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?
(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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