Question

Suppose that we need to develop a model to help us to allocate a resource between two time periods: Period 1 and Period 2. Demand can be characterized in both periods as: P = 20 – 2Q. In both periods the marginal cost of extracting the resource is constant at $4 per unit. The total quantity of the resource that is available to allocate between the two periods is 18 units (in other words, Q1 + Q2 <= 18). Assume that the discount rate is 5%. What is the marginal user cost?

Suppose that the discount rate changes to 10%, now. What is the change in marginal user cost?

Answer 1

1.The marginal benefits for both periods is same and equal to P=20-2Q (the inverse demand curves). Thus, for period 1, marginal benefit = 20-2Q1 and for period 2, it is 20-2Q2.

The Marginal costs are same, MC1=MC2= 4

The Marginal user cost is the marginal net benefit forgone in a period due to scarcity.

Marginal net benefit in period 1=>MNB1= MB1-MC1 => MNB1= 20-2Q1-4 => MNB= 16-2Q1.

Similarly, MNB2= 16-2Q2.

Dynamic efficiency requires that present value of marginal net benefits be same in both periods. That is, Pv(MNB1) =Pv(MNB2).

Present value of first period is same as current value of MNB1 because we don't discount present.

Pv(MNB2) = 16-2Q2/(1+0.05) = 16-2Q2/1.05 = 15.238-1.9047Q2.

Equating, Pv(MNB1) =Pv(MNB2)

16-2Q1=15.238-1.9047Q2 => 0.762= 2Q1-1.9047Q2 => (0.762+1.9047Q2)/2= Q1.

Putting this in resource constraint, we get, Q1+Q2 8

(0.762+1.9047Q2)/2 +Q2 18 => 0.762+1.9047Q2+2Q218*2

=>3.9047Q235.238 => Q2 9.024 and thus Q1 8.976

(assuming we utilize all resource in two periods)

Marginal user cost in period 1= MNB1= 16-2Q1= 16-2*8.976= 16-17.952=(-1.952)

In period 2, MUC= 16-2*9.024= 16- 18.048=(-2.048)

2. If we let discount rate equal to 10%

Then, Pv(MNB2) = 16-2Q2/(1+0.1) = 16-2Q2/1.1 = 14.54-1.81Q2

Equating Present values of MNB1 and MNB2,

16-2Q1= 14.54 - 1.81Q2 => 1.46 +1.81Q2= 2Q1 => Q1=(1.46+1.81Q2)/2

Putting in resource constraint we get,

(1.46+1.81Q2)/2 +Q2 18 => 1.46+ 1.81Q2 +2Q236

=> 3.81Q2 34.54 => Q2 9.065 and Q1 8.935

Marginal user cost in period 1 = 16-2(8.935) = 16-17.87 = (-1.87)

Marginal user cost in period 2= 16-2(9.065)= 16-18.13 = (-2.13)

[Assuming resources are being used in both periods, thus considering equality in the resource constraint]

The Marginal user cost in period 1 is less at 5% discount rate than 10% discount rate whereas marginal user cost in period 2 is higher with 5% discount rate than 10% discount rate(-2.048>-2.13).

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