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A competitive dividend economy specifies the joint law of total output, an information filtration, utility functions, discount factors, initial ownership shares and trading restrictions. Its derived objects are a common ex-dividend price, individually optimal consumption and share holdings satisfying ${\sum }_j{C}_t^j=d_t$ and ${\sum }_j{\theta }_t^j=1$. Each investor obeys the discrete dividend budget equation and, at an unconstrained interior optimum, marginal utility pricing in a dividend economy. A common price is required; a unique common state-price density or arbitrary contingent transfers require extra market completeness or implementability hypotheses.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

A consumption satisfaction stock smooths past consumption by ${\xi }_t=e^{-\lambda t}{\xi }_0+{\int }_0^t{e}^{-\lambda (t-s)}{c}_{s}ds$. Unlike multiplicative habit utility, the running utility function may depend on this stock alone. A unit of present consumption raises future consumption satisfaction while costing one unit of financial portfolio wealth.

A self-financing investor with consumption satisfies $X=H\cdot P$ and $dX=H\cdot dP-cdt$. The Itô product rule and $d[Y,X]=H\cdot d[Y,P]$ give the displayed equation. For a positive local martingale deflator $Y$, the first term is a stochastic integral against the vector of deflated asset-price local martingales, not their current levels.

When a productive stock pays $d_t$ before trading at ex-dividend price $S_t$, an investor entering with ${\theta }_t$ shares receives $d_t{\theta }_t$ and sells or buys shares to leave with ${\theta }_{t+1}$. The displayed budget constraint accounts for the available consumption and the cost of the new holding. Here ${\theta }_t$ is known before date $t$, while $C_t$ and ${\theta }_{t+1}$ may depend on the information revealed at that date.

Consider an investment-consumption problem with an independent one-way rate change of intensity $\lambda$ from $r_1$ to $r_0$. For a stock of constant drift $\mu$ and positive volatility $\sigma$, set $B_j=r_j+(\mu -r_j{)}^2/(2R{\sigma }^2)$, ${\gamma }_0=[\rho -(1-R)B_0]/R$, and $D_1=\rho +\lambda -(1-R)B_1$. In the finite-value regime ${\gamma }_0,D_1>0$, the prechange consumption coefficient is the unique positive root displayed above. The left side increases continuously from zero to infinity. Before the change, the value is ${\gamma }_1^{-R}{w}^{1-R}/(1-R)$, the stock fraction is $(\mu -r_1)/(R{\sigma }^2)$, and consumption is ${\gamma }_{1}w$. At the change these become the ordinary Merton consumption-investment problem controls for $r_0$. The jump term in the Hamilton-Jacobi-Bellman equation is $\lambda (V_0-V_1)$; knowing only the marginal law of the change time does not justify that term without a conditional intensity assumption.

With positive welfare weights $a_j$, maximize ${\sum }_j{a}_j{\beta }_j^t[-e^{-{\gamma }_j{c}_j}/{\gamma }_j]$ subject to ${\sum }_j{c}_j=d$. Interior first-order conditions give $a_j{\beta }_j^t{e}^{-{\gamma }_j{c}_j}=\nu$. Put $\Gamma =({\sum }_j{\gamma }_j^{-1}{)}^{-1}$, $\log \bar{\beta }=\Gamma {\sum }_j{\gamma }_j^{-1}\log {\beta }_j$ and $\log A=\Gamma {\sum }_j{\gamma }_j^{-1}\log a_j$. ThenIf consumption must be nonnegative, the interior formula is replaced by $c_j=\max \{0,(\log a_j+t\log {\beta }_j-\log \nu )/{\gamma }_j\}$, with $\nu$ determined by resource clearing. These are Pareto allocations; realizing them as a competitive equilibrium with one productive asset requires feasible financing. For common ${\beta }_j=\beta$ and initial ownership $\Gamma /{\gamma }_j$, constant holdings and $C_t^j=(\Gamma /{\gamma }_j)d_t$ give such an equilibrium whenever fundamental prices and the utility sums are finite.

