Both population and the work force grow at the rate of n = 1% per year in a closed economy. Consumption is C = 0.5(1 - t)Y, where t is the tax rate on income and Y is total output. The per-worker production function is y = 8vk, where y is output per worker and k is the capital–labor ratio. The depreciation rate of capital is d = 9% per year. Suppose for now that there are no government purchases and the tax rate on income is t = 0.

a. Find expressions for national saving per worker and the steady-state level of investment per worker as functions of the capital–labor ratio, k. In the steady state, what are the values of the capital– labor ratio, output per worker, consumption per worker, and investment per worker?

b. Suppose that the government purchases goods each year and pays for these purchases using taxes on income. The government runs a balanced budget in each period and the tax rate on income is t = 0.5. Repeat Part (a) and compare your results.