Reference no: EM13741245

<p>here is a worker with talent θ. There is a market of potential employerswho are willing to pay a wage equalling the talent level. The worker knows thetalent, but the employers do not. They believe talent to be normally distributedwith mean μθ and precision (which is defined as one divided by the variance)τ = 1 . The worker has the option to make an effort to get educated. Effort is θ σθ2chosen from the real numbers. Effort is invisible to anyone except the worker, but higher effort leads (probabilistically) to a higher grade s. Specifically, the grade is normally distributed with mean equal to the effort, and precision τs. The employers can see the grade and use Bayes' rule to estimate the effort.Bayes' rule: If the previous belief about effort is normally distributed withmean μe and precision τe and the grade is normally distributed with mean e andprecision τs, then after observing grade s, the new estimate of effort is normallydistributed with mean μeτe+sτs and precision τe + τs. τe +τsLinear transformation of normal distribution: If a variable x is nor- mally distributed with mean μx and variance σx2, then the linearly transformed variable ax + b is normally distributed with mean aμx + b and variance a2σx2.Suppose the employers believe that all workers choose their effort as a linear function of their talent: e = k1 + k2θ. Here, k1 is a fixed real number and k2 > 0.(1) Find the mean and variance of the normally distributed belief about the effort before the grade is seen.(2) Find the mean and variance of the normally distributed belief about the effort after the grade is seen.(3) Find the mean and variance of the normally distributed estimate of the talent after the grade is seen.The employers are risk-neutral and thus pay a wage equal to the mean of their belief about the talent. Wage is random, because it depends on the random grade (through the random mean of the belief). The worker's benefit from the random wage is: the expected the wage times b, with b > 0. If the wage is linear in the grade, then to find the expected wage, simply replace the grade in the wage by the mean of the grade.The worker maximizes the difference between the expected wage and the cost of effort. The cost of effort e for a worker with talent θ is c(e - αθ), with c positive, strictly increasing and strictly convex and α > 0.(4) Write down the maximization problem of a worker with talent θ and solve it for e. Find k1, k2 by equating the optimal e with k1 + k2θ for each θ.Effort should turn out to be a linear function of the talent, which justifies the belief of the employers that effort is linear in talent.(5) How does effort change in the parameters α, μθ, τθ, τs, b?(6) Write the expected payoff (benefit minus cost) of the worker when effort is chosen optimally. Denote this payoff by V . How does the expected payoff change in the parameters α, μθ, τθ, τs, b?3 Learning from data[6 marks]A worker would like to know how wage responds to grades. The assumptionis that wage is linear in the grade (w = q1 + q2s).(1) The worker has two observations: one person with wage w1 and grades1, another with different w2 and different s2. Find q1 and q2.(2) A third observation is added: w3 together with s3. Assume w3 is different from w1,w2 and s3 from s1,s2. The worker fits a line to the three observations by minimizing the sum of the squared vertical distances between the line and the data points. Write the minimization problem and solve for q1 and q2. The vertical distance between a line y = ax + b and a point (x0, y0) is |y0 - ax0 - b|. (3) The worker assumes that q2 ≥ 0. Solve the previous minimization prob-lem again, with this constraint added.</p>