<p>As discussed in comments above, this should be <span class="math-container">$U(c)=-\frac{1}{2}(c^* - c)^2$</span>. The economic intuition is that c* is the "bliss" consumption level.  It is probably your first glimpse at the idea of "satiation", which is an extension of the idea of diminishing marginal utility which allows discussion of situations where additional consumption actually produces disutility.  The most obvious example is the consumption of food, where c* would be the point at which you get "full".  But there are other contexts involving the study of externalities, where satiation is important, such as in environmental economics.  </p>

<p>The utility function isn't that interesting by itself, but remember that utility maximization is done subject to a budget constraint so that in general c* will not always be attainable.  If you haven't yet, you will eventually see this utility function in the context of lifetime consumption and savings decisions, usually as a lead-in to Friedman's permanent income hypothesis.</p>

CC BY-SA 4.0
2019-10-11T13:58:40.287Z
2019-10-11T18:02:33.203Z
2019-10-11T18:02:33.203Z
heh
<p>Remember that utility function only matters <em><a href="https://en.wikipedia.org/wiki/Ordinal_utility" rel="noreferrer">ordinally</a></em>... that is, you only care if <span class="math-container">$u(A) &gt; u(B)$</span>, and <span class="math-container">$u(A) = 100$</span> with <span class="math-container">$u(B) = 10$</span> is the same thing as <span class="math-container">$u(A) = -1000$</span> and <span class="math-container">$u(B) = -1000.0001$</span> (both have <span class="math-container">$u(A) &gt; u(B)$</span>.)</p>

CC BY-SA 4.0
2019-10-10T15:14:55.647Z
2019-10-10T15:14:55.647Z
Art
<p>To see if this function makes economic sense we must recall what the definition of economic rationality is:</p>

<p>A consumer is said to have rational preferences if: 
1) preferences are complete (any 2 bundles can be compared)
2)preferences are transitive (in this case we know if <span class="math-container">$-\frac{1}{2}(c^1-c)&gt;-\frac{1}{2}(c^2-c)$</span> and <span class="math-container">$-\frac{1}{2}(c^2-c)&gt;-\frac{1}{2}(c^3-c)$</span>
Then <span class="math-container">$-\frac{1}{2}(c^1-c)&gt;-\frac{1}{2}(c^3-c)$</span></p>

<p>Thus our consumers preferences are rational and are consistent with economic theory.</p>

CC BY-SA 4.0
2019-10-11T17:46:09.330Z
2019-10-11T17:46:09.330Z
EconJohn
<p>In a problem set, I found a strange utility function: <span class="math-container">$U(c)=-1/2(c^* - c)^2$</span>, where <span class="math-container">$c^* =$</span> positive constant level of consumption. Does this function have economic sense? </p>

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2019-10-10T20:23:51.707Z
Are you sure it's not $U(c)=-1/2(c* - c)^2$?  I ask because that looks an awful lot like a standard "bliss consumption" utility function used in macroeconomics problems. The factor of 1/2 is there so that the first derivative works out cleanly.
heh
CC BY-SA 4.0
2019-10-11T09:24:50.270Z
Yes, of course. You are right! Now I have understood the economics meaning of this function form, but I don't know how to interpret the possible value of c, such as the range of consumption levels that makes economic sense. I think there is just one level of consumption, namely c=c*, where utility equals 0. In fact, each increase in consumption level makes the consumer less satisfied.
user24514
CC BY-SA 4.0
2019-10-11T13:53:20.903Z
OK, having confirmed this, I'll give you a proper answer below.
heh
<microeconomics><utility>
utility function always negative
user24514
# utility function always negative

(Asked by: username_0 on 2019-10-10)

In a problem set, I found a strange utility function: $U(c)=-1/2(c^\* - c)^2$, where $c^\* =$ positive constant level of consumption. Does this function have economic sense?

**Tags:** microeconomics, utility
> **Comments:**
> **username_1**: Are you sure it's not $U(c)=-1/2(c* - c)^2$?  I ask because that looks an awful lot like a standard "bliss consumption" utility function used in macroeconomics problems. The factor of 1/2 is there so that the first derivative works out cleanly.  
> **username_0**: Yes, of course. You are right! Now I have understood the economics meaning of this function form, but I don't know how to interpret the possible value of c, such as the range of consumption levels that makes economic sense. I think there is just one level of consumption, namely c=c*, where utility equals 0. In fact, each increase in consumption level makes the consumer less satisfied.  
> **username_1**: OK, having confirmed this, I'll give you a proper answer below.  

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## 3 answers

### Answer by username_1 on 2019-10-11. Score: 0 | Accepted answer

As discussed in comments above, this should be $U(c)=-\frac{1}{2}(c^\* - c)^2$. The economic intuition is that c\* is the "bliss" consumption level. It is probably your first glimpse at the idea of "satiation", which is an extension of the idea of diminishing marginal utility which allows discussion of situations where additional consumption actually produces disutility. The most obvious example is the consumption of food, where c\* would be the point at which you get "full". But there are other contexts involving the study of externalities, where satiation is important, such as in environmental economics.

The utility function isn't that interesting by itself, but remember that utility maximization is done subject to a budget constraint so that in general c\* will not always be attainable. If you haven't yet, you will eventually see this utility function in the context of lifetime consumption and savings decisions, usually as a lead-in to Friedman's permanent income hypothesis.


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### Answer by username_2 on 2019-10-10. Score: 5

Remember that utility function only matters *[ordinally](https://en.wikipedia.org/wiki/Ordinal_utility)*... that is, you only care if $u(A) > u(B)$, and $u(A) = 100$ with $u(B) = 10$ is the same thing as $u(A) = -1000$ and $u(B) = -1000.0001$ (both have $u(A) > u(B)$.)


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### Answer by username_3 on 2019-10-11. Score: 0

To see if this function makes economic sense we must recall what the definition of economic rationality is:

A consumer is said to have rational preferences if:
1) preferences are complete (any 2 bundles can be compared)
2)preferences are transitive (in this case we know if $-\frac{1}{2}(c^1-c)>-\frac{1}{2}(c^2-c)$ and $-\frac{1}{2}(c^2-c)>-\frac{1}{2}(c^3-c)$
Then $-\frac{1}{2}(c^1-c)>-\frac{1}{2}(c^3-c)$

Thus our consumers preferences are rational and are consistent with economic theory.


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