For terminal logarithmic utility $a\log V_T$ and running logarithmic utility $b\log c_t$, with $a,b,w_0>0$, constant Black-Scholes model coefficients give $\lambda =(a+bT)/w_0$ and $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$, where $\xi$ is the state-price density. The optimal terminal wealth and consumption are $a/(\lambda {\xi }_T)$ and $b/(\lambda {\xi }_t)$. Their discounted budget adds to $(a+bT)/\lambda =w_0$. The concave supporting-tangent inequality, integrated over time, proves optimality; the Itô formula gives the displayed dollar stock holding. The time-dependent consumption-to-wealth ratio reflects both remaining consumption opportunities and the terminal utility weight.

For constant relative risk aversion utility with parameter $\gamma >0$, terminal weight $\eta >0$ and utility discount rate $\delta$, in a Black-Scholes model put $\lambda =(\mu -r)/\sigma$, $k=[\delta +(\gamma -1)(r+{\lambda }^2/(2\gamma ))]/\gamma$ and $b(\tau )={\int }_0^{\tau }{e}^{-ks}ds+{\eta }^{1/\gamma }{e}^{-k\tau }$. The displayed optimal consumption and dollar stock position follow by inverse marginal utility and the state-price budget constraint. For $\gamma \ne 1$ the value is $b(T-t{)}^{\gamma }{W}_t^{1-\gamma }/(1-\gamma )$ with discount measured from $t$. Finite horizons do not require $k>0$. The concave supporting-tangent inequality and replication of the optimal budget prove global optimality, rather than only a formal differential-equation solution.

If consumption contributes $c(V_{\xi }-V_w)$ to the Hamilton-Jacobi-Bellman equation and has no finite upper bound, finite value requires $V_{\xi }\le V_w$. Strict inequality makes zero consumption optimal locally. Equality describes an active transfer boundary, often reached through singular consumption control.

For $0<R<1$, nonnegative consumption with state-price budget $\mathbb{E}\int Z_t{c}_{t}dt\le x$ satisfies this bound under CRRA utility. Apply Holder inequality on probability-times-time measure to $(Zc{)}^{1-R}$ and $e^{-bt}{Z}^{R-1}$, with conjugate exponents $1/(1-R)$ and $1/R$. If the price integral $D$ is finite, equality forces full budget use and $c_t=(x/D)e^{-bt/R}{Z}_t^{-1/R}$. Financing this process is a separate feasibility issue, so the equality pattern alone does not assert attainability in an incomplete market.

The additional utility function obtained per additional unit of consumption or portfolio wealth. For a differentiable utility function it is its derivative; for a concave utility function it is nonincreasing. In a complete-market allocation, equating marginal utility to a multiplier times the state-price density compares the utility benefit of a payoff with its financing cost. The concave supporting-tangent inequality makes this comparison a global optimality certificate rather than only a local first-order condition.

For concave increasing utility, the concave supporting-tangent inequality gives $e^{-bt}[U({\hat{c}}_t)-U(c_t)]\le Y_t({\hat{c}}_t-c_t)$. If the proposed consumption exhausts the state-price budget and every competitor has no larger budget, integration proves optimality. The objective must be well defined; a positive discount rate and finite utility bounds are a convenient sufficient condition.

In the infinite-horizon Merton consumption-investment problem with constant relative risk aversion utility, this is the optimal ratio of consumption to wealth. A finite value requires ${\gamma }_M>0$. The state-price budget constraint then fixes consumption's multiplier because $\mathbb{E}[{\xi }_t(y{e}^{\rho t}{\xi }_t{)}^{-1/R}]=y^{-1/R}{e}^{-{\gamma }_{M}t}$, whose time integral is $y^{-1/R}/{\gamma }_M$. When this integral diverges, the value is $+\infty$ for $0<R<1$ and $-\infty$ for $R>1$.

