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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

1 Answer

1 vote

Final Answer:

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^0.33or 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.

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