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 begin by restating the problem setup and then examine each proposed price equation for_pt_t_ in light of the stated utility function u(c) = c^α (with α > 0, presumably). The investor chooses θ in period t, paying p_t θ, so in period t she has consumption c_t = e_t − p_t θ. In period t+1 she receives x_{t+1} = p_{t+1} + d_{t+1} (the payoff from the asset) and her consumption in period t+1 is c_{t+1} = e_{t+1} + x_{t+1} (assuming she does not sell the asset at t+1 for another price). The objective is max_θ [ u(c_t) + β E_t u(c_{t+1}) ]. The asset price p_t should be consistent with the intertemporal tradeoff implied by this optimization, typically tying p_t to the expected discounted marginal utility in period t+1 times the payoff x_{t+1}, i.e., a form like p_t = E_t [ β u'(c_{t+1}) x_{t+1} / u'(c_t) ] in a standard, smooth-utility, perfect-hedge setting. Because the options as given in the multiple-choice list mix various combinations of c_t, x_t, and constants in ways that do not align with this standard structure, each option should be carefully checked for dimensional consistency and for whether it uses x_t (the current-period payoff) versus x_{t+1} (the future payoff), and for whether the utility derivatives appear in the correct way to reflect the Euler equation (which in this discrete-time, CRRA-like form would involve powers of c_t and c_{t+1} in a ratio, multiplied by the payoff x_{t+1}). With that framework in mind, we analyze each option one by one.

Option 1: p_t = E_t[ β ( c_t α α ) x_t + 1 ]
- This candidate uses c_t with a malformed exponent expression “α α” which is ambiguous and does not match the u'(c) form or any clean interpretation from the utility specification. It also uses x_t (the current-period payoff) rather than the future payoff x_{t+1}, which is inconsistent with the intertemporal pricing that would tie today's price to the expected future payoff. The addition of +1 inside the expectation is also not aligned with a standard present-value relation in this setup. Overall, this option misplaces the marginal-utility weighting, misuses the payoff term, and introduces an unclear exponent structure, making it incorrect.

Option 2: p_t = E_t[ β ( c_t + 1 c_t ) 1 − α x_t + 1 ]
- This expression contains a clearly broken form “c_t + 1 c_t” which makes no sense (likely a formatting issue, but as written it is not dimensionally coherent or interpretable). It also uses x_t (the current-period payoff) and includes a “1 − α x_t” term that does not reflect any standard Euler-like relation between prices, marginal utility, and payoffs. The presence of a scalar 1 and the subtraction of α x_t inside the bracket without a clear normalization or division by u'(c_t) further signals that this option cannot be correct in the intended model.

Option 3: p_t = E_t[ β c_t + 1 c_t x_t + 1 ]
- This form appears to mix terms with very odd structure: “β c_t” plus “1 c_t x_t” plus a constant 1. There is no explicit dependence on u'(c_{t+1}) or the marginal value of future consumption, and the term x_t is again the current-period payoff, not the future payoff. The combination lacks the necessary multiplicative relation between the payoff and the discounted marginal utility of future consumption, so it does not reflect the standard asset-pricing relation in this two-period problem. Hence it is incorrect.

Option 4: p_t = E_t[ β( c_t + 1 c_t ) α − 1 x_t + 1 ]
- This option includes a problematic composite: “β( c_t + 1 c_t ) α” which again suggests a malformed expression (the inner sum is ill-defined), followed by “− 1 x_t + 1” which subtracts the current payoff scaled by 1 and then adds 1. The overall form neither mirrors the typical Euler-condition-based price formula nor cleanly incorporates u'(c_{t+1})/u'(c_t) factors. The inconsistent use of c_t, x_t, and the weird placement of α and −1 indicate this is not the correct intertemporal pricing equation.

Option 5: p_t = E_t[ β( c_t + 1 α α ) x_t + 1 ]
- This option once again contains an unclear “1 α α” term inside the parentheses, which does not correspond to a meaningful function of c_t under u(c) = c^α. It uses x_t (the current-period payoff) rather than x_{t+1}, and the structure does not reflect the required weighting by the marginal utility of future consumption. The presence of +1 inside the expectation is not consistent with the standard discounted, marginal-utility-weighted payoff relationship. Therefore, this option is also not correct.

In summary, each candidate option presents structural issues: mismatched use of current vs. future payoffs (x_t vs x_{t+1}), unclear or invalid exponent/power notation on c_t, and missing or improperly placed marginal-utility terms that would arise from the Euler equation in a CRRA-like two-period setting. The correct form in this model would typically tie p_t to an expected, β-weighted ratio of marginal utilities evaluated at c_{t+1} and c_t, multiplied by the payoff that materializes in period t+1 (i.e., involving x_{t+1} rather than x_t) and adjusted to reflect the asset’s payoff x_{t+1} directly if the asset is a single-period holding-to-maturity that pays x_{t+1} in period t+1. None of the provided options neatly encapsulates that structure; each one contains inconsistencies that render it incorrect.

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