Question

Question 15: All Parts

15. Suppose your goal this year is to produce 6,400 bushels of organic corn. Suppose your production technology has the following relationship for producing bushels of corn Q - f(LT LT where Q is the number of bushels of corn you produce, L is the number of labor hours you utilize, and T is the number of tractor hours you utilize. You know that the cost per hour of labor is $20 and the cost per hour of tractor time is $500. Assume that you are a cost minimizing producer. (30 points) Theoretically, tell me how you would figure out what your minimum cost is for producing a quantity of 6,400 bushels of corn a. b. How much money will you need to ask from your banker to achieve your goal? (Note: when given your particular production function, the optimal amount of input formulas are LP and T- Q, where pu is the cost of labor and pr is the cost of tractor time.) Please sketch a graph of this solution. Include the isoquant and iso-cost line. c. d. Suppose after reviewing your business plan, your banker tells you she is only willing to give you $8,000. Theoretically, tell me how you would figure out what your maximum production is when you have only $8,000 to work with and you are a cost minimizer. What is the maximum amount you can produce? (Note: when given a particular expenditure E and you are trying to maximize output, the optimal e. amount of input formulas are L- and T- 2 where pt is the cost of 2 labor and pr is the cost of tractor time.) f. Please sketch this new answer on the graph from answer c. Please include the isoquant and the iso-cost line.

Answer 1

Q^bar = 6400, Q = L middot T, P_L = 20, P_T = 500 Now total production cost = C = L middot P_L + T middot P_T = 20 L + 500 T Now Min C st Q^bar = > 6400 = L middot T 80 L = 20 L + 500 T + lambda [6400 - LT] Foc: partial differential L/ partial differential L = > 20 - lambda T = 0, partial differential L/ partial differential T = 500 - lambda L = 0, So 20/T = 500/L Thus 2L = 50 T, = > L = 25 T from production function 6400 = 25 middot T^2, t^2 = 256, T^* = 16, 80 L^* = 400 Now total money Needed = C = 20 L + 500 T = (20 times 400) + (500 times 16) = 8000 + 8000 = 16,000 graph rightarrow Iso quantity (let L on vertical axis) So L = 6400/T,