Explicitly reference any theorem or definition from the lecture notes which you appeal to when answering this question.

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Explicitly Reference Any Theorem Or Definition From The Lecture Notes Which You Appeal To When Answering This Question 1 (50.18 KiB) Viewed 55 times

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Explicitly reference any theorem or definition from the lecture notes which you appeal to when answering this question. Marks will be deducted for failing to do so. Consider a firm which produces a good, y, using two inputs or factors of production, Xi and X2. The firm's production function, which describes the mathematical relationship between the inputs xi and X2 and output y, is given by y = f(x1,x2) = x2 + x? where f: R: R. Consider the set D = {x1,x2) € R3_lx2 + x?? 2 yo}. That is, D is the set of all (x1,x2) € R}which, given (1), produces at least output level yo. Dis known as the upper contour set associated with output level yo. (a) Determine the degree of homogeneity of the production function given by (1). Show all steps in deriving your answer. No marks will be awarded for an unsupported answer. - = E (b) Prove that the production function y = x2 + x)2 is strictly concave on R++. (c) Prove that the set D = {(x1,x2) € Rxbx2 + x?? 2 yo} is a convex set. Hint 1: Assume that x = (x1,x2) € Dand v = (v1.V2) E D and prove that z = 4x + (1 - 2)v D for any 0<<1.
1/2 (d) Let So = {(x1,x2) € R$_[x2 + x2 = yo}. That is, So is the set of all combinations of (x1,x2) that produce exactly output level yo. Economists call the isoquant associated with output level yo. The equation 1/2 x1 + x2 = yo, implicitly defines xi as a function of x2. i) Derive the slope of the isoquant for yo. That is, derive dx2 dx ii) Derive d x2 dx iii) What do you conclude regarding the slope and curvature of the isoquant for yo? Briefly explain.