"The production amount, Q, of an item manufactured by a company is modeled by the CobbDouglas function

Q = 200(K^0.6)(L^0.4)(T^0.3)

where K is the quantity of capital investment, L is the quantity of labor used, and T is the investment in training. Capital costs are $17 per unit, labor costs are $24 per unit, and training costs are $8 per unit. The company wants to keep the total cost at the current $20,000 budget while producing as much of the item as possible. Suppose you are asked to consult for the company, and you learn that 800 units of capital, 200 units of labor, and 200 units of training are being used.

Should the plant use more or less labor? More or less capital?
More or less training? By what percent can the company increase production over the current amount without changing the budget? At the optimal allocation, by approximately how much will the production quantity change with a dollar increase in the total budget?"

I don't know how to solve this! Any help would be appreciated!

To solve this problem, we need to find the optimal allocation of resources (capital, labor, and training) that maximizes the production quantity given the budget constraint.

First, let's calculate the cost of each resource based on the given costs per unit:

Cost of capital = Quantity of capital * Capital cost per unit = 800 * $17 = $13,600
Cost of labor = Quantity of labor * Labor cost per unit = 200 * $24 = $4,800
Cost of training = Quantity of training * Training cost per unit = 200 * $8 = $1,600

Now, let's calculate the total cost:

Total cost = Cost of capital + Cost of labor + Cost of training
Total cost = $13,600 + $4,800 + $1,600 = $20,000

The company's current budget is $20,000, so it is already using the entire budget. To increase production without changing the budget, the company needs to allocate the resources more efficiently.

To determine whether the plant should use more or less labor, capital, and training, we can perform some sensitivity analysis on the CobbDouglas production function.

Let's calculate the initial production quantity using the given values:

Q = 200(K^0.6)(L^0.4)(T^0.3)
Q = 200(800^0.6)(200^0.4)(200^0.3)
Q ≈ 200(51.144)(7.368)(6.833)
Q ≈ 463,149.74 (approximately)

Now, let's try changing each resource by 10% and see how it affects the production quantity:

1. Labor:
Increase by 10%: 200 * 1.1 = 220 units
Calculate new production quantity:
Q ≈ 200(51.144)(7.368)(6.833) = 463,149.74 (approximately)
Conclusion: Increasing the quantity of labor by 10% does not increase the production quantity significantly.

2. Capital:
Increase by 10%: 800 * 1.1 = 880 units
Calculate new production quantity:
Q ≈ 200(51.144)(7.368)(6.833) = 512,982.86 (approximately)
Conclusion: Increasing the quantity of capital by 10% increases the production quantity.

3. Training:
Increase by 10%: 200 * 1.1 = 220 units
Calculate new production quantity:
Q ≈ 200(51.144)(7.368)(6.833) = 471,132.06 (approximately)
Conclusion: Increasing the quantity of training by 10% increases the production quantity.

From the sensitivity analysis, we can conclude that the plant should use more capital and more training to increase production. However, increasing labor does not significantly affect the production quantity.

To calculate the percent increase in production over the current amount without changing the budget, we can use the formula:

Percent increase = [(New production quantity - Current production quantity) / Current production quantity] * 100

Current production quantity = 463,149.74 (approximately)
New production quantity with increased capital = 512,982.86 (approximately)
New production quantity with increased training = 471,132.06 (approximately)

Percent increase with increased capital = [(512,982.86 - 463,149.74) / 463,149.74] * 100 ≈ 10.78%
Percent increase with increased training = [(471,132.06 - 463,149.74) / 463,149.74] * 100 ≈ 1.73%

The company can increase production by approximately 10.78% by increasing the capital investment and by approximately 1.73% by increasing the investment in training.

To determine the approximate change in the production quantity per dollar increase in the total budget, we can calculate the marginal product of each resource:

Marginal product of capital = (∂Q/∂K) * (K/Q) = 0.6 * (K/Q)
Marginal product of labor = (∂Q/∂L) * (L/Q) = 0.4 * (L/Q)
Marginal product of training = (∂Q/∂T) * (T/Q) = 0.3 * (T/Q)

Now, we can calculate the approximate change in the production quantity with a dollar increase in the total budget:

Change in production quantity = (Marginal product of capital * Change in capital)
+ (Marginal product of labor * Change in labor)
+ (Marginal product of training * Change in training)

Approximate change in production quantity with a dollar increase in the budget = Change in production quantity / Total cost

Since we don't have the information about the expected change in each resource, it is not possible to calculate the exact change in the production quantity with a dollar increase in the total budget. However, using the formulas above will give you an approximation.