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
2
𝑐
𝑡
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
)
=
𝑐
𝑡
−
𝛼
2
𝑐
𝑡
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 to keep the relationships clear. The investor chooses θ to buy at price p_t in period t, yielding c_t = e_t − p_t θ in the first period. In period t+1, consumption is c_{t+1} = e_{t+1} + θ x_{t+1}, where x_{t+1} = p_{t+1} + d_{t+1}. The objective is max_θ [ u(c_t) + β E_t u(c_{t+1}) ], with u(c) = c − (α/2) c^2 so u'(c) = 1 − α c.

Step 1: Compute the first-order condition with respect to θ.
- The first-period consumption c_t depends on θ as c_t = e_t − p_t θ, hence ∂c_t/∂θ = − p_t.
- The second-period consumption c_{t+1} depends on θ as c_{t+1} = e_{t+1} + θ x_{t+1}, hence ∂c_{t+1}/∂θ = x_{t+1}.
- Differentiate the objective:
  ∂/∂θ [ u(c_t) + β E_t u(c_{t+1}) ] = u'(c_t) (− p_t) + β E_t [ u'(c_{t+1}) x_{t+1} ].

Step 2: Set the derivative to zero for the FOC.
-0 = − p_t u'(c_t) + β E_t [ x_{t+1} u'(c_{t+1}) ].
- Rearranging gives p_t = β E_t [ x_{t+1} u'(c_{t+1}) ] / u'(c_t).

Step 3: Substitute the specific form of the derivative u'(c) = 1 − α c.
- Thus the equation becomes p_t = β E_t [ x_{t+1} (1 − α c_{t+1}) ] / (1 − α c_t).

Step 4: Express c_t and c_{t+1} in terms of observable/choice variables.
- c_t = e_t − p_t θ.
- c_{t+1} = e_{t+1} + θ x_{t+1}.
- Therefore the right-hand side involves θ through c_t and c_{t+1}, making the equation implicit in p_t and θ together: p_t = β E_t [ x_{t+1} (1 − α (e_{t+1} + θ x_{t+1})) ] / (1 − α (e_t − p_t θ)).

Step 5: Compare with each provided option.
- Option 1: p_t = E_t [ β (1 − α c_t + 1 1 − α c_t) x_t + 1 ]. This expression mixes constants in a way that does not align with the derived structure p_t = β E_t [ x_{t+1} (1 − α c_{t+1}) ] / (1 − α c_t). In particular, the term inside the expectation does not reflect the correct dependence on c_{t+1} or the division by (1 − α c_t). Overall, it is not consistent with the FOC we derived.
- Option 2: p_t = E_t [ β (α c_t^2) x_t + 1 ]. Here the presence of α c_t^2 multiplied by x_t and added constant 1 inside the expectation bears little resemblance to the correct form β E_t [ x_{t+1} (1 − α c_{t+1}) ] divided by (1 − α c_t). The quadratic dependence on c_t and the missing denominator structure indicate this is incorrect.
- Option 3: p_t = E_t [ β (c_t + 1/(2α)) x_t + 1 ]. This mixes c_t with a constant offset and multiplies by x_t, but the correct expression involves (1 − α c_{t+1}) inside the expectation and a division by (1 − α c_t); the proposed form ignores the c_{t+1} inside the multiplier and lacks the proper normalization by u'(c_t). It is thus incorrect.
- Option 4: p_t = E_t [ β (c_t + 1/c_t) x_t − α x_t + 1 ]. This form introduces an odd combination of c_t and 1/c_t, plus a linear term in x_t, which does not match the derived FOC structure with u'(c) factors and the x_{t+1} term. The algebraic pattern is inconsistent with p_t = β E_t [ x_{t+1} (1 − α c_{t+1}) ] / (1 − α c_t).
- Option 5: p_t = E_t [ β c_t + (1/c_t) x_t + 1 ]. This option again fails to reproduce the necessary form that includes the (1 − α c_{t+1}) term inside the expectation and the division by (1 − α c_t). The placement of c_t and x_t inside a sum, rather than in a product with the appropriate u'(·) terms, makes it incorrect.

Step 6: Synthesize what we know.
- The mathematically correct expression from the standard two-period Merton-style optimization is p_t = β E_t [ x_{t+1} (1 − α c_{t+1}) ] / (1 − α c_t), with c_t = e_t − p_t θ and c_{t+1} = e_{t+1} + θ x_{t+1}. This captures the marginal utility of wealth in both periods and the asset payoff weighted by its impact on future consumption.
- All given options either misplace the x-term, drop the division by (1 − α c_t), misstate the dependence on c_{t+1}, or otherwise fail to reflect the derivative structure we derived.

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