Questions
Single choice
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?
Options
A.𝑝
𝑡
=
𝔼
𝑡
[
𝛽
(
𝑐
𝑡
+
1
−
𝛼
)
2
(
𝑐
𝑡
−
𝛼
)
2
𝑥
𝑡
+
1
]
B.𝑝
𝑡
=
𝔼
𝑡
[
𝛽
(
𝑐
𝑡
+
1
𝑐
𝑡
)
−
𝛼
𝑥
𝑡
+
1
]
C.𝑝
𝑡
=
𝔼
𝑡
[
𝛽
𝑐
𝑡
+
1
𝑐
𝑡
𝑥
𝑡
+
1
]
D.𝑝
𝑡
=
𝔼
𝑡
[
𝛽
(
𝑐
𝑡
+
1
2
𝛼
)
𝑥
𝑡
+
1
]
E.𝑝
𝑡
=
𝔼
𝑡
[
𝛽
(
𝑐
𝑡
+
1
−
𝛼
𝑐
𝑡
−
𝛼
)
𝑥
𝑡
+
1
]
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Step-by-Step Analysis
We begin by restating what is being asked and what the candidate formulas look like.
The problem asks for the equation for the price of the asset in period t as a function of tomorrow’s payoffs, given the investor’s quadratic utility u(c)=1/2 (c−α)^2 and the intertemporal optimization framework. Among the answer options, each candidate expresses p_t as a discounted expectation E_t[ ... ] of a function that involves c_t, α, x_t or x_{t+1}, and constants, inside a β multiplier, plus possibly a +1 term.
Option 1: p_t = E_t[ β( c_t+1 − α c_t − α ) x_t + 1 ]
This candidate uses x_t (current period payoff) inside the expectation, and includes a +1 outside the product. Given the setup, the asset price in period t should reflect the discounted expected marginal utility of tomorrow’s payoff plus any certainty-equivalent normaliza......Login to view full explanationLog in for full answers
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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 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 ) = 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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