Pigeon peas are a legume prized by Trinidadians as a delicacy. However, they are
ID: 1240290 • Letter: P
Question
Pigeon peas are a legume prized by Trinidadians as a delicacy. However, they are also a fertilizerintensive crop. Agricultural runoff from fields there the peas are grown causes eutrophication of rivers,and poisons the fragile mangrove ecosystems along TrinidadVs coasts. Denote pigeon pea farmersV use of
fertilizer by X. FarmersV profit per acre from the sale of pigeon peas is directly related to the amount of
fertilizer input they use, and is given by
? = (10X^0.5)-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. What is
the definition of ??
(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 Trinidad 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 ??
(v) Using Excel, draw graphs of ? and ? as functions of X, and include them with your solution.
Explain briefly why ? changes as it does in the graph. Is there level of X for which ? = 0? What
does that tell you about what this point means?
Explanation / Answer
too many qns in a single post