Exercise 2.17. * We now derive the CES production function following the method in the original article by Arrow, Chenery, Minhas and Solow (1961). These authors noted that a good empirical approximation to the relationship between income per capita and the wage rate was provided by an equation of the form y = αw−σ, where y = f (k) is again output per capita and w is the wage rate. With competitive markets, recall that w = f (k) − kf0 (k). Thus the above equation can be written as y = α ¡ y − ky0 ¢−σ , where y = y (k) ≡ f (k) and y0 denotes f0 (k). This is a nonlinear first-order differential equation. (1) Using separation of variables (see Appendix Chapter B), show that the solution to this equation satisfies y (k) = h α1/σ + c0k σ−1 σ i σ σ−1 , where c0 is a constant of integration. (2) How would you put more structure on α and c0 and derive the exact form of the CES production function in (2.37)?