Question

Consider a non-renewable resource that can be consumed today (period 1) or tomorrow (period 2). Moreover, this resource has a fixed supply of 5 units. Assume that the inverse demand for this resource in each period is given by P1 = 25 - 5Q1 P2 = 25 - 5Q2 and assume that the marginal extraction cost of extracting the resource is constant in both periods at $5. Finally, assume that the social discount rate is 25% (i.e., r=0.25). What is the efficient level of consumption in period 1? Please round your final answer to two decimal places if necessary. As with the homework problems, please also round your intermediate values (e.g., quantities and prices that are not your final answer) to two decimal places when calculating your final answer.

Answer 1

The efficient level of consumption in period 1 is
`Q_1 =4` units.

To find the efficient level of consumption in period 1, we need to compare the marginal benefit of consumption (given by the inverse demand curve) with the marginal cost of extraction.

The total benefit from consuming in period 1 is the area under the demand curve up to the quantity consumed. The total cost is the quantity consumed times the constant marginal cost of $5.

The inverse demand in period 1 is given by
`P_1 =25- 5Q_1 .`

The total benefit (TB) is the integral of this demand function:

TB=
`\int\limits^(Q_1)_0 {(25 - 5 Q_1)} \, dQ_1`

TB=
`[25 Q_1- (5)/(2)Q(2)/(1) ]^(Q_1) _0`

TB=25
`Q_1` −
`(5)/(2)`Q
`(2)/(1)`

Now, the social planner maximizes the net benefit, which is the total benefit minus the total cost:

NB=TB−TC

NB=(25Q_1 −
`(5)/(2)`Q
`(2)/(1)` )−5Q_1

NB=20
`Q_1` −
`(5)/(2)`Q
`(2)/(1)`

​To find the efficient level of consumption, we need to find the quantity that maximizes net benefit. Take the derivative of the net benefit with respect to Q_1 and set it equal to zero:

`(dNB)/(dQ_1)`​ =
`20- 5Q_1​ = 0`

Solving for Q_1:

`5Q_1 =20\\Q_1 =4`

So, the efficient level of consumption in period 1 is
`Q_1 =4` units.