The complete market makes every state-contingent consumption vector attainable. Its budget is ${\sum }_s{p}_s{c}_s=100$, and expected logarithmic utility under the equally likely states is $\frac{1}{4}{\sum }_s\log c_s$, with $c_s>0$. Increasing utility makes the budget bind. For a Lagrange multiplier $\eta$, the first-order conditions areSubstitution in the budget gives $1/\eta =100$, so $\eta =0.01$ andEvery state receives an expenditure $p_s{c}_s=25$, rather than equal consumption. Strict concavity of logarithmic utility makes this feasible stationary point the unique global maximizer. The physical probabilities enter the utility objective; the supporting prices enter the budget, and they must not be interchanged.

The current price of one unit of certain future consumption is the sum of the Arrow state prices, namely $0.9$. A unit-price risk-free asset yielding gross return $R$ must satisfy $0.9R=1$. HenceNormalize the Arrow state prices by their sum, or multiply them by $R$, to obtain the risk-neutral probabilities:They are positive and sum to one. Any terminal payoff $Y$ has current price ${\sum }_s{p}_s{Y}_s=R^{-1}{\mathbb{E}}_{Q}Y$. In particular any asset with current price one has ${\mathbb{E}}_Q{r}_i=R$. These pricing probabilities need not equal the physical probabilities governing the investor's utility.

Use the physical dynamics of the Black-Scholes model, $d{S}_t/S_t=\mu dt+\sigma d{W}_t$, with bank rate $\rho$ and $\sigma >0$. Write $\lambda =(\mu -\rho )/\sigma$ for the market price of risk. If $w_t$ is portfolio wealth, ${\pi }_t$ the dollar amount in stock, and $c_t\ge 0$ the consumption rate, the budget dynamics areAdmissible controls are progressively measurable, locally integrable in the required drift and diffusion norms, and keep $w_t\ge 0$. Such a solvency restriction excludes doubling-type strategies. In a complete Brownian market, the investment-consumption problem is to maximizewhere the utility functions are increasing and concave. A terminal-only problem omits the integral. Strict concavity gives uniqueness of optimal consumption and terminal wealth when an optimum exists.

The state-price density isThe Itô formula givesThus $H_t{w}_t+{\int }_0^t{H}_s{c}_{s}ds$ is a nonnegative local martingale, hence a supermartingale, yielding the state-price budget constraintConversely, a nonnegative consumption plan and terminal payoff $\xi$ satisfying budget equality and the necessary integrability can be financed in the Brownian complete market. DefineThe Brownian martingale representation theorem for the total future budget supplies a stochastic-integral coefficient; comparing it with $H_t(\sigma {\pi }_t-\lambda w_t)$ defines ${\pi }_t$. This reconstructs the wealth equation with $w_t\ge 0$. Localization gives the representation for integrable positive budgets when an $L^2$ assumption is not available.

For smooth strictly concave utilities with the usual marginal-utility range, let $I_u=(u^{\prime }{)}^{-1}$ and $I_U=(U^{\prime }{)}^{-1}$ be the inverse marginal utility functions. The Lagrange-multiplier solution iswhere $y>0$ makes the budget an equality. This is a global argument, not merely formal differentiation: the concave supporting-tangent inequality giveswith the analogous terminal inequality. Taking expectations, integrating, and applying the budget bound makes the difference in objectives nonpositive. Replication of the candidate budget proves attainability and optimality whenever these expectations are well-defined and finite. If the marginal-utility ranges or integrability conditions fail, existence must be established separately rather than inferred from the first-order equations.

