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THEOREM 3.3 Consider the investors problem in the binomial asset pricing world with $U(x) = \frac{1}{1-\gamma}x^{1-\gamma}$, for $\gamma > 0$. Then the optimal fraction of wealth in stocks and bonds are given by $k_s = \frac{(1+r)(P^{-1/\gamma} - 1)}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$ and $k_b = \frac{u - P^{-1/\gamma}d}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$ where $P = \frac{q(1+r-d)}{p(u-1-r)} = \frac{\tilde{p}\tilde{q}}{\tilde{p}\tilde{q}}$ where $\tilde{p} = \frac{1+r-d}{u-d}$, and $\tilde{q} = 1 - \tilde{p}$. The optimal amount of consumption in the up and down states are $c_u = \frac{(1+r)(u-d)P^{-1/\gamma}X_0}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$ and $c_d = \frac{(1+r)(u-d)X_0}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$. The Lagrangian multiplier is $\lambda = (p c_u^{-\gamma} + q c_d^{-\gamma})(1+r)X_0$. Exercise 11. Prove the above theorem.

          THEOREM 3.3 Consider the investors problem in the binomial asset pricing world with $U(x) = \frac{1}{1-\gamma}x^{1-\gamma}$, for $\gamma > 0$. Then the optimal fraction of wealth in stocks and bonds are given by
$k_s = \frac{(1+r)(P^{-1/\gamma} - 1)}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$ and $k_b = \frac{u - P^{-1/\gamma}d}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$
where $P = \frac{q(1+r-d)}{p(u-1-r)} = \frac{\tilde{p}\tilde{q}}{\tilde{p}\tilde{q}}$ where $\tilde{p} = \frac{1+r-d}{u-d}$, and $\tilde{q} = 1 - \tilde{p}$. The optimal amount of consumption in the up and down states are
$c_u = \frac{(1+r)(u-d)P^{-1/\gamma}X_0}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$ and $c_d = \frac{(1+r)(u-d)X_0}{u - (1+r) + P^{-1/\gamma}(1+r-d)}$.
The Lagrangian multiplier is
$\lambda = (p c_u^{-\gamma} + q c_d^{-\gamma})(1+r)X_0$.
Exercise 11. Prove the above theorem.

$THEOREM 3.3 Consider the investors problem in the binomial asset pricing world with U(x) = (1)/(1-γ)x^1-γ, for γ > 0. Then the optimal fraction of wealth in stocks and bonds are given by ks = ((1+r)(P^-1/γ - 1))/(u - (1+r) + P^-1/γ(1+r-d)) and kb = (u - P^-1/γd)/(u - (1+r) + P^-1/γ(1+r-d)) where P = (q(1+r-d))/(p(u-1-r)) = (p̃q̃)/(p̃q̃) where p̃ = (1+r-d)/(u-d), and q̃ = 1 - p̃. The optimal amount of consumption in the up and down states are cu = ((1+r)(u-d)P^-1/γX0)/(u - (1+r) + P^-1/γ(1+r-d)) and cd = ((1+r)(u-d)X0)/(u - (1+r) + P^-1/γ(1+r-d)). The Lagrangian multiplier is λ = (p cu^-γ + q cd^-γ)(1+r)X0. Exercise 11. Prove the above theorem.$

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