Question

# Consider the Production Function Q = 19L^2–7L^3 where Q = output and L = labor units.(a)...

Consider the Production Function Q = 19L^2–7L^3 where Q = output and L = labor units.(a) Derive the Marginal Product (MP) Function and the Average Product (AP) Function.(b)Determine the point of inflection of the Production Function.(c)Graph the Production Function in the first quadrant and directly below it graph the MP and AP functions.(d)The Labor market is efficient and the market supply of Labor is W=7 +0.5L. How much labor (how many labor units) would the producer hire and what would be the corresponding output?

We have a Production Function Q = 19L^2–7L^3

(a) Marginal Product (MP) Function = dQ/dL = 19*2L - 7*3L^2 which becomes MPL = 38L - 21L^2.

Average Product (AP) Function = Q/L = 19L - 7L^2

(b) Point of inflection of the Production Function has dMPL/dL = 0. This gives 38 - 42L = 0 or L = 0.905

(c) Graph is provided below

(d) The Labor market is efficient and the market supply of Labor is W = 7 +0.5L. Labor demand is MPL = 38L - 21L^2. Market is in equilibrium when W = MPL

7 + 0.5L = 38L - 21L^2

21L^2 - 37.5L + 7 = 0

L = 1.57

Wage rate = 7+1.57 = \$8.57