Consider a two-period expected utility maximization problem with quadratic utility: 4 [ - 1/ (0 - 2 + 1)²] (T max_U = − 12 (ē − c₂)² + ßE₁ s.t. Ct + at = Yt Ct+1 = Yt+1 + (1+r) at where is constant. Assume 3 = 1 and r=0 for simplicity. First period income, yt, is certain and equals y: Yt = y Second period income, yt+1, will be equal to either y + e or y-e with equal probability of 3t+1=y+e with probability 1/2 y-e with probability 1/2 -ÏN
1. Use budget constraints to express consumption levels, ct and ct+1. (Hint: Use income conditions given above in the budget constraint. Notice that there are two possible states in the second period.)
2. Rewrite the utility maximization problem as choosing the optimal at alone. (Hint: Replace ct and ct+1 in the utility function with your answers from point 1. Use probabilities to derive the expected value in the utility function. Remember that a random variable that takes values x1 in state one with probability p and x2 in state two with probability 1 − p has the expected value E [x] = p.x1 + (1 − p).x2)
3. Derive the first order condition and find the optimal value of savings, at. (Hint: The only control (choice) variable is at)
4. Does household accumulate precautionary savings to self-insure against the scenario of low income in the second period? Why or why not?
![Problem 1
Consider a two-period expected utility maximization problem with quadratic utility:
4 [ - 12 ( ² - 9₁4+ 1)²]
max U- - 1/2 (c-ct)² + BEt
Ct, Ct+1,at
s.t.
Ct + at = Yt
Ct+1=Yt+1+(1+r) at
where & is constant. Assume 3 = 1 and r=0 for simplicity. First period income, yt, is certain and equals y:
Yt = Y
Second period income, yt+1, will be equal to either y + e or y-e with equal probability of
Yt+1 =
y+e with probability 1/2
y-e
with probability 1/2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71673091-805c-48ce-80ce-588078013db7%2F2fe91a46-af01-462b-b9bd-fe091d0052c9%2Frz62ptq_processed.png&w=3840&q=75)
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