Answer to Question #298599 in Microeconomics for Jabesa

Question

Consider a firm who involved in investment of coffee production short run with a function given by:Q=10KL-4L²-2K² where Q is quantity of coffee production, L is labor input and K is fixed capital input (K=8).

a) determine the average product of labor APL function

b) at what level of labor does the total output of coffee yield reach the maximum?

c) what will be the maximum achievable amount of cut-flower production?

Answer 1

$\text{a.}\text{substituting for the value of K in Q,}\\ \text{we have,}\\ \text{Q}=100(8)\text{L}-4(L)^2-2(8)^2\\ \text{Q}=800\text{L}-4L^2-128\\ \, \text{APL}=\dfrac{Q}{L}\\ \text{APL}=\dfrac{800L-4L^2-128}{L}\\ APL=800-4L-\dfrac{128}{L}\\ \text{b.}\, \text{Output is maximized at MPL=0}\\ MPL=\dfrac{dQ}{dL}\\ MPL=800-8L\\ \text{set MPL=0}\\ 800-8L=0\\ \text{collect like terms;}\\ 8L=800\\ \text{divide both sides by 8, we have;}\\ L=100\\ \text{at L=100, output is maximized.}\\ \text{c.}\, \text{To get the maximum achievable output,}\\ \text{insert L=100 to production function.}\\ \text{We then have,}\\ Q=800(100)-4(100)^2-128\\ Q=39,872.\\ \text{At the maximum value L=100,}\\ \text{the maximum achievable quantity is Q=39,872 }$