Question

An Individual lives for two periods, 1 and 2. In the first he works and earn an income of M. In the second he is retired and has no income His/her life time utility is a function of how much he consumes in the two periods. Cydenotes consumption in period 1 and 2 consumption in period 2. (Hint: If you want to, you can view and treat C and C2 as any pair of "goods", eg, good x and y). He/her uses one part of M to buy Ci in period 1 and saves the rest to buy C2 in period 2. This means that C1=M-S, where S denotes how much the individual saves of his earnings M. The price of both C and C2 is equal to 1 The amount he saves earns an interest rate of r implying that C2=(1+r)S The individuals budget restriction could for instance hence be written as: M = Cy+S or M=Cy+Cz/(1+). a. Draw a graph showing the individual's budget line (.e. combinations of C and C2 where the individual uses all of M for consumption in the two periods). What is the slope of the budget line? Now, the individual has an utility function U=Inc, +(1-2)inc, where 0<d<1. The term d denotes a discount factor, ie that the individual values consumption in period 2 lower than consumption in period 1 b. Derive the demand functions for C, and C2 as a function of Mr and d. C. Assume that the interest rate is 5% (r=0.05), the discount rate is 10% (d=0.1) and that the individual has an earning in period 1 of 380.000$. How much will he consume in period 1 and how much will he save? d. Assume that the interest rate increases to 10% (r=0.1). How will this affect his savings and total utility?

Answer 1

* Ansulxi Given intormation Cl- M-s (2 lits M:ctc2 =) (2=4148) M-(1+8) CI (1+) *lal 1 C Keton slope: +1+0) M (9) * U= linct Eind) en C₂ u= lncitli-d) In{(1+2)(M-C2)} differentiatis du dci 무 ' + (1-d) a+8)(m-c.) (-(1+8)) = 0 d M-C1) VICI lond) (M-C) =) M-ci =) C-cid cicl-d +1)M CS PZ For minimum utility E's 대 2-d

C2= (1+r){ (mm) C2 = (1+3) M20) C2 : MZI+CI-d) a (2-d) Demand function *(0) 20.1 Ci will remain same So Saling heill also demain Same i.e $180 since ci is inderendent of 'G' C2 = M(1+0)(-d) (2-0) C2- 380 (told co-on) 2-01 C2- 380 (1.1) (0.a) 1.9 (2= 380 (0.99) 1.9 C2- 376.2 1:9 C2- 1198 la Inca ti-d) Inca dh(200) + (-on) In (198) 10.05

carlian wele halle rio.os Ci - 200 C2 - 380 CI+0.05)(1-0:1) 2 -Oil 380 (los) 10.9.) С Ca 1.9 C2 380 (0:945) 1.9 Cz 35901 . 1.9 C2 189 U- Inzo0 +(1-0. In 189 Us 10:0) Total utility will increase from 10.01 to 10.05

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