Question 4 [16 points] Consider an economy with n = 3 goods. The utility function of the representative consumer of the economy is U(X1, X2, X3) = 21 X2 X3 where x, is the amount of good i (assume x1, x2, x3 are all positive). (a) [4 points] Making a suitable monotonic transformation, show that the preference of the consumer can be presented by a utility function w(x1, x2, x;) that is separable in X1, X2, X3. For the rest of the question, work with the function u(x1, x2, x3) obtained in (a). The economy has L units of labour that the representative consumer supplies. There are three sectors 1, 2, 3 in the economy, with sector i producing good i. Each sector has a competitive fringe of firms. Initially firms in all sectors use the existing technology (traditional mode) in which 1 unit of labour can produce 1 unit of good and the wage is 1. There is a new technology (modern mode) in which 1 unit of labour can produce t > 1 units of the good. The modern mode pays higher wage 1 + v. It is known that _ = 400, f = 4 and 1 + v = 2 (so that v = 1). (b) [12 points] Suppose sectors 2,3 are non-industrialized (that is, all firms in sectors 2,3 use traditional mode) and sector 1 is industrialized (that is, all firms in sector 1 use modern mode). Showing all steps of your work, determine (i) the income of economy and (ii) the demand for each of the goods 1, 2, 3