Consider an economy with n = 3 goods. The utility function of the

representative consumer of the economy is

U(x1, x2, x3) = x1

1/2 x2

1/4 x3

1/4

where xi 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 u(x1, x2, x3) 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 L = 400, t = 4 and 1 + v = 2 (so that v = 1).

(b) [10 points] Suppose sectors 1,2 are non-industrialized (that is, all firms in sectors 1,2 use

traditional mode) and sector 3 is industrialized (that is, all firms in sector 3 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.