Suppose a product can be produced using virgin ore at a marginal cost of M*C_{1} = 0.5Q_{1} and with recycled materials at a marginal cost given by M*C_{2} = 5 + 0.1Q_{2} Suppose the inverse demand curve was given by P = 20 - 0.5(Q_{1} + Q_{2}) Under efficiency, how many units of the product would be produced with virgin ore and how many units with recycled materials?

To determine the efficient allocation of production between virgin ore (\(Q_1\)) and recycled materials (\(Q_2\)), we need to equalize the marginal costs with the marginal benefit derived from the inverse demand curve.

### Inverse Demand Curve
The inverse demand curve is given by:
\[ P = 20 - 0.5(Q_1 + Q_2) \]

### Marginal Costs
The marginal cost for producing with virgin ore (\(Q_1\)) is:
\[ M C_{1} = 0.5Q_{1} \]

The marginal cost for producing with recycled materials (\(Q_2\)) is:
\[ M C_{2} = 5 + 0.1Q_{2} \]

### Efficient Production
Efficiency requires that the marginal cost equals the marginal benefit (price) for both types of production. The marginal benefit is derived from the inverse demand curve:

\[ P = 20 - 0.5(Q_1 + Q_2) \]

Set the marginal costs equal to the price:

1. For virgin ore:
\[ 0.5Q_1 = 20 - 0.5(Q_1 + Q_2) \]

2. For recycled materials:
\[ 5 + 0.1Q_2 = 20 - 0.5(Q_1 + Q_2) \]

### Solving the Equations

1. **From the equation for virgin ore:**
\[ 0.5Q_1 = 20 - 0.5Q_1 - 0.5Q_2 \]
\[ 0.5Q_1 + 0.5Q_1 = 20 - 0.5Q_2 \]
\[ Q_1 = 20 - 0.5Q_2 \]

2. **From the equation for recycled materials:**
\[ 5 + 0.1Q_2 = 20 - 0.5Q_1 - 0.5Q_2 \]
\[ 5 + 0.1Q_2 + 0.5Q_2 = 20 - 0.5Q_1 \]
\[ 5 + 0.6Q_2 = 20 - 0.5Q_1 \]
\[ 0.5Q_1 = 20 - 5 - 0.6Q_2 \]
\[ 0.5Q_1 = 15 - 0.6Q_2 \]
\[ Q_1 = 30 - 1.2Q_2 \]

Now, we have two equations for \(Q_1\):
\[ Q_1 = 20 - 0.5Q_2 \]
\[ Q_1 = 30 - 1.2Q_2 \]

Set these equations equal to each other to find \(Q_2\):
\[ 20 - 0.5Q_2 = 30 - 1.2Q_2 \]
\[ 0.7Q_2 = 10 \]
\[ Q_2 = \frac{10}{0.7} \]
\[ Q_2 \approx 14.29 \]

Substitute \(Q_2\) back into the first equation to find \(Q_1\):
\[ Q_1 = 20 - 0.5(14.29) \]
\[ Q_1 = 20 - 7.145 \]
\[ Q_1 \approx 12.86 \]

### Conclusion
Under efficiency:
- The quantity produced with virgin ore (\(Q_1\)) is approximately \(12.86\) units.
- The quantity produced with recycled materials (\(Q_2\)) is approximately \(14.29\) units.

Do you want details or have any questions?

Here are some additional questions related to this topic:

1. How would the production quantities change if the marginal cost for recycled materials increased?
2. What would happen to the efficiency if the marginal cost of virgin ore was constant instead of increasing with quantity?
3. How does the inverse demand curve affect the allocation of production between virgin ore and recycled materials?
4. What are the environmental implications of producing more with recycled materials versus virgin ore?
5. How would a subsidy for recycled materials impact the efficient allocation?
6. What role does consumer preference play in determining the optimal production mix?
7. Can we derive the total cost function from the given marginal cost functions?
8. How would a tax on virgin ore production impact the quantities produced?

**Tip:** Always check the units and ensure consistency throughout your calculations to avoid errors.

Explore efficient production allocation between virgin ore and recycled materials using microeconomic principles. Learn how to determine production quantities based on marginal costs and an inverse demand curve. Case study analysis includes calculations and implications for economic efficiency.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation