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Cobb-Douglas production function mathematical relationships

This is an analytical exercise from my macroeconomics book chapter 4, The Theory of Economic Growth. I need help in answering these questions in order to be better understand this model.

(1) Consider the Cobb-Douglas production function

Y/L = (K/L)^α (E)^1-α

Show that multiplying both sides of this expression by L yields
Y= Kα (EL)1-α

I'm confused because when I multiply both sides by L I get
Y= Kα (E)1-α
Should not both L's cancel out if you multiply both sides by L?. The only way to get (EL)1-α is by doing this

Y/L(L) = (K/L)α (L) (E)1-α (L) ??

(2) Using the production function as expressed in question (1), graph the relationship it implies between Y and K. Graph Y on the horizontal axis and K on the vertical axis. In constructing your graph, analytically calculate intercept and slope, and be sure your graph accurately reflects your calculations. Note that the slope of this relationship corresponds to the marginal product of capital.


Solution Preview

While it may appear that the L's should both cancel, the math is a little more complicated because exponents are involved. Remember that an L with an exponent will not be canceled out by one without an exponent, and that negative exponents are those on the bottom of a quotient. For more on this, see ...

Solution Summary

The output elasticities of labor and capital for the Cobb-Douglas production function and relationship between Y and K. (Graph not given, but described.)