We are given two functions:

1. The Cobb-Douglas Production function: $P(L, K) = 23L^{0.5}K^{0.5}$ where $L$ represents units of labor, and $K$ represents units of capital.

2. The cost function: $C(L, K) = 300L + 200K$

The goal is to produce 16,000 units of output and find the allocation of labor and capital that minimizes the cost. To solve this, we will need to use a system of equations, incorporating both the production constraint and the cost minimization approach.

Step 1: Express the production constraint

We are given that the production target is 16,000 units: $23L^{0.5}K^{0.5} = 16,000$ Divide both sides by 23: $L^{0.5}K^{0.5} = \frac{16,000}{23} \approx 695.65$ Square both sides to eliminate the square roots: $LK = (695.65)^2 \approx 483,918.63$ This is the first equation: $LK = 483,918.63$

Step 2: Cost Minimization via Lagrange Multiplier

To minimize the cost, we use the Lagrange multiplier method. Let the Lagrangian be: $\mathcal{L}(L, K, \lambda) = 300L + 200K + \lambda(23L^{0.5}K^{0.5} - 16,000)$ We now compute the partial derivatives and set them equal to zero.

Partial derivatives:

1. $\frac{\partial \mathcal{L}}{\partial L} = 300 + \lambda \cdot \frac{23}{2} L^{-0.5} K^{0.5} = 0$
2. $\frac{\partial \mathcal{L}}{\partial K} = 200 + \lambda \cdot \frac{23}{2} L^{0.5} K^{-0.5} = 0$
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = 23L^{0.5}K^{0.5} - 16,000 = 0$

By simplifying and solving this system of equations, we can find optimal values for $L$ and $K$.

Step 3: Solve for $L$ and $K$

From the two equations involving $\lambda$, we derive a relationship between $L$ and $K$.

Let's solve this step fully.

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