Assignment:
Q1. Let Y=L^αK^1-α, 0<α<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 α and (1-α)
Q2. 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.
Q3. 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).
Provide complete and step by step solution for the question and show calculations and use formulas.