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MPK, steady-state, golden-rule

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(question is attached below) please finish a,b,c,d questions (with equations)

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Determine production function.

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Here are the answers to your questions.

Question a
I'll use here the first approach suggested by the hint. The production function is.

Y = A*K^(3/10)*L^(7/10)

Taking the derivative with respect to K, we get:

dY/dK = (3/10)*A*K^(-7/10)*L^(7/10)

Now let's multiply and divide by K:

dY/dK = (3/10)*A*K^(-7/10)*L^(7/10)*K/K

= (3/10)*A*K^(3/10)*L^(7/10)/K

But now, notice that A*K^(3/10)*L^(7/10) is equal to Y. Therefore we get:

dY/dK = (3/10)*Y/K

Since we know that K/Y=3 (because capital to output ratio is 3), then Y/K = 1/3, so we get:

dY/dK = 1/10

Question b
In the steady state, the growth of output should equal (n+g). Here's the mathematical proof. I will use the following notation: the derivative of a variable X with respect to time will be simply X'. So when I write Y', I will be talking about dY/dt, ie the derivative of output with respect to time.

In the Solow model with change in technology (as in this question) the steady state is assumed to be reached when the output per "effective labour" is growing at rate zero. Just as Y/L denotes capital-per-labour unit, Y/(A*L) is the capital-per-effective-labor. Intuitively, labour is adjusted by the technology level, hence the term "effective" labour. So we have that, in the ...

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