ECON1008 - Maths For Economics II - Stationary Points - Economics Assignment Help

Coursework
Answer ALL questions.
SHOW ALL THE STEPS IN YOUR SOLUTIONS.

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1. Find the stationary points of the following functions, and determine whether each is a minimum, maximum, or a saddle point.
(a) f(x, y) = x 3 + y 3 − 3x − 3y
(b) f(x, y) = (2xy + y 2 )e x

2. A firm’s production function is given by Q = 2L 1/2 + 3K1/2 , where Q, L and K denote the number of units of output, labour and capital. Labour costs are $2 per unit, capital costs are $1 per unit and output sells at $8 per unit. Find the maximum profit and the values of L and K at which it is achieved.

3. A firm’s production function is given by Q = 80KL. Unit capital and labour costs are $2 and $1 respectively. The firm is contracted to provide 4000 units of out- put and wants to fulfill this contract at minimal cost. What is the minimal cost?

4. Use the total differential to find the derivative for the following implicit function: 3x 3 − y 2 = 20.

5. A consumer’s utility function is given by U(x1, x2) = 2x1x2 + 3x1 where x1 and x2 denote the number of items of the two goods. Each item costs $1 for good 1 and $2 for good 2.
(a) Use the Lagrange multiplier method to find the maximum value of U if the consumer’s income is $83.
(b) Show that the ratio of marginal utilities is equal to the ratio of prices at the optimal solution.

6. Find the slope of an indifference curve for the utility function u(x, y) = √ x + y.

7. Solve the following constrained optimization problem and find the optimal value of z (a) by the method of direct substitution and (b) by the method of implicit differentiation.
z = x + 2xy subject to the constraint x + 2y = 5 

8. Solve the following constrained optimization problem and find the optimal value of z (a) by the method of implicit differentiation and (b) by the Lagrange multiplier method.
z = 6x − 3x 2 + 2y subject to the constraint y − x 2 = 2

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