Question

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?

 

Answer 1

(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.