 Suppose a depletable resource has a constant marginal extraction cost of $5 and the maximum
amount anyone is willing to pay for the resource is $10. Suppose also that a renewable resource
with a constant marginal extraction cost of $8 is a perfect substitute for the depletable resource.
Will the depletable resource be exhausted? If so, at what point will the resource be exhausted?

To determine whether the depletable resource will be exhausted and, if so, when it will happen, we need to consider the economic dynamics involved in substituting between the depletable and renewable resources. Here are the key details provided:

1. **Marginal extraction cost of the depletable resource (C_d):** $5
2. **Maximum willingness to pay for the depletable resource (P_max):** $10
3. **Marginal extraction cost of the renewable resource (C_r):** $8
4. **Renewable resource is a perfect substitute for the depletable resource.**

### Economic Analysis:

1. **Price Dynamics:**
   - Since the renewable resource is a perfect substitute and costs $8 to extract, the depletable resource price cannot exceed $8 once the renewable resource is available. This is because consumers would switch to the cheaper renewable resource once the depletable resource price exceeds $8.

2. **Exhaustion Condition:**
   - The depletable resource will be exhausted if its marginal extraction cost plus any scarcity rent equals the price at which consumers are willing to switch to the renewable resource. This switch happens when the price of the depletable resource reaches $8.

### Scarcity Rent:

The scarcity rent is the difference between the price of the depletable resource and its marginal extraction cost. This rent increases as the resource becomes scarcer, driving up the price.

\[ \text{Scarcity Rent} = P - C_d \]

### Transition Point:

1. **Initial Price and Marginal Extraction Cost:**
   - Initially, the depletable resource is extracted at $5 per unit. As the resource is used, its price will rise due to increasing scarcity rent.

2. **Price at Transition:**
   - When the price of the depletable resource reaches $8 (the marginal extraction cost of the renewable resource), consumers will switch to the renewable resource.

3. **Exhaustion Point:**
   - The depletable resource will be exhausted at the point where its price, due to scarcity, reaches $8.

### Detailed Calculation:

- The maximum amount anyone is willing to pay for the depletable resource is $10.
- The marginal extraction cost of the depletable resource is $5, so the maximum possible scarcity rent is $10 - $5 = $5.
- However, since the renewable resource costs $8, the transition price for the depletable resource will be $8, not $10.

\[ \text{Scarcity Rent at Transition} = $8 - $5 = $3 \]

### Conclusion:

- The depletable resource will be exhausted at the point where its price reaches $8.
- This occurs because, at $8, consumers will switch to the renewable resource, which is a perfect substitute and costs the same to extract.

**Summary:**
- The depletable resource will indeed be exhausted.
- It will be exhausted at the point where its price, due to increasing scarcity rent, reaches $8.

Would you like more details or have any questions?

Here are 8 related questions:
1. How is scarcity rent calculated in the context of depletable resources?
2. What factors could influence the rate at which the depletable resource is exhausted?
3. How would an increase in the marginal extraction cost of the renewable resource affect the exhaustion point?
4. What role does consumer willingness to pay play in resource allocation?
5. How could technological advancements impact the dynamics between depletable and renewable resources?
6. What are some real-world examples of perfect substitutes in natural resource markets?
7. How does the concept of opportunity cost apply to resource extraction decisions?
8. What are the implications of resource exhaustion for long-term economic planning?

**Tip:** When analyzing resource allocation, consider both the economic and environmental impacts of extraction and consumption patterns to achieve sustainable outcomes.

Explore the economic analysis of depletable resource exhaustion and renewable substitution. Understand the conditions under which a depletable resource will be exhausted and at what price point. Includes detailed calculations and economic dynamics.

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