1. a) A firm produces two goods x and y. The demand function for x is x=160-0.25Px And the demand function for y is y=240-0.5Py. The total cost of the firm is TC=x2 + xy + y2 + 45 And the firm should produce a maximum of 50 units of both X and Y. So now optimize the firm's profit subject to x + y=50. Use the Lagrange Multiplier method of constrained optimization and find all the optimal values. (Hint: Find the profit (TR-TC), then Lagrangian Function of Profit) b) Now use the Bordered Hessian method to show that all the optimal values that you find in (a) maximize the profit. (Hint: Please do not repeat the necessary conditions of optimization for bordered Hessian and the Crammer's Rule. Just construct the matrix with sufficient conditions of optimization and show that the conditions are satisfied.) c) Define shadow price and the effect on shadow price in optimization using this context. If now this firm can produce a maximum of 51 units of both X and Y. Please briefly discuss how it will affect a firm's decision? 1. a) A firm produces two goods x and y. The demand function for x is x=160-0.25Px And the demand function for y is y=240-0.5Py. The total cost of the firm is TC=x2 + xy + y2 + 45 And the firm should produce a maximum of 50 units of both X and Y. So now optimize the firm's profit subject to x + y=50. Use the Lagrange Multiplier method of constrained optimization and find all the optimal values. (Hint: Find the profit (TR-TC), then Lagrangian Function of Profit) b) Now use the Bordered Hessian method to show that all the optimal values that you find in (a) maximize the profit. (Hint: Please do not repeat the necessary conditions of optimization for bordered Hessian and the Crammer's Rule. Just construct the matrix with sufficient conditions of optimization and show that the conditions are satisfied.) c) Define shadow price and the effect on shadow price in optimization using this context. If now this firm can produce a maximum of 51 units of both X and Y. Please briefly discuss how it will affect a firm's decision