Question

Can someone please solve parts c, d, e, and h ASAP? This is due tomorrow

2. (30 pts) Consider a production function given by: Q = 12(KL)2 – L 4 . a. (5 pts) Does this production function exhibit increasing, constant, or decreasing returns to scale? b. (5 pts) Derive the equation for the average product of labor and the marginal product of labor? c. (4 pts) Let K = 2. Find the level of L at which the marginal product of labor is the highest. d. (3 pts) Let K = 2. Find the level of L at which the marginal product of Labor is zero. 2 ECON 340 Fall 2019 Andres J Vargas e. (3 pts) Let K = 2. Based on you previous answers: Over what range of labor usage do increasing marginal returns exist? Over what range of labor usage do decreasing marginal returns exist? Over what range of labor usage do negative marginal returns exist? f. (4 pts) Let K = 2. Find the level of L at which the average product of labor is the highest. g. (3 pts) Let K = 2. At what level of L does the marginal product of labor equal the average product of labor? h. (3 pts) Let K = 2. Graph the average product of labor and the marginal product of labor identifying as many points as you can.

Answer 1

Q = 12(KL)² - L⁴

(c) When K = 2,

Q = 12(2L)² - L⁴

Q = 12 x 4L² - L⁴

Q = 48L² - L⁴

MPL = dQ/dL = 96L - 4L³

MPL is maximized when dMPL/dL = 0.

dMPL/dL = 96 - 12L² = 0

12L² = 96

L² = 8

L = 2.83

(d) When K = 2, MPL = 96L - 4L³

96L - 4L³ = 0

4L x (24 - L²) = 0

Either L = 0

Or (24 - L²) = 0, or L² = 24, or L = 9.80

(e)

Increasing (decreasing) marginal returns exist when MPL is increasing (decreasing), i.e. dMPL/dL is positive (negative).

Negative marginal returns exist when MPL is negative.

dMPL/dL = 96 - 12L²

dMPL/dL > 0 when (96 - 12L² ) > 0 or 12L² < 96, or L < 2.83.

dMPL/dL < 0 when (96 - 12L² ) < 0 or 12L² > 96, or L > 2.83.

MPL < 0 when (96L - 4L³) < 0, or 4L³ > 96L, or L² > 24, or L > 4.90.

(h) When K = 2,

MPL = 96L - 4L³

APL = Q/L = 48L - L³

Data table used:

L	MPL	APL
0	0	0
0.5	47.5	23.875
1	92	47
1.5	130.5	68.625
2	160	88
2.5	177.5	104.375
3	180	117
3.5	164.5	125.125
4	128	128
4.5	67.5	124.875

Graph: