2. Mike lives for seven periods, t = 1, 2, 3, 4, 5, 6, 7. In period 7, he enjoys the fruits of the human capital, e, that he has amassed up to that point. In period 7 his instantaneous utility for human capital is u7(e) = e. Mike does not derive utility from his human capital in periods 1 through 6, nor does his stock of human capital depreciate in any way. At the beginning of period 1, he starts off with a stock of 250 units of human capital. During periods 1 through 6 Mike has three options: Option A Write no documents Option B Write documents in Microsoft Word Option C Write documents in ATEXusing Overleaf Mike starts period 1 in the default option A, which does not builds human capi- tal. Option B builds 5 units of human capital per period (up to period 6, i.e., no 1 additional human capital is built in period 7), and Mike can switch costlessly from option A to option B at the beginning of any period. Option C builds 40 units of human capital per period (up to period 6), but it takes a one-time immediate effort cost of 90 units to switch from option A to option C (because it is painful and tricky to set up LATEX). Suppose that there is no way to switch between options B and C, and that once Mike makes any switch he cannot switch again. Mike is a hyperbolic discounter with = /and 6 = 1. He cannot commit his future behavior. He decides at the beginning of each period, including period 1, whether to switch from option A to some other option, and if so, to which other option. If he switches at the beginning of period t, he will build human capital at the rate given by the new option during period t, e.g., if he switches to option C in period 3, he will build 40 units of human capital in periods 3, 4, 5, and 6. (a) [14pts] Suppose Mike only has access to options A and B. If he is naive, when does he switch to option B, if at all? What about if he is sophisticated? (b) [14pts] Suppose Mike only has access to options A and C. If he is naive, when does he switch to option C, if at all? What about if he is sophisticated? (e) [12pts] Suppose Mike has access to all three options. Show that if he is naive he waits until period 4 and then switches to option B at the beginning of that period. Explain intuitively why he waits so long to switch to a superior option when he could have costlessly switched all along. Show that if Mike were sophisticated he would switch to option C in period 1.