Question

1)

Using the firm's information shown below answer the following questions:

Q= f(K, L) =10KL

w=$25

r=$40

MPk= 10L

MPl= 10K

A)

in the short run if the firm's capital is fixed at 25 units, how much will it cost them to produce q=500

B)

Will the firm do in the long run to minimize the cost of producing q=500

C)

What are the firms cost saving from the short-run is (2 when Q=100, 4 when Q=200 and 10 when Q= 500) to the long run is ( 3.87 when Q=100, 5.47 when Q=200, and 8.6 when Q=500)

Answer 1

a)

Given

Q=10KL

On rearranging, we get

L=0.1Q/K

TC=r*K+w*(0.1*Q/K)

Set K=25, w=$25 and K=40

SRTC=25*40+2.5*Q/25=1000+0.10*Q

Now set Q=500

TC=1000+0.10*500=$1050

b)

Cost minimization requires

MPL/MPK=w/r

10K/10L=25/40

K/L=0.625

K=0.625*L

Given

Q=10KL

Set K=0.625*L

Q=10*0.625*L*L

L^2=(Q/6.25)

L=(Q/6.25)^0.5

K=0.625*L=0.625*(Q/6.25)^0.5

Set Q=500

L=(500/6.25)^0.5=8.9443

K=0.625*L=0.625*8.9443=5.5901

In the long run K should be reduced to 5.59 units and L should be increased to 8.94 units

C)

LRTC=w*L+r*K

LRTC=25*(Q/6.25)^0.5+40*0.625*(Q/6.25)^0.5=50*(Q/6.25)^0.5

If output is 100 units

SRTC=1000+0.1*Q=1000+0.1*100=$1010

LTRC=50*(Q/6.25)^0.5=50*(100/6.25)^0.5=$200

Cost savings=1010-200=$810

If output is 200 units

SRTC=1000+0.1*Q=1000+0.1*200=$1020

LTRC=50*(Q/6.25)^0.5=50*(200/6.25)^0.5=$282.84

Cost savings=1020-282.84=$737.16

If output is 500 units

SRTC=1000+0.1*Q=1000+0.1*500=$1050

LTRC=50*(Q/6.25)^0.5=50*(500/6.25)^0.5=$441.21

Cost savings=1020-441.21=$602.79