Answer to Question #267657 in Microeconomics for Jessica

Question #267657

suppose the production function for trucks is given by: q=kl+6l2 -(1/3)l3

where q represents the weekly quantity of trucks produced, k represents weekly capital input, and l represents weekly labor input.

 

a. Suppose k = 45; at what level of labor input does this average productivity reach a maximum? How many trucks are produced at that point?

 

b. Again assuming that k = 45, at what level of labor input does the total production reach a maximum? How many trucks are produced at that point?

 


1
Expert's answer
2021-11-18T10:23:59-0500

(a)

"q=kl+6l^2-(\\frac{1}{3})l^3"

"k=45"

"AP_l=\\frac{q}{l}"

"=\\frac{kl+6l^2-\\frac{1}{3}l^3}{l}"

"=k+6l-\\frac{1}{3}l^2"

"\\frac{dAP_l}{dl}=6-\\frac{2}{3}l"

"6-\\frac{2}{3}l=0"

"l=9."

"q=45(9)+6(9^2)-\\frac{1}{3}(9^3)=648"

Labor input=9.

No. of trucks produced=648.


(b)

"q=45l+6l^2-\\frac{1}{3}l^3"

Total production reaches maximum when marginal productivity of labor equals to zero.

"MP_l=\\frac{dq}{dl}"

"=45+12l-l^2=0"

"\\implies l=15"

"q=45(15)+6(15^2)-\\frac{1}{3}(15^3)"

"q=900"

Labor input=15.

No.of trucks produced=900.


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