Consider the following principal agent problem with adverse selection. A firm faces a worker who may be one of two types, with equal probabilities. The firm's profits from a type i worker are given by i = ei Si, i = 1, 2, where e; is the effort supplied by a type i worker and s; is the payment to a type i worker. The cost function of the more productive worker (type 1) is given by c₁ = el and the cost function of the less productive worker (type 2) is given by c₂ = 2e2. The utility function of a worker of type i is given by: u; = c; and his opportunity cost utility is u= 0. Find the solution to the firm's problem (assuming that effort is observable and contractible). = s1/2 - -
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- Suppose that the expected value of weekly profits for an ice cream shop, before paying the manager, Amy, is where e is Amy's weekly avertime hours. Amy is risk-neutral but incurs a cost for working overtime. Thus, tatal expected surplus is What level of effort maximizes total surplus? The value of overtime that maximizes total surplus is e-hours. (Enter your response rounded to one decimal place.) E(x)=500+10c C(e)=² E(S)-[(x)-C(e).3. A risk-neutral principal hires an agent to work on a project at wage w. The agent's utility function is: v(w)-g(e), where v(w)= Jw and g(e)=e/2 The agent can choose one of two possible effort levels, e¡ = 4 or e, = 6 . If the agent chooses effort level e, = 4 the project yields 100 with probability 1/4 and 0 with probability 3/4. If he chooses e, = 6 the project yields 100 with probability ½ and 0 with probability ½. The reservation utility of the agent is 0. (a) Suppose the effort level chosen by the agent is observable by the principal. Find the contract chosen by the principal. Show graphically in terms of contingent utilities v(w,00) and v(w.)When the second order derivative of a function is greater than zero than the agent is risk lover. question; Asses the risk attitude of an agent represented by the expected utility function u(x)= 2x2-5. However my course material writes that this agent is risk neutral because it is affine. My question is that whys is this so despite the fact that the second order derivative is '4' which is >0. Kindly explain this to me with complete steps.
- QUESTION 4 Mrs. Obaatanpa has a wealth of Ghe 3,500 for one year. There is a 35% probability that she will get sick and she estimates her loss from the illness to be Gh 1,600. Her utility function is given as U(Y) = VY, where Y is the amount of wealth she has. a) Comment on her utility function. Is she risk-neutral? b) Estimate the risk that she faces and explain. c) What is her expected utility? d) Suppose that she can buy an insurance policy that will cover the entire loss, what is the maximum premium she would be willing to pay?A firm's revenue R is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of effort e e [0, 0). The choice of effort affects revenue so that: E(R|e) = e and Var(R|e) = 1 where E(R|e) and Var(R|e) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level e. The employer cannot observe the level of effort exerted by the employee. The employer wants to design a wage contract w based on the revenue and considers only contracts of the form: W-α + βR, and so the employee is guaranteed a payment a and then a bonus payment BR which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w gives expected utility: 1 Eu(w\e) = E(w\e) - jeV ar(w\e) – c(e) pV. where E(wle) and Var(w|e) denote the expected value and variance of the contract, re- spectively, conditional on effort e, p is a parameter of risk aversion, and c(e) denotes the…QUESTION 12 Suppose that a worker value jobs by both the wage rate and the workplace collegiality. The woker's utility is strictly increasing in the wage rate but strictly decreasing in the chance of being bullied in the workplace. The utility function of the worker is U = wa (1- ), where a = 0.5, w is the wage rate and b is the probability that a worker is bullied in a workplace. Suppose for a typical job A, the chance of getting bullied is 0.01, and the wage rate is 100. Which of the following statements is correct? If job B offer w = 121 and b = 0.04, the worker would prefer job B to job A %3D If job B offer w = 144 and b = 0.09, the worker would prefer job B to job A %3D If job B offer w = 80 and b = 0, the worker would prefer job B to job A If job B has bullying probablity b = 0.04, the worker is indifferent between job A and job B. Then, the compensating wage differential of job B is at least 25. %3D If job B offer w = 80 and b = 0, the worker would be indifferent between job A…
