Now that you’ve studied lecture 4, please answer the following questions using R and email the R script to me by the end of business on the 28th of March.

Questions

  1. Graph the production function \(y = 0.4x + 0.09x^2 - 0.003x^3\) for values of \(x\) between 0 and 20. Derive and graph the corresponding \(MPP\) and \(APP\). What is the algebraic expression for the elasticity of production in this case? Is the elasticity of production constant or variable for this function? Explain.

  2. Suppose that the coefficients or parameters of a production function of the polynomial form are to be found. The production function is \(y = ax + bx^2 + cx^3\)
    where
    \(y =\) corn yield in bushels per acre
    \(x =\) nitrogen application in pounds per acre
    \(a\), \(b\) and \(c\) are coefficients or unknown parameters
    The production function should produce a corn yield of 150 bushels per acre when 200 pounds of nitrogen is applied to an acre. This should be the maximum corn yield (\(MPP = 0\)). The maximum \(APP\) should occur at a nitrogen application rate of 125 pounds per acre. Find the parameters a, b and c for a production function meeting these restrictions.
    Hint: First find the equation for \(APP\) and \(MPP\), and the equations representing maximum \(APP\) and zero \(MPP\).Then insert the correct nitrogen application levels in the three equations representing \(TPP\), maximum \(APP\) and zero \(MPP\). There are three equations in three unknowns (\(a\), \(b\), and \(c\)).

  3. Suppose that the output sells for $5 and the input sells for $4. Recreate the following table in R and calculate the \(VMP\) and \(AVP\).

  4. Why are the values for \(VMP\) between two numbers?

  5. In Problem 4, what appears to be the profit-maximizing level of input use? Verify this by calculating \(TVP\) and \(TFC\) for each level of input use as shown in the table.