A detailed example is constant relative risk aversion utility $u(c)=c^{1-\gamma }/(1-\gamma )$ and $U(w)=\eta w^{1-\gamma }/(1-\gamma )$, with $\gamma >0$, $\gamma \ne 1$, $\eta >0$, and $\delta \ge 0$. PutThe optimal plans from the marginal-utility equations are $c_t^{\ast }=(y{e}^{\delta t}{H}_t{)}^{-1/\gamma }$ and ${\xi }^{\ast }={\eta }^{1/\gamma }(y{e}^{\delta T}{H}_T{)}^{-1/\gamma }$. The normal distribution of $W_t$ gives, for $a=1-1/\gamma$,The budget therefore becomes $y^{-1/\gamma }b(T)=w_0$, determining the multiplier explicitly. Independent Brownian increments in the conditional budget formula give $w_t^{\ast }=c_t^{\ast }b(T-t)$. The diffusion coefficient of $c_t^{\ast }$ is $(\lambda /\gamma )c_t^{\ast }$, so matching the wealth diffusion determines the dollar stock holding. Consequently the finite-horizon power-utility investment and consumption optimum isAll candidate moments are finite on a finite horizon, since powers of a lognormal state-price density have finite moments. The budget and tangent argument above therefore verify optimality in this example, including admissibility and positive wealth.

For another verification of the formulas, the Hamilton-Jacobi-Bellman equation for a value function with discount measured from the current time isSubstitute $V(w,t)=b(T-t{)}^{\gamma }{w}^{1-\gamma }/(1-\gamma )$. The two concave maximizations give the displayed controls, and the remaining equation is $b^{\prime }(\tau )=1-kb(\tau )$ with $b(0)={\eta }^{1/\gamma }$, exactly solved by the formula above. ThusFor $k=0$, interpret the integral as $\tau$, so $b(\tau )=\tau +{\eta }^{1/\gamma }$. Unlike the infinite-horizon Merton consumption-investment problem, finite horizons do not require $k>0$. Consumption vanishes in the terminal-only formulation; then the integral term in $b$ is absent and the optimal stock fraction remains $(\mu -\rho )/(\gamma {\sigma }^2)$. Logarithmic utility is handled directly with inverse marginal utility $1/z$ and the same wealth-to-consumption formula at $\gamma =1$, with $k=\delta$.

Write $T=t_0$, and specify the physical Black-Scholes model as $d{S}_t/S_t=\mu dt+\sigma d{W}_t$, with constant $\sigma >0$ and bank rate $\rho$. The physical drift $\mu$ is needed for an expected utility calculation; a pricing model specified only under a risk-neutral measure does not determine the investor's preferences over risky positions. PutHere $\vartheta$ is the market price of risk, $Z_T$ is the density of the risk-neutral measure $Q$, and $\xi$ is the state-price density. The Girsanov theorem makes $W_t^Q=W_t+\vartheta t$ a Brownian motion under $Q$.

For a dollar holding ${\pi }_t$ in the stock and consumption rate $c_t$, the portfolio wealth equation isSince $d{\xi }_t=-\rho {\xi }_{t}dt-\vartheta {\xi }_{t}d{W}_t$, integration by parts gives the useful deflated wealth equation with consumptionFor strategies with a true deflated gain martingale, terminal wealth $X=V_T$ and consumption obey the state-price budget constraintFor nonnegative wealth and consumption, the admissibility condition can instead give a budget inequality; increasing utilities use the full budget at an optimum. Conversely, suitable integrable payoff and consumption plans with this budget are financeable in the Brownian complete market. To construct the strategy, setBy the Brownian martingale representation theorem, write $d{M}_t=H_{t}d{W}_t$. The required dollar stock holding is ${\pi }_t=(H_t/{\xi }_t+\vartheta V_t)/\sigma$; substitution in the deflated gain equation verifies financing. Remaining wealth goes into the bank account. For square-integrable data the representation holds directly, and localization handles the customary integrable admissible data.

First take no consumption. For an increasing strictly concave differentiable utility function whose marginal derivative can be inverted, let $I_u=(u^{\prime }{)}^{-1}$. Choose $\lambda >0$ so thatThis candidate is not merely a first-order guess. The concave supporting-tangent inequality gives, pointwise,Taking expectations and using the budget proves $\mathbb{E}u(X)\le \mathbb{E}u(X^{\ast })$ for every feasible competitor. Finance the optimizer by its risk-neutral pricing value $V_t=e^{-\rho (T-t)}{\mathbb{E}}_Q[X^{\ast }\mid {\mathcal{F}}_t]$, using the preceding representation to find the holdings.