- Exercise 3.17 (competitive product market interac- tions). There is a mass 1 of identical entrepreneurs with the variable-investment technology described in Section 3.4. The representative entrepreneur has wealth A, is risk neutral, and is protected by limited liability. Denote the average investment by I and the indi- vidual investment i (in equilibrium i = I by symme- try but we need to distinguish the two in a first step in order to compute the competitive equilibrium). A project produces Ri units of goods when successful and 0 when it fails. The probability of success is pH in the case of good behavior (the entrepreneur receives no private benefit) and pL = pH − ∆p in the case of misbehavior (the entrepreneur then receives private benefit Bi). Assume that it is optimal to induce the entrepreneur to behave. The market price of output is P = P(Q), with P' 0, where Q is aggregate production (with P(Q) tend- R ≡ RS…A risk-averse agent, Andy, has power utility of consumption with riskaversion coefficient γ = 0.5. While standing in line at the conveniencestore, Andy hears that the odds of winning the jackpot in a new statelottery game are 1 in 250. A lottery ticket costs $1. Assume his income isIt = $100. You can assume that there is only one jackpot prize awarded,and there is no chance it will be shared with another player. The lotterywill be drawn shortly after Andy buys the ticket, so you can ignore therole of discounting for time value. For simplicity, assume that ct+1 = 100even if Andy buys the ticket How large would the jackpot have to be in order for Andy to play thelottery? b) What is the fair (expected) value of the lottery with the jackpot youfound in (a)? What is the dollar amount of the risk premium that Andyrequires to play the lottery? Solve for the optimal number of lottery tickets that Andy would buyif the jackpot value were $10,000 (the ticket price, the odds of winning,and Andy’s…Scenario 17.5 Consider the following information: Income to the firm from workers who sell door-to-door Bad Luck Good Luck Low Effort (e = :0) $5,000 $7,000 High Effort (e = 1) $7,000 $13,000 Cost of effort: c = $2500e Probabilities: Bad luck = .75; Good luck = .25 A principal-agent problem arises in the situation in Scenario 17.5 because: O the principal can measure effort and output; the agent can measure only output. O the principal can measure only effort, and the agent can measure only output. the principal can measure only output, and the agent can measure effort and output. neither the principal nor the agent can measure effort. O neither the principal nor the agent can measure output.
- Consider the following example. A risk-neutral worker can choose high or low effort. The worker's outside option is 0. The manager cannot observe the worker's action, but the manager can observe the realized revenue for the firm (either $100 or $200). The probability of each revenue depends on the worker's effort: Low effort: cost of effort : $0 probability of low revenue ($100): 75% probability of high revenue ($200) : 25% High effort: cost of effort : $11 probability of low revenue ($100): 25% probability of high revenue ($200) : 75% The manager offers a contract which gives the worker a flat wage of $10 and a bonus of $20 if revenue is high. Given this payment scheme, the worker will put in ✓effort. The contract (is/is not) ✓incentive compatible. The firm's expected profit is $ The firm is considering an investment that would increase worker morale. By making work more enjoyable, the program would reduce the worker's cost of effort from $11 to $9. If it costs the firm $20 to…Can you explain how Constant Relative Risk Aversion utility function should be understood and how it works mathematicallyCharles is participating in an experiment. His payoff in the experiment is tied to his effort e doing a mundane task. There is also some risk involved by design-there is a chance p that Charles is going to get a fixed payment L regardless of his effort. Charles' payoff is thus: with probability p w.e with probability 1- p Charles has to pay a cost C, which increases with his effort. First, let us assume that Charles' utility is the expected payoff net of this cost: U(e) = pL + (1 – p)we – c(e) Derive the first order condition with respect to e. b. How doesp affect Charles' effort e? c. How does L affect e?