Note: how can i solve this? especially (c) and (d) if possible beware that this is a adverse selection problem. 6. A worker with a marginal value product of can earn w, = k√e in self-employment. Marginal value products vary across the population accordingto the uniform distribution, i.e.,0~U[0, 2].Firms cannot measure individual productivity, so all workers are paid a wageequal to the average of the marginal value products of those employed.&(a Find the condition for k under which every worker will choose self-employment. Also, find the condition for k under which all the workerswill choose to be employed.Assume that k = 1. And, each worker can take a free test which can onlyte whether his marginal value product is in [0,1] or in [1,2]. Who will takethe test and who will not? Also, who will be'self-employed and who will beemployed?(c) Assume that k = 1. And, each worker can take a test at the expenseof c which can reveal his marginal value product exactly. Who will take thetest and who will not? Also, find the condition for c under which there existsome workers who will choose to take the test.(d) Continuing from (c), the test is not perfect in the sense that if a workerwith takes the test, the test reveals that the worker's marginal value productis 0+d with probability 1/2 and 0 - d with probability 1/2. Will more orless workers take the test than in (c)? (Hint: After the test result is revealed,each worker can decide whether to be self-employed or employed.)

Adverse selection refers to a situation where one party in a transaction has more information than the other, leading to suboptimal outcomes. In this case, the workers have private information about their marginal value products, and the employer cannot observe this information directly.Who Will Take the Test:Workers with a true MVP in the range [0,1] would find it beneficial to take the test if Condition for c:The condition is c≤k.So, workers with a cost of taking the test (c) less than or equal to k will choose to take the test. The introduction of a test adds a layer to the adverse selection problem. Workers with MVPs in the range [0,1] have an incentive to take the test if the cost c is less than the potential gain from revealing their information.The imperfect test complicates the adverse selection problem further. Now, even if a worker takes the test and reveals their information, there is uncertainty about the true MVP due to the test's imperfection.This imperfection may lead to a more cautious approach by workers. More workers might still take the test compared to a situation with no test, but the uncertainty introduced by the imperfect test might reduce the overall willingness of workers to reveal their information.In adverse selection scenarios, the challenge is often to design mechanisms that mitigate the information asymmetry. This can involve creating incentives for truthful revelation or finding ways to make private information less relevant to decision-making. In this case, the imperfect test adds an element of risk and further complicates the decision-making process for both workers and employers.Decision-Making After the Test:After the test, a worker with a revealed MVP of v+d will choose self-employment if kv+d≥k1.A worker with a revealed MVP of 0−d will choose self-employment if .The probabilities of these outcomes depend on the distribution.Who Will Take the Imperfect Test:Workers with a true MVP in the range might still find it beneficial to take the test if the expected benefit from the imperfect test justifies the cost c.Let's calculate the expected benefit:Decision Rule:Workers will take the imperfect test if the expected benefit is greater than the cost (c):Simplify the Inequality:This inequality depends on the specific values of v, d, and k, but the general structure involves comparing the expected benefits from the two potential outcomes of the test to the cost of taking the test. In summary, more workers might take the imperfect test in Part (d) compared to the perfect test in Part (c) if the expected benefit from the imperfect test outweighs the cost c. The decision to take the test will depend on the specific values involved in the inequality.C - workers with a cost of taking the test (c) less than or equal to k will choose to take the test. The workers whose cost of taking test is higher than k , will not take the test.d- More workers might take the imperfect test in Part (d) compared to the perfