# Bliss point hamiltonian function

I have the following utility function

$$u(c_t)=g(c_t-b)^2$$ for constants $$g,b>0$$

Edited: in order to be bliss utility function, It must be g is negative. But in the question g is given to be positive. I don’t know so. Please help me solving this question according to true assumption of g. (I don’t know which one true. )

I want to do the following

$$\max_{c_t} \int_0^{\infty} u(c_t)e^{-pt}$$

Subject to $$\dot{k_t} =Ak_t^a -c_t-(n+\delta)k_t$$ $$k_0>0$$ $$c_t>0$$

I set up hamiltonian function

$$H= g(c_t-b)^2 +q_t [Ak_t^a -c_t-(n+\delta)k_t]$$

where q is costate variables. $$c$$ is control variables and $$k$$ is State variables.

Focs

W.r.t $$c$$ : $$c_t[2g(c_t-b)-q_t]=0$$

W.r.t $$k$$: $$q_t[aAk_t^{a-1}-(n+\delta)]=-[\dot{q_t}-pq_t]$$

W.r.t. $$q_t$$ : $$\dot{k_t} =Ak_t^a -c_t-(n+\delta)k_t$$

Transversality condition $$\lim q_t e^{-pt}k_t=0$$ When I solve them, I found

$$\frac{\dot{q_t}}{q_t} = -[aAk_t^{a-1}-(n+p+\delta)]$$

And $$\frac{\dot{q_t}}{q_t}= \frac{\dot{c_t}}{c_t-g}$$

But I could not proceed after that. Because the denominator of the fraction is $$c_t-g$$ .

What should I get the results (equations) that I need to draw phase diagram? They are Euler equation and other differential equations.

Please help me at this point. If you have any material related to this, please share it with me. Thanks.

• "phase diagram" in what coordinate system? Seems weird that you already got 4 upvotes but no one thought to ask this... – Giskard Nov 25 '19 at 22:53
• I assume $g$ is negative here? Otherwise that's not a bliss point utility function. – corran_horn Nov 26 '19 at 2:30
• Can you show your solution? I just want to find euler equation @corran_horn – B11b Nov 26 '19 at 6:55
• No no I don't want you to draw phase diagram. I just ask for finding euler equation and other necessary differential equations in this question. @giskard – B11b Nov 26 '19 at 6:57
• Also c k graph. y-axis consumption (c). x-axis is capital (k) @Giskard – B11b Nov 26 '19 at 9:21