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PC is examining a dynamic Ivlirrlees taxation problem. In essence, the productivity [3 of the agents may change over time. Let the agent's utility be it {c}, and it is continuously differentiable, strictly increasing, and strictly concave. Let the discount factor be ,3 and the interest rate be r, which are both constant over time. PC solved the government's optimization problem. For any two adjacent periods, t and t + 1, he found the following relationship between consumption in t and future expected consumption in t + l for an agent Twith productivity at in t : _1_=E _1__ mite. ' (1+r)u'(cn1 (91+lll Here, Cf (Hr) denotes the consumption of an agent with productivity H; in t, and E; denotes the expectation operator conditioned on the information in t. Given the relationship that PC found, prove that it is optimal to introduce a savings tax. Hint: The government inim- dnces :1 savings tax 1' :2: 0 if 11' (Cr (Hrll : 13 (1 + "l (1 TlEr [n'(cri1(8r+1})]