A closed economy has a constant return to scale production technology as following : Y = zF(K, N), where Y is the aggregate output of consumption goods, z is the level of productivity, K denotes aggregate capital stocks, and N is the labor size. The capital stock accumulation function is: K 0 = (1 )K + I where K0 represents the aggregate capital stock of tomorrow, K denotes the aggregate capital stock of today, is the depreciation rate of capital, and I is today's aggregate investment. The rate of growth in the population equals to n which is a constant over time. The government purchases G units of consumption goods in the current period, where G = gN and g is a positive constant. The government finances its purchases through lump-sum taxes on consumers, where T denotes total taxes, and the government balances its budget each period, so that 2 G = T. consumers consume a constant fraction of disposable income. i.e. C = (1s)(Y T), where s is the saving rate and C is the aggregate consumption, with 0 < s < 1 and C > 0.

(a) Soppose T = 0.

i. Derive the capital/labor ratio of tomorrow as a function capital/labor ratio of today.

ii. Suppose that the population growth rate n increases. Analyze, using diagrams and equations, what will happen to the steady-state capital per worker k, the steadystate output per worker y, and the steady-state consumption per worker c.

(b) Suppose 0 < T < Y and Y = zF(K, N) = zk.

i. Derive the capital/labor ratio of tomorrow as a function capital/labor ratio of today.

ii. Under what circumstances, there will be a steady state.

iii. Assume this country is in the steady state before an exogenous shock happens. Now, s, saving rate, permanently increases, graphically and intuitively explain the short run change in k (capital/labor ratio), i (investment/capital ratio), c (consumption/labor ratio), and long run as well.