Investigate the role of ?, the Lagrange multiplier, in constrained optimization problems. Corn is a fertilizer-intensive crop. Agricultural runoff from fields where corn is grown causes eutrophication of rivers, and poisons the nearby ecosystems. Denote corn farmers' use of fertilizer by X. Farmers' profit per acre from the sale of corn is directly related to the amount of fertilizer input they use, and is given by:
? = 10 X^(1/2) - X
Assume that the cost of damage D to surrounding ecosystems from fertilizer runoff increases quadratically with fertilizer use: D = X^2. In this situation, define social welfare as: W = ? - D.
(i) Write down the Lagrangian for this problem, using ? to indicate the Lagrange multiplier. Define what ? is and briefly explain how it applies to this problem.
(ii) Suppose that fertilizer use is unregulated. How much fertilizer will farmers use? What will their profit be? How much damage will be caused? What is the net societal benefit?
(iii) Now suppose that the local government attempts to limit the environmental damage from runoff to 16 units. How much fertilizer will farmers use now? What will their profit be? What is the net societal benefit? What is the value of ??
(iv) In response to pressure from environmentalists the government now tightens the maximum damage standard to 4 units. How much fertilizer will farmers use now? What will their profit be? What is the net societal benefit? What is the value of ??