Econ 6976: Homework # 1, Dr. Usip, Economics

Econ 6904 Homework # 2 Dr. Usip

Mathematical Economics Problems

1. Finding the total derivative of functions.
    a. find dz/dy where z = f(x,y) = 2x + xy - y² and x = g(y) = 3y²

    b. find dz/dt where z = f(x,y) = x² - 8xy - y³ and x = g(t) = 3t, and y = h(t) = 1- t

2. Finding the derivative of the exponential and logarithmic functions.
    a. y = 3e^(2-t2)
    b. y = x²e^2x
    c. y = ln(3x ÷(1 + x))
d. y = 5x⁴lnx²

3. Unconstrained Optimization
    a. Suppose y = x³ + 6x² + 7. Find the relative maxima and minima of y by the second derivative test.

    b. Find the extreme value(s) of this function and determine whether they are maxima or minima
       i. Z = x² + xy + 2y² + 3                           ii. Z = -x² + xy - y² + 2x + y

    c. A firm produces and sells two products (Q₁, Q₂) under circumstances of pure competition.
       Accordingly, the firm's revenue function is given as: R = P₁Q₁ + P₂Q₂.
       Suppose the firm's cost function is also given as: C = 2Q₁² + Q₁Q₂ + 2Q₂²
        i. Formulate and write the profit (π) function for this firm
       ii. Find the profit maximization output levels (Q₁*, Q₂* ) that the firm should produce if
           P₁ = 12/unit and P₂ =$18/unit.
      iii. What is the firm's profit?
      iv. Does your result in part iii represent the maximum profit? Verify by using the second order test/condition.

4. Constrained Optimization
     a. Use the Lagrange-multiplier method to find the stationary values of Z = x-3y-xy, subject to x + y = 6
     b. Given U = (x+2)(y+1) and P_x = $4, P_y = $6 and Income = $130
       i. Write the Lagragian function
      ii. Find the optimal levels of purchase of each good in the consumption basket.
      iii. Is the second-order test (sufficient condition) satisfied? Please verify.

Basic Statistics Problems: Applications using Eviews
Consider the following dataset:

Wife's Age	22	27	25	32	34	25
Husband's Age	24	33	28	30	40	25

1. Use the mean age to explain the distribution of age among the wives vis-a-vis the husbands
2. Use the median age to explain the distribution of age among the wives vis-a-vis the husbands
3. What are the social implications of your findings in parts1 and 2 above?
4. Is the spread in the age of the wives greater or smaller than the spread in the age of their
husbands? Please explain your answer.
5. Which husband's age would you consider an outlier, if any?
6. Which wife's age would you consider an outlier, if any?

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