Two firms X and Y produce the samecommodity. Due to the productionconstraints, each firm is able to producepackages of 1, 3 and 5 units of the product.The cost of producing qx units for firm X is< [6 +2xq – 2qx + 5], and firm Y has theidentical cost function < [6 +2yq – 2qy + 5]for producing qy units. p is the price of oneunit for firm X. We assume that the marketis in equilibrium. The outcomes are theprofits of the firm shown in the form of amatrix A = [aij] (pay-off matrix). Write(i) a11, (ii) a22, (iii) a21, if the demandfunction D(p) is given by D(p) = 50 – p.

(i) To find a11, we need to calculate the profit of firm X when it produces 1 unit and firm Y produces 1 unit.

The cost function for firm X is given by [6 + 2xq - 2qx + 5], where qx represents the quantity produced by firm X. In this case, qx = 1.

The cost function for firm Y is identical, [6 + 2yq - 2qy + 5], where qy represents the quantity produced by firm Y. In this case, qy = 1.

To calculate the profit, we need to subtract the cost from the revenue. The revenue is given by the demand function D(p) multiplied by the price p.

Given that the demand function is D(p) = 50 - p, and assuming the market is in equilibrium, the price p is equal to the cost of production.

So, for firm X, the revenue is (50 - p) * p = (50 - (6 + 2(1) - 2(1)(1) + 5)) * (6 + 2(1) - 2(1)(1) + 5) = (50 - 9) * 9 = 41 * 9 = 369.

The cost for firm X is [6 + 2(1)(1) - 2(1)(1) + 5] = 6 + 2 - 2 + 5 = 11.

Therefore, the profit for firm X when it produces 1 unit and firm Y produces 1 unit is a11 = 369 - 11 = 358.

(ii) To find a22, we need to calculate the profit of firm X when it produces 3 units and firm Y produces 3 units.

Using the same calculations as above, the revenue for firm X is (50 - p) * p = (50 - (6 + 2(3) - 2(3)(3) + 5)) * (6 + 2(3) - 2(3)(3) + 5) = (50 - 33) * 33 = 17 * 33 = 561.

The cost for firm X is [6 + 2(3)(3) - 2(3)(3) + 5] = 6 + 18 - 18 + 5 = 11.

Therefore, the profit for firm X when it produces 3 units and firm Y produces 3 units is a22 = 561 - 11 = 550.

(iii) To find a21, we need to calculate the profit of firm X when it produces 1 unit and firm Y produces 3 units.

Using the same calculations as above, the revenue for firm X is (50 - p) * p = (50 - (6 + 2(1) - 2(1)(1) + 5)) * (6 + 2(1) - 2(1)(1) + 5) = (50 - 9) * 9 = 41 * 9 = 369.

The cost for firm X is [6 + 2(1)(1) - 2(1)(1) + 5] = 6 + 2 - 2 + 5 = 11.

Therefore, the profit for firm X when it produces 1 unit and firm Y produces 3 units is a21 = 369 - 11 = 358.