Question

Consider the function U defined on R^2, where U(x,y) = sq rt (3x + y). Which of the following statements is true? 1. U is strictly concave 2. U is strictly quasi-concave 3. U is both strictly concave & strictly quasi-concave 4. U is both concave & quasi

Please show me how to solve.. thank you.

Answer 1

See below.

Explanation:

Consider the function `U` defined on `RR^2`, where `U(x,y) = sqrt (3x + y)`. Which of the following statements is true?
1. U is strictly concave
2. U is strictly quasi-concave
3. U is both strictly concave & strictly quasi-concave
4. U is both concave & quasi

Considering `p = (x,y)` and `f(p) = sqrt(<< c, p >>)`

we have that the hypograph of `f(p)` is convex if

`f((1-lambda)p_1+lambda p_2) ge (1-lambda)f(p_1)+lambda f(p_2)` with `lambda in [0,1]`

In our case we have

`sqrt((1-lambda)p_1+lambda p_2) ge (1-lambda) sqrt(<< c, p_1>>) + lambda sqrt( << c, p_2 >>)`

Squaring both sides

`(1-lambda)<< c, p_1>> +lambda << c,p_2>> ge (1- lambda)^2<< c, p_1 >> + lambda^2 << c, p_2 >> +2 lambda(1-lambda) f(p_1)f(p_2)` or

`lambda(1-lambda)<< c, p_1 >> + lambda(1-lambda) << c, p_2 >> ge 2 lambda(1-lambda) f(p_1)f(p_2)` or

`<< c, p_1 >> + << c, p_2 >> ge 2 f(p_1)f(p_2)` or

`<< c, p_1 >> + << c, p_2 >> ge 2 sqrt(<< c, p_1 >> xx << c,p_2 >>)`

and this is true for all `<< c, p_1 >> ge 0, << c, p_2 >>ge 0`

and as we know `sqrtabs(3x+y) ge 0` so we have that the hypograph for `f(p)` is quasi-convex.

The conclusions about the correct answer are left as an exercise.

NOTE