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?

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
)
=
𝑐
𝑡
𝛼
𝛼
 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 candidate answers to ensure clarity about what we are evaluating.
- The investor chooses theta in period t to maximize a two-period utility that depends on current consumption c_t and expected future utility from c_{t+1}.
- The utility function is u(c) = c^α, with α > 0 (and presumably less than 1 to reflect risk preferences, though the exact range isn’t specified here). The question asks for the equation that prices the asset in period t as a function of tomorrow’s payoffs.
- The answer options present candidate expressions for p_t as an efficiently expected discounted value that depends on ct, ct+1, x_t, etc., with β as the discount factor and the payoff structure for x_{t+1} given by x_{t+1} = p_t+1 + d_t+1 (i.e., the asset’s payoff depends on the price today plus a dividend).

Option 1: p_t = E_t[ β ( ct + 1/ct ) 1 − α x_t + 1 ]
- This expression mixes ct with a term 1/ct in a way that is not dimensionally consistent with the usual pricing of assets derived from the Euler equation under u(c) = c^α. It also subtracts α x_t without a properly specified aggregation of tomorrow’s payoff x_{t+1}, and the presence of ct + 1/ct multiplied by 1 is unclear and does not align with the standard structure from the intertemporal choice problem.
- Why it’s incorrect: the combination of ct, 1/ct, a simple −α x_t, and a +1 term does not reflect the appropriate marginal utility timing and the dependence on tomorrow’s payoff structure in the Bellman or Euler equation context. It lacks the explicit α and β interplay seen in the correct discrete-time asset pricing derivation.

Option 2: p_t = E_t[ β ct + 1/ct x_t + 1 ]
- This option presents a linear dependence on ct and an extraneous product of (1/ct) with x_t, which is dimensionally inconsistent with the standard intertemporal optimization that would involve ct and ct+1, not x_t directly multiplied by 1/ct. It also arbitrarily adds a +1 term that does not correspond to the normalized present-value pricing equation.
- Why it’s incorrect: it fails to correctly incorporate the future consumption level ct+1 or the stochastic payoff x_{t+1}; it uses x_t instead of x_{t+1} and misplaces the discounting through β in a way that contradicts the stated problem structure.

Option 3: p_t = E_t[ β ( ct α α ) x_t + 1 ]
- This option appears to distort the functional form by placing α-parameters in a way that produces ( ct α α ), which is not a standard operation; and it multiplies by x_t rather than incorporating the tomorrow’s payoff x_{t+1} and the asset’s role in balancing current and future utilities. The presence of a plain +1 addition again lacks a clear failed-to-price component of future consumption or payoff.
- Why it’s incorrect: the algebraic form is not consistent with the typical Euler equation for u(c) = c^α and the asset-pricing relation, and it uses x_t instead of x_{t+1} in a way that misrepresents the payoff linkage.

Option 4: p_t = E_t[ β ( ct + 1/ct ) α − 1/x_t + 1 ]
- This option keeps the key structural elements that align with the problem: a discounted (via β) intertemporal trade-off reflecting marginal utility, a term resembling (ct + 1/ct) raised to a power α, and a term involving the payoff metric with a negative contribution of 1/x_t plus a +1 offset, which can be interpreted as a normalization or payoff adjustment consistent with the given u(c) = c^α framework.
- Why it’s correct (in the context of the provided answer): the expression captures the idea that the asset price today should reflect the discounted expected marginal utility of consumption tomorrow, incorporating the specified u(c) form and the stochastic payoff x_{t+1} through its relationship to p_t+1 and dividends, in a way that mirrors the typical intertemporal Euler equation with a CRRA-like power α applied to ct and ct+1. The −1/x_t term can be seen as accounting for how tomorrow’s payoff interacts with today’s state through the asset’s payoff mechanism, consistent with the problem’s specified payoff structure. Overall, this option best matches the expected balance of current and future utilities under the given functional form.

Option 5: p_t = E_t[ β ( ct + 1/α α ) x_t + 1 ]
- Here, the term ( ct + 1/α α ) is ill-posed as written; 1/α α collapses to 1 if interpreted naively, which would reduce the expression to β( ct + 1 ) x_t + 1, a product of current consumption and the payoff x_t with discounting, which does not reflect the standard intertemporal decision structure or the specified u(c) = c^α form. It also uses x_t instead of x_{t+1} in a way that breaks the asset’s forward-looking pricing relation.
- Why it’s incorrect: the mis-specified algebra, improper use of x_t vs x_{t+1}, and the incorrect combination of terms with respect to the discounting and the utility power α make this option inconsistent with the model’s derivation.

Summary of evaluation: among the five options, only option 4 aligns with the problem’s use of u(c) = c^α, the intertemporal Bellman-like pricing structure, and the dependence on tomorrow’s payoff through the asset’s price in period t. All other alternatives involve improper indexing (using x_t instead of x_{t+1}), incorrect algebraic forms for combining ct, ct+1, and payoffs, or misplacement of the discounting and the α-power in ways that contradict the stated setup.

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