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- A concave function is quasiconcave. A convex function is quasiconvex.
mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/qcc/tMathematical methods for economic theory - University of Toronto
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A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$. Another representation of quasi-concave functions: $f(x)$ is called quasi- concave if and only if
- mathematical economics
a function f2: A → defined on the convex set A ⊂ n. Then f2...
- functions
The function $f(x)=\frac2x$ is both quasi-concave and...
- mathematical economics
A weaker condition to describe a function is quasiconvexity (or quasiconcav-ity). Functions which are quasiconvex maintain this quality under monotonic transformations; moreover, every monotonic transformation of a concave func-tion is quasiconcave (although it is not true that every quasiconcave function
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In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (,) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints.
Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
convex function is quasiconvex. Every c. Prove that the following statements are equivalent. The function f : C R is quasiconvex. →. (b) For all α R, the sublevel set x C : f(x) α is convex. ∈ { ∈ ⩽ } 7–2. (c) For all α R, the strict sublevel set x C : f(x) < α is convex. ∈ { ∈ }
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May 27, 2023 · a function f2: A → defined on the convex set A ⊂ n. Then f2 is said to be quasi-concave if every upper contour set of f2 is convex. i.e., S(α) = {x ∈ S ∣ f2(x) ≥ α} is a convex set ∀ α ∈. The function f: R2 + → R defined as f(x, y) = 3xy is quasi-concave but not concave.
Jan 15, 2019 · The function $f(x)=\frac2x$ is both quasi-concave and quasi-convex, because: $$f(tx_1+(1-t)x_2)\ge \min\{f(x_1),f(x_2)\}, t\in[0,1] \ \text{(quasi-concavity)}\\ f(tx_1+(1-t)x_2)\le \max\{f(x_1),f(x_2)\}, t\in[0,1] \ \text{(quasi-convexity)}$$
