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?

Question

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?

BlackTom AI · Accepted Answer

We start by restating the problem setup and the given answer choices in a clear way to compare them.
- The investor chooses θ in period t to maximize the intertemporal utility: u(c_t) + β E_t[u(c_{t+1})], where c_t = e_t − p_t θ and c_{t+1} = e_{t+1} + θ x_{t+1}.
- The utility function is quadratic: u(c) = (1/2)(c − α)^2.
- The payoff in the second period is x_{t+1} = p_{t+1} + d_{t+1} and the price paid for the asset in period t is p_t.
- The question asks for the equation of the asset price p_t in period t as a function of tomorrow’s payoffs, given the stated setup.

Now, I’ll analyze each answer choice one by one, explaining why it is or isn’t consistent with the model and the standard intertemporal optimization with a quadratic utility.

Option A: p_t = E_t [ β ( c_t + 1 − α c_t − α ) x_t + 1 ]
- To assess this, note that an asset price in period t should reflect the trade-off between today’s consumption and the expected discounted marginal utility of tomorrow’s consumption, given the optimization over θ. In a quadratic utility, marginal utility is u'(c) = c − α. The interior solution (FOC) typically ties p_t to the expected discounted payoff times a term that involves the change in marginal utility with respect to consumption, evaluated at the two periods. The expression inside the expectation here, ( c_t + 1 − α c_t − α ), corresponds to (c_t + 1) − α(c_t + 1), which equals (1 − α)(c_t + 1). However, the appearance of x_t (as opposed to x_{t+1} or a direct payoff structure like p_{t+1} + d_{t+1}) and the standalone +1 term inside the product are not aligned with the standard Euler equation for this setup. The presence of x_t rather than the tomorrow’s payoff x_{t+1} or an explicit link to p_{t+1} + d_{t+1} makes this form suspect as a correct pricing equation given the problem's notation. Overall, this option blends the consumption terms with a payoff variable in a way that does not cleanly reflect the t-to-(t+1) intertemporal trade-off implied by the problem’s structure.

Option B: p_t = E_t [ β ( c_t + 1/2 α ) x_t + 1 ]
- This option introduces 1/2 α inside the bracket with c_t, but then multiplies by x_t. The 1/2 α is not dimensionally consistent with the derivative u'(c) = c − α, which would involve c_t − α, not c_t plus a constant scaled by α. Moreover, the payoff term is again x_t rather than the tomorrow’s payoff x_{t+1} or its direct monetary component p_{t+1} + d_{t+1}. The combination here does not align with the standard characteristic of an asset pricing equation under a quadratic utility, which should feature a linear dependence on (c_t − α) and the stochastic payoff in period t+1. Therefore, this form is not a correct derivation from the model.

Option C: p_t = E_t [ β c_t + 1 c_t x_t + 1 ]
- This option has a confusing structure: it mixes terms like β c_t, a separate 1 c_t x_t (which would be c_t x_t), and a constant 1. The asset price in period t should reflect the expected discounted marginal benefit of shifting consumption today into tomorrow via θ, which—under u(c) = (1/2)(c − α)^2—would involve (c_t − α) and the tomorrow’s payoff. A product like c_t x_t inside the expectation without a clear link to the tomorrow’s marginal utility is not consistent with the standard Euler-type relationship that prices today’s payoff with tomorrow’s uncertain payoff. In short, the mixed product structure here does not reflect the correct separation between present marginal utility and tomorrow’s payoff dynamics.

Option D: p_t = E_t [ β ( c_t + 1 − α c_t − α ) x_t + 1 ]
- This option is structurally identical to Option A up to a small rearrangement, but importantly still uses x_t (the current or per-period payoff variable) rather than x_{t+1} (the tomorrow’s payoff). The inner term ( c_t + 1 − α c_t − α ) simplifies to (1 − α)(c_t + 1). While this simplification makes the algebra marginally cleaner, the conceptual issue remains: the asset price in period t should reflect the intertemporal substitution between c_t and c_{t+1} through the tomorrow’s payoff, and the argument of the expectation should hinge on the tomorrow’s payoff and the correct marginal utility difference. Using x_t instead of x_{t+1} does not capture the correct dependence on tomorrow’s uncertain payoff in this model, and so this option cannot be the correct price expression derived from the stated setup.

Summary across options:
- All options attempt to link p_t to an expected discounted payoff times a function of current consumption c_t, but only a correct derivation would appropriately couple p_t to the tomorrow’s payoff variable (x_{t+1} or the random component of c_{t+1}) and include the marginal utility term (u'(c_t) or its equivalent) in a way consistent with the quadratic u. The given candidate in the problem statement uses a form β ( c_t + 1 − α c_t − α ) x_t + 1, which can be seen as β (1 − α)(c_t + 1) x_t + 1 after simplification, yet it still relies on x_t rather than the tomorrow’s payoff x_{t+1}, which is at odds with standard intertemporal pricing logic for this two-period model.

Concluding thought (without stating the final answer): the correct option should reflect the Euler-type intertemporal condition for p_t that ties today’s price to the expected tomorrow’s payoff and the change in marginal utility across periods, specifically involving (c_t − α) and the tomorrow’s payoff variable. The other options fail to correctly represent this structure in their treatment of c_t, α, and the payoff variable, either by misplacing the payoff term (using x_t instead of x_{t+1}), inserting inconsistent constants, or producing algebraic forms that do not match the u'(c) = c − α relationship inherent to the quadratic utility function.

View Explanation

Log in for full answers

Similar Questions

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler