# mathematical economics calculus matrix algebra and optimization problems

**Please See Attachment for clarity – All work/steps to solution must be shown **

- Simplify the following expression so that it contains only a single exponent of
*x*[8 points]

- Differentiate the following function with respect to
*x*[8 points]:

- Evaluate the derivative ofat
*x = 2*[8 points]*.*

- [8 points] Find Y* (equilibrium output) from the following [Hint:
*Y=(Y*]:^{1/2})^{2}

- [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?

a.

b.

- [8 points] Find the critical values of the following and use the N
^{th}derivative test to determine if the values determine a maximum, minimum or inflection point.

- [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]
*ADâ€™* - [2 points]
*Dâ€™C* - [2 points]
*Eâ€™D* - [2 points]
*FBâ€™* - [2 points]
*A*^{-1}C - [2 points]
*B*^{–}^{1}

- [8 points] Consider the following system of equations:

Find *x*_{1} using Cramerâ€™s rule.