Cobb-Douglas Production Function Properties

andrew

Right, I'm not sure how to insert LaTEX stuff into posts, so this might seem a little messy.

I'm studying for a test tomorrow. Our lecturer likes to not include some information on slides, instead just writing 'show,' and the book isn't any help. Please, could someone fill in for me the missing 'show' bits, or point me in the right direction, in the following cases.

For a Cobb-Douglas Production function: A(K^α)(L^β)

MRTS = (β/α)(K/L)

Show that this is the case.

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Marginal Product of Capital:

α.APk (that K is a subscript of P)

Show that this is the case

I thought that it'd just be δY/δK which gives αA(K^α-1)(L^β) so I don't know where Pk comes from or what it is. The same is the case with Marginal product of labour.

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Marginal Product of Labour:

β.APl (The L is a subscript of P)

Show that this is the case.

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Euler's Theorem:

MPkK + MPlL = (ε)Y [the first k and the first l are subscripts]

Where ε is the degree of homogeneity. Show.


Thanks for any help that can be provided.

andrew

If someone is looking at this post thinking 'I know that shit, I'll just reply later' ....please reply now.

Économiste Monétaire

I just saw your post now, it's usually a good idea to ask a question a few days before an exam rather than on a Sunday night before it .

Question one, the MRTS is simply the MPL/MPK.

[latex]\displaystyle \frac{\beta AK^{a}L^{\beta - 1}}{a AK^{a - 1}L^{\beta}}[/latex]

Simply move the exponents on L below, and K above to get

[latex]\displaystyle \frac{\beta AK^{a - a %2B 1}}{a AL^{\beta - \beta %2B 1}}[/latex]

which is just
[latex]\displaystyle \frac{\beta K}{a L}[/latex] .

The second question I assume he's taken P to be = (K^α)(L^β) and the [latex]\displaystyle P_{k}[/latex] is just the partial derivative of P with respect to k. So your answer is correct. The same method is used for question 3.

Question 4 isn't in my head at the minute, I'll post up in a few mins if/when I remember it.

Économiste Monétaire

Bah, I should have remembered question 4...

Take an example like this:

[latex]\displaystyle Y = L^{0.7}K^{0.3}[/latex]

[latex]\displaystyle MP_{l} = 0.7L^{0.7 - 1}K^{0.3}[/latex]

[latex]\displaystyle MP_{k} = 0.3L^{0.7}K^{0.3 - 1}[/latex]

[latex]\displaystyle [MP_{l} \times L] + [MP_{k} \times K] = \epsilon Y[/latex]

=> [latex]\displaystyle (0.7L^{0.7 - 1}K^{0.3})L + (0.3L^{0.7}K^{0.3 - 1})K = \epsilon Y[/latex]

=> [latex]\displaystyle 0.7Y + 0.3Y = 1 Y[/latex]

[latex]\displaystyle \epsilon = 1[/latex] i.e. the degree of homogeneity is 1.

Does that help?

andrew

Thanks a lot. The test went pretty excellently.

Économiste Monétaire

No problemo. I'm glad your exam went well .