A firm uses capital and labor to produce output according to the production func

Question

A firm uses capital and labor to produce output according to the production function q= 4rad(KL), for which MPL = 2rad(K/L) and MPK = 2rad(L/K).

a) If the wage w=$4 and the rental rate of capital r=$1, what is the least expensive way to produce 16 units of output?

b) What is the minimum cost of producing 16 units?

c) Show that for any level of output q, the minimum cost of producing q is$q.

d) Explain how a 10% wage tax would affect the way in which the firm chooses to produce any given amount of output.

Explanation / Answer

given,

q = 4sqrt(KL) , MPL = 2sqrt(K/L) , MPK = 2sqrt(L/K)

a) for producing least expensive way to produce 16 units of output , condition is

MPL/MPK = w/r

=>2sqrt(K/L)/2sqrt(L/K) = 4/1

=> sqrt(K/L)/sqrt(L/K) =4

=>sqrt( K/L*K/L) =4

=> K/L =4 ----(1)

as given q=16 units,

so 16 =4sqrt(KL)

=> sqrt(KL) =4

=> KL =16 ----- (2)

so solving equations (1) and(2)

we get (4L)*L =16

=> L=2 , so putting L=2 in equation (1) ,

we get K/2 =4

=> K=8

so least expensive way is use K=8 and L=2 for producing 16 units

b)minimum cost of producing 16 units = w*L+r*K

=4*2+1*8

=16

=$16

so minimum cost of producing 16 units =$16

c)as cost =w*L+r*K

and as w=4 , r=1

so cost = 4L+K

so for producing q units, q = 4*sqrt(KL)

=> sqrt(KL) = q/4

=> KL = q^2/16 ----(3)

and for minimum cost MPL/MPK = w/r

=> 2sqrt(K/L)/2sqrt(L/K) = 4/1

=> K/L =16 ----(4)

so solving (3) and (4)

we get 16L*L =q^2/16

=> q^2 =256L^2

=> q=16L

and substituting q=16L in (3), we get q=3q/4

so as cost =4L+K

= 4*(q/16) +3q/4

=q

so minimum cost =$q for producing q units

d)if there is 10% wage tax,

then wage = 4*(1-0.1)

=$3.6

so as MPL/MPK =w/r

so 2sqrt(K/L)/2sqrt(L/K) = 3.6/1

=> K/L =3.6

and as KL = q/16

solving this we get optimum L = sqrt( q/16*3.6) and K = sqrt( 3.6q/16)

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