie 1|In'rillage, is so busy writing prelim exam questions that talking to him about keeping up his gardening is no use. Thus, we can safely assume that Emma takes e_.- as well as phpg, and is as given. Since Emma diligently studies microeconomic theory for the prelims, she is eager to maximize her utility function subject to constraints [1] and {2]. This yields demand functions 11m,pg,w,e_i] and Igp1,Pg,w,B_I-} as well as her optimal desired amount of public good e{p1,p3, to, e_.-}. a.) Write down her Kuhn-TuckerLagrangian [ignore nonnegativity constraints]. b.} Derive the KuhnTucker rst-order conditions [ignore nonnegativity constraints]. c.) Use the Kuhn-Tucker conditions and the assumptions that the solution is interior, that it is unique, and that constraints {1} and {2} are satised with equality to derive | solar system. One feature of these aliens is that they can read immediately the utility function of others. (Although this sounds quite useful, it is rather a curse.) Anyway, as a proof of this claim we print here Emma's utility function: u( T1, 12, e) = 11 + (02, 0.) (2 ) -5(12,5)83 ( 52 (3) where az, a, > 0 and B = bea bee bae ) is symmetric positive definite. I know, you surely must think "Wow" but let's focus again on the prelim exam. Assume that solutions are interior and that constraints are satisfied with equality. Write out the system of equations from problem c.) for Emma's utility function. e.) Provide an interpretation of the partial derivatives #2(PI.Paw,e-) and De-1 De-1 and their signs. f.) Compute dra(p1,pa,w.=_) and De( p1, pa,we_) De_1 g.) Assume by 2 0. Derive the signs of Uzz(PL.12,,e-1) - and h.) Assume now ba Anje -2mink/N 1=0 and inverse transform given by N-1 fin] = = > F[hjelmink/N k=0 Define the power, P, of a periodic sequence, f[n], of period N by N-1 (a) For each fixed & show that the periodic sequence * FluJezwink/N has power | FI? / N2. [5 marks] (b) If gin] is a periodic sequence of period N with N-point DFT G show that N-1 V - 1 1=0 [5 marks] (c) Show that the power of the periodic sequence f[n] is equal to # EN IFI?. [5 marks]