Consider the following representative agent model. The representative consumer has preferences given by (c, l)= c+ pl

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 = zK^1'-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.

Required:
Solve for all prices and quantities in a competitive equilibrium (there consider).