Question

Explain why a production possibilities curve is concave brainly a answer

Answer 1

Explanation:

A real-valued function {\displaystyle f} f on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any {\displaystyle x} x and {\displaystyle y} y in the interval and for any {\displaystyle \alpha \in [0,1]} \alpha \in [0,1],[1]

{\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)} {\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}

A function is called strictly concave if

{\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)\,} {\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)\,}

for any {\displaystyle \alpha \in (0,1)} \alpha \in (0,1) and {\displaystyle x\neq y} x\neq y.

For a function {\displaystyle f:\mathbb {R} \to \mathbb {R} } f:{\mathbb {R}}\to {\mathbb {R}}, this second definition merely states that for every {\displaystyle z} z strictly between {\displaystyle x} x and {\displaystyle y} y, the point {\displaystyle (z,f(z))} {\displaystyle (z,f(z))} on the graph of {\displaystyle f} f is above the straight line joining the points {\displaystyle (x,f(x))} (x, f(x)) and {\displaystyle (y,f(y))} {\displaystyle (y,f(y))}.

ConcaveDef.png

A function {\displaystyle f} f is quasiconcave if the upper contour sets of the function {\displaystyle S(a)=\{x:f(x)\geq a\}} S(a)=\{x:f(x)\geq a\} are convex sets.[2]:496