Question

A firm’s production function is q = 10KL with per unit input prices for labor w = 3 and capital r = 2. Support your answers with a graph of isoquant-isocosts.

a. Calculate the least-cost input combination of L and K to produce 60 units of output.

b. Suppose the wage decreases to $2. How does this affect input use holding constant output at 60?

c. What are the total costs of producing the two output levels in parts (a) and (b)?

Please complete all parts and show work, thanks!!!

Answer 1

a)

Given

q=10KL

Marginal product of labor=MPL=dq/dL=10K

Marginal product of capital=MPK=dq/dK=10L

Cost minimization requires

MPL/MPK=w/r

10K/10L=3/2

2K=3L

K=1.5 L

Now desired output is 60 units i.e. q=60

So,

q=10KL

60=10KL

KL=6

Set K=1.5L

1.5L*L=6

L^2=4

L=2 [Required (Cost minimizing) units of labor]

K=1.5L=1.5*2=3 [Required (Cost minimizing) units of capital]

b)

Since input price ratio has changed, optimal combination of L and K will change.

Now w=2,

Cost minimization requires

MPL/MPK=w/r

10K/10L=2/2

K=L

Now desired output is 60 units i.e. q=60

So,

q=10KL

60=10KL

KL=6

Set K=L

L*L=6

L^2=6

L=2.45 [Required (Cost minimizing) units of labor]

K=L=2.45 [Required (Cost minimizing) units of capital]

c)

In case of part a, total cost is given

TCa=wL+rK=3*2+2*3=12

In case of part b, total cost is given

TCb=wL+rK=2*2.45+2*2.45=9.80