A technology firm produces two types of processors. Let x be the number of units of...

   

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A technology firm produces two types of processors. Let x be the number of units of processor A and x2 the number of units of processor B. Let w and w be the unit price of processor A and processor B respectively. We know that the firm has to produce a total of q units. We also know that the firm's production function is g(x1, x2). We do not know how x1 and x2 enter the production function. What we know is that g(x1, x2) is a concave and increasing gi(x1, x2) function. That is > 0 for i = 1,2 for all values of (x1, x2). The firm's objective is ; to minimise the cost of production. Please answer the following (a) Write down this optimisation problem. [2 marks] (b) Write down the Lagrangian and the optimality conditions. (c) If (x1, x2) satisfies the optimality conditions, is it necessarily a solution of the problem? [4 marks] Justify your answer carefully. [4 marks] [Total: 10 marks] A technology firm produces two types of processors. Let x be the number of units of processor A and x2 the number of units of processor B. Let w and w be the unit price of processor A and processor B respectively. We know that the firm has to produce a total of q units. We also know that the firm's production function is g(x1, x2). We do not know how x1 and x2 enter the production function. What we know is that g(x1, x2) is a concave and increasing gi(x1, x2) function. That is > 0 for i = 1,2 for all values of (x1, x2). The firm's objective is ; to minimise the cost of production. Please answer the following (a) Write down this optimisation problem. [2 marks] (b) Write down the Lagrangian and the optimality conditions. (c) If (x1, x2) satisfies the optimality conditions, is it necessarily a solution of the problem? [4 marks] Justify your answer carefully. [4 marks] [Total: 10 marks]