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.

(iii) Assume that government spending G is 10, the representative consumer receives the profits that you calculated above, earns hourly wage w = 10, and has h = 16 hours to divide between leisure and labor. Calculate how many hours of work the representative consumer would like to supply in the market.

(iv) In the economy described above, (i) the government budget is balanced, (ii) the representative consumer maximizes lifetime utility given the budget constraint and w, and (iii) the firm maximizes profits given the production function and w. Is this a competitive equilibrium? Why or why not?