= Solution
<Concavity> gives the supporting-tangent inequality
$$
U(\hat c_t)-U(c_t)\leq U'(c_t)(\hat c_t-c_t).
$$
Multiply by the <discount factor> $e^{-bt}$ and use the first-order condition $e^{-bt}U'(c_t)=Y_t$. This gives the pointwise <marginal-utility verification of optimal consumption> inequality
$$
e^{-bt}[U(\hat c_t)-U(c_t)]\leq Y_t(\hat c_t-c_t).
$$
Apply the budget inequality from part (d) to the competing admissible strategy, and use equality for the proposed one:
$$
\mathbb E\int_0^\infty Y_t\hat c_tdt\leq Y_0X_0
=\mathbb E\int_0^\infty Y_tc_tdt.
$$
The two weighted <consumption> integrals are finite, so their difference is <integrable>. Under the usual positive discount-rate assumption $b>0$, the utility integrals are also <integrable>: $U(0)\leq U(x)\leq L$ for a finite upper bound $L$, and $\int_0^\infty e^{-bt}dt=1/b$. Integrating the tangent inequality and taking <expectations> therefore gives
$$
\boxed{\mathbb E\int_0^\infty e^{-bt}U(c_t)dt
\geq\mathbb E\int_0^\infty e^{-bt}U(\hat c_t)dt.}
$$
Economically, both consumers face the same state-price budget, and the candidate spends it exactly where its discounted marginal utility equals the state price. This is <utility duality with martingale deflators> in its <consumption> form.
A positive $b$, or another hypothesis making the infinite-horizon objectives well defined and permitting this integration, is needed. The printed question does not specify the sign of $b$. Bounded utility alone does not ensure that an undiscounted infinite time integral exists: a bounded integrand can have both infinite positive and negative parts. Thus the conclusion is established under the standard discount convention $b>0$, and also whenever the displayed objectives satisfy the stated <integrability> conditions; without either convention the literal infinite-horizon comparison need not be a defined mathematical expression.
This can occur within an admissible financial model, not just for an abstract bounded integrand. Take $U(x)=1-2e^{-x}$ and $b=0$. Choose a deterministic smooth nonnegative $c$ with successive unit-length plateaus alternating between zero and the integer $n$, and connect them over intervals whose lengths have finite sum. Then $\int_0^\infty2c_te^{-c_t}dt<\infty$: the high-plateau contributions sum to $\sum_{n\geq1}2ne^{-n}$ and the transition integrand is bounded by $2/e$. Set
$$
Y_t=2e^{-c_t},\qquad B_t=Y_0/Y_t,\qquad
X_t=\frac1{Y_t}\int_t^\infty Y_sc_sds.
$$
The bank account is a positive deterministic <Itô process> and $Y_tB_t=Y_0$, so $Y$ is a <state-price density>. Holding $X_t/B_t$ bank units finances consumption with nonnegative wealth and exact budget equality. Also $U'(c_t)=Y_t$. Nevertheless every zero plateau contributes $-1$ to the utility integral, while each sufficiently high plateau contributes a fixed positive amount. Both its negative and positive parts are infinite, so the undiscounted objective is undefined. This establishes why the missing discount or objective-integrability hypothesis is substantive.
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