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Assume that a five-year, government-issued coupon bond at par value NOK 100,000 and 10% coupon rate was issued today. The bond pays one annual coupon at the end of each year, while the par value is redeemed exactly five years from today including the last coupon payment. In the market, the bond is considered default-free. Government-issued, zero-coupon bonds (Bo,r) with maturities ranging from exactly one to five years from today (T = 1, 2, ...,5) are currently trading at the NOK-prices listed below. Assume that the bonds are default-free. Maturity Market price 1 year 93,023.16 2 year 84,167.99 3 year 75,131.42 4 year 67,073.49 5 year 60,017.97 To the following questions, please provide numerical answers based on the unbiased expectations theory of the term structure of interest rates: (a) (7 points) Calculate the current, risk-free spot-rates for one, two, three, four, and five years from today. (b) (7 points) Calculate the one-year forward rates for year two, year three, year four and year five. (c) (8 points) Calculate the equilibrium market price of the five-year, government-issued coupon bond issued today. (d) (8 points) What must be the equilibrium rate of interest on a one-year investment (or loan) in year five?Question 4 In ECN200A we assumed that consumers have preferences over consumption bundles. In many contexts though it is more natural to think that consumers have preferences over characteristics of consumption bundles like "lots of vitamins", "gluten free", "lots of horse power", "no tail pipe emissions", "many mega pixels" etc. In the following I outline an alternative model of consumer theory, in which preferences are defined over characteristics rather than consumption bundles. You are right, we have never discussed it in class. But there is no need to freak out since we know all the tools that are required to think about such a model. Goods are indexed by f = 1, ..., L and characteristics are indexed by i = 1, ..., I. We denote by one > 0 the quantity of characteristics i possessed by one unit of good f. If denotes the quantity of good f. z, is the quantity of characteristic i. We let z = (21, ..., 27) and assume that z e Z C R4. Moreover, as usual we let r = (21, .... II) EX CR. We assume 2 = 2_ died for i = 1, ..., I. This is the amount of characteristic z; derived from a bundle of goods I = (11, ..., IL). We arrange A = (ait)izl,..If=1.,L into a matrix, in which rows refer to characteristics and columns to goods. a.) Consider a binary relation _ on the space of characteristics, Z. State conditions on _ that are sufficient for the existence of a utility function over characteristics u : Z - R that represents _. b.) Given prices of goods p = (p1, ....PL) > > 0 and wealth w 2 0, define the budget set on the characteristics space by Kow,A := {= EZ : there exist r ( X s.t. = = Ar, p- r Sw}. Show that for any p > > 0 and w 2 0, the budget set Kpa,A is convex. c.) Assume that u is monotone and continuously differentiable. Consider the consumer problem max u(z) s.t. z E Kpu,A. Show that necessary conditions for a utility maximum are pe 2 du(z) for { = 1, .... L. where A is the Lagrange multiplier w.r.t. to the budget constraint p. r S w. d.) Provide an economic interpretation of the term 1 du(z) A dziQuestion 3 Emma is a first-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 w. She spends it on coffee and gardening. Let z, denote the amount of coffee and ro the amount of gardening and let p, and p, denote the corresponding unit prices. Her budget constraint is given by Pili + patz w. (1) Coffee is really a private good in the sense that she is the sole beneficiary of caffeine in her coffee (unless she calls up in panic her fellow student in the middle of the night because she cannot solve her ECN200A 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 Village. Denote by e the total externality or public good created from gardening in the community. Her utility function u(11, 12, e) is concave and continuously differentiable with a strictly positive gradient on the interior of its domain. The total externality depends in part on Emma's gardening r2 and on the externality created by the gardening of others, denoted by e-;. It is assumed to satisfy ese-ito12 (2) for some parameter a satisfying o > 0. 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 Village, 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 p1. p2, and w 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 I,(p1, paw, e_;) and 12(p1, P2, w, e_;) as well as her optimal desired amount of public good e(P1 , P2, w, e-;). a.) Write down her Kuhn-Tucker-Lagrangian (ignore non-negativity constraints). b.) Derive the Kuhn-Tucker first-order conditions (ignore non-negativity 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 satisfied with equality to derive a system of three equations and three unknowns that does not involve multipliers and whose solution defines ri(p1, pz, w, e-;), 2(p1, p2, w, e_.) and e(p1, p2, w, e-;). (No need to solve it.) d.) If you think that Professor Schipper acts in a strange way sometimes, this is not just due to being an economic theorist. The secret is he comes from another