Question

The Cobb-Douglas model of production in an economy is P(L, K) = b["Kl-a where • Pis the total production (the monetary value of all goods produced in a year) • L is the amount of labor (the total number of person-hours worked in a year) • K is the amount of capital invested (the monetary worth of all machinery, equipment, and buildings) • band a are constants which characterize the particular economy. Suppose that a manufacturer uses the Cobb-Douglas model for its own yearly production, and determines from past data that if Pis measured in millions of dollars, L is measured in thousands of person-hours, and K is measured in millions of dollars, then b= 1.49 and a = 0.6. Suppose also that currently the manufacturer is operating with L = 30 and K = 8, but the labor force is decreasing at a rate of 2000 person-hours per year while invested capital is increasing at a rate of $500,000 per year. What is the current rate of change of the manufacturer's annual production, in millions of dollars per year? Enter a numerical value without units, rounded to three decimal places.

Answer 1

Given :- P- total production (millions of dollars) L amount of Labour (thousands of person-hours) K- amount of capital invested (millions of dollars) L = 30; k=8. x=0.6, b=1.49 OL =-2000 peron-hours per yr. at =-2 thousands of person - hours/yr ak $ 500,000/yo at = $0.5 million per year P=blak'd

1-2 al U al & bla-1 k at + (1-2) b Ld K (1-2-DƏK at at = (0.6) (1:49) [30) 04 (8504 (-2) (1-0.6-1 +(1-06)(1.49) (30) 0.6 0.5 ( 8 ) = -1.05379226721 + 0.658620167 = -0.39517210021 -0.395 Aus
Hence, in this way this question can be easily solved.