Consider an Infinite Horizon Model where households devote their entire unit of time to the labor market (l_t=L_1=1) and where the instantaneous household's utility function is given by u_t=logc_t. Also, the production function is given by Y_t=AK_t^ L_t^(1-) . Assume = 1.

a. Write the household's maximization problem and obtain its optimality conditions;

b. Write the firm's maximization problem and obtain its optimality conditions;

c. Find the steady state values for the endogenous variables.

d. What is the value function for the consumer?