For a merely concave differentiable utility function, the general formulation is to select $X^{\ast }(\omega )$ from the maximizers of $u(x)-\lambda {\xi }_T(\omega )x$ over its wealth domain, and choose the multiplier to satisfy the budget. If increasing utility is not assumed, the equality-budget multiplier may have either sign; the positive-multiplier inverse formula above is for increasing utility. Flat portions allow multiple maximizers; boundaries require one-sided inequalities rather than equality of marginal derivatives. Existence of a multiplier, integrability and an admissible maximizer must be checked. Concavity alone does not guarantee a well-posed problem: with unrestricted dollar positions, linear utility $u(x)=x$ and $\mu >\rho$ permits unbounded expected terminal wealth. The explicit utilities below satisfy the needed conditions in their stated domains.

For exponential utility $u(x)=(1-e^{-ax})/a$ on $\mathbb{R}$, $u^{\prime }(x)=e^{-ax}$ and $I_u(y)=-\log y/a$. ThereforeThe budget is equivalently ${\mathbb{E}}_Q{X}^{\ast }=w_0{e}^{\rho T}$, since ${\xi }_T=e^{-\rho T}dQ/dP$. Under $Q$, ${\mathbb{E}}_Q{W}_T=-\vartheta T$, giving $\log \lambda =\rho T-{\vartheta }^{2}T/2-a{w}_0{e}^{\rho T}$. HenceMatching the Brownian coefficient of $dV$ with $\sigma {\pi }_t$ gives the finite-horizon exponential-utility portfolioHold ${\pi }_t^{\ast }/S_t$ shares and invest $V_t-{\pi }_t^{\ast }$ in the bank account. This is a dollar amount, not a fixed wealth fraction. The Gaussian exponential formula also gives the optimal expected utilityThis calculation uses unrestricted wealth and an admissibility class permitting the displayed Gaussian terminal wealth, with integrable deflated gains and the relevant exponential moment. A nonnegative-wealth constraint would change the optimizer; no such constraint is imposed on this exponential-utility example.

For both terminal and running utility function, choose $X$ and a nonnegative adapted consumption rate $c$ to maximize ${\mathbb{E}}_P[u(X)+{\int }_0^{T}v(c_t)dt]$ subject to the combined budget. Pointwise maximization of $u(x)-\lambda {\xi }_{T}x$ and $v(c)-\lambda {\xi }_{t}c$ gives, for interior strictly concave cases,Apply the concave supporting-tangent inequality to $u$ and to $v$ at every time, then integrate and take expectations. The difference in objectives is at most $\lambda$ times the difference in budget expenditures, which is nonpositive. This proves optimality, not just the marginal conditions. The financing value isThe question's consumption rate $R_t$ is denoted $c_t$ here to keep it distinct from interest-rate notation.

For logarithmic utility with $u(x)=a\log x$, $v(x)=b\log x$, assume $a,b,w_0>0$ and positive wealth and consumption. The marginal conditions and budget giveEach state-price-weighted payment is deterministic, so conditional valuation immediately gives $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$. Put $A_t=a+b(T-t)$. Since the Itô formula gives $d(1/{\xi }_t)=(1/{\xi }_t)[(\rho +{\vartheta }^2)dt+\vartheta d{W}_t]$, this value satisfiesComparison with the wealth equation verifies the complete finite-horizon logarithmic investment and consumption strategy:Thus the optimal share holding is ${\pi }_t^{\ast }/S_t$, and the remaining value $V_t-{\pi }_t^{\ast }$ is in the bank account; borrowing or shorting is allowed if that value or the stock holding is negative. Terminal wealth is positive and consumption is positive throughout. Finally $\mathbb{E}\log {\xi }_t=-(\rho +{\vartheta }^2/2)t$, so the optimal objective, if desired explicitly, isFinite-horizon lognormal moments ensure the budget and expected utilities are finite, completing the feasibility and optimality checks.

