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Unformatted text preview: l—LnijLFS to 1(th reclaims 8b" Exercise 8. 6 (a) Si=marginal propensity to save; TELmarginal income tax rate; I‘=marginal propensity to invest. (b) Writing the equilibrium condition as F0”; Go) — SO”) : TO”) 10’) 6'9 _ O: we find
that F has continuous partial derivatives and g; = .5" + T’ — 1’ 5'5 0. Thus the implicit
function theorem is applicable. The equilibrium identity is: S(Y")+T{Y*)—I(Y")Gg _=.
D. (c) By the implicitfunction rule, we have dY' _ —1 _ I
((100) "“ ‘s’+r‘*—r* * awn—r ) 0 As increase in Go will increase the equilibrium national income. 47 (a) HP; Yo,To) = D (P. Yo) ~ Seem = 0 (b) F has continuous partial derivatives, and P}: = Dp — Sp 95 [11 thus the implicitfunction theorem is applicable. The equilibrium identity is: DU”, YD) — S{P",Tg) E 0. {C} By the implicitfunction rule: 610' _~ _ Dug ap' __ _ —Svh
(6Y0) — grasp. > 0 ("an") " Din—5p. > 0 An increase in income or taxes will raise the equilibrium price. (d) The supply function implies Q’ = S(P*, To}; thus : ﬁfi. (gig) ) 0. The demand function implies Q‘ = D {P*, Y0); thus = ail—Pg. (K) C 0. Note: To use the demand function to get would be more complicated. sinco Yo
has both direct and indirect eﬁects on Q2. A similar complication arises when the suppl}r function is used to get the other comparativestatic derivative. 3. Writing the equilibrium conditions as F1 (PJQiYD:TUl Dipeyol—Q=0 F2(P,Q;%.To) = scam—on D —1
We ﬁnd [J] = P 2 Sp — DP 7E 0. Thus the implicitfunction theorem still appliee, Sp —1
and we can write the equilibrium identities DUD‘J’iil—Q+ E 0
5(P‘rTﬁ) _ (2‘‘ E D
Total differentiation yields
Blood?" — dQ‘ = LDyndYQ
SpdP'" — dQ‘ = —SdeTg When Y0 is disequilibrating factor ((1% = D), we have
DP *1 (3%) = Sp —1 0 48 Q: Ul 3p _ Dy 3 ‘ _ Dy 3 .
Thus (an) " 3p.—o,,. ) 0 and " spifpp—P. 3" 0
When To is the disequilibrium factor (oil/'0 = 0): we can similarly get (%i;;) = 5—1?— ) U
[3 J" ‘— I" ~ —SvD.
and (63%;) = —’l‘—’—— <0 sip—op. (a) 6.0/8.3” < D, and 3Dfato > U
(bl FipﬁuiQsol = Digital * Q50 = U {c} Since the partial derivatives of F are all continuous, and Fp = 3—? 5:6 0, the implicit function theorem applies. (d) To ﬁnd (gig), use the implicitfunction rule on the equilibrium identity D(P‘,to) —
Q50 5 0.. to set in
U ‘2) es in
at“ an An increase in consumers‘ taste will raise the equilibrium price >0 0.:
lo (a) Yes.
(b) Is)” + L (i)
(c) We can take the two equilibrium conditions as the equilibrium FI = U and F2 : 0, respectively. Since the Jacobian is nonzero: 6F” 3!” I r
— F.— 1 H c —r m .—. 6:; 8:2 2 f = L’(1 h 0’) + kl” < 0
BY 3i k L the implicit—function theorem applies‘ and we have the equilibrium identities Y‘—C(Y*)—I{i‘)—Gg a o
kY’+L(i")—~Msg ; 0 with M50 as the disequilibrating factor‘ we can get the equation ar‘
1 — C” —I’ 467:3; 2 1
. 6:"
L: Li 6—6; 0
This yields the results
a)” L’ 62" k
— = ~— 0 cl = —— 0
(3G0) Hi i a” (6%) EJ! ’ 49 (d) f’(s:) 2 6:: u 6 = 0 iff x =1; f(1] = —1 is a relative minimum. (a) The ﬁrst equation stands, but the second equation should be changed to
kY—Msg =0 or kY+Lo—ﬂ/Iso (b)
1—0' —I’
k 0 m“ = = M“ JJ’ has a smaller numerical value than M.
(c) Yes. (d) With Mso changing, we have 1—0! hr!
1: 0 Thus (er/anew) = I’/ iJl' > o and (Bf/611450} = (1 — C’)/ M < 0. Next, with G9 changing, we have
1— C" —I’ {BY‘/3Gg) w l
k 0 (8%“ [509) 0 Thus (av/ace) = 0 and (as/airing = —k/ W > 0 (BYVE‘Msg) Z 0
(ﬂaw/6111139) 1 (e) Fiscal policy becomes totally ineffective in the new model. {f} Since J’ is numerically smaller than H], we find that F/ lJlI > F/ IJI. Thus monetary policy becomes more effective in the new model. ...
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This note was uploaded on 12/12/2011 for the course ECON 101 taught by Professor Bi during the Spring '11 term at York University.
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