Answer to Question #106420 in Operations Research for simran

Question

Two firms X and Y produce the same commodity. Due to production constraints, each firm is

able to produce ,1 3 and 5 units. The cost of producing x units for firm X is

6x^2 - qx +5 and firm Y has identical cost function

6y^2 − qy +5 for producing y units. p is the price of one unit for firm X . We assume that the market is in equilibrium.

The outcomes are the profits of the firm shown in the form of a matrix { } A = aij . Write (i) a_11

(ii) a_22 (iii) a_21 , if demand function D( p) is given as D( p) = 50 − p .

Answer 1

Solution:

$$A_{ij}=\Sigma TR- \Sigma TC$$

$$TC_x=6x^2-qx+5, FC=5, VC = 6x^2-qx$$

$$TC_y=6y^2-qy+5, FC=5, VC = 6y^2-qy$$

$$TR=pq$$

For A11

$$1+1=50-p$$

$$p=48$$

$$TR=48 \times 2=96$$

$$TC=TC_x+TC_y$$

$$TC_x=6x^2-x+5; TC_y=6y^2-y+5$$

$$A_{11}=96-6(x^2+y^2)-(x+y)-10=86 -6(x^2+y^2)-(x+y)$$

For A22:

$$3+3=50-p; p=44$$

$$A_{22}=264-6(x^2+y^2)-3(x+y)-10=254 -6(x^2+y^2)-3(x+y)$$

For A21:

$$3+1=50-p; p=46$$

$$A_{21}=184-6(x^2+y^2)-3-y-10=174 -6(x^2+y^2)-3x-y.$$