Question

3. A small manufacturing firm in Vatuwaqa produces two goods, X and Y for the domestic market.The selling price of Good X is $54 and Good Y is $52. The manager has estimated the total cost function to be ?? = 3?^2 + 3?? + 2?^2 − 100. Compute the quantity of the two goods the firm should produce in order to maximize profits. Show all working.

4. A firm produces two goods, X and Y. The demand function for each good is given as : ?? = 36 − 3? ; ?? = 56 − 4?. Based on the given information: a) Determine an expression for total revenue. b) Compute the quantity of good X and Y the firm should produce and sell to maximize total revenue. c) Compute the maximum revenue.

5. Nancy has a utility function ? = 10?^0.2?^0.8 . c denotes packets of chocolates, w denotes bottles of wine. Based on the given information: a) Derive an expression for marginal utility of chocolates and wine. b) Derive an expression for marginal rate of substitution.

Answer 1

(3)

Total revenue (TR) = 54x + 52y

Profit (Z) = TR - TC = 54x + 52y - (3x² + 3xy + 2y² - 100) = 54x + 52y - 3x² - 3xy - 2y² + 100

Profit is maximized when (Z/x) = 0 and (Z/y) = 0.

Z/x = 54 - 6x - 3y = 0

6x + 3y = 54.........(1)

Z/y = 52 - 3x - 4y = 0

3x + 4y = 52.........(2)

Multiplying (2) by 2,

6x + 8y = 104........(3)

6x + 3y = 54..........(1)

(3) - (1) yields: 5y = 50

y = 10

x = (52 - 4y)/3 [From (2)] = [52 - (4 x 10)]/3 = (52 - 40)/3 = 12/3 = 4

NOTE: As per Answering Policy, 1st question is answered.