I was studying about the Solow Model from Acemoglu. One of the properties of the production function $Y = F(A,K,L)$ is that it exhibits diminishing marginal products. That is, $F_{KK} < 0$ and $F_{LL} < 0$.
Can we also say that $F_{LK} < 0$?
1 Answer
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You can't just say that from the diminishing marginal product assumption. $F_{KK}<0$ and $F_{LL}<0$ does not imply that $F_{LK} <0$.
In fact, a standard textbook Cobb-Douglas production function $F= AK^{\alpha}L^{1-\alpha}; 0<\alpha<1$, which is often used together with Solow Swan model will have positive cross derivative, given the parameter restrictions, if any capital and labor is employed:
$$F_{LK}= (1-\alpha)\alpha\frac{A}{(KL)^\alpha} >0 $$
This is despite that the same function has diminishing marginal product for both capital and labor (you can verify that yourself).
Of course, you can play with the model by using different production functions, however, assuming $F_{LK}<0$ is from economic perspective very strange.
$F_{LK}<0$ says that when you increase both the amount of labor and capital in your factory at the same time marginal output declines. For example, if you have sewing factory adding both more sewing machines and more seamsters to operate the sewing machine would result in less shirts produced on the margin. This is very implausible, perhaps there are some special circumstances where you want to assume that but it will not be generally assumed in macroeconomic models that approximate production of whole economy.
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$\begingroup$ "$F_{LK}<0$ says that when you increase both the amount of labor and capital in your factory at the same time output declines." Can you please explain why this says what you claim it says, and why it is not just a second order effect? By the same logic $F_{LK}=0$ would imply that increasing both inputs has no effect on the output, but $F(K,L) = K + L$ is a trivial counterexample to this. In fact, the less conventional $F(K,L) = K + L - 1/(KL)$ is a counterexample to your claim under most parameter combinations. $\endgroup$– GiskardCommented Mar 23 at 12:30
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1$\begingroup$ @Giskard cross-derivative $f_{xy}$ tells you how the slope of x changes with respect to changes in y. $f_{xy}<0$ implies that with more y the slope of x becomes less steep. In a context of production that means that when we add more labor the marginal productivity of capital declines $\endgroup$– 1muflon1 ♦Commented Mar 23 at 12:37
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$\begingroup$ I also think this is what it means, but you wrote something completely different: "when you increase both the amount of labor and capital in your factory at the same time output declines." $\endgroup$– GiskardCommented Mar 23 at 12:39
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1$\begingroup$ New version of the statement however is unclear, as there are several possible margins you can think of. Losing the final paragraph is the simplest solution to the entire problem. $\endgroup$– GiskardCommented Mar 23 at 12:45
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1$\begingroup$ Thank you 1muflon1 and @Giskard. The answer and the discussion that followed was helpful to me. $\endgroup$– guestCommented Mar 23 at 14:36