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# prelims_Macro Prelim ANSWERS Sept 2004 - ”go'0 ﬂ/d/M...

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Unformatted text preview: ”go/'0 ﬂ/d/M ﬂMM—f 857/? 55’ Questions from 200E material, September 2004 Macro prelim la. All I am looking for here is a discussion of how the strength of the income and substitution effects are inﬂuenced by the persistence of the technology shock. Speciﬁcally, the greater the persistence of a shock, the greater the implied income effect. This does indeed have implications for the ampliﬁcation mechanism: this is primarily due to the response of labor so with greater income effects, the ampliﬁcation mechanism will be dampened. For propagation, itshould ﬁrst be noted that a typical RBC model does not have very strong propagation mechanisms. . .but investment will be affected by the degree of persistence. Here, high persistence implies a strong substitution effect (higher expected MPK) so this will cause investment to increase. To the extent that capital formation does propagate shocks, this will help in this dimension. lb. Here is the dynamic programming formulation of the problem (I express everything in real terms, but nominal would also work): Vth = (mat? [U913 )+ W — h, )+ 5% M, H P, 111+] M T - M +3. —-'—‘-‘++—’— + h ——' I‘[ 1:: 1)“ Cf f( f) 3] [Mn 1: ] +y, —-—+——C, P: P. where A, is the Lagrange multiplier on the budget constraint and y, is the Lagrange multiplier on the CIA constraint. Tf 2 MM is the lump sum monetary transfer. The FOC’s are: (l) c,:U,'—zi.,—y,=0 (2) h,: ~W,'+/i.,ﬁ'=0 P r+l . . L_ L: (3) Mi' ABVIH[ ] AP! 0 Eqs (2) represents the standard labor-leisure tradeoﬁ‘. By the Envelope Theorem, we have V: 2 it, + y, . Updating this and using cq.(l) permits eq(3)'to be written as: , P Ar=ﬂIt+l I Rn That is, in this model the Lagrange multiplier on real wealth is not equal to today’s MU of consumption but, instead, the MU that-a dollar provides next period. Using this in eq. (2), we obtain: (4) W: = 130:. if: PM This also makes sense given the CIA censtraint: Here the MPL of labor is affected by inﬂation since the proceeds can not be spent until next period (and therefore it is next period’s MU of consumption that is relevant. _ — P l . [n steady-state, c = f (h) and FL 2 1— . Using this in eq. (4) yields the condition that H] + [1 determines steady-state labor (here the terms in parentheses denote the arguments of the functions): _ . _ 1 _ W' 1~ h = ' h —— ’ h ( ) mm (who Taking the total differential (arguments are suppressed — the parentheses denote multiplication) — l — l — _ W”(—dh)= ﬁU”—f‘2dh + ﬂU’—f'h'h — guru + ﬁt) ”an 1+ ,u 1+ y Concavity of the utility functions and the production function establish: £<0 dﬂ Hence, money is not superneutral because of the inﬂation tax on labor. 2. (a&b) The maximization problem is max }U(cl ,] + ,8 E[U(c2, )] subject to: (at, ,bnzl. ,c 2.. 77x: :0]: + prb: +932: €21+I : bl +zr(qr+1 + (1 _U 1+1) Substituting the second constraint into preferences and then setting up a Simple Lagrangian yields the FOC’s: (I) p.U'(c1.)= ﬁE[U'(c2...)l (2) q,U'(cl. ) = M10 '(62... 19m +(1- nlx... )1 These are standard intertemporal efﬁciency conditions representing the MC=MB tradeoffs implied by each asset. c. A recursive competitive equilibrium is characterized by the following conditions: 1. b, = 0 V t . (Since all young are identical and the old do not want to save, there will be no bond trade in equilibrium.) 2. z' z I V I . (The young will purchase all the equity ﬁom the old each period.) Note that these imply: cl , + (:2, = x, -- aggregate demand equals aggregate supply. This isjust Walras Law — if asset markets clear, the ﬁnal market (goods) must clear. 3. There are two functions, p(x:) and 9(a) that satisfy equations (1) and (2) when these are evaluated at the market clearing quantities. These conditions are: Pkg)” fault—'90:.) =55: U! q(xt+l)+ (1”? 1+] ‘—v—“' ‘—v——’ "It 62M q(x.)U’ rpm-9(a) =ﬁ 0' qr(xi+1)+(l-7!)Ii+r (9(x,+1)+(1-ﬁln+l) Cl, 62:9! d. Note that due to the i.i.d. assumption, the RHS of both expressions are constant. Then taking the derivative with respect to x! establishes the result. -e. The intuition is straightfonvard — since consumption when old is the return to equity, the covariance between the MU of consumption and the return must be negative. According to the CCAPM, this implies a positive risk premium on equity. I—u/ (“TV 470; __[_____ --;--A C e I A pp Cant—cl 3 ﬁCl-ﬂ") C”— ng-u . (1- V 2.1-V‘ CH— 'T TSC'E" 7 HY“ 1 Eff; CH:- : l-rﬁ [Tl/- b) 751731,— CO/USUHPTK’A/ " Q l/UCC“ *ELP) = Niﬂi—a: PM Nf:;)(”?i:’) a) 5530.] [bx/Jug WGDMEUHPDb/ll; ﬂafE/VDDWHE/U-r WWW 2'” 4L ”VJ {[113 1 TV‘ K:- L:‘ i—ﬁ 9:44.0ch coug_ uuu_ Cf :NCIt-+'T;CL{~ V "dz/V1? -r- 1-!“ B Q My (-—«t-)——-— (riﬂe AIOT‘ (JflelJe‘ Ctr "3 CaH. = V b :(ﬂ’g; {' A/t’ ...
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prelims_Macro Prelim ANSWERS Sept 2004 - ”go'0 ﬂ/d/M...

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