Question

The inverse demand curve for a product using resource X is given by P = 60 – 0.2Q and the cost of production is constant at MC = 10 1. Find the static equilibrium for this product (recall at equilibrium Supply equates Demand and in this case Supply = Marginal Cost) (call this t0). 2. Draw the static equilibrium. 3. What is the value of consumer surplus from consumption in t0? (recall, this is simply the area under the demand curve and above the price, which now equates Marginal Cost) Imagine that resource X is infinite in supply or perfectly renewable and that the demand and supply curves are unchanged between years. 4. How much would be produced/consumed in the following year (called t1)? Because Marginal Cost is constant in this example, it is possible to simply subtract MC from the demand curve to establish the Marginal Net Benefits (i.e. the Consumer Surplus) in each time period. This effectively creates a Marginal Net Benefit curve that shifts downwards from the initial demand curve but remains parallel. Thus, the Marginal Net Benefits are given by MNB = 60 – 0.2Q-10 = 50-0.2Q Now imagine that there is not enough of resource X to support static efficient consumption in both time periods (i.e. t0 and t1). More specifically, assume that there is a total stock such that only 400 units can be consumed altogether. 5. How much is consumed in t0? 6. How much is left to consume in t1? You may find this distribution is quite uneven with most consumption occurring in t0. 7. How much is the MNB of the following year (period t1)? (note that you need to discount the future value of MNB in period t1 into present value). Hint: take the MNB formulae for t1 – presently assumed the same as for t0 – and divide by 1.1 (because the discount rate is 10%). Now that you have the MNB for t1 you should be able to solve to find the dynamically efficient output and consumption between the two time periods. Remember that to be dynamically efficient the MNB in t0 must equal the discounted MNB for t1. (Hint: this is done algebraically by setting the two equations equal and solving for Q. The solution for Q is the amount consumed in t0 so the difference between this and 400 is the amount consumed in t1). 8. What is the dynamically efficient consumption/production in t0? 9. What is the dynamically efficient consumption/production in t1? 10. When future benefits are discounted, is the dynamically efficient consumption of the current generation higher or lower than that which occurs without discounting? 11. What does this mean about higher discount rates (i.e. will more be consumed now or later)?

Answer 1

1. At Equilibrium, demand is equal to supply. Here, since supply is 10, equilibrium will be at

10=60-.2Q

OR Q=250.

2. Static equilibrium graph=

3. The area is 1/2*(50*250)= 6250.

4. There are total of only 400 units to be consumed. 250 have already been consumed. This leaves us with 150 remaining units. Since consumer surplus, MNB = 60 – 0.2Q-10 = 50-0.2Q, we get

Consumer surplus at t1= 50-.2*150 = 20.

Now the curve shits downwards parallely, this means the area under the surplus curve is now 20 but the proportions remain the same (i.e., if price is x, Q would be 5x). So, we get

1/2*(x*5x)=20 OR 5x²=40 OR x²=8 or x=2.83.

5. 250 units is consumed at t0.

6. 150 units is consumsed at t1.

7. The formula for MNB at t0 is 50-0.2Q₀. Discounted MNB at t1 is (50-0.2Q₁)/1.1. We also have a reource constraint, which is Q₁=400-Q_0.

The two MNBs should be equal for dynamic efficient allocation. Hence

50-0.2Q₀=(50-0.2Q₁)/1.1

OR 50-0.2Q₀= 45.45-.182Q₁

OR 4.55=0.2Q₀-.182Q₁

OR 4.55=0.2Q₀-.182(400-Q₀)

OR Q₀~202.

8. dynamically efficient consumption/production in t0 is ~202.

9. dynamically efficient consumption/production in t1 is ~198.

10. Its asking whether 202/1.1² (which is ~167) is higher than 198/1.1 (which is 180). Its not. So dynamically efficient consumption of the current generation is lower.

11. More will be consumed later.