Question

Catalina Films produces video shorts using digital editing equipment, KK, and editors L. The firm has the production function =30K0.670.33Q=30K0.67L0.33, where Q is the hours of edited footage. The wage is $25$25, and the rental rate of capital is $50$50. The firm wants to produce 3,000 units of output at the lowest possible cost.

a. The constrained optimization problem is minK,LC=minK,LC=

RK+WLs.t.3,000=30K0.67L0.33

.

b. The Lagrangian of this cost‑minimization problem is minK,L,λL(K,L,λ)=minK,L,λL(K,L,λ)=

50K+25L +λ(30K0.67L0.33)

.

c. Rounding to the nearest integer, the first‑order conditions are

∂L∂K∂L∂K =

= 0

∂L∂L∂L∂L =

= 0

∂L∂λ∂L∂λ =

= 0

d. The cost‑minimizing quantities of capital and labor that solve the first‑order conditions are K=K=

and L=L=

.

e. What is the total cost of producing 3,000 units?

$ 0

$ 200

$ 7,500

$ 5,000

f. What is the value of λ, or marginal cost, that satisfied the first‑order conditions?

$ 5

$ 2.5

$ 50

$ 25

Answer 1

a. The constrained optimization problem is minK LC=minK LC= 50K + 25L s.t.
`3000 = 30K^0.67 L^0.33.`

`The Lagrangian of this cost-minimization problem is minK LλL K L λ)=minKLλ 50K + 25L + λ(30K^0.67 L^0.33).`

c. Rounding to the nearest integer the first-order conditions are
`∂L/∂K = 0∂L/∂L = 0 ∂L/∂λ = 0.`

d. The cost-minimizing quantities of capital and labor that solve the first-order conditions are K = [value] and L = [value].

e. The total cost of producing 3,000 units is $ value.

f. The value of 0
`=30K^0.67L^0.33`or marginal cost that satisfies the first-order conditions is $[value].

Step-by-step explanation:

Catalina Films aims to minimize costs while producing 3000 units expressed as the optimization problem minK LC=50K+25L subject to the production function 300
`0=30K^0.67L^0.33.` The Lagrangian for this cost-minimization problem is formed including the Lagrange multiplier
`(λ)` to incorporate the constraint into the objective function.

By differentiating the Lagrangian with respect to K L and
`(λ)`and setting the derivatives equal to zero, the first-order conditions are obtained. Solving these conditions yields the cost-minimizing quantities of capital (K) and labor (L). The values obtained for K and L are then inserted into the production function to calculate the total cost of producing 3,000 units.

Finally the Lagrange multiplier
`(λ)`represents the marginal cost. It is found by substituting the optimal values of K and L into the first-order conditions. The rounded value of
`λ` gives the marginal cost. In this way Catalina Films can determine the most cost-efficient combination of capital and labor to achieve its production goal while considering the constraints imposed by the production function.