Two-period Ramsey optimal taxation. Consider a two-period economy in which the government must collect taxes in order to nance government purchases 1 and 8'2. Suppose that the govemment cannot levy lump-sum taxes; instead, all taxes are in the form of proportional consumption taxes (i.e., sales taxes). When taxes are pro- portional, one general lesson we've seen is that the timing of taxes does matter (for consumption and national savings).10 What it means that the timing of taxes \"matters\" is that consumption decisions of the economy are affected by the tax rates that the government levies. In order to raise the revenue needed to pay for 8| and 82, the government does have to levy some taxes: an interesting question is what is the optimal tax rates for the government to set. Here we mean \"optimal\" in the sense that the government ultimately cares about the representa- tive consumer's lifetime utility, which is known as a \"Ramsey government problem." The Ramsey govemment problem essentially tries to answer the following question: What tax rates should the government set? Assume the following: a. b. E. f. The real interest rate between period 1 and period 2 is zero (i.e., r, : 0). The representative agent has no control over his real income 3!. or }'2. The consumption tax rates in the two periods are denoted If and rg'. The representative consumer starts with zero initial assets (on = 0); thus the LBC ofthe consumer is [l + fitm + (I + the; = y: + 3:3. The government starts with zero initial assets [bu = 0); thus the LBC of the gov- ernment is g. +32 : rf'c. +rc3. The lifetime utility function ofthe consumer is total, {:3} : 1n c. + lncg. There are three steps to computing the Ramsey-optimal tax rates. a. The first step is to determine the consumer's optimal choices of c: and (:3 as func- tions of y1, yz, fl, and If. In setting up the appropriate Lagrangian, solve for the optimal choices of consumption in period 1 and consumption in period 2 as a function of these four objects. Next, for a moment, suppose that the government did have the ability to levy lump- sum taxes. If it could levy lump-sum taxes, then the government's LBC would be g1 +552 = T, + T3 (i.e., these are the some 31 and 82 as above) and the consumer's LBC would be (3. +1". +c2 + T2 = y: + ya, where T1 and T3 denote lump-sum taxes in periods 1 and 2. b. In the case oflump-sum taxes, what would be the consumer's optimal choices of consumption in period 1 and period 2? {Set up a Lagrangian here if you need, but if you are able to logically determine the optimal quantities of consumption here, you may do so.) Compare your solutions for optimal period-1 consumption and optimal period-2 consumption in parts a and b above- Solve for the tax rates rf' and :5 that equate these two different sets of choices. These tax rates you nd, which are the optimal tax rates, should be functions of El, )'2, 31, and 32 and no other variables. How do the two tax rates compare to each other