Cost Minimization
1. An orange goes through four major steps on its way from grove to grocery store at the Sunkist Citrus Processing Plant in the southern San Joaquin Valley: washing, waxing, grading (inspection), and packaging. The facility's production technology can be represented by the 
following production function:
q=F(L,K)=10L^(1/2)K^(1/2)
Where q denotes the amount of boxes of oranges produced in an hour, L denotes the number of workers, and K denotes the capital (machinery) used in production. The wage rate for a factory worker is $20/hour and the rental rate of capital is $80/hour.
A. What are the cost minimizing quantities of labor and capital when the factory produces 100 boxes of oranges each hour? What are the facility's total costs under this scenario? Use the Method of Lagrange Multipliers.
B. Calculate the values for MPL and MPK when evaluated at the optimal solution in Part A. Then calculate the MRTS and verify (numerically) that the tangency rule holds.
C. Illustrate the optimal input combination found in Part A using a diagram that contains both an isoquant and isocost curve.
D. If the factory's manager decides to increase output to 140 boxes per hour and capital remains fixed (since the decision is made abruptly), how much labor will the facilityrequire in order to produce at minimum cost? What will the facility's total costs be?
E. How will the optimal level of labor and capital be affected in the long run if Sunkist decides to maintain production at 140 boxes? What will resultant total costs be?