Assume a dynamically effecient allocation of resources. Suppose a depletable resource has a constant marginal extraction cost of $9 and the maximum amount anyone is willing to pay for the resource is $15

In the context of a dynamically efficient allocation of a depletable resource, we need to consider the intertemporal allocation of the resource over multiple periods. This involves determining the optimal consumption path that maximizes the present value of net benefits derived from the resource over time.

### Problem Setup
- **Marginal extraction cost**: $ C = \$9 $
- **Maximum willingness to pay**: $ P_{\text{max}} = \$15 $
- **Interest rate (discount rate)**: Assume an interest rate $ r $ for discounting future benefits (we'll denote this as $ r $).

To find the dynamically efficient allocation, we need to consider the Hotelling rule, which states that the marginal net benefit (price minus marginal cost) should rise at the rate of interest over time.

### Hotelling Rule
The marginal net benefit $ (P_t - C) $ at any time $ t $ should increase at the rate of interest $ r $:
\[ P_{t+1} - C = (P_t - C)(1 + r) \]

### Step-by-Step Solution
1. **Initial Conditions**:
   - **Today**: The price $ P_0 $ is such that $ P_0 = 15 $.

2. **Marginal Net Benefit Today**:
   \[ P_0 - C = 15 - 9 = 6 \]

3. **Future Marginal Net Benefit**:
   According to the Hotelling rule, the future marginal net benefit should be:
   \[ (P_0 - C)(1 + r) = 6(1 + r) \]

4. **Price in Future Period**:
   Let $ P_1 $ be the price in the next period:
   \[ P_1 - C = 6(1 + r) \]
   \[ P_1 - 9 = 6(1 + r) \]
   \[ P_1 = 9 + 6(1 + r) \]
   \[ P_1 = 15 + 6r \]

### Interpretation and Allocation
Given that the resource is depletable, the dynamically efficient allocation implies that we use some amount of the resource today and save some for the future, ensuring that the marginal net benefit rises at the rate of interest.

#### Example with a Specific Interest Rate
Suppose the interest rate $ r $ is 5% (or 0.05):

\[ P_1 = 15 + 6 \times 0.05 \]
\[ P_1 = 15 + 0.3 \]
\[ P_1 = 15.3 \]

### Conclusion
- **Today (Period 0)**: Use the resource up to the point where the price is $15, with a marginal extraction cost of $9, giving a marginal net benefit of $6.
- **Next Period (Period 1)**: The price adjusts to $15.3 to account for the interest rate, ensuring the marginal net benefit grows as expected.

This simplified model assumes a constant interest rate and marginal extraction cost. Real-world scenarios might involve more complex factors like varying interest rates, changes in extraction technology, or external market conditions.

Do you want details or have any questions?

Here are some additional questions related to this topic:

1. How would a change in the interest rate affect the dynamically efficient allocation?
2. What impact would technological advancements in extraction have on the allocation?
3. How does the assumption of a constant marginal extraction cost simplify the model?
4. What are the environmental considerations in the dynamically efficient allocation of a depletable resource?
5. How would the introduction of a substitute resource affect the price trajectory over time?
6. Can the Hotelling rule be applied to renewable resources? Why or why not?
7. What role does government policy play in the management of depletable resources?
8. How does uncertainty in future demand and supply conditions impact the efficient allocation?

**Tip:** Understanding the Hotelling rule is crucial for analyzing the optimal use of non-renewable resources over time, considering both economic and environmental factors.

Explore the dynamically efficient allocation of depletable resources using the Hotelling rule. Understand how to maximize present value of net benefits over time with a constant marginal extraction cost of $9 and a maximum willingness to pay of $15. Learn the implications of interest rates on resource allocation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation