Problem 11 (10 pts) Consider a simple technology search problem. In each period (the period is in discrete time), the decision maker (DM) can either engage in production using the current technology or waiting for a new technique. If the DM accepts the current technology, then the production starts and the revenue of each future period will be fixed and equals the value of the technology; otherwise, there will be no revenue at current period and a new technology will appear at next period. Assume that the value of technology a in each period follows a period-invariant uniform distribution on [0,a]. Given the exponential discount factor (0,1), what is the optimal cutoff for the DM (the optimal cutoff is the minimal value of the technology that the DM will accept) to maximize the expected discounted revenue? - Hint 1: For exponential discount factor, the current value of revenue x that appears at any future period t>0, is tx. - Hint 2: The probability density function for uniform distribution on [0,a] is f(x)={a1,0,[0,a],xa. - Hint 3: Try to derive two value functions: one to model the original problem and another to model the value on the acceptance of certain technology. Problem 11 (10 pts) Consider a simple technology search problem. In each period (the period is in discrete time), the decision maker (DM) can either engage in production using the current technology or waiting for a new technique. If the DM accepts the current technology, then the production starts and the revenue of each future period will be fixed and equals the value of the technology; otherwise, there will be no revenue at current period and a new technology will appear at next period. Assume that the value of technology a in each period follows a period-invariant uniform distribution on [0,a]. Given the exponential discount factor (0,1), what is the optimal cutoff for the DM (the optimal cutoff is the minimal value of the technology that the DM will accept) to maximize the expected discounted revenue? - Hint 1: For exponential discount factor, the current value of revenue x that appears at any future period t>0, is tx. - Hint 2: The probability density function for uniform distribution on [0,a] is f(x)={a1,0,[0,a],xa. - Hint 3: Try to derive two value functions: one to model the original problem and another to model the value on the acceptance of certain technology