- #1
Sidney
- 12
- 0
I've been reading a book on economics and they defined a homogeneous function as: ƒ(x1,x2,…,xn) such that
ƒ(tx1,tx2,…,txn)=tkƒ(x1,x2,…,xn) ..totally understandable.. they further explained that a direct result from this is that the partial derivative of such a function will be homogeneous to the degree k-1.They proved this by simply differentiating both sides of the equation. My problem arises when they differentiate the left hand side (with respect to the first argument as an arbitrary choice). They say the partial differential(of the LHS wrt x1) is:
(∂ƒ(tx1,tx2,…,txn)/∂x1).t
my question is where does the t come from.. ..please bear with me
ƒ(tx1,tx2,…,txn)=tkƒ(x1,x2,…,xn) ..totally understandable.. they further explained that a direct result from this is that the partial derivative of such a function will be homogeneous to the degree k-1.They proved this by simply differentiating both sides of the equation. My problem arises when they differentiate the left hand side (with respect to the first argument as an arbitrary choice). They say the partial differential(of the LHS wrt x1) is:
(∂ƒ(tx1,tx2,…,txn)/∂x1).t
my question is where does the t come from.. ..please bear with me