Final answer:
The optimal input mix of harvester machine hours and farmworker labor hours to produce 100 bushels of corn is found using Lagrange multiplier technique. By setting up the Lagrangian function, incorporating the production function with cost constraints, and solving the resulting equations for maximum production, the optimal values of K (capital input) and L (labor input) can be determined. Without specific values for K and L, a definitive answer cannot be provided.
Step-by-step explanation:
Optimal Input Mix for Producing 100 Bushels of Corn
To find the optimal input mix of harvester machine hours (K) and farmworker labor hours (L) for producing 100 bushels of corn, we need to solve the problem with the help of the Lagrange multiplier technique. The farmer's production function is given by Q(K,L) = 2K0.67L0.33, and the goal is to maximize this output subject to a cost constraint, which is the sum of the costs of K and L given their respective rates ($20 and $10).
First, we must write the Lagrangian function which incorporates the production function and the cost constraint:
L(K, L, λ) = 2K0.67L0.33 - λ (20K + 10L - C), where λ is the Lagrange multiplier and C is the total cost we aim to determine for producing 100 bushels.
We would then take partial derivatives of the Lagrangian with respect to K, L, and λ, and set them equal to zero to find the points of maximum output for the given cost. Solving these equations would provide us with the values of K, L, and λ that maximize the production of corn given the cost constraints. However, since the values of K and L aren't specified here, we cannot give a definitive answer.