The solution provided is based on the principle of dynamic efficiency for a depletable resource, where the goal is to maximize the present value of the net benefits from the resource. Here's a step-by-step explanation of the solution: The inverse demand function is given as P=25-0.4q, where P is the price and q is the quantity. The marginal supply cost (MSC) is constant at £5. The total quantity to be allocated is 40 units, and it needs to be divided between two periods, period 1 and period 2. To find the dynamic efficient allocation, we equalize the present value of the marginal net benefits (MNB) from each period, which is the price minus the marginal cost. The present value of the marginal net benefits for each period can be expressed as: • For period 1: PV(MNB1)=(25-0.4q1-5)(1+r) • For period 2: PV(MNB2)=(25-0.492-5) Because the total benefit must be maximized over both periods, we set PV(MNB1) equal to PV(MNB2), and solve for 1q1 and q2 with the constraint that q1+q2=40. The equation to be solved is (1+r)=25-0.4q2-5, where r is the discount rate (0.15 in this case). To solve for q1 and q2, we manipulate the equations as follows: •Expand the left side: 20(1+0.15)-0.4q1 (1+0.15)=20-0.4q2 • Simplify both sides: 23-0.46q1-20-0.4q2 By substituting the constraint q2=40-q1 into the equation, we solve for q1 and then use it to find q2. The result of this process gives the allocation of the resource to each period. The calculated allocation is approximately 14.44 units for period 1 and 25.56 units for period 2. This C C Suppose the inverse demand function for a depletable resource is linear, P = 25 - 0.4q, and the marginal supply cost is constant at £5. i. If 40 units are to be allocated between two periods in a dynamic efficient allocation, how much would be allocated to period 1 and how much to period 2 when the discount rate is r = 0.15? Show your working
The solution provided is based on the principle of dynamic efficiency for a depletable resource, where the goal is to maximize the present value of the net benefits from the resource. Here's a step-by-step explanation of the solution: The inverse demand function is given as P=25-0.4q, where P is the price and q is the quantity. The marginal supply cost (MSC) is constant at £5. The total quantity to be allocated is 40 units, and it needs to be divided between two periods, period 1 and period 2. To find the dynamic efficient allocation, we equalize the present value of the marginal net benefits (MNB) from each period, which is the price minus the marginal cost. The present value of the marginal net benefits for each period can be expressed as: • For period 1: PV(MNB1)=(25-0.4q1-5)(1+r) • For period 2: PV(MNB2)=(25-0.492-5) Because the total benefit must be maximized over both periods, we set PV(MNB1) equal to PV(MNB2), and solve for 1q1 and q2 with the constraint that q1+q2=40. The equation to be solved is (1+r)=25-0.4q2-5, where r is the discount rate (0.15 in this case). To solve for q1 and q2, we manipulate the equations as follows: •Expand the left side: 20(1+0.15)-0.4q1 (1+0.15)=20-0.4q2 • Simplify both sides: 23-0.46q1-20-0.4q2 By substituting the constraint q2=40-q1 into the equation, we solve for q1 and then use it to find q2. The result of this process gives the allocation of the resource to each period. The calculated allocation is approximately 14.44 units for period 1 and 25.56 units for period 2. This C C Suppose the inverse demand function for a depletable resource is linear, P = 25 - 0.4q, and the marginal supply cost is constant at £5. i. If 40 units are to be allocated between two periods in a dynamic efficient allocation, how much would be allocated to period 1 and how much to period 2 when the discount rate is r = 0.15? Show your working
Microeconomics A Contemporary Intro
10th Edition
ISBN:9781285635101
Author:MCEACHERN
Publisher:MCEACHERN
Chapter11: Resource Markets
Section: Chapter Questions
Problem 4QFR
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My question is attached below along with the answer from the expert. However, i dont see how the equation is solved at the end. I can see how you get to the point where 23 - 0.46q1 = 20 - 0.4q2 and q2 = 40 - q1 but don't see how this system of equations is solved to give the answer q1=14.44 and q2=25.56.
Transcribed Image Text:The solution provided is based on the principle of dynamic efficiency for a depletable resource, where the
goal is to maximize the present value of the net benefits from the resource. Here's a step-by-step
explanation of the solution:
The inverse demand function is given as P=25-0.4q, where P is the price and q is the quantity.
The marginal supply cost (MSC) is constant at £5.
The total quantity to be allocated is 40 units, and it needs to be divided between two periods, period 1 and
period 2.
To find the dynamic efficient allocation, we equalize the present value of the marginal net benefits (MNB)
from each period, which is the price minus the marginal cost.
The present value of the marginal net benefits for each period can be expressed as:
• For period 1: PV(MNB1)=(25-0.4q1-5)(1+r)
• For period 2: PV(MNB2)=(25-0.492-5)
Because the total benefit must be maximized over both periods, we set PV(MNB1) equal to PV(MNB2), and
solve for 1q1 and q2 with the constraint that q1+q2=40.
The equation to be solved is (1+r)=25-0.4q2-5, where r is the discount rate (0.15 in this case).
To solve for q1 and q2, we manipulate the equations as follows:
•Expand the left side: 20(1+0.15)-0.4q1 (1+0.15)=20-0.4q2
• Simplify both sides: 23-0.46q1-20-0.4q2
By substituting the constraint q2=40-q1 into the equation, we solve for q1 and then use it to find q2.
The result of this process gives the allocation of the resource to each period.
The calculated allocation is approximately 14.44 units for period 1 and 25.56 units for period 2. This
C
C
Transcribed Image Text:Suppose the inverse demand function for a depletable resource is linear, P = 25 - 0.4q, and the marginal supply cost is
constant at £5.
i. If 40 units are to be allocated between two periods in a dynamic efficient allocation, how much would be allocated to
period 1 and how much to period 2 when the discount rate is r = 0.15? Show your working
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