Consider an economy with identical households who are distributed on the unit interval. Social planner faces the following problem: subject to max Btu(ct) t=0 (1) Ct +kt+1 = [(1 Xt)kt] + (1 8)kt (2) Ct, kt+1 0 and ko > 0 given where a (0, 1) and Xt [0, 1] is the fraction of capital allocated to improve environmental quality. Assume a typical utility function: strictly concave, strictly increasing, and (at least) twice differen- tiable. Suppose that the planner solves the above problem taking Xt as given; it is controlled by the Environmental Protection Agency (EPA). 1 1. Setup the Lagrangian of the planner's problem and derive the conditions that characterize opti- mal path of this economy. (10 points) 2. Suppose that Xt = X at the steady state. What is the steady state level of capital (k) in this economy? Describe the effect of x on k and provide your economic reasoning. (10 points) 3. Suppose that the EPA decides to increase Xt from x to x' where x'> x because their concern for the environment has increased. Assume that the EPA announces at t = 0 that Xt = x' is implemented from t = 0 and the change is permanent (hence unanticipated permanent policy change). Assume that it is t = 0 now and the economy is at the steady state. As was learned at the class, assume that income effect dominates substitution effect. What happens to this economy? Go as far as you can in describing the behavior of capital and consumption from t > 0 using diagrams. (20 points) 4. An economist A argues that increasing x discourages the firm's incentive to invest hence invest- ment to output ratio (i/y) at the steady state will decrease but instead consumption to output ratio (c/y) at the steady state will increase. Do you agree with Economist A? Please be as formal as you can in articulating your answer. (20 points)