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

The question presents an intertemporal optimization with a two-period horizon, where the utility is u(c_t) = c_t^α (up to a normalization that is unspecified in the prompt) and the investor purchases θ shares at price p_t in period t, facing a payoff x_{t+1} = p_{t+1} + d_{t+1} in period t+1. The task asks for the equation for p_t as a function of tomorrow’s payoffs, given the specified u(c_t). The answer options show several algebraic forms for p_t that mix c_t, x_t, β, and α in various ways. Without reconstructing the entire derivation from scratch, we can nonetheless evaluate the options for internal consistency and alignment with the typical intertemporal Euler equation structure. The key idea in these problems is that the no-arbitrage/optimality condition ties the current price to the expected discounted marginal utility of tomorrow’s consumption, often leading to an expression like p_t = E_t[ β (u'(c_{t+1}) / u'(c_t)) x_{t+1} ], possibly with additional terms depending on the precise model (cash-on-hand, budget constraints, and the payoff composition). With a CRRA-like or power utility u(c) ∝ c^α (ignoring scaling constants), u'(c) ∝ c^{α-1}, so the ratio u'(c_{t+1})/u'(c_t) tends to (c_{t+1}/c_t)^{α-1}, which would appear in the price equation. Now, let’s examine the given options one by one.

Option 1: p_t = E_t[ β ( (c_t + 1) / c_t ) α − 1/x_t + 1 ]
- This form inexplicably places α as a multiplicative factor outside a ratio that already contains c_t; the presence of −1/x_t is suspicious since the price expression should depend on tomorrow’s payoffs, not inverses of tomorrow’s payoff quantity in a subtractive way that is dimensionally inconsistent with a discounted expected marginal utility term. Moreover, the structure "(c_t + 1)/c_t" suggests an additive +1 inside the consumption term, which is unusual unless there is an endowment or a specific normalization; the combination inside the expectation looks dimensionally and algebraically inconsistent with standard Euler-equation-derived prices. Overall, this option mixes terms in a way that typically would not arise from a clean no-arbitrage condition.

Option 2: p_t = E_t[ β c_t + 1/c_t x_t + 1 ]
- This formulation puts β times c_t plus a term involving (1/c_t) x_t, which lacks the usual ratio of marginal utilities or a clear dependence on tomorrow’s consumption levels. The appearance of c_t without any consistency with marginal utilities and the separate additive terms (including a +1) disrupts the standard intertemporal linkage. Also, x_t is tomorrow’s payoff, so including it inside the current-period expectation without a proper transformation (e.g., via u'(c_{t+1})/u'(c_t)) is not aligned with the usual derivation. In short, the combination here seems ad hoc rather than following from the model’s Euler equation.

Option 3: p_t = E_t[ β ( (c_t + 1)/αα ) x_t + 1 ]
- This option introduces a double α (αα) in the denominator, which is unusual and likely a notational slip. The term (c_t + 1) divided by αα is unclear in meaning; moreover, it multiplies x_t directly, whereas the standard form would involve a ratio of marginal utilities or a function of c_t and c_{t+1}, not a simple product with x_t. The +1 inside the expectation is again a nonstandard additive constant. Overall, this option appears inconsistent with the typical Euler-type relation and contains ambiguous constants.

Option 4: p_t = E_t[ β ( (c_t αα) x_t + 1 ) ]
- Here the product c_t αα with x_t inside the expectation looks like a misplacement of the α parameters and a misunderstanding of how tomorrow’s payoff enters the pricing equation. There is no explicit division by c_t or a marginal utility ratio; the expression instead forms a simple linear combination that doesn’t reflect the intertemporal trade-off between present and future consumption under the given utility form. The structure is overly simplistic relative to the fairly intricate intertemporal optimization described, making this option unlikely to be correct.

Option 5: p_t = E_t[ β ( (c_t + 1)/c_t ) 1 − α x_t + 1 ]
- This option resembles a variant of a discounting formula, but again the pattern contains a − α x_t term subtracted directly from the payoff term, which lacks a clear derivation from the Euler equation using u'(c) ∝ c^{α-1}. The presence of a standalone 1 inside the square brackets plus the −α x_t together imply a dimensionally inconsistent combination of terms. The −α x_t term is particularly problematic because the payoff x_t is tomorrow’s random variable, and the price expression would typically involve E_t of a function of x_{t+1} (or x_t under a one-period shift) scaled by marginal utilities, not a direct subtraction by a scaled x_t inside the expectation.

Across all options, a common issue is that none of the forms cleanly reproduces the standard intertemporal pricing condition that would tie p_t to the expected discounted marginal utility of tomorrow’s consumption, typically involving a ratio like u'(c_{t+1})/u'(c_t) multiplied by tomorrow’s payoff measure, and expressed as an expectation under the information available at time t. Moreover, the prompt’s stated u(c_t) = c_t^α (with or without a normalization) strongly suggests a CRRA-like structure, which would produce a clear marginal-utility-based pricing form, not the ad hoc algebra seen in these alternatives.

In light of these observations, none of the options appears to be a clean derivation from the described optimization problem under the implied utility specification. The most plausible route to a correct form would require a precise derivation where p_t equals E_t[ β (u'(c_{t+1}) / u'(c_t)) x_{t+1} ], with u'(c) ∝ c^{α-1} for u(c) ∝ c^α (up to constants). If we enforce that structure, the resulting expression would involve a ratio of future to current marginal utilities times tomorrow’s payoff, rather than the inconsistent combinations shown here.

If you must choose among these options under exam conditions, you would look for the form that at least preserves the idea of an expected discounted future marginal utility weighting tomorrow’s payoff and avoids nonsensical additive constants or inverses of tomorrow’s payoff. However, given the explicit terms in these options, a confident selection cannot be made as correct without additional clarification or a corrected derivation in the prompt.

BU.232.630.W4.SP25 Quiz 2

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