<p>There isn’t a general answer (beside the pure mathematical answer) because the meaning will be different in different fields. E.g. in theory of production interpretation would be different (if you for example examine the cross derivative of production function).</p>
<p>Generally cross-derivative  <span class="math-container">$f_{xy}’’$</span> tells you how the slope <span class="math-container">$x$</span> changes with respect to changes in <span class="math-container">$y$</span>.</p>
<p>When it comes to the utility <span class="math-container">$u(c,l)$</span> the cross derivative <span class="math-container">$u_{cl}’’$</span> tells you how much the marginal utility of <span class="math-container">$c$</span> changes when you change amount of <span class="math-container">$l$</span>.</p>

CC BY-SA 4.0
2023-05-06T10:16:29.750Z
This is an interesting question and your answer which I've bookmarked is very useful. Would you be willing to expand on this> E.g. to discuss the production function case you mentioned, or more generally taking mixed partials with respect to a profit function and what they tell us about the relationship between inputs. I'm happy to write this as a post for you to answer if you prefer?
CormJack
CC BY-SA 4.0
2023-05-06T10:46:35.883Z
@CormJack Hi, the interpretation of cross-partial derivative of profit function $\Pi^{''}_{q_1,q_2}$ is that it tells you have marginal profit of product 1 changes when you increase more of product 2. For a production function such as $F^{''}_{L,K}$ tells you how marginal product of labor changes when you add more capital
1muflon1
CC BY-SA 4.0
2023-05-06T11:18:44.730Z
Thanks this is very helpful! I tried to use this information to solve my underlying problem, but I still wasn't entirely clear. I have attached it here if you have the time: https://economics.stackexchange.com/questions/55294/mixed-partial-derivatives-in-profit-function
CormJack
CC BY-SA 4.0
2021-12-06T10:18:02.370Z
2021-12-06T10:24:35.377Z
2021-12-06T10:24:35.377Z
1muflon1
<p>Suppose I have a utility function <span class="math-container">$u(c,l)$</span>, where c is consumption and l is leisure. What does <span class="math-container">$u_{cl}$</span> mean in economic terms for example, or in general what is the economic meaning of cross derivatives?</p>

CC BY-SA 4.0
2021-12-06T20:19:29.543Z
https://economics.stackexchange.com/questions/5340/what-is-a-substitute-complement-in-terms-of-mixed-partial-derivatives
Jesper Hybel
<mathematical-economics>
What is the economic intuition behind cross derivatives?
Ali
# What is the economic intuition behind cross derivatives?

(Asked by: username_0 on 2021-12-06)

Suppose I have a utility function $u(c,l)$, where c is consumption and l is leisure. What does $u\_{cl}$ mean in economic terms for example, or in general what is the economic meaning of cross derivatives?

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: https://economics.stackexchange.com/questions/5340/what-is-a-substitute-complement-in-terms-of-mixed-partial-derivatives  

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## 1 answers

### Answer by username_2 on 2021-12-06. Score: 3

There isn’t a general answer (beside the pure mathematical answer) because the meaning will be different in different fields. E.g. in theory of production interpretation would be different (if you for example examine the cross derivative of production function).

Generally cross-derivative $f\_{xy}’’$ tells you how the slope $x$ changes with respect to changes in $y$.

When it comes to the utility $u(c,l)$ the cross derivative $u\_{cl}’’$ tells you how much the marginal utility of $c$ changes when you change amount of $l$.

> **Comments:**
> **username_3**: This is an interesting question and your answer which I've bookmarked is very useful. Would you be willing to expand on this> E.g. to discuss the production function case you mentioned, or more generally taking mixed partials with respect to a profit function and what they tell us about the relationship between inputs. I'm happy to write this as a post for you to answer if you prefer?  
> **username_2**: @username_3 Hi, the interpretation of cross-partial derivative of profit function $\Pi^{''}_{q_1,q_2}$ is that it tells you have marginal profit of product 1 changes when you increase more of product 2. For a production function such as $F^{''}_{L,K}$ tells you how marginal product of labor changes when you add more capital  
> **username_3**: Thanks this is very helpful! I tried to use this information to solve my underlying problem, but I still wasn't entirely clear. I have attached it here if you have the time: https://economics.stackexchange.com/questions/55294/mixed-partial-derivatives-in-profit-function  

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