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# Cobb-Douglas

1. Let Y=L^&#945;K^1-&#945;, 0<&#945;<1; let y=lnY, l=lnL, k=lnK.
a.Show that this production function is linear in its natural logarithms.
b.Show that the instantaneous percentage change in out put is the weighted sum of the instantaneous change in L and K, with weights &#945; and (1-&#945;)

2. Suppose that the production function in Question 1 represents macroeconomic output and suppose further that capital and labor grow at different rates according to the equations L=Loe^nt, K=Koe^mt. Derive dY/dt and d(Y/L)/dt.

3. Consider the amount of money, M, invested at an instantaneous rate of interest r. Show that the number of years, n, that it will take for the balance to grow to 2M is approximately 72/r%. (Consider r to be r/5).

#### Solution Preview

1. Let Y=L^&#945;K^1-&#945;, 0<&#945;<1; let y=lnY, l=lnL, k=lnK.
a.Show that this production function is linear in its natural logarithms.
Ln Y = ln (L^&#945; K^1-&#945;)
Ln Y = ln (L^&#945;) +ln(K^1-&#945;)
Ln Y =&#945; ln (L) + (1-&#945;)ln K
i.e.,
y = &#945; l + (1-&#945;) k
this is a linear function

b. Show that the instantaneous percentage change in out put is the weighted sum of the instantaneous change in L and K, with weights &#945; and (1-&#945;)

From: y = &#945; l + (1-&#945;) k, we take total ...

#### Solution Summary

The solution answers 3 questions related to economics. The questions are stated below. The answers are well explained and thorough.

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