In the process of producing 1 hormone-free, stress-free, happy cow, the New Age Cattle Ranch produces 1 ton of natural, organic plant fertilizer.Demand for the ranch's cattle is given by P_c = 2,000 - 20Q_c, and demand for fertilizer is given by P_f = 500 - 4Q_f.The cost of producing Q cow-fertilizer bundles is C = 5,000 + 400Q.To maximize profits, the ranch should A)produce 40 cow-fertilizer units, sell 40 cattle, and sell 40 tons of fertilizer. B)produce 40 cow-fertilizer units, sell 40 cattle, and sell 12.50 tons of fertilizer. C)produce 62.50 cow-fertilizer units, sell 40 cattle, and sell 62.50 tons of fertilizer. D)produce 43.75 cow-fertilizer units, sell 43.75 cattle, and sell 43.75 tons of fertilizer. E)produce 50 cow-fertilizer units, sell 50 cattle, and dispose of all the fertilizer in an environmentally friendly manner.

Question

Quizplus · Accepted Answer

To maximize profits, the ranch should use the first order conditions, which are:

MU(Qcow)/P(Qcow) = MU(Qfertilizer)/P(Qfertilizer)
where MU is marginal utility and P is price

Taking the partial derivatives of the demand functions with respect to Qcow and Qfertilizer, we get:
MU(Qcow) = 2000 - 40Qfertilizer
MU(Qfertilizer) = 500 - 4Qcow

Setting these equal and solving for Qcow, we get:
2000 - 40Qfertilizer = 500 - 4Qcow
Qcow = 375 - 0.1Qfertilizer

Taking the derivative of the cost function with respect to Qcow and setting it equal to the marginal revenue from selling a cow (which is its price), we get:
5000 + 400Qcow' = 3750 - 0.5Qcow
Qcow' = -0.015Qcow + 9.375

Using the production constraint (1 ton of fertilizer produced for each cow-fertilizer unit), we can express Qfertilizer in terms of Qcow:
Qfertilizer = Qcow

Substituting this into the demand equation for fertilizer and solving for Qcow, we get:
500Qcow - 4Qcow^2 = 4000
Qcow = 43.75

Using the production constraint, we get:
Qfertilizer = Qcow = 43.75

Therefore, the ranch should produce 43.75 cow-fertilizer units, sell 43.75 cattle, and sell 43.75 tons of fertilizer to maximize profits. This choice corresponds to option D.

In the Process of Producing 1 Hormone-Free, Stress-Free, Happy Cow