Question

A firm produces two commodities, A and B. The inverse demand functions are:

pA =900−2x−2y, pB =1400−2x−4y

respectively, where the firm produces and sells x units of commodity A and y units of commodity B. Its costs are given by:

CA =7000+100x+x^2 and CB =10000+6y^2

where A, a and b are positive constants.

(a) Show that the firms total profit is given by:

π(x,y)=−3x^2 −10y^2 −4xy+800x+1400y−17000.

(b) Assume π(x, y) has a maximum point. Find, step by step, the production levels that maximize profit by solving the first-order conditions. If you need to solve any system of linear equations, use Cramer’s rule and provide all calculation details.

(c) Due to technology constraints, the total production must be restricted to be exactly 60 units. Find, step by step, the production levels that now maximize profits – using the Lagrange Method. If you need to solve any system of linear equations, use Cramer’s rule and provide all calculation details. You may assume that the optimal point exists in this case.

(d) Report the Lagrange multiplier value at the maximum point and the maximal profit value from part (c). No explanation is needed.

(e) Using new technology, the total production can now be up to 200 units (i.e. less or equal to 200 units). Use the values from part (c) and part (d) to approximate the new maximal profit.

(f) Calculate the true new maximal profit for part (e) and compare with its approximate value you obtained. By what percentage is the true maximal profit different from the approximate value?

Answer 1