Consider the following representative agent model. The representative consumer has preferences given by u(c, l) = c + βl
where c is consumption, l is leisure, and β > 0. The consumer has an endowment of one unit of time and ko units of capital. The representative firm has a technology for producing consumption goods, given by
Y = zKαLα-a
where Y is output, z is total factor productivity, K is the capital input, L is the labor input, and 0 < a < 1. The market real wage is w and r denotes the rental rate on capital.
Solve for all prices and quantities in a competitive equilibrium (there consider).