Question

Question 1 (32 marks) Consider a firm which produces a good, y, using two inputs or factors of production, x1 and x2. The firm's production function describes the mathematical relationship between inputs and output, and is given by (a) Derive the degree of homogeneity of the firm's production function. 4 marks) (b) The set is the set of combinations of (xi,x2) which produce output level yo.S is a level curve of f and is referred to by economists as the isoquant associated with output level yo. The isoquant implicitly defines x2 as a function of x1 i) Use implicit differentiation to derive the slope of the isoquant f3 - yo- That is, derive (Note that along a given isoquant Ay-0). (3 marks) i) Use implicit differentiation to rivefor the isoquant (3 marks) iii) What conclusions do you draw regarding the slope and curvature of the soquant? Briefly explain (2 marks) i) Derive the Hessian matrix associated with the firm's production function (4 marks) ii) State a sufficient condition(s) for (1) to be a strictly concave function. (2 marks) iii) Derive a restriction on the parameters α and β which ensures that (1) is a strictly concave function (4 marks) (d) Let S* is the set of combinations of (xi,x2) which produce at least output level yo. Economists refer to S as the upper contour set associated with output yo. Assume that That is 2o and y 2 yo i) Let z = (zi,22) E R What must be true for (1 mark) ii) Let =Ax+(1-1), where 0

Answer 1

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