# Kuhn Tucker Maximization

I have to maximize following expected utility function using Kuhn tucker conditions - 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 - My answer doesn't match the solution provided. Can someone please look at my solution and tell me what did I do wrong?

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 $$\begin{eqnarray*} q_t^e \in \begin{cases} \{1\} & \text{if } z_t > 1 \\ \{0\} & \text{if } z_t < 1 \\ [0, 1] & \text{if } z_t = 1 \end{cases} \end{eqnarray*}$$