Question

Suppose that a household has a utility function and intertemporal budget constraint as follows: U(C1,C,) - (cº:" + Bc2:5)1-y U(C1,C2) = - 1- ITBC: C1 + = yı + 1+1 a) Determine the marginal rate of substitution for this utility function and derive the Euler equation faced by this consumer (define the Lagrangian and then obtain first order conditions as we did it in the lecture). Explain the intuition of the Euler equation. b) Find a solution for optimal consumption in both period 1 and 2 as a function of the parameter ß as well as the variables r, y1 and yz. c) What is the ceteris paribus effect on consumption today (C1) and tomorrow (C2) of the following shocks: (i) An increase in ß. (ii) A decrease in lifetime income. (iii) An increase in y. d) Is there a condition where C1=Cz? If so, what value must ß assume?

Answer 1

u CG,CI) = (9.5+ BC-5) ?-? 1-Y a MROSE MRS = OU OG Əylacz W T Bees J6530 zs (95 +265-7, 8 165166-05 Now at Eqm MRS 12 = (1+r) CH 1 C2 = B2 (ro)? Euler Pan And Buz = 409,69) + a Famoa +6] + (1+0) 4, +42] L= u CG, (2) + XE (+8) Yity2 -(1+2) 4-ca] Foco ah = l - (+8)x=0 allace= u -A=0 80 from foc? " , = (1+8) # 1044 = (1tos Now Cg = 82 G (148)2 from ITBC: (140) 4+ Cg = (1+r) YtY2 (148) at B2C CHO)2 = (1+8Mity2 GE (1+0) Yity2 L (to) +82 (140)2! * = B2 (48) 2 (1+0) YtY2 - L(+8) + B² (148)2 GM = ß2 (178) (148) YtY2] (o ( 1+ B2 lito)] -

on CX = ß2 (148) [(1+r)y, tY2] [It B² (Ho) ] OB B1, 2G/3B o, c (falls 2428 = (to [(10) Yi+Yz] S [14B2 (170)]2B - B2C &ß (140)] [1tß2 (148) 2 = (1+0){1+8) y, ty,] [2Bt 233 (148) - 2 83 (148] [It B2 (Ho)]² 28 (148) [(Hr) % t/2] 70, 80 as ß1, Cap. [1+ B2 (148)]2 (ii) Life time Income both ca & CA (MICH) +Y2) a ↑ Ano uti dihy as t us not No effect on GA & Co. there in expression of GA & Con d) 9=62, then (18+12 = ß2 (1+0) (17811772) [Uto) + B2 (170)2] [+B2 (140) = B2 (itor) to) B [1tr12] 12 - Hotel Ane