kindly

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 re. She spends it on coffee and gardening. Let I1 denote the amount of coffee and I2 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 ca'eine in her coffee [unless she calls up in panic her fellow student in the middle of the night because she cannot solve her ECNEUUA homework problem]. In contrast, gardening creates a positive cidernality 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 1'2 and on the externality created by the gardening of others.| denoted by e_.-. It is assumed to satisfy e i c_.- + 1121:; [2} for some parameter :1 satisfying or 3: Cl. Emma does not think that she can affect the level of externalities provided by others. For instanceJ Professor S-chipper1 who also lives in the Aggie "I-'rillage.I 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 re 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 I1[p1,pg,w,e_;} and Ig{p1,pg,w,e_.-} as well as her optimal desired amount of public good efp11p3,1u,e_.-}. a.) Write down her KuhnTuckerLagrangian [ignore nonnegativity constraints]. b.) Derive the KuhnTucker rst-order conditions [ignore nonnegativity constraints]. c.) Use the KuhnTucker conditions and the assumptions that the solution is interior, that it is unique1 and that constraints {1} and {2} are satised with equality to derive a system of three equations and three unknowns that does: not involve multipliers and whose solution denes I1{p11p2,w,e_.-]1 IngLp-g, w,e_.-:I and e[m,pg,w,e_; . 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.) (72 ) -2(12, ()B (72 ), (3) where az, a. > 0 and B = (ba bee 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 253(PI.P2,",_) and De( P1,P2.We_1) De-1 De-1 and their signs. f.) Compute #23(P1.P3,10,=_ Delp1, pa,we_) de_1 and de_1 g.) Assume by 2 0. Derive the signs of Uzz(PI.pa,w,=_) and De( p1.P2, w, de-1 h.) Assume now ba IE that represents 3;. b.} Given prices of goods 13 = [p],...,pg,]l Lib-3: I] and wealth to 13 ll, dene the budget set on the characteristics space by Kplw:={zE3: thereexistIEXst z=}L-i:,p-Iu.l}. Show that for anyr p 3*} D and er 2 ll, the budget set Kama\"- is convex. c.} Assume that u. is monotone and continuously differentiable. Consider the consumer problem may ufz] at. z E me. Show that necessary conditions for a utility maximum are 1 ' an \"21(25\" 323:!) forf= ,...,L. 1Where J. is the Lagrange multiplier w.r.t. to the budget constraint 1: - I 5 w