Question

Doug lives for two periods. In the first period of his life he earns income Y1. The value of Y1 was determined by your student number. In the second period of his life, Doug is retired and does not earn any income. Doug’s decision is how much of his period one income should he save (S) in order to consume in period two. For every dollar that Doug saves in period one he has (1 + r) dollars available to spend in period two, where r is the interest rate.

Doug gets utility from consumption in period one (given by C1) and in period two (given by C2) according to the following utility function:

U(C1, C2) = ln(C1) + β ln(C2)

where the value of β was determined by your student number. Given this utility

function Doug’s marginal utility from consumption in period one is given by: MUC1 = ∂U / ∂C1 = 1 / C1

and his marginal utility from consumption in period two is given by:

MUC2 = ∂U / ∂C2 = β / C2

The parameter β describes how impatient Doug is. The lower the value of β the more he prefers consumption in the present (period one) to consumption when retired (period two).

(a) What is Doug’s budget constraint in period one? What is his budget constraint in period two?

(b) Combine these two budget constraints in order to have a single budget constraint that relates C1 and C2 to Doug’s income and the interest rate.

(c) Assume that Doug is currently saving exactly 40% of his income in period one and that the interest rate is 50%, that is r = 0.50.

i. How much does he consume in each period?

ii. Calculate the slope of Doug’s indifference curve at this point. Is the slope of his indifference curve at this point greater than, or less than the slope of his budget constraint?

iii. At this point could Doug raise his utility by increasing or decreasing the amount that he saves? Carefully explain your answer.

(d) Solve for Doug’s optimal choice of savings, and how much to consume in periods one and two.

Where:

β = 0.9

Y1= 20,000

Answer 1

u = lng + B In Ca al dn period. The can consume or tave. So G + s = 7, -0 Lie consumption & sowing a numel _ 0.is period 1 Budget constraint for period a he has to rely on the interest 4 the amount of his saving in period 1 - Ca= (ltr)s 0 ② is period a budget constraint. b) Substituting valuse of si from 10. G = (Hr) S = (1+r)(y-2) fat , 7 Intertempored Budget . contraint c) i S - 40% of % = Gouby, 82 507 = 10.50 i so, c = 4 = 1,-0.40% = 0.60% (2 0.60% ] Ca = (1+).5 = (1+r) 0.40 V - (1.50)(0.40)% = 0.60% slope of Ic = mua a la i mula

[92=6= 0.box] slope I c = 1/2 slope of Rudget constraint = (1+r) =(1.50) There's no way to figure out which has - greater slupe. but if slope of IC > slope of Budget line -- > 150 then ] A if B its (IC ir flatler) - ß= t ( equel) ir lese of slope u I l > slope of Budget line. Sie metal (ltr) then the Coub should increase the wonsumption in period I because the willingness to po for period 1 consumption is a reater than The market rate so, a increuse a decrease till willingnen to pay for G is equal to morket -tate Case I slope of Il slope of Budget line B < (177)

Then Consumption decrease 4 to pay for in period 1 should full l. e .. 6 Increase as willingness G is less than the market to be constraint BG -13=0.9 %= 20,000 At optimal slope of TC = slope of Budget C = (tr) C = (Hr) BG G=(1+0.5 (0.9)ą C2 = 1.350, substituting in Intertempored budget constraint 4 + Ctra Co + 1.354 320,000 (tot) 1.9 c = 20,00 1.9 [liz 1359.1.35 (20600) = 27000 1 1.9 1.9 S= 20000 -4.2 19 (2000)-(Zcord 1.9 0.9 (20,000) = lobo 1.9 174