Question

8 Consider the following 3-input version of a Cobb-Douglas production function y Axxx A 0, 0a, B, y < 1 Find the first- and second-order partial derivatives, and determine the signs. What is the economic interpretation of the signs of these derivatives?

Answer 1

Answer:

Given,
y = Arq2 23
   where A>0,
Ο <α, β, γ <1

First order partial derivatives with respect to x₁, x₂ and x₃:

and

дуо в aArx -(Artix7) = aAr4-175x3 = = дхідх ay

and

ду а — (Ахзх;) = BAxx-x) = - ВАхах до до Т2

дуо - Ах, тях;) = .Аххях; -=- 1 у Аххх дурдаха уу T3

Now, second order partial derivatives with respect to x₁, x₂ and x₃:

д?у о , ді?ді = -a.Az-175x3 = (а — 1)a.Az-2x, x3 = (а — 1) — a.Art.r

$=>\frac{\partial ^2y}{\partial x_1^2}=(\alpha -1)\frac{\alpha y}{x_1^2}<0$

Similarly,

BArg xjх; = — 3 Axt-x) = (8 – 1) BAx-2x) = (в – 1) для дх2

>器-(4- 10

д? уд a = a Ax* 1923-1 = (у — 1) 1.Az zar3-2 = (у — 1) 1 Ал

$=>\frac{\partial ^2y}{\partial x_3^2}=(\gamma -1)\frac{\gamma y}{x_3^2}<0$ because

Ο <α, β, γ <1

The positive signs of the first order partial derivatives mean that y is rising when x₁, x₂ and x₃ increases.

But the negative signs of the second order partial derivatives mean that y is rising with decreasing rate.