I have to maximize following expected utility function using Kuhn tucker conditions -

enter image description here

Since expected utility function are increasing $C_{1,t}$ and $C_{2,t}$ so constraints (i) and (ii) will hold with equality. Thus, I substituted these constraints into the objective function.

Note: Here, only $q_t^e$ is the only variable and expected utility function is maximized subject to $q_t^e$

For simplification, I denoted M= (1-$q_t^e$)$w_t$ + $q_t^e$$w_t$$z_t$

This made $C_{1,t}$ = nM and $C_{2,t}$= (ARθ/$z_t$)M

I also broke down constraint iii into two constraints $q_t^e$≤1 and -$q_t^e$≤0

I set up the legrange:

L = (1-π)[${(nM)}^{-γ}$]/(-γ) + π[${((ARθ/z_t)M)}^{-γ}$]/(-γ) + $λ_1$(1-$q_t^e$) + $λ_2$(0+$q_t^e$)

Differentiating with respect to $q_t^e$

(1-π)[${(nM)}^{-γ-1}$][n($z_t$-1)] + π[${((ARθ/z_t)M)}^{-γ-1}$][(ARθ/$z_t$)($z_t$-1)] = $λ_1$ - $λ_2$

$λ_1$(1-$q_t^e$) =0; $λ_1$≥0

$λ_2$(0+$q_t^e$) = 0; $λ_2$≥0

Case 1: $λ_1$=0 and $λ_2$

I got (1-π)[${(nM)}^{-γ-1}$][n($z_t$-1)] + π[${((ARθ/z_t)M)}^{-γ-1}$][(ARθ/$z_t$)($z_t$-1)] =0

Which boils down to M=0

which gave me - - > $q_t^e$ = 1/(1-$z_t$)

But solution provided is of the following form -

enter image description here

My answer doesn't match the solution provided. Can someone please look at my solution and tell me what did I do wrong?


1 Answer 1


After substituting for $c_{1t}$ and $c_{2t}$, the problem can be rewritten as : \begin{eqnarray*} \max_{q_t^e} \ & \frac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma} \left((1-q_t^e) + z_tq_t^e\right)^{-\gamma} + \frac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} \left((1-q_t^e) + z_tq_t^e\right)^{-\gamma} \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}

which is same as \begin{eqnarray*} \max_{q_t^e} \ & \alpha \left((1-q_t^e) + z_tq_t^e\right)^{-\gamma} \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*} where $\alpha = \dfrac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma} + \dfrac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} $

Observe that this is equivalent to solving : \begin{eqnarray*} \max_{q_t^e} \ & (1-q_t^e) + z_tq_t^e \\ \text{s.t. } & q_t^e \in [0, 1]\end{eqnarray*}

This is an objective that is linear in choice variable $q_t^e$, which is increasing in $q_t^e$ when $z_t > 1$, decreasing in $q_t^e$ when $z_t < 1$, and is a constant for $z_t = 1$. Consequently, the solution is \begin{eqnarray*} q_t^e \in \begin{cases} \{1\} & \text{if } z_t > 1 \\ \{0\} & \text{if } z_t < 1 \\ [0, 1] & \text{if } z_t = 1 \end{cases} \end{eqnarray*}

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    – studentp
    Commented Jan 8, 2020 at 14:29

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