Question

Consider a firm which produces a good, y, using two factors of production, xi and x2. The firm's production function is 1/2 1/4 = xi X2. (4) Note that (4) is a special case of the production function in Question 1, in which 1/4 1/2 and B a = Consequently, any properties that the production function in Q1 has been shown to possess, must also be possessed by the production function defined in (4). The firm faces exogenously given factor prices, W1, W2 € R.t where wiis the price of a unit ofxi and w2 is the price of a unit of x2.The firm's total cost of production, TC, is TC = wix1 + w2X2, (5) where xi denotes the number of units of factor 1 used in production and x2 denotes the number of units of factor 2 used in production. Therefore, the function TC is the objective function for this problem. For any given level of output y that the firm produced, it wishes to choose the levels of xi and x2 which minimize the cost of producing that level of output. That is, the firm's optimization problem is min (wixi + w2x2) X1x2 subject to 1/2 1/4 (6) (a) i) What assumptions about the properties of (5) and (6) guarantee that there is a global solution to the firms minimization problem? Briefly explain. (2 marks) i Does (5) satisfy the required assumption(s)? Briefly explain. (2 marks) ii Does (6) satisfy the required assumption(s)? Briefly explain. Hint: Draw a graph of S* for a given level of output, bearing in mind that fi > 0,f2 > 0. Use the graph to argue that (6) does or does not satisfy the required assumption(s) (2 marks) iv) What is the degree of homogeneity of the production function f(x1,x2) = x1^x^^? Briefly explain

Answer 1

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