Economic and Game Theory
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(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.