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)=
1

2
(ct−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 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)=
1

2
(ct−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 setup to anchor the analysis: the investor with utility u(c) = 1/2 (c − α)^2 chooses θ to maximize the present value of utility, given ct = et − p t θ and ct+1 = et+1 − p t θ + x t+1 − p t+1 θ? (Note: payoffs in period t+1 depend on the asset price and the payoff dt+1, with xt+1 = pt+1 + dt+1.) The question asks for the price p t as a function of tomorrow’s payoffs, derived from a first-order condition (FOC) in this quadratic utility setting.

Option A: pt = E t [ β ( ct+1 − α ) xt+1 / ( ct − α ) ]
- This form mixes ct+1 and ct in a ratio with α and multiplies by E t[β xt+1]. The structure resembles a ratio of marginal utilities evaluated at two consecutive periods, but a clean FOC for a quadratic utility typically yields a linear relationship in the payoff xt+1 when solving for pt. The division by (ct − α) in the denominator creates a potential division-by-zero issue and does not mirror the standard Euler equation form derived from the quadratic utility given. This option seems structurally off because the expected payoff xt+1 is being scaled by a ratio of marginal utilities that itself depends on ct and ct+1, which would complicate rather than linearize the price expression.

Option B: pt = E t [ β ( ct+1 / ct − α ) xt+1 ]
- Here, the term inside the expectation is β times a ratio ( ct+1 / ct ) minus α, all multiplying xt+1. The expression ct+1/ct is not the standard marginal-utility ratio arising in the Euler equation for a quadratic utility with additive separable consumption across periods. Including −α outside the ratio further disrupts the usual linear relationship with xt+1. This option seems dimensionally inconsistent with the typical asset-pricing Euler equation, and the exact placement of α does not align with the quadratic utility clouding the multiplier on xt+1. Overall, this option does not reflect the conventional derivation path.

Option C: pt = E t [ β ( ct+1 − α ) / ( ct − α ) · xt+1 ]
- This expression features a clean ratio of marginal utilities: ( ct+1 − α ) is proportional to marginal utility in period 2, and ( ct − α ) is proportional to marginal utility in period 1, with the quadratic u(c) leading to marginal utility proportional to ( c − α ). The Euler equation for a two-period model with certainty equivalent structure under a quadratic utility often yields p t equals the expected discounted payoff times the ratio of marginal utilities across periods, precisely like [ (ct+1 − α) / (ct − α) ] times xt+1, all multiplied by β and expectation E t. This matches the standard intuition: the price today equals the expected discounted payoff weighted by how much the marginal utility changes from today to tomorrow. The presence of xt+1 as the payoff component aligns with the second-period random payoff, and the β factor is the time discount. The structure is consistent with a first-order condition derived from max θ [ u(ct) + β E t u(ct+1) ] with u(c) = 1/2 (c − α)^2.

Option D: pt = E t [ β ct+1 × ct × xt+1 ]
- This is a highly nonlinear product of ct+1, ct, and xt+1, lacking the subtractive α terms that naturally arise in the marginal utilities for a quadratic utility function of the form (c − α)^2. It would imply a price that depends on the product of consumptions in both periods, which is not consistent with the Euler-equation derivation for this utility specification. Moreover, there is no division by ct or ct+1, which would be expected if we were expressing the ratio of marginal utilities (both containing ct − α and ct+1 − α). This option therefore misplaces the mathematical structure that stems from the FOC.

Option E: pt = E t [ β ( ct+1 − α )^2 / ( ct − α )^2 × xt+1 ]
- This option introduces squared marginal utilities, effectively squaring the ( ct − α ) terms. While the square could appear if one were dealing with a second-order condition or a more complex utility transformation, the standard linearization in the Euler equation for this quadratic utility does not yield squared ratios; it yields a simple ratio of first powers: ( ct+1 − α ) / ( ct − α ). The squared form exaggerates sensitivity to consumption levels and is not consistent with the straightforward derivative of u(c) = 1/2 ( c − α )^2 with respect to c. Hence this option misrepresents the derivative structure that would arise from the FOC.

Putting the pieces together, the option that matches the conventional Euler-equation derivation for this setup is the one that uses the marginal-utility ratio ( ct+1 − α ) / ( ct − α ) multiplied by xt+1 and scaled by β under the conditional expectation E t. This corresponds to Option C.

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