mathematical economics calculus matrix algebra and optimization problems

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  1. Simplify the following expression so that it contains only a single exponent of x [8 points]
  1. Differentiate the following function with respect to x [8 points]:
  1. Evaluate the derivative ofat x = 2 [8 points].
  1. [8 points] Find Y* (equilibrium output) from the following [Hint: Y=(Y1/2)2]:
  1. [8 points] Find the maximum profit and the quantity (Q) that will maximize the profit function below.Show evidence that profit is indeed maximized:
  • [8 points] Find the price elasticity of demand of the following:
  • [8 points total] Find the derivatives of the following by first taking the natural logs:
  • [ points] Given the following matrices:
  • [8 points]Assume a stock price is a positive function of earnings per share [E], a negative function of bond interest rate [i], and a positive function of inflation [Ï€] and GDP [Y].The function is denoted:S = S(E, i, Ï€, Y).To complicate matters GDP is a negative function of bond interest rate.Write the total differential equation.What can you say about the total derivative of S with respect to the bond interest rate?



  1. [8 points] Find the critical values of the following and use the Nth derivative test to determine if the values determine a maximum, minimum or inflection point.
  1. [8 points] Find the values of x, y and that maximize the following utility function subject to the budget constraint: ;

Find, or if it does not exist state that it does not exist:

  • [2 points] AD’
  • [2 points] D’C
  • [2 points] E’D
  • [2 points] FB’
  • [2 points] A-1C
  • [2 points] B1
  1. [8 points] Consider the following system of equations:

Find x1 using Cramer’s rule.

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