Question

For Question 12 to 15, let the utility function of the household be U(c,d) = ln(c) + Bln(c'), where B is a parameter between 0 and 1, and assume that there is always an interior solution to the household's problem. 12. What is the marginal rate of substitution of current consumption for future con- sumption MRS given this utility function? How does it change with c and c'? 13. Solve the household's optimization problem with the lifetime budget constraint. That is, find the household's optimal current consumption c and future consump- tion d' in terms of (y, yt,',r, b). 14. From your answer to the last question, what is the household's savings s in the current period? Your answer should be in terms of (y,y',t,t',r,b). 15. The parameter p in the utility function governs how patient the household is. From your answer to the last two questions, how do the household's current consumption c, future consumption c', and savings s respond to an increase of B (i.e., when the household becomes more patient)? (Hint: The value of increases with x when x is between 0 and 1.).

Answer 1

UCC, C") = ln(C) + B ln (C) C = Cevrent Consumption c = Future Consumption. Q To find MRS (C = ? Let the budget constracht be, Сс lt n y = curent income y = ficture ? r = current rate of antoest i tr Constructing the Lagrangian, 2 = ln C + B in C thLg + 2 y = c-c7 Om + - = 0 5 : = < -> (1) de din > B (138) = ->(2) до С " 1th From (I) and 2). o " Marginal Rate of BC | BC B C 7 Substitution between and C. Higher the c' - Heigher the MRS c, c. Higher the c - lower the MRS. c.

Q2 From above optimisation process we got (1+0) c = BC 1 th) an e- a G =CB Clth) (or art] Optimal value of a we have the enterteinposal budget consteacht C+C =gt for C + eR(Ita) = y + y - ltr ITH CT CB =gt y th c (1+B) = y + gn c = tey + b To = B g (1H2) + g] ITB Q3 g = caş =s=cry It ltr.

QH. et ß lß and If value of ß in seases then for c= y + y of the denominator will be - Itß (178) (HB) 7 larger so value of C will be smaller. for c= y B (ITR) + By > Here, both numatorerise IP lt B and denominator shall rise The value of B chocases with B when B is between oan =) with rise ehvalue of ß the value of Swill be lower. ITB B with rise u for S = tegin ito