2. In one of the first lessons of the class, we talked about the intertemporal tradeoff between capital and consumption. We can think about that more carefully now, and tie a bunch of threads together. The economy of West Haven exists for two time periods. Its utility function is U = 10 C and the interest rate is r=10%, so the present value of all social welfare in West Haven is W = 10 C1 +1 1.1 10 C2, where C1 is consumption in the first period and C2 is consumption in the second period. West Haven starts with total capital of K1 = 9. Depreciation is 100%, so absent any investment, K2 = 0. (That is, West Haven must entirely re-build its capital stock each time period). Aggregate output is determined by the production function Y = 3 K. Disposable income (Yd = Y-T) is split between savings and consumption, Y - T = C + S, where the savings rate is 0.1 (households save 10% of each dollar of disposable income). Savings equals investment, so any period-1 savings increases period-2 capital (S1 = I1 = K2). Initially there are no taxes, T = 0, so Yd = Y. a. What is output in period 1, Y

1? How much is saved S1 and how much is consumed C1 in period 1?

b. How much capital will there be in period 2? What will output Y2 and consumption C2 be in period 2?

c. What the present value of all social welfare in West Haven be, in this scenario? Now, the government imposes a tax. They set T = 3 in time period 1. All tax revenues go into capital investment, in addition to any private savings (so that S1 + T = I1 = K2). There is no tax in time period 2.

d. How much is privately saved S1 and how much is consumed C1 in period 1 under the tax?

e. Given the tax, how much capital will there be in period 2? What will output Y2 and consumption C2 be in period 2?

f. What will be the present value of all social welfare in West Haven, with the tax?

g. Did the tax increase or decrease the present value of social welfare? Explain, using the concept of consumption smoothing.