Question

1. Suppose two landowners have land that can be set aside for habitat preservation purposes, although at a cost. Farm 1’s marginal costs from setting aside habitat is MC₁ = 2q₁, where q₁ is the units of land set aside. Farm 2’s marginal costs from setting aside habitat is MC₂ = q₂, where q₂ is the units of land set aside.

For parts a, b, and c, below, please assume that the government wants to preserve 1200 units of land from these two landowners, that is, q1 + q2 = 1200.

1. What is the cost-effective (meaning minimal-cost) allocation of habitat preservation between the two landowners? Calculate and show this graphically using the appropriate graph. Label (do not calculate) the associated total cost.  

2. Identify graphically, and explain (both algebraically and intuitively), the efficiency loss (here, meaning increase in total costs) that would occur if only Farm 1’s land was used to satisfy the habitat goal.

2. For the cost-effective allocation computed in part a, what are the imputed marginal benefits of habitat preservation, and why?

Answer 1

A) From the question, it is clear that the marginal cost of 1 unit of land is twice more in Farm 1 than in Farm 2 since MC1 = 2q1 and MC2 = q2. So for instance, if a 1 unit of land preveration costs 10 units in Farm 2, then the same 1 unit of land preservation will cost 20 units in Farm 1. Having cleared this, we can now compute the cost effective allocation of habitat preservation of the two landowners. Minimial cost is basically the minimized total cost. So this boils down to a simple cost minimization problem where we differentiate cost with q and set it equal to 0. We can compute the Total Cost (TC) which is equal to TC = TC1 + TC2. We can then subsitute q1 = 1200 - q2 in the TC equation and then differentiate it with q2 to get the minimum cost. We compute the TC of each Farm by simply integrating it with dq1 and dq2 respectively. But we know that differentiating TC with q, we will get MC which we then set to 0 for minimization, and since we have been given MC in the question, we can use it directly.

So, we set MC1 = 0 and substitute q1 = 1200 - q2 in it. Therefore, we get:

MC1 = 0 = 2(1200 - q2) = 2400 - 2q2. This implies q2* = 2400/2 = 1200. Putting this back in the government constraint, we get q1* = 1200 - q2* = 1200 - 1200 = 0. So, MC2 = 1200 and MC1 = 0. Total cost for Farm 2 = 1200 x 1200 = 1440000 and Total cost for Farm 1 = 0.

The x axis measures the land unit and y axis measures the costs. The MC1 curve is twice steeper than MC2 curve. The total costs has been marked as the horizontally dashed part of MC2 curve.

b) If only Farm 1's land were to be used to preservation, then the marginal costs would be 2 times 1440000 = 2880000. So the increase in Total costs would be of the amount of 1440000. This has been shown as the vertically dashed part of the MC1 curve.

c) The imputed marginal benefits from the 1200 units of land used in habitat preservation is the increase in flora and fauna, afforestation and increase in better environmental outcomes.