Please Solve it Step by Step

Advanced Micrp

Consider the model of overconfidence from the lecture. Time is discrete and infinite: t{1,2,}. There is an overconfident agent with true ability a who believes that his 1/2 ability is a+. There is an unknown fundamental , which the agent initially believes is equal to ^0>0. Denote the agent's estimate of in period t as ^t. Each period the agent exerts effort et and observes output Qt(et,a,)=a++et21et2 We assume that in every period 1. given his current belief about the agent exerts effort et which maximizes Qt under the assumption that his ability is a+, 2. the agent observes actual Qt(et,a,) and as it does not coincide with his expectations he recalculates his belief about based solely on this observation assuming that his ability is a+. (a) What is an optimal et given ^t1 ? (b) Calculate ^t as a function of ^t1. (c) Suppose ^0=, i.e., the agent's initial belief about is correct. Illustrate that asymptotically the agent's belief about approaches some 0. Denote the agent's estimate of in period t as ^t. Each period the agent exerts effort et and observes output Qt(et,a,)=a++et21et2 We assume that in every period 1. given his current belief about the agent exerts effort et which maximizes Qt under the assumption that his ability is a+, 2. the agent observes actual Qt(et,a,) and as it does not coincide with his expectations he recalculates his belief about based solely on this observation assuming that his ability is a+. (a) What is an optimal et given ^t1 ? (b) Calculate ^t as a function of ^t1. (c) Suppose ^0=, i.e., the agent's initial belief about is correct. Illustrate that asymptotically the agent's belief about approaches some