1. Suppose a Pharmaceutical company\'s production function is given by: q-10LK-0
ID: 1114373 • Letter: 1
Question
1. Suppose a Pharmaceutical company's production function is given by: q-10LK-09K2-0.3L. where q represents the weekly quantity of pills produced and K and L are weekly capital and labor inputs respectively. a. Suppose that K-10. What is the average productivity of labor? At what level of labor does this average productivity reach its maximum? What is the level of output at this point? b. Continuing to assume K- 10, what is the marginal productivity of labor MPL? At what level of labor inputs does MP-0? What is the maximum output that can be produced with K = 10?Explanation / Answer
a)
q = 10LK - 0.9K2 - 0.3L2
K= 10
APL = q/L
= (10LK - 0.9K2 - 0.3L2 )/L
= 10K - 0.9K2/L - 0.3L
K = 10
APL = 10(10) - 0.9(10)2/L - 0.3L
= 100 - 90/L - 0.3L
dAPL/dL = 90/L2 - 0.3
Put 90/L2 - 0.3 = 0
L = (300)1/2
L = 17.32
d2APL/dL2 = - 180/L3 < 0
so at L = 17.32 APL is maximum
Now
q = 10LK - 0.9K2 - 0.3L2
= 10(17.32)10 - 0.9(10)2 - 0.3(17.32)2
= 1552.00528
b) q = 10LK - 0.9K2 - 0.3L2
MPL = dq/dL
= 10K - 0.6L
MPL = 10K - 0.6L
at K = 10
MPL = 10(10) - 0.6L
= 100 - 0.6L
100 - 0.6L = 0
L = 166.66
d2q/dL2 = - 0.6 < 0
so MPL is maximum at L = 166.66
Maximum output q =10(166.66)10 - 0.9(10)2 - 0.3(166.66)2
= 8243.33
c)
q = 10LK - 0.9K2 - 0.3L2
if production point is given by q = f(k,L) and all inputs are multiplied by positive constant then
f(tk,tL) = tf(k,L) implies CRS
f(tk,tL) > tf(k,L) implies IRS
f(tk,tL) < tf(k,L) implies DRS
q(tk,tL) = 10(tK)(tL) - 0.9(tK)2 - 0.3(tL)2
= t210kL - 0.9K2t2 - 0.3t2L2
= t2(10LK - 0.9K2 - 0.3L2 )
= t2q
Since q(tk,tL) > tq(k,L)
Therefore proctuion function exhibits increasing returns to scale
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