Now that you’ve studied lecture 5, please answer the following questions using R and email the R script to me by the end of business on the 5th of May.

Questions

  1. Suppose that the following combinations of \(x_1\) and \(x_2\) all produce 100 bushels of corn.

    1. Calculate the \(MRS_{x_1x_2}\) and the \(MRS_{x_2x_1}\) at each midpoint.
    2. Suppose that the price of \(x_1\) and \(x_2\) is each a dollar. What combination of \(x_1\) and \(x_2\) would be used to achieve the least-cost combination of inputs needed to produce 100 bushels of corn?
    3. Suppose that the price of \(x_2\) increased to $2. What combination of \(x_1\) and \(x_2\) would be used to produce 100 bushels of corn?
    4. If the farmer was capable of producing 100 bushels of corn when the price of \(x_1\) and \(x_2\) were both $1, would he or she necessarily also be able to produce 100 bushels of corn when the price of \(x_2\) increases to $2? Explain.

  1. For the production function \(y = 3x_1 + 2x_2\), find
    1. The \(MPP\) of \(x_1\).
    2. The \(MPP\) of \(x_2\).
    3. The marginal rate of substitution of \(x_1\) for \(x_2\).
       
  2. For the production function \(y = x_1^{0.5}x_2^{0.333}\), find
    1. The \(MPP\) of \(x_1\).
    2. The \(MPP\) of \(x_2\).
    3. The marginal rate of substitution of \(x_1\) for \(x_2\).
       
  3. Assume that a farmer has budget of $200 to buy inputs. What is the slope of the isocost line when
    1. \(Px_1 = \$1; Px_2 = \$2.00\)?
    2. \(Px_3 = \$1; Px_2 = \$1.75\)?
       
  4. Answer these two questions regarding input relationships:
    1. What is the difference between the marginal rate of substitution and the elasticity of substitution between two inputs?
    2. What is the difference between Inputs which are substitutes or complements and the idea of substitution between Inputs?
    3. Can two Inputs be both complements and substitutes at the same time?