题目
BU.232.630.W1.SP25 Quiz 2 solutions
单项选择题
Consider an investor who lives 2 periods (t and t+1). She has income et in the first period and et+1 in the second period. The investor buys θ shares of an asset at price pt in the first period, and receives an uncertain payoff xt+1=pt+1+dt+1 in the second period. Her investment decision is thus described by the following maximization problem: max θ[u(ct)+β𝔼t(u(ct+1))]. Assuming that the utility function is u(ct)= c α t α what is the equation for the price of the asset in period t as a function of tomorrow’s payoffs?
选项
A.pt=𝔼t[β(
ct+1
ct
)α−1xt+1]
B.pt=𝔼t[β(
c
α
t
α
)xt+1]
C.pt=𝔼t[β(
c
α
t+1
α
)xt+1]
D.pt=𝔼t[β
ct+1
ct
xt+1]
E.pt=𝔼t[β(
ct+1
ct
)1−αxt+1]
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标准答案
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思路分析
We start by identifying the intertemporal optimization structure and the relevant first-order condition, then evaluate each answer choice against that condition.
First, note how consumption evolves. In period t, the investor spends p_t on θ shares, so her consumption is c_t = e_t − p_t θ. In period t+1, she consumes c_{t+1} = e_{t+1} + θ x_{t+1}, where x_{t+1} is the asset payoff in period t+1.
The objective is max over θ of u(c_t) + β E_t[u(c_{t+1})]. The first-order condition with respect to θ (the asset holdings) equates the marginal benefit of increasing θ to its marginal cost today, taking into account future utility:
−p_t u'(c_t) + β E_t[ u'(c_{t+1}) · ∂c_{t+1}/∂θ......Login to view full explanation登录即可查看完整答案
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类似问题
Question textThis question applies to parts 1-10. It contains drop-down multiple choice and numerical questions. Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible states of nature: a good weather state (G), a fair weather state (F), and a bad weather state (B). The state at date zero is known. There is one non-storable consumption good, apples. There is one representative consumer in the economy. The endowment of apples at time 0, is 2. At time 1 the endowment of apples is state-dependent. The physical probabilities, π, and state-dependent endowments, e, of the states at time 1 are given in the table: [table] State | Probability | Endowment G | 0.4 | 4 F | 0.3 | 2 B | 0.3 | 1 [/table] The expected utility is given by [math: u(c0)+β[πG⋅u(cG)+πF⋅u(cF)+πB⋅u(cB)],] u(c_0) +\beta[\pi_G\cdot u(c_G) + \pi_F\cdot u(c_F) +\pi_B\cdot u(c_B)], where the instantaneous utility function is: [math: u(c)=ln(c)]u\left ( c\right ) =\ln (c) (natural logarithm). The consumer’s time discount factor, β, is 0.95. Note: round your answers to 2 decimal places if necessary. 1) At least how many different securities is required for this market to be complete? Answer 1 Question 8[select: , 0, 1, 2, 3, none of these answers].For the rest of this question, consider Arrow-Debreu securities are available. 2) Compute the equilibrium fair weather state price: Answer 2 Question 8[input] 3) Compute the equilibrium traded quantity of the fair weather atomic (Arrow-Debreu) security: Answer 3 Question 8[input] 4) Compute the equilibrium quantity consumed in the bad weather state: Answer 4 Question 8[input] 5) In this Arrow-Debreu economy, maximization of expected utility reflects the assumption of: Answer 5 Question 8[select: , price is taken as given, the prices are maximised, market clearing, agents are selfish, none of these answers] 6) To solve for the Arrow-Debreu equilibrium, we need to do the following EXCEPT: Answer 6 Question 8[select: , set up the Lagrangian, find the first-order-condition for consumptions, find the first-order-condition for asset holdings, find the first-order-condition for asset prices, none of these answers] 7) The agent in this economy is Answer 7 Question 8[select: , risk-averse, risk neutral, risk loving, none of these answers] 8) The stochastic discount factor of the good weather state is Answer 8 Question 8[select: , equal to 0.95, less than 0.95, more than discount factor, none of these answers] 9) Assume now that the instantaneous utility is [math: u(c)=10c]u\left ( c\right ) =10 c and all other parameters remain the same. Compute the discount factor: Answer 9 Question 8[input] 10) Assume again that the instantaneous utility is [math: u(c)=10c]u\left ( c\right ) =10 c and all other parameters remain the same. Compute the risk premium of the good weather atomic (Arrow-Debreu) security: Answer 10 Question 8[input]
Consider an investor who lives 2 periods (t and t+1). She has income et in the first period and et+1 in the second period. The investor buys θ shares of an asset at price pt in the first period, and receives an uncertain payoff xt+1=pt+1+dt+1 in the second period. Her investment decision is thus described by the following maximization problem: max θ[u(ct)+β𝔼t(u(ct+1))]. Assuming that the utility function is u(ct)=ct− α 2 c 2 t what is the equation for the price of the asset in period t as a function of tomorrow’s payoffs?
Consider an investor who lives 2 periods ( 𝑡 and 𝑡 + 1 ). She has income 𝑒 𝑡 in the first period and 𝑒 𝑡 + 1 in the second period. The investor buys 𝜃 shares of an asset at price 𝑝 𝑡 in the first period, and receives an uncertain payoff 𝑥 𝑡 + 1 = 𝑝 𝑡 + 1 + 𝑑 𝑡 + 1 in the second period. Her investment decision is thus described by the following maximization problem: 𝑚 𝑎 𝑥 𝜃 [ 𝑢 ( 𝑐 𝑡 ) + 𝛽 𝔼 𝑡 ( 𝑢 ( 𝑐 𝑡 + 1 ) ) ] . Assuming that the utility function is 𝑢 ( 𝑐 𝑡 ) = 1 2 ( 𝑐 𝑡 − 𝛼 ) 2 what is the equation for the price of the asset in period 𝑡 as a function of tomorrow’s payoffs?
Consider an investor who lives 2 periods ( 𝑡 and 𝑡 + 1 ). She has income 𝑒 𝑡 in the first period and 𝑒 𝑡 + 1 in the second period. The investor buys 𝜃 shares of an asset at price 𝑝 𝑡 in the first period, and receives an uncertain payoff 𝑥 𝑡 + 1 = 𝑝 𝑡 + 1 + 𝑑 𝑡 + 1 in the second period. Her investment decision is thus described by the following maximization problem: 𝑚 𝑎 𝑥 𝜃 [ 𝑢 ( 𝑐 𝑡 ) + 𝛽 𝔼 𝑡 ( 𝑢 ( 𝑐 𝑡 + 1 ) ) ] . Assuming that the utility function is 𝑢 ( 𝑐 𝑡 ) = 𝑐 𝑡 𝛼 𝛼 what is the equation for the price of the asset in period 𝑡 as a function of tomorrow’s payoffs?
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