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Respond to the question: Properties of CobbDouglas Production fu?

 01/21/2001 07:27 PM by Brandon; Pages of Nicholson's Microeconomic Theory
Thank you very much Rodrigo! I've read the pages in the test you recommended (well, it's high time...) The climbing analogy made the concept easier to understand. I know this forum relates to Game Theory more but I certainly hope

 12/18/2000 10:32 PM by Rodrigo; On Young's theorem and production functions
In words, this means:
(1) An additional unit of K makes the MPL vary. Compute the amount of this variation.
(2) An additional unit of L makes the MPK vary. Compute the amount of this variation.
(3) Now, it just happens that both amounts computed above are the same.

Don't think in terms of isoquants. They won't help you. The best way to understand this symmetry property is by thinking of its geometrical interpretation (by the way, this symmetry property is known as Young's theorem). Consider a function of two variables. Specifically, consider a "smooth" production function F(K,L). Let's do a little geometry.

Why is it that we compute the first-order derivatives of F with respect to each of its variables K and L? In other words, MPK and MPL?

The answer is: Because at any given point (K*,L*), the first-order derivatives (MPK and MPL) help us find out the best LINEAR approximation to F at that given point. This is what we all learn when we study calculus, isn't it? It is always useful to think of first-order derivatives as the slopes of the best LINEAR functions that approximate F.

Let's move a little bit further. Why is it that we compute the second-order derivatives of F at a given point (K*,L*)?

The answer is: Because they help us find out the best QUADRATIC approximation of F at that given point. The graph of a QUADRATIC approximation to F at a given point looks like the graph of a parabola in two variables (it looks like a cup or a perfectly symmetric mountain).

Now, forget F, and let's focus on the best QUADRATIC approximation, the parabola with two variables (K and L). Visualize this parabola as a mountain, a very regular one.

Finally, I recommend you to read the following pages of:

Walter NICHOLSON: "Microeconomic Theory", 7th edition (1998), The Dryden Press, pages 31 and 32 (in chapter 2)

All what we said above about "visualizing" the best QUADRATIC approximation to F at some given point will be useful when you read the pages I recommended. On those pages, the author explains the Young's theorem by a mountain-climbing analogy. Together with the visualization exercise we just did, those pages will help a lot.

Rodrigo.

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 12/18/2000 03:55 AM by Brandon; Yes, I understand now
Thanks a lot Rodrigo! I understand now. If that were to be put into words, what would that mean? Does it mean that an additional unit of K would be just as efficient as a unit of L on the exact opposite end of the isoquant? [View full text and thread]

 12/12/2000 09:54 PM by Rodrigo; On a property of the Cobb-Douglas
Hi, Brandon: Any function with continuous second-order derivatives satisfy this property: their cross second-order derivatives are equal. Such functions may be called "smooth functions". The Cobb-Douglas production function is one of [View full text and thread]

 12/12/2000 10:17 AM by Brandon; Properties of Cobb-Douglas Production function
Hi! I came across this portion under the chapter of "Production function" which reads, "An increase in labour input has the same impact on the marginal & average products of capital as does an increase in capital inputs on the [View full text and thread]