Consider the following one-period model. Assume that the consumption good is produced by a linear technology: Y = zN D where Y is the output of the consumption good, z is the exogenous total factor productivity, N D is the labour hours. Government has to finance its expenditures, G, using a tax on the representative firm. The government collects t units of consumption goods from the firm for each unit of labor it employs (0 < t < 1). There is no other tax in the economy. The firm is owned by the representative consumer who is endowed with h hours of time she can allocate between work, NS and leisure, l. Preferences of the representative consumer are: U(c, l) = ln c + ln l

(a) Write down the definition of a competitive equilibrium for the above economy.

(b) Show that the Walras' law holds for this economy.

(c) Solve for the leisure, l, the consumption, c, employment, N, wage rate, w, tax rate, , and output, Y in equilibrium. (d) Solve for the optimal allocation of leisure, l, the consumption, c, employment, N, output, Y . Contrast these quantities with those in competitive equilibrium from (2c). Is the competitive equilibrium identical to the optimal allocation? Explain why (not).