Question

craft corp's production function is given by q = 5min(K, 2L), where q is the quantity produced and K and L are the amounts of capital and labor input. Input prices are r = 2 and w = 6.

a) find the long run cost function

b) draw the isoquants and isocost and show the equilibrium when q = 2000. what is the total cost to produce q = 2000? what is the labor cost to produce q = 2000? what is the capital cost to produce q = 2000?

c) supposed in the short run K is fixed @ 20. find the short run cost function. represent the short-run and long-run cost functions in a diagram together.

Answer 1

Craft crop's production function is given as,

q = 5.min{K, 2.L}

Now, when K > 2.L we get,

q = 5.(2.L)

or, L = q/10..........(1)

And, when K < 2.L we get,

q = 5.K

or, K = q/5...........(2)

Wage rate is w = 6 and rent is r = 2.

Hence, the long run cost function is

LC = w.L + r.K

or, LC = 6.L + 2.K

or, LC = 6.(q/10) + 2.(q/5)

or, LC = 3.q/5 + 2.q/5

or, LC = q

(a) The long-run cost function is

LC = q

(b) The isoquant at q = 2000 is

min{K, 2.L} = 2000

or, min{K, 2.L} = 400

And, the isocost curve is

LC = q = 2000

or, 6.L + 2.K = 2000

or, 3.L + K = 1000

The equilibrum will be along the line

K = 2.L

Putting this in the isocost line we get,

3.L + 2.L = 1000

or, L* = 200

and, K* = 2.L* = 2×200

or, K* = 400

The equilibrum is at (L*, K*) = (200, 400)

The following diagram shows the equilibrum K and L.

The labor cost to produce q = 2000 is

= w.L*

= 6×200

= 1200

The labor cost to produce q = 2000 is 1200.

The capital cost to produce q = 2000 is

= r.K*

= 2×400

= 800

The capital cost to produce q = 2000 is 800.

(d) If K = 20 in the short run, we get

q = 5.min{20, 2.L}

Now, in the short run,

When, L > 10 we get,

q = 5×20 = 100

And, when L < 10 we get,

q = 5×2.L = 10.L

or, L = q/10

Hence, the short run cost function is

SC = w.L + r.K

or, SC = 6.(q/10) + 2×20

or, SC = 3q/5 + 40

The following diagram shows the long run and short run cost function together.

Hope the solutions and diagrams are clear to you my friend.