ECO 201: Mathematics for Business and Economics

Assignment 3 (Section 2 & 3)

Date of Submission: December 4, 2010

Total Marks: 40

1. Given the following functions;

a. Show whether the functions are increasing or decreasing. (1+1)

b. Show whether the functions are concave or convex? (1+1)

c. Draw a graph for the functions. (0.5+0.5)

2. A firm has the following cost and demand functions:

a) Formulate the total-revenue function R in terms of Q. (1)

b) Formulate the profit function (π), solve for the profit maximizing level of output Q* at which the function is optimized. (2)

c) Comment whether the profit is maximized or minimized. What is the maximum profit? (2)

3. For each of the following functions:

i.

ii.

- Find the critical points at which the function may be optimized. (1+1)

- Determine whether at these points the function is maximized, is minimized, is at an inflection or is at a saddle point. (1.5+1.5)

4. Optimize the following function and comment about the findings. (5)

5. A firm produces two goods in pure competition and has the following total revenue and total cost functions:

Use Cramer’s rule for the first order condition to solve for and Hessian for the

second order condition and comment. (2.5+2.5)

6. Given the utility function U = xy+x+2y+2 and Px = 4, Py = 6 and B=130

a) Construct the Lagrangian function and find the utility maximizing values of x*, y* and λ*. Find the maximum utility U*. (2)

b) What is the economic interpretation of the Lagrangian Multiplier λ*? (1)

c) Is the second order sufficient condition for maximum satisfied? (2)

7. Minimize the cost of 434 units of production for a firm when Q = 10K0.7L0.1 and Pk = 28, PL=10 by finding the critical values and using the bordered Hessian to satisfy the 1st order necessary and 2nd order sufficient condition. (5)

8. A producer has the possibility of discriminating between the domestic and foreign market for a product where the demands respectively are:

What price will the producer charge in order to maximize profit?

a) With discrimination between markets. (2)b) Without discrimination between markets. (2)c) Compare the profit differential between discrimination and nondiscrimination.(1)

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