A rare earth metal, Scandium, is difficult to find in significant concentrations in the Earth's crust....

 

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A rare earth metal, Scandium, is difficult to find in significant concentrations in the Earth's crust. It is estimated that there are two generations worth of the metal remaining (time period 1 and 2). The demand curve for Scandium is P = 9 -0.3q and the total amount remaining is 30 units (q₁ + 92 = 30). For this problem, assume a constant marginal cost of extraction: MC extraction = 2. Solving for a dynamic efficiency problem, we seek to maximize the net present benefit subject to the constraint of only having 30 units remaining. The solution to this problem contains three equations and three unknowns (91, 92, A) with a discount rate of 6%. 1) Marginal Benefit with respect to quantity in time period 1 (q₁): 9 -0.3q₁ - 2 - A = 0 2) Marginal Benefit with respect to quantity in time period 2 (92): 9-0.39₂-2 1.06 3) Marginal Benefit with respect to marginal user cost (): 30-9₁-92 = 0 Use these equations and solve for each unknown to the hundredths place: 91 = 92 = A* = Now given these answers, solve for the price in each time period: P₁ = P₂ = - λ = O A rare earth metal, Scandium, is difficult to find in significant concentrations in the Earth's crust. It is estimated that there are two generations worth of the metal remaining (time period 1 and 2). The demand curve for Scandium is P = 9 -0.3q and the total amount remaining is 30 units (q₁ + 92 = 30). For this problem, assume a constant marginal cost of extraction: MC extraction = 2. Solving for a dynamic efficiency problem, we seek to maximize the net present benefit subject to the constraint of only having 30 units remaining. The solution to this problem contains three equations and three unknowns (91, 92, A) with a discount rate of 6%. 1) Marginal Benefit with respect to quantity in time period 1 (q₁): 9 -0.3q₁ - 2 - A = 0 2) Marginal Benefit with respect to quantity in time period 2 (92): 9-0.39₂-2 1.06 3) Marginal Benefit with respect to marginal user cost (): 30-9₁-92 = 0 Use these equations and solve for each unknown to the hundredths place: 91 = 92 = A* = Now given these answers, solve for the price in each time period: P₁ = P₂ = - λ = O