Consider a one-period economy with a single representative consumer, a single representative firm and the government. The representative consumer derives utility from consumption c and leisure l:

u (c, l) = ln c + ln l (1)

The firm produces output Y using capital K and labor N according to

Y = zKaN 1a (2)

where z is the total factor productivity and a is the Cobb-Douglas parameter. The firm maximizes profits which are then transferred to the representative consumer. The government balances the budget using lump-sum taxes T on the representative consumer to finance government spending G. The hourly wage in this economy is w and the consumer has h hours to divide between leisure and labor. Assume that h = 16.

(i) Write down the consumer's budget constraint and the firm's profits function.

(ii) Assume that w = 10, z = 20, a = 0.3, and K = 1. Calculate the number of hours that the firm would like to hire and the profits of the representative firm.