term1 Definition1term2 Definition2term3 Definition3
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Consider a person who has the following exponential inter-temporal utility function,
U(C0, C1,,C2) = ln C0 + δ ln C1 + δ2 ln C2 , where δ =0.8
Ct is the amount of consumption, measured in dollars, that they get in period t, and δ < 1 is the individual’s discount factor (that is, how much they discount future consumption, relative to current time 0 consumption.
Find the PV of $100 for periods 0,1, and 2.
r=(1-0.8)/0.8=0.25
Year 0: PV0: 100/(1+0)0= $100
Year 1: PV1: 100/(1+0.25)^1= $80
Year 2: PV2: 100/(1+0.25)^2= $64
Suppose this person has $60 in period 0. How much should they consume in each period? Show your math. (Set the discounted marginal utility of consumption between period 0 and 1 equal, and also the discounted marginal utility of consumption between period 1 and 2 equal. That gives you 2 equations and 3 unknowns. The third equation comes from the constraint. Recall that the derivative of ln x is 1/x.)
Constraint: C0 + C1 + C2 = 60
MU_0=MU_1
1/C_0 =(0.8) 1/C_1
MU_1=MU_2
(0.8) 1/C_1 =(〖0.8〗^2 ) 1/C_2 simplifies to: 1/C_1 =(0.8) 1/C_2
For C0: 1/C_0 =0.8/C_1
so,C_0=C_1/0.8
For C2: 1/C_1 =0.8/C_2
so,C_2=0.8*(C_1)
Now we can plug in C0 and C2 back into our constraint equation: C_0+C_1+C_2=60
C_1/0.8+C_1+(0.8)*(C_1 )=60
Now we can solve for C1:
C_1/0.8+C_1+(0.8)(C_1 )=60
C_1=19.6721
Now we can solve for C0:
C_0=C_1/0.8,
so C_0=19.6721/0.8 C_0=24.5901
Now we can solve for C2:
C_2=0.8(C_1 ),
so C_2=0.8(19.6721) C_2=15.7376
Lifetime Consumption plan in period 0:
C0= 24.5901
C1= 19.6721
C2= 15.7376
Check your answer to b), by comparing the present discounted value of utility of a person who follows the consumption plan that you derived with the following suboptimal plan: C0 = 84, C1 = 84, C2 = 84. (Hint: You'll need a few decimal places.)
ln(24.59) + 0.8*ln(19.67) + 0.64*ln(15.74) = 7.35
ln(20) + 0.8*ln(20) + 0.64*ln(20) = 7.31
For a person with quasi-hyperbolic inter-temporal utility function:
U = ln C0 + δ (ln C1 + ln C2) where δ = 0.7
What is the PV of $100 for periods:0,1, and 2
Year 1: PV1: 100/(1+0.42)^1= ~$70
Year 2: PV2: 100/(1+0.42)^1= ~$70
NOTICE THAT PERIODS 1 and 2 ARE THE SAME WITH HYPERBOLIC DISCOUNTING
Using the definition of MRS, explain what we mean by time inconsistent preferences in this model. Hint: “MRS means … In period 0 the MRS between … and …. is X. But in period 1…”
The MRS is the ratio of marginal utilities, or the slope of the indifference curve. The MRS measures the rate at which a person is just willing to trade one good for another. Here the goods are consumption in different time periods. In period 0 the MRS between periods 1 & 2 is one to one. However, in period 1, the MRS between periods 1 & 2 is 0.7 to 1. The MRS changes depending on which time period one is in, meaning that the preferences are time inconsistent.
Suppose that the contract in part a) actually cost money. In other words you have to pay someone $X, in period 0), to have the contract enforced. How much would you pay? Set up the equation correctly, but you don't have to solve it.
Set U with commitment and the consumption numbers planned for in period 0, less x the commitment cost, equal to the U using the consumption numbers that will be used without commitment:
ln(25-x) + 0.7 (ln(17.5) + ln(17.5)) = ln(25) + 0.7 (ln(20.59) + ln(14.41))
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