Final answer:
The Cobb–Douglas production function is maximized by using the method of Lagrange multipliers, setting up a Lagrange function, taking its partial derivatives, and solving the resulting equations simultaneously. Upon a marginal change in the cost constraint, the change in Q is approximated by the value of the Lagrange multiplier from the initial optimization. For the exact new maximum value, the system must be re-solved using the updated constraint.
Step-by-step explanation:
To solve the optimization problem for the Cobb–Douglas production function using Lagrange multipliers, we first set up our Lagrange function:
L = 20 K1/2 L1/2 + λ(64 - K - 4L)
We then take partial derivatives of L concerning K, L, and λ and set them equal to zero to find our critical points:
- ∂L/∂K = 10K-1/2L1/2 - λ = 0
- ∂L/∂L = 10K1/2L-1/2 - 4λ = 0
- ∂L/∂λ = 64 - K - 4L = 0
By solving these equations simultaneously, we find the values of K, L, and λ that maximize Q.
For part b, we reassess the change in the maximum value of Q as the cost constraint changes from 64 to 65. The new constraint is K + 4L = 65, which implies a marginal change. The Lagrange multiplier from the first part of the problem, λ, tells us how the optimal value of the production function, Q, changes about changes in the constraint. So, the change in Q will be approximately equal to λ.
To find the new maximum, we re-resolve the system with the updated constraint.