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Consider the Cobb-Douglas production function of three variables, $w=f(x, y, z)=C x^{\alpha} y^{\beta} z^{\gamma} .$ Show that $(\mathrm{a})$ $f(d x, d y, d z)=d^{\alpha+\beta+\gamma} f(x, y, z),(\mathrm{b})$ $x f_{x}(x, y, z)+y f_{y}(x, y, z)+z f_{z}(x, y, z)=(\alpha+\beta+\gamma) f(x, y, z)$ (c) in particular, when $\alpha+\beta+\gamma=1,$ give an economic interpretation of these results.

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Show that the Cobb-Douglas…

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So we're given the relation p equals b times l to the Alfa chimes kay to the beta. And we want to prove that oh, times dp do you l close k times d p d k is equal to Alfa flows beta times p So first we, um, should find these partial derivatives. So first, let's find DPD. L So we're gonna use of this first equation here and differentiate with respect to l. So we're keeping everything else as a constant. So when we differentiate, uh, D p d l Alfa comes outside. So we have Alfa Beta, her Alfa B and K to the beta. And then we subtract one from the power. So we're left with L to the Alfa minus one and next. It's fine. DP decay. So we're gonna differentiate again, Um, the equation above. But this time we're going to keep que const er case variable and everything else constant Scylla Beta is gonna come up front. So we have beta times be l to the Alpha on K to the beta minus one. So now let's plug these into our equation. So let's rewrite the left hand side. We have L Times uh, blue. So we have else. The awful minus one. Alfa be Kate of the beta, plus que times what we found here in green. So let's put the case together. So, Kato, the bait of on this one, and then we have beta B and else the Alfa. So let's group these together. So we have l to the Alfa Times Alfa be Kate of the beta, and then we have cage of the beta. They'd, uh, be l to the Alfa. So now let's see what we can factor that we can take out. Uh, L b and Kate up to the beta k to the beta. It's okay. Be out of the Alfa. So are the Alfa be okay to the beta, And then we're left with on Alfa plus a beta inside, which is almost what we want to get. We want a novel post beta time, Pete. So we have the awful plus beta, but we know that P is equal to be l've the Alfa Kato the beta, Which is exactly what we have right here. So it can replace us with a P. And this is exactly what we wanted to get. No

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