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6 Cards in this Set
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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 |
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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 |
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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.
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 |
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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 0: PV0: 100/(1+0)0= $100 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 |
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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. |
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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)) |
