Question

A company has a linear cost function C(K,L)=K+2L and a linear production function F(K,L)=3K+L. The company has signed a contract, saying that it will produce an amount Y=20. The two factors of production K and L cannot be negative. (a) Formulate the firm's minimum cost problem as a formal constrained optimization problem

Answer 1

The firm's minimum cost problem is formulated by minimizing the cost function C(K,L) = K + 2L, subject to the constraint F(K,L) = 3K + L = 20 and the non-negativity conditions on K and L.

Step-by-step explanation:

The student's question involves formulating a minimum cost problem given a linear cost function and linear production function. The firm is tasked with producing a fixed output of Y=20 units. The cost function C(K,L) represents the total costs in terms of capital (K) and labor (L), while the production function F(K,L) represents the total output. This problem is a classic constrained optimization problem where the firm must minimize costs subject to producing a certain output level, with the additional constraints that the production inputs (K and L) cannot be negative.

To formally set up the problem, we define an objective function which is the cost function in this scenario, C(K,L) = K + 2L. The constraint is that the production function F(K,L) = 3K + L must equal the target production level Y=20. Mathematically, this can be represented as:

• Minimize C(K,L) = K + 2L
• Subject to F(K,L) = 3K + L = 20
• With K, L >= 0