<p>I have the following utility function </p>

<p><span class="math-container">$$u(c_t)=g(c_t-b)^2$$</span> for constants <span class="math-container">$g,b&gt;0$</span> </p>

<p>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. )</p>

<p>I want to do the following</p>

<p><span class="math-container">$$ \max_{c_t} \int_0^{\infty} u(c_t)e^{-pt}$$</span></p>

<p>Subject to <span class="math-container">$$\dot{k_t} =Ak_t^a -c_t-(n+\delta)k_t$$</span> <span class="math-container">$$k_0&gt;0$$</span> <span class="math-container">$$c_t&gt;0$$</span></p>

<p>I set up hamiltonian function </p>

<p><span class="math-container">$$H= g(c_t-b)^2 +q_t [Ak_t^a -c_t-(n+\delta)k_t]$$</span></p>

<p>where q is costate variables. <span class="math-container">$c$</span> is control variables and <span class="math-container">$k$</span> is State variables. </p>

<p>Focs</p>

<p>W.r.t <span class="math-container">$c$</span> : <span class="math-container">$$c_t[2g(c_t-b)-q_t]=0$$</span></p>

<p>W.r.t <span class="math-container">$k$</span>: <span class="math-container">$$q_t[aAk_t^{a-1}-(n+\delta)]=-[\dot{q_t}-pq_t]$$</span></p>

<p>W.r.t. <span class="math-container">$q_t$</span> : <span class="math-container">$$\dot{k_t} =Ak_t^a -c_t-(n+\delta)k_t$$</span></p>

<p>Transversality condition <span class="math-container">$$\lim q_t e^{-pt}k_t=0$$</span>
 When I solve them, I found </p>

<p><span class="math-container">$$\frac{\dot{q_t}}{q_t} = -[aAk_t^{a-1}-(n+p+\delta)]$$</span></p>

<p>And <span class="math-container">$$\frac{\dot{q_t}}{q_t}= \frac{\dot{c_t}}{c_t-g}$$</span></p>

<p>But I could not proceed after that. Because the  denominator of the fraction is <span class="math-container">$c_t-g$</span> .</p>

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

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

CC BY-SA 4.0
2019-11-25T22:53:01.647Z
"phase diagram" in what coordinate system? Seems weird that you already got 4 upvotes but no one thought to ask this...
Giskard
CC BY-SA 4.0
2019-11-26T02:30:03.407Z
I assume $g$ is negative here? Otherwise that's not a bliss point utility function.
corran_horn
CC BY-SA 4.0
2019-11-26T06:55:10.933Z
Can you show your solution? I just want to find euler equation @corran_horn
studentp
CC BY-SA 4.0
2019-11-26T06:57:08.690Z
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
studentp
CC BY-SA 4.0
2019-11-26T09:21:06.090Z
Also c k graph. y-axis consumption (c). x-axis is capital (k) @Giskard
studentp
<macroeconomics><mathematical-economics><self-study>
Bliss point hamiltonian function
studentp
# Bliss point hamiltonian function

(Asked by: username_0 on 2019-11-25)

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.

**Tags:** macroeconomics, mathematical-economics, self-study
> **Comments:**
> **username_1**: "phase diagram" in what coordinate system? Seems weird that you already got 4 upvotes but no one thought to ask this...  
> **username_2**: I assume $g$ is negative here? Otherwise that's not a bliss point utility function.  
> **username_0**: Can you show your solution? I just want to find euler equation @corran_horn  
> **username_0**: 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  
> **username_0**: Also c k graph. y-axis consumption (c). x-axis is capital (k) @Giskard  

---

## 0 answers

