Assume the inverse demand function for natural gas, a depletable resource, is P = 15-0.250. Assume the natural gas will be consumed over two periods and the discount rate is 5% (meaning r = 0.05). The marginal cost to supply natural gas is $3 per unit. If there are 100 units of natural gas available to be allocated between the two periods, how many units would be allocated in period 1 and how many would be allocated in period 2 in a dynamically efficient allocation

To determine the dynamically efficient allocation of natural gas over the two periods, we need to maximize the present value of net benefits (consumer surplus minus cost) from consumption in both periods. Let's outline the steps for this process.

### Given:
- Inverse demand function: $ P = 15 - 0.25Q $
- Discount rate: $ r = 0.05 $
- Marginal cost: $ MC = 3 $
- Total units available: $ 100 $ units
- Time periods: $ t = 1, 2 $

### Step 1: Calculate the present value of marginal net benefit (PVMB)

The net benefit in each period is given by the consumer surplus minus the cost. The consumer surplus is the area under the demand curve above the price level, up to the quantity consumed.

First, let's rewrite the inverse demand function for both periods:
\[ P_1 = 15 - 0.25Q_1 \]
\[ P_2 = 15 - 0.25Q_2 \]

Where $ Q_1 $ and $ Q_2 $ are the quantities consumed in periods 1 and 2, respectively. Note that $ Q_1 + Q_2 = 100 $.

### Step 2: Calculate the marginal net benefit (MNB)

MNB in each period is the price minus the marginal cost:
\[ MNB_1 = (15 - 0.25Q_1) - 3 \]
\[ MNB_2 = (15 - 0.25Q_2) - 3 \]

### Step 3: Set up the intertemporal condition for dynamically efficient allocation

In a dynamically efficient allocation, the present value of marginal net benefit in period 1 should equal the present value of marginal net benefit in period 2:
\[ MNB_1 = \frac{MNB_2}{1 + r} \]

### Step 4: Substitute and solve the equation

Substitute the expressions for $ MNB_1 $ and $ MNB_2 $:
\[ (15 - 0.25Q_1 - 3) = \frac{(15 - 0.25Q_2 - 3)}{1.05} \]

Simplify the equation:
\[ 12 - 0.25Q_1 = \frac{12 - 0.25Q_2}{1.05} \]

Multiply both sides by 1.05 to clear the fraction:
\[ 1.05 \times (12 - 0.25Q_1) = 12 - 0.25Q_2 \]
\[ 12.6 - 0.2625Q_1 = 12 - 0.25Q_2 \]

Rearrange the equation to solve for $ Q_2 $ in terms of $ Q_1 $:
\[ 0.2625Q_1 - 0.25Q_2 = 0.6 \]
\[ 0.25Q_2 = 0.2625Q_1 - 0.6 \]
\[ Q_2 = \frac{0.2625Q_1 - 0.6}{0.25} \]
\[ Q_2 = 1.05Q_1 - 2.4 \]

### Step 5: Use the resource constraint to solve for $ Q_1 $ and $ Q_2 $

Since the total quantity is 100 units:
\[ Q_1 + Q_2 = 100 \]

Substitute $ Q_2 $ from the equation above:
\[ Q_1 + 1.05Q_1 - 2.4 = 100 \]
\[ 2.05Q_1 - 2.4 = 100 \]
\[ 2.05Q_1 = 102.4 \]
\[ Q_1 = \frac{102.4}{2.05} \]
\[ Q_1 \approx 49.95 \]

Now, substitute $ Q_1 $ back to find $ Q_2 $:
\[ Q_2 = 100 - Q_1 \]
\[ Q_2 = 100 - 49.95 \]
\[ Q_2 \approx 50.05 \]

### Conclusion

- The quantity allocated in period 1: $ Q_1 \approx 49.95 $
- The quantity allocated in period 2: $ Q_2 \approx 50.05 $

This allocation maximizes the present value of net benefits from the consumption of natural gas over the two periods given the constraints.

Would you like more details or have any questions?

Here are 8 related questions:
1. How is the concept of dynamic efficiency different from static efficiency?
2. What role does the discount rate play in determining the dynamically efficient allocation?
3. How would a change in the marginal cost affect the allocation between periods?
4. What if the total available quantity of natural gas was different?
5. How would a higher discount rate affect the allocation between the two periods?
6. What is the significance of the marginal net benefit in this context?
7. How would the allocation change if there were more than two periods?
8. Can this model be applied to other depletable resources?

**Tip:** Understanding the role of discount rates is crucial in intertemporal allocation problems as they influence how future benefits and costs are valued in present terms.

Explore the optimal intertemporal allocation of natural gas over two periods using dynamic efficiency. Learn how to maximize net benefits considering the inverse demand function, marginal costs, and a discount rate of 5%. Detailed step-by-step solution included.

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