题目
题目

BU.232.630.F3.SP25 Quiz 2 2025 - all questions

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)=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?

选项
A.pt=𝔼t[β( ct+1 ct )−αxt+1]
B.pt=𝔼t[β( 1−αct+1 1−αct )xt+1]
C.pt=𝔼t[β ct+1 ct xt+1]
D.pt=𝔼t[β(αc 2 t )xt+1]
E.pt=𝔼t[β( c 2 t+1 α )xt+1]
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思路分析
We start by restating the core setup and the FOC that governs the price today. - The investor chooses θ to maximize E[ u(c_t) + β u(c_{t+1}) ], subject to c_t = e_t − θ p_t and c_{t+1} = x_{t+1} − θ p_{t+1}. The utility is u(c) = c − (α/2) c^2, so the marginal utility is u'(c) = 1 − α c. - The first-order condition with respect to θ equates the marginal cost today to the expected discounted marginal benefit tomorrow, taking into account how c_t and c_{t+1} change with θ: ∂/∂θ [ −p_t u(c_t) + β E_t[u(c_{t+1})] ] = 0 ⇒ −p_t u'(c_t) + β E_t[ u'(c_{t+1}) ∂c_{t+1}/∂θ ] = 0. - Since ∂c_{t+1}/∂θ = −p_{t+1}, the condition becomes p_t u'(c_t) = β E_t[ p_{t+1} u'(c_{t+1}) ]. - Substituting u'(c) = 1 − α c gives p_t (1 − α c_t) = β E_t[ p_{t+1} (1 − α c_{t+1}) ]. - Solving for p_t in terms of tomorrow’s payoffs yields a pric......Login to view full explanation

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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 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?

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 2 𝑐 𝑡 2 what is the equation for the price of the asset in period 𝑡 as a function of tomorrow’s payoffs?

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