Question

The production function of a certain country is given by Q = f(K,L) = 90K^1/3,L^2/3 where Q is the number of output produced in units of millions , K is the capital expenditures in units of $1 million and L is the size of labor force in thousands of worker – hours .

1. Find the level of output if the capital expenditure is Tsh 27 million dollars and the labor level is 8,000 workers – hours.
2. At the same level of the capital expenditure and labor level in part (i), compute the marginal productivity of labor , interpret the results.

Answer 1

Given Q = f(K,L) = 90K^1/3,L^2/3 where K is capital expenditure in units of 1 million and L is size of labor in thousand of worker hours.

1. Given K = 27  and L = 8 (as L is in thousand , hence 8000 worker hours = 8000/1000 = 8 L)

Putting this in production function:

=> Q = f(K,L) = 90K^1/3*L^2/3

=> Q = 90 * (27)^2/3 * (8)^2/3

=> Since 27 = 3³ and 8 = 2³ hence,

=> Q = 90 * (3)² * (2)²

=> Q = 90 * 9 * 4

=> Q = 3240 million units.

Answer: Hence total production of Q is 3240 million units.

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2. Marginal productivity of labor is the additional increase in production due to one unit increase in labor. Or in this case, the additional increase in Q due to 1000 increase in labor since 1 uni of L = 1000 labor.

Mathematically it is obtained by differentiating the production function with respect to Q.

Thus MP_L = marginal product of labor = dQ/dL

=> dQ/dL = d(90K^1/3,L^2/3 )/dL

=> MP_L = 90K^1/3 *2/3L^1-2/3

=> MP_L = 180/3 * K^1/3*L^-1/3

=> MP_L = 60 * K^1/3/L^1/3

Putting the value of K = 27 and L = 8

=> MP_L = 60 * 27^1/3/8^1/3

=> MP_L = 60 * 3/2 (As 27 = 3³ hence 3 = 27^1/3 and similarly 2 = 8^1/3)

=> MP_L = 90

Thus this suggests that if the L goes up by 1 or labor goes up by 1000, the quantity produced will go up by 90 million units.