<p>For a "well-behaved" bivariate utility function as regards utiliy maximiaztion, we need the Hessian matrix of partial derivatives to be negative definite, which translates into</p>

<p><span class="math-container">$$U_{cc}&lt;0, \;\;\;U_{ll}&lt;0,\;\;\; U_{cc}U_{ll} - \left[U_{cl}\right]^2 &gt;0$$</span></p>

<p><span class="math-container">$$\implies |U_{cl} | &lt; \sqrt{U_{cc}U_{ll}}$$</span></p>

<p>The first two conditions you have by assumption, the third condition poses a constraint on the relative magnitude in absolute value of the cross-partial, although not on its sign: it has to be smaller in absolute terms than the geometric average of the own-second derivatives, at least at the optimal point.</p>

<p>I notice that in many research papers it is assumed that the cross-partial is zero (utility additive in consumption and leisure).</p>

<p>Nevertheless, this is a good opportunity to apply some economic/behavioral thinking: what would it mean, as regards the relation between consumption and leisure to have <span class="math-container">$U_{cl} &lt;0$</span>? What would it mean to have <span class="math-container">$U_{cl} &gt;0$</span> ?</p>

CC BY-SA 4.0
2019-02-20T21:35:09.267Z
2019-02-20T21:35:09.267Z
Alecos Papadopoulos
<p>No.</p>

<p>In general, based on the information you have, there are no results you can infer for the partial derivatives <span class="math-container">$U_{cl}$</span> / <span class="math-container">$U_{lc}$</span>. In this context they could be anything.</p>

<p>As to what you are allowed to assume is a different story. You can assume anything in general, of course. Nevertheless, there is no way to check if your assumptions would be correct in this case.</p>

CC BY-SA 4.0
2019-02-20T21:11:00.363Z
2019-02-20T21:11:00.363Z
BB King
<p>My question concerns a household's utility function U(c,l). We're told that the partial derivatives dU/dc and dU/dl > 0, and the second order partial derivatives are negative. Now, can I assume anything about the sign of the partial derivatives Ucl (or Ulc)?</p>

<macroeconomics><microeconomics>
Can I make this assumption about the utility function?
BorionWaffles
# Can I make this assumption about the utility function?

(Asked by: username_0 on 2019-02-13)

My question concerns a household's utility function U(c,l). We're told that the partial derivatives dU/dc and dU/dl > 0, and the second order partial derivatives are negative. Now, can I assume anything about the sign of the partial derivatives Ucl (or Ulc)?

**Tags:** macroeconomics, microeconomics

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## 2 answers

### Answer by username_1 on 2019-02-20. Score: 1

For a "well-behaved" bivariate utility function as regards utiliy maximiaztion, we need the Hessian matrix of partial derivatives to be negative definite, which translates into

$$U\_{cc}<0, \;\;\;U\_{ll}<0,\;\;\; U\_{cc}U\_{ll} - \left[U\_{cl}\right]^2 >0$$

$$\implies |U\_{cl} | < \sqrt{U\_{cc}U\_{ll}}$$

The first two conditions you have by assumption, the third condition poses a constraint on the relative magnitude in absolute value of the cross-partial, although not on its sign: it has to be smaller in absolute terms than the geometric average of the own-second derivatives, at least at the optimal point.

I notice that in many research papers it is assumed that the cross-partial is zero (utility additive in consumption and leisure).

Nevertheless, this is a good opportunity to apply some economic/behavioral thinking: what would it mean, as regards the relation between consumption and leisure to have $U\_{cl} <0$? What would it mean to have $U\_{cl} >0$ ?


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### Answer by username_2 on 2019-02-20. Score: 1

No.

In general, based on the information you have, there are no results you can infer for the partial derivatives $U\_{cl}$ / $U\_{lc}$. In this context they could be anything.

As to what you are allowed to assume is a different story. You can assume anything in general, of course. Nevertheless, there is no way to check if your assumptions would be correct in this case.


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