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 i. To allocate 40 units between two periods in a dynamic efficient allocation, we need to find the optimal quantities that equate the present value of marginal net benefits (MNB) in each period. Given the inverse demand function P = 25-0.4q and the marginal supply cost of £5, the marginal net benefit in period t is MNBt = (25-0.4qt-5)(1 + r)t-1. The dynamic efficient allocation must satisfy the necessary condition PV(MNB1) = PV(MNB2) =λ, where A is the present value of marginal user cost. Let q1 and q2 be the quantities allocated to periods 1 and 2, respectively. Then, the present value of marginal net benefits in each period is: PV(MNB1)=(25-0.4q1-5) (1+r) = PV(MNB2) (25 0.4q2 -5) Since PV(MNB1) PV(MNB2) =λ, we have: = (25-0.4q1-5)(1+r) = 25-0.4q2 -5 => Solving this system of equations, we get: q1=14.44 (rounded to two decimal places) q2=25.56 (rounded to two decimal places)
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 i. To allocate 40 units between two periods in a dynamic efficient allocation, we need to find the optimal quantities that equate the present value of marginal net benefits (MNB) in each period. Given the inverse demand function P = 25-0.4q and the marginal supply cost of £5, the marginal net benefit in period t is MNBt = (25-0.4qt-5)(1 + r)t-1. The dynamic efficient allocation must satisfy the necessary condition PV(MNB1) = PV(MNB2) =λ, where A is the present value of marginal user cost. Let q1 and q2 be the quantities allocated to periods 1 and 2, respectively. Then, the present value of marginal net benefits in each period is: PV(MNB1)=(25-0.4q1-5) (1+r) = PV(MNB2) (25 0.4q2 -5) Since PV(MNB1) PV(MNB2) =λ, we have: = (25-0.4q1-5)(1+r) = 25-0.4q2 -5 => Solving this system of equations, we get: q1=14.44 (rounded to two decimal places) q2=25.56 (rounded to two decimal places)
Chapter16: Labor Markets
Section: Chapter Questions
Problem 16.7P
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How is the equation solved on the bottom line in picture 2? I recieved an answer as attached on what the answer is to the question but I can't understand the solving of the equation.
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
Transcribed Image Text:i. To allocate 40 units between two periods in a dynamic efficient allocation, we need to find the optimal
quantities that equate the present value of marginal net benefits (MNB) in each period. Given the inverse
demand function P = 25-0.4q and the marginal supply cost of £5, the marginal net benefit in period t is
MNBt = (25-0.4qt-5)(1 + r)t-1. The dynamic efficient allocation must satisfy the necessary condition
PV(MNB1) = PV(MNB2) =λ, where A is the present value of marginal user cost.
Let q1 and q2 be the quantities allocated to periods 1 and 2, respectively. Then, the present value of
marginal net benefits in each period is:
PV(MNB1)=(25-0.4q1-5) (1+r)
=
PV(MNB2) (25 0.4q2 -5)
Since PV(MNB1) PV(MNB2) =λ, we have:
=
(25-0.4q1-5)(1+r) = 25-0.4q2 -5 =>
Solving this system of equations, we get:
q1=14.44 (rounded to two decimal places)
q2=25.56 (rounded to two decimal places)
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