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On the Curvature of Homogeneous Functions

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Abstract

Consider a quasiconcave, upper semicontinuous and homogeneous of degree \(\gamma \) function f. This paper shows that the reciprocal of the degree of homogeneity, \(1/\gamma \), can be interpreted as a measure of the degree of concavity of f. As a direct implication of this result, it is also shown that f is harmonically concave if \(\gamma \le -1\) or \(\gamma \ge 0\), concave if \(0\le \gamma \le 1\) and logconcave if \(\gamma \ge 0\). Some relevant applications to economic theory are given. For example, it is shown that a quasiconcave and homogeneous production function is concave if it displays nonincreasing returns to scale and logconcave if it displays increasing returns to scale.

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Notes

  1. \(\mathbb {R} _{++}\equiv \left\{ y\in \mathbb {R}\mid y>0\right\} \).

  2. Strictly speaking, Crouzeix [7] proves this result in terms of convexity. Here, I phrase the results in terms of concavity because I apply the results to economics in which functions are more commonly assumed (quasi)concave in order to guarantee a maximum. Moreover, it should be noted that Crouzeix [7] also shows that f is a concave function when f is nonpositive for every x that belongs to the relative interior of \(\mathcal { X}\) (i.e., the interior of \(\mathcal {X}\) in the affine hull of \(\mathcal {X}\) ).

  3. I discuss the case \(f=0\) in Remark 3 but leave the remaining case \(f<0\) to future research.

  4. Balogh and Ewerhart [3] survey the origins of \(\rho \)-concavity and give Martos [14] credit of having introduced \(\rho \)-concavity. Martos [14] is written in Hungarian, where \(\rho \)-concavity is originally called \(\omega \)- concavity. Balogh and Ewerhart [3] give translations of relevant sections in Martos [14] into English. \(\rho \) (or \(\omega \))-concavity has also been referred to as \(\alpha \)-concavity in the mathematics literature (e.g., [13, 17]).

  5. But see e.g., [13].

  6. Specifically, by Jensen’s inequality, \(\partial M_{\rho }\left( y_{1},y_{2}\right) /\partial \rho \ge 0\). See also Hardy et al. ([10], Ch.2.9, p.26). And see e.g., ([5], Chapter III) where weighted power means and their properties are studied thoroughly.

  7. Indeed, the cases \(\lambda =0\) and \(\lambda =1\) trivially satisfy the definition of \(\rho \)-concavity regardless of whether the function is quasi-concave, homogeneous or upper semicontinuous. To see this consider \( \lambda =0\) (\(\lambda =1\) follows analogously), in which case the left-hand side of (1) in Definition 1.2 becomes \(f\left( \lambda x_{1}+\left( 1-\lambda \right) x_{2}\right) =f\left( x_{2}\right) \), while the right-hand side becomes \(\left[ \lambda f\left( x_{1}\right) ^{\frac{1}{ \gamma }}+\left( 1-\lambda \right) f\left( x_{2}\right) ^{\frac{1}{\gamma }} \right] ^{\gamma }=f\left( x_{2}\right) \), which of course trivially satisfies (1).

  8. See Section 3.5 in [4] for a detailed discussion of the properties of logconvex/logconcave functions and [2] about logconcave/logconvex probability distributions.

  9. This also follows as a special case of the results in Theorem 3.1 in Jehle and Reny [11] and [8, 15].

  10. See e.g., Bagnoli and Bergstrom [2] for conditions on distributions which guarantee logconcavity.

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  1. Research Institute of Industrial Economics (IFN), P.O. Box 55665, 102 15, Stockholm, Sweden

    Per Hjertstrand

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  1. Per Hjertstrand
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Correspondence to Per Hjertstrand.

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Communicated by Juan-Enrique Martinez Legaz.

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I would like to thank two reviewers for comments that greatly improved the paper, Magnus Henrekson and Gunnar Rosenqvist for encouragement, and Torsten Sö derbergs stiftelse for funding.

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Hjertstrand, P. On the Curvature of Homogeneous Functions. J Optim Theory Appl 198, 215–223 (2023). https://doi.org/10.1007/s10957-023-02249-6

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