The equalities are valid at a differentiable interior optimum under the intended concave formulation, but not under the printed assumptions alone. To derive the intended result, differentiate the finite conditional sum in the Bellman objective. For an interior portfolio fraction with positive invested wealth, its derivative givesThe consumption derivative gives $U^{\prime }(c_n^{\ast })=E_n[V_{n+1}^{\prime }(w_{n+1}^{\ast })G_{{\theta }_n^{\ast }}]$. Since $G_{\theta }=r+\theta (X-r)$, the first equality cancels its risky part and yields the second requested equality. The envelope derivative with respect to wealth gives $V_n^{\prime }(w_n^{\ast })=E_n[V_{n+1}^{\prime }(w_{n+1}^{\ast })G_{{\theta }_n^{\ast }}]$, giving the third. Thus, under these added hypotheses,The first prices the excess stock payoff using future marginal utility; the second equates marginal utility from consuming now to that from saving in the bank; the third identifies the marginal value of wealth with that common quantity. At portfolio or consumption boundaries these are replaced by one-sided optimality inequalities. Finiteness of the sample space justifies differentiating a finite sum; it does not ensure differentiability of a supremum or an interior optimum.

A concrete counterexample satisfies the printed convex assumptions and attains every supremum. Take one remaining decision, a one-point sample space, a deterministic stock return $X=x>r>1$, $U(c)=c^2$, and controls in $[0,w]\times [0,1]$. The optimum is $c^{\ast }=0,{\theta }^{\ast }=1$, since the objective $c^2+x^2(w-c{)}^2$ is convex in $c$ and its larger endpoint value is $x^2{w}^2$. Then $w_{n+1}^{\ast }=xw$ and $V_{n+1}^{\prime }=2xw$. The first claimed expectation is $2xw(x-r)>0$; the second compares $2rxw>0$ to $U^{\prime }(0)=0$; the third compares $2rxw$ to $V_n^{\prime }(w)=2x^{2}w$. All three fail. Consequently the requested equalities cannot be proved as printed; the derivation above supplies their precise intended interior version.

Here $U^{\prime }(x)=x^{-R}>0$ and $U^{\prime \prime }(x)=-R{x}^{-R-1}<0$: the specified utility is strictly concave, contrary to the word “convex” in part (i). Assume its expected values are finite on the admissible set. Define $G_{\theta }=r+\theta (X-r)$ and choose a constant portfolio fraction ${\theta }_{\ast }$ maximizing $E[U(G_{\theta })]$. Equivalently maximize $E[G_{\theta }^{1-R}]$ for $0<R<1$ and minimize it for $R>1$. An interior optimum satisfiesFor a constrained endpoint optimum the corresponding derivative has the appropriate one-sided sign. The objective is concave in $\theta$ and strictly concave if $P(X\ne r)>0$, so this condition or its endpoint version determines the optimum. Put $\kappa =E[G_{{\theta }_{\ast }}^{1-R}]$ and $\gamma ={\kappa }^{1/R}>0$.

Backward induction with independence of the returns gives $V_n(w)=A_n{w}^{1-R}/(1-R)$ and $A_N=1$. Maximizing the consumption expressiongives $c^{-R}=A_{n+1}\kappa (w-c{)}^{-R}$, henceWriting $b_n=A_n^{1/R}$ yields $b_N=1$ and $b_n=1+\gamma b_{n+1}$. Therefore the fully explicit horizon dependence isUse $b_n=(1-{\gamma }^{N-n+1})/(1-\gamma )$ when $\gamma \ne 1$, and $b_n=N-n+1$ when $\gamma =1$. The investor consumes this fraction, invests the remaining wealth with the same optimal portfolio mix at each date, and consumes all remaining wealth at $N$. If the stock is identical to the bank almost surely, every portfolio fraction is equivalent.