<p>To solve (a), define <span class="math-container">$f(c)=u'(c)$</span>, thus the absolute risk aversion <span class="math-container">$\alpha$</span> becomes:</p>
<p><span class="math-container">$$\frac{f'(c)}{f(c)}=-\alpha,$$</span>
<span class="math-container">$$\frac{d}{dc}[\ln(f(c))]=-\alpha,$$</span>
<span class="math-container">$$d[\ln(f(c))]=-\alpha dc,$$</span>
<span class="math-container">$$\int d[\ln(f(c))]=\int-\alpha dc,$$</span>
<span class="math-container">$$\ln(f(c))=-\alpha\int dc,$$</span>
<span class="math-container">$$\ln(f(c))=-\alpha c+\beta_1,$$</span>
<span class="math-container">$$f(c)=\beta_2e^{-\alpha c},\;\;\text{with }\beta_2=e^{\beta_1}&gt;0$$</span>
<span class="math-container">$$f(c)=\beta_2e^{-\alpha c}.$$</span></p>
<p>We know that <span class="math-container">$u'(c)=f(c)$</span> thus</p>
<p><span class="math-container">$$\frac{du}{dc}=\beta_2e^{-\alpha c}$$</span>
<span class="math-container">$$\int du=\int\beta_2e^{-\alpha c}dc$$</span>
<span class="math-container">$$u(c)=-\frac{\beta_2}{\alpha}e^{-\alpha c}+\beta_3.$$</span></p>
<p>If <span class="math-container">$\alpha&gt;0$</span>, which is not stated explicitly but may be derived in the case that <span class="math-container">$u'(c)&gt;0$</span> and <span class="math-container">$u''(c)&lt;0$</span>, then we note that <span class="math-container">$-e^{-\alpha c}$</span> represents the same preferences as <span class="math-container">$u(c)$</span> since it is an increasing transformation of <span class="math-container">$u(c)$</span>.</p>
<p>To solve (b), a variation of a similar trick may work.</p>

CC BY-SA 4.0
2024-09-02T03:40:42.207Z
Thank for the ingenious solution, it is really ridiculous of Cochrane because how are we even expected to come up with a solution like this for an introductory text in asset pricing? I am really flabbergasted.
KaiSqDist
CC BY-SA 4.0
2024-09-01T18:21:59.207Z
2024-09-01T18:35:08.300Z
2024-09-01T18:35:08.300Z
manifold
<p>I went through the first chapter of Cochrane's Asset pricing and am completely stuck at the first question. I admit that I am an economics idiot, but there is totally no structure to this problem, like how do I even start?</p>
<p>Even the solution set says it is too obvious. I would appreciate any nudge in the right direction or a full solution is VERY MUCH appreciated (although not required).</p>
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CC BY-SA 4.0
2024-09-01T18:07:28.527Z
Taking a look at the definition, seems we are looking for u(c). Take the coefficients of risk aversion, try to integrate them 1 and 2 times respectively and solve for u(c).
T123
<utility><asset-pricing><consumption>
Proof of utility function for CARA [1(a)] and CRRA equality with power coefficient [1(b)]
KaiSqDist
# Proof of utility function for CARA [1(a)] and CRRA equality with power coefficient [1(b)]

(Asked by: username_0 on 2024-09-01)

I went through the first chapter of Cochrane's Asset pricing and am completely stuck at the first question. I admit that I am an economics idiot, but there is totally no structure to this problem, like how do I even start?

Even the solution set says it is too obvious. I would appreciate any nudge in the right direction or a full solution is VERY MUCH appreciated (although not required).

[![enter image description here](https://i.sstatic.net/HlKtrNtO.png)](https://i.sstatic.net/HlKtrNtO.png)

[![enter image description here](https://i.sstatic.net/6HSTYYMB.png)](https://i.sstatic.net/6HSTYYMB.png)

**Tags:** utility, asset-pricing, consumption
> **Comments:**
> **username_1**: Taking a look at the definition, seems we are looking for u(c). Take the coefficients of risk aversion, try to integrate them 1 and 2 times respectively and solve for u(c).  

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## 1 answers

### Answer by username_2 on 2024-09-01. Score: 2 | Accepted answer

To solve (a), define $f(c)=u'(c)$, thus the absolute risk aversion $\alpha$ becomes:

$$\frac{f'(c)}{f(c)}=-\alpha,$$
$$\frac{d}{dc}[\ln(f(c))]=-\alpha,$$
$$d[\ln(f(c))]=-\alpha dc,$$
$$\int d[\ln(f(c))]=\int-\alpha dc,$$
$$\ln(f(c))=-\alpha\int dc,$$
$$\ln(f(c))=-\alpha c+\beta\_1,$$
$$f(c)=\beta\_2e^{-\alpha c},\;\;\text{with }\beta\_2=e^{\beta\_1}>0$$
$$f(c)=\beta\_2e^{-\alpha c}.$$

We know that $u'(c)=f(c)$ thus

$$\frac{du}{dc}=\beta\_2e^{-\alpha c}$$
$$\int du=\int\beta\_2e^{-\alpha c}dc$$
$$u(c)=-\frac{\beta\_2}{\alpha}e^{-\alpha c}+\beta\_3.$$

If $\alpha>0$, which is not stated explicitly but may be derived in the case that $u'(c)>0$ and $u''(c)<0$, then we note that $-e^{-\alpha c}$ represents the same preferences as $u(c)$ since it is an increasing transformation of $u(c)$.

To solve (b), a variation of a similar trick may work.

> **Comments:**
> **username_0**: Thank for the ingenious solution, it is really ridiculous of Cochrane because how are we even expected to come up with a solution like this for an introductory text in asset pricing? I am really flabbergasted.  

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