<p>You have to be careful when comparing different utility functions.</p>
<p>For example the function
<span class="math-container">$$
v(C_A) = - \frac{1}{C_A},
$$</span>
and the function:
<span class="math-container">$$
u(C_A) + u(C_B) = - \frac{1}{C_A} - \frac{1}{C_B},
$$</span>
cannot simply be compared. The first represents preferences over one good, while the second gives preferences over two goods.</p>
<p>So if you say:
<span class="math-container">$$
u(C_A) + u(C_B) &lt; v(C_A),
$$</span>
you are in fact comparing two different utility functions.</p>
<p>If you want to know what happens if you only consume one of the goods, you need to compare, for example <span class="math-container">$u(C_A) + u(C_B)$</span> with <span class="math-container">$u(C_A) + u(0)$</span>.</p>
<p>Also if you increase <span class="math-container">$k$</span> in the function:
<span class="math-container">$$
-\frac{1}{C_A} - \frac{k}{C_B},
$$</span>
you are in fact making <span class="math-container">$C_B$</span> more valuable. To see this let's compute the marginal utility of <span class="math-container">$C_B$</span> relative to the marginal utility of <span class="math-container">$C_A$</span>:
<span class="math-container">$$
\frac{MU_B}{MU_A} = \frac{\dfrac{k}{C_B^2}}{\dfrac{1}{C_A^2}} = k \frac{C_A^2}{C_B^2},
$$</span>
which does increase in <span class="math-container">$k$</span>. So the higher <span class="math-container">$k$</span> the higher the value of an additional unit of <span class="math-container">$B$</span> compared to an additional unit of <span class="math-container">$A$</span>.</p>

CC BY-SA 4.0
2021-05-18T11:35:04.560Z
2021-05-18T11:35:04.560Z
tdm
<p>Suppose our utility function is the usual CRRA utility with <span class="math-container">$\gamma=2$</span> so that:
<span class="math-container">$$u(C) = \frac{C^{1-\gamma}}{1-\gamma} = -\frac{1}{C}$$</span></p>
<p>Now suppose there are 2 goods, A and B, available for consumption. The consumption of A and B are denoted <span class="math-container">$C_A, C_B$</span>. How do I compute the final utility?</p>
<p>For example, if I just add the utility, then the summed utility will be less than consuming either one of the goods, because the utility function is negative.
<span class="math-container">$$u(C_A) + u(C_B) = -\frac{1}{C_A} -\frac{1}{C_B} &lt; u(C_A), u(C_B)$$</span></p>
<p>Suppose also that we want to make good B &quot;more valuable&quot;. ie. 1 unit of good B should be worth more than 1 unity of good C. How would I reflect this in the utility function? A simple scaling with a value greater than 1 won't work, again because of the negative utility:
<span class="math-container">$$u(C_A) + u(C_B) = -\frac{1}{C_A} -k \cdot\frac{1}{C_B}$$</span>
where <span class="math-container">$k&gt;1$</span></p>
<p>Thanks in advance.</p>

<utility><consumption>
Adding scaled CRRA utility
user34870
# Adding scaled CRRA utility

(Asked by: username_0 on 2021-05-18)

Suppose our utility function is the usual CRRA utility with $\gamma=2$ so that:
$$u(C) = \frac{C^{1-\gamma}}{1-\gamma} = -\frac{1}{C}$$

Now suppose there are 2 goods, A and B, available for consumption. The consumption of A and B are denoted $C\_A, C\_B$. How do I compute the final utility?

For example, if I just add the utility, then the summed utility will be less than consuming either one of the goods, because the utility function is negative.
$$u(C\_A) + u(C\_B) = -\frac{1}{C\_A} -\frac{1}{C\_B} < u(C\_A), u(C\_B)$$

Suppose also that we want to make good B "more valuable". ie. 1 unit of good B should be worth more than 1 unity of good C. How would I reflect this in the utility function? A simple scaling with a value greater than 1 won't work, again because of the negative utility:
$$u(C\_A) + u(C\_B) = -\frac{1}{C\_A} -k \cdot\frac{1}{C\_B}$$
where $k>1$

Thanks in advance.

**Tags:** utility, consumption


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## 1 answers

### Answer by username_1 on 2021-05-18. Score: 2

You have to be careful when comparing different utility functions.

For example the function
$$
v(C\_A) = - \frac{1}{C\_A},
$$
and the function:
$$
u(C\_A) + u(C\_B) = - \frac{1}{C\_A} - \frac{1}{C\_B},
$$
cannot simply be compared. The first represents preferences over one good, while the second gives preferences over two goods.

So if you say:
$$
u(C\_A) + u(C\_B) < v(C\_A),
$$
you are in fact comparing two different utility functions.

If you want to know what happens if you only consume one of the goods, you need to compare, for example $u(C\_A) + u(C\_B)$ with $u(C\_A) + u(0)$.

Also if you increase $k$ in the function:
$$
-\frac{1}{C\_A} - \frac{k}{C\_B},
$$
you are in fact making $C\_B$ more valuable. To see this let's compute the marginal utility of $C\_B$ relative to the marginal utility of $C\_A$:
$$
\frac{MU\_B}{MU\_A} = \frac{\dfrac{k}{C\_B^2}}{\dfrac{1}{C\_A^2}} = k \frac{C\_A^2}{C\_B^2},
$$
which does increase in $k$. So the higher $k$ the higher the value of an additional unit of $B$ compared to an additional unit of $A$.


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