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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?

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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 $l$.

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  • $\begingroup$ 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? $\endgroup$
    – CormJack
    May 6, 2023 at 10:16
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    $\begingroup$ @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 $\endgroup$
    – 1muflon1
    May 6, 2023 at 10:46
  • $\begingroup$ 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: economics.stackexchange.com/questions/55294/… $\endgroup$
    – CormJack
    May 6, 2023 at 11:18

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