1. Answer the following questions based on the following production function: y = f (z) = 5z* + 300z*. Please show how you found your answer. (70 Points) A. What is the marginal physical product (MPP) function for this production function? B. In terms of inputs, show where the MPP function is at a maximum. C. Demonstrate that MPP is at a maximum by using derivatives. D. What is the highest achievable MPP? E. What is the average physical product (APP) function for this production function? F. In terms of inputs, show where the APP function is at a maximum. G. Demonstrate that APP is at a maximum by using derivatives. H. What is the highest achievable APP? I. In terms of inputs, where is the maximum point on the production function? J. Demonstrate that the maximum point that you found for the production function is truly a maximum. K. What is the highest output achievable? L. Where does Stage | of the production function turn into Stage Il of the production function (Please give me the actual x value)? M. Where does Stage Il of the production function turn into Stage Il of the production function (Please give me the actual x value)? N. Where does the Law of Diminishing Marginal Returns take effect (Please give me the actual x value)? 2. Answer the following questions based on the following production function: y=rf (ml , 32'2) = m% + 10z a:% + 20x5. Please show how you found your answer. (60 Points) A. What is the marginal physical product (MPP,4) function for this production function with respect to input 17 B. What is the marginal physical product (MPP,5) function for this production function with respect to input 27? C. What is the extrema point to this production function? (Note, the extrema point is made up of x4 and x together and should represent the input combination that gives the maximum amount of output.) D. What is the maximum you can achieve for output? E. Find the isoquant for any given output level? (Note: make sure you find the one that makes economic sense.) F. If output is equal to 115 and input 1 is equal to 2, how much input 2 do you need? G. Calculate the marginal rate of technical substitution (MRTS) using the fact that MRTS is equal to the negative ratio of the MPP's. H. Calculate the marginal rate of technical substitution (MRTS) in terms of inputs by taking the derivative of the isoquant you found in part E. I. Given the inputs from part F and the MRTS equations you found in parts G and H, what is the MRTS? (You shou get a single number.) J. Please explain what part | means in an economic sense? K. Assuming that x4 is fixed at an input level of 3, what is the highest achievable output? L. Demonstrate that x, in part k is the highest achievable output by using derivatives. 1l 1 3. Answer the following questions based on the following production function: y = f (:El S .fL'g) = 120;1:12 ;1:23 . Please show how you found your answer. (55 Points) A. What is the marginal physical product (MPPy) function for this production function with respect to input 1? B. What is the marginal physical product (MPPy) function for this production function with respect to input 27 C. Find the isoquant for any given output level. D. If output is equal to 4320 and input 1 is equal to 36, how much input 2 do you need? E. Calculate the marginal rate of technical substitution (MRTS) using the fact that MRTS is equal to the negative ratio of the MPP's. F. Calculate the marginal rate of technical substitution (MRTS) in terms of inputs by taking the derivative of the isoquant you found in part C. G. Given the inputs from part D and the MRTS equations from Part E and F, what is the MRTS? (You shou get a single number.) H. Please explain what part G means in an economic sense? . What are the returns to scale for this production function? 1 1 1 J. Suppose you had a new production function thatis y = f (@1, 22, 23) = 120:1:13 9325 :13'3g , what are the returns to scale of this production function? K. Assuming that x5 is the dependent variable, find the isoquant for any given output level for the production function in part I. 4. Suppose you have 100 acres of land to allocate to corn and soybeans. The production function for corn is 1 = me , Where Y is the amount of bushels of corn and x4 is the amount of land used for corn. You 1 also know that the production function for soybeans is Yo = 81:1025 , where Y5 is the amount of bushels of soybeans and x; is the amount of land used for soybeans. (40 Points) A. Please find the production possibility frontier (PPF) using soybeans as the dependent variable and corn as the independent variable? B. How many bushels of soybeans can you produce if you produced 72 bushels of corn? C. What is the maximum bushels of soybeans you can produce (Hint: set corn to zero)? D. What is the maximum bushel of corn you can produce (Hint: set soybeans to zero)? E. Please sketch the PPF with corn on the x-axis and soybeans on the y-axis. Please include information from parts B, C, and D on your graph. F. What is the marginal rate of product transformation (MRPT) for the PPF? G. What is the slope, i.e, MRPT, of the PPF when corn = 72,i.e, Y1 =72. H. Please explain what part G means in an economic sense