Thanks What\'s lambda A Got to Do With It? This question asks you to investigate

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What's lambda A Got to Do With It? This question asks you to investigate the role of lambda, the Lagrange multiplier, in constrained optimization problems. Pigeon peas are a legume prized by Trinidadians as a delicacy. However, they are also a fertilizer-intensive crop. Agricultural runoff from fields there the peas are grown causes eutrophication of rivers, and poisons the fragile mangrove ecosystems along Trinidad's coasts. Denote pigeon pea farmers' use of fertilizer by X. Farmers' profit per acre from the sale of pigeon peas is directly related to the amount of fertilizer input they use, and is given by pi =10X1/2-X Assume that the cost of damage D to surrounding ecosystems from fertilizer runoff increases quadratically with fertilizer use: D = X2. In this situation, define social welfare as: W = pi - D. Write down the Lagrangian for this problem, using lambda to indicate the Lagrange multiplier. Define what lambda is and briefly explain how it applies to this problem. 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? 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 lambda? 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 lambda? Using Excel, draw graphs of pi and lambda as functions of X, and include them with your solution. Explain briefly why lambda changes as it does in the graph. Is there level of X for which lambda = 0? What does that tell you about what this point means?

Explanation / Answer

i)don't know

ii)

they go for maximum profit

==>pi'=0

==>(10/2)x^(-1/2) -1=0

==>x^(1/2)=5

==>x=25

D=25^2=625

pi=10*5-25=25

W=pi-D=25-625=-600

iii)D=16==>x=4

profit pi= 10*4 -16=24

w=pi-D=24-16=8

iv)x^2=4==>x=2

profit=20-2=18

w=18-4=14

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