Solution to Problem Set 21 Within this context, Xt+n/Xt= 0.6, r = −0.05. Plug in the formula withdiscrete rate, i.e.Xt+n= Xt(1 + r)nwe haveXt+nXt= (1 + r)nlnXt+nXt= n ln(1 + r)n = lnXt+nXtln(1 + r) =ln 0.6ln 0.95≈ 9.962 (1)lnx4y3(xy2)2= ln(x4+2y4−3) = 6 ln x + ln y(2)exp(logbb2) = exp(2 logbb) = exp(2)(3)log927 =log327log39=323x = 2y − 4z + 3a + 2b (1)y = 2x + 2z − 2a (2)z = x + y − b (3)(1)+(2):x + y = 2(x + y) − 2z + a + 2bx + y = 2z − a − 2b (4)Plug (4) in (3)z = (2z − a − 2b) − bz = a + 3b (5)1(1)-(2):x − y = −2(x − y) − 6z + 5a + 2bx − y = −2z +53a +23b (6)((4) + (6)) ×12:x =13a −23b (7)((4) − (6)) ×12:y = 2z −43a −43b = 2(a + 3b) −43a −43b =23a +143b (8)Therefore, the closed form for x, y, z are (5),(7) and (8):x =13a −23by =23a +143bz = a + 3bCalculate the difference quotient with respect to a and b:∆x =13∆a −23∆b∆y =23∆a +143∆b∆z = ∆a + 3∆bSince a and b increases by 2 and 3 respectively and simultaneously, plug in∆a = 2 and ∆b = 3, we have∆x =13× 2 −23× 3 = −43∆y =23× 2 +143× 3 =463∆z = 2 + 3 × 3 =
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