Cobb-Douglas Production Function Calculator - Cost Minimization Problem
Cobb-Douglas Production Function Calculator - Cost Minimization Problem helps solving the cost minimization problem given Cobb-Douglas Production Function
What is Cobb-Douglas Production Function?
In economics, a production function represents the relationship between the output and the combination of factors, or inputs, used to obtain it.
The Cobb-Douglas production function is a particular form of the production function. It is widely used because it has many attractive characteristics.
The Formula of Cobb-Douglas production function
The basic form of the Cobb-Douglas production function is as follows:
Q(L,K) = A Lβ Kα
Where:
Q is the quantity of products.
A is a positive constant (Called Total Factor Productivity).
L is the quantity of labor.
K is the quantity of capital.
α is the output elasticity of capital.
β is the output elasticity of labor.
α and β are constants between 0 and 1.
Cobb-Douglas Production Function Calculator - Cost Minimization Problem
Solving Cost Minimization Problem given Cobb-Douglas Production Function
The goal: minimize total cost:
TC = PL L + PK K
Constraint: produce amount Qo = Q(L,K)
Where:
PL = per unit cost of Labor
Pk = per unit cost of Capital
Constraint: produce amount Qo = Q(L,K)
Key relationships:
(1) Tangency Condition (tc): MPL / MPK = (∂Q/∂L)/(∂Q/∂K) = PL / PK
(2) Output Constraint: Qo = Q(L,K)
Step 1: take partial derivatives of Q to get the tangency condition (tc):
MPL / MPK = PL / PK
Step 2: rearrange the tangency condition to express K as the dependent variable.
Step 3: plug the expression for K into the output constraint to solve for L.
Step 4: plug the solution for L into the formula for K derived in Step 2 to solve for K.
Step 5: Plug your solutions for L and K into the cost equation (TC = PL L + PK K ) to find out the minimum cost of producing Q.
To check your answers:
Is the tangency condition met?
Are you producing your targeted level of output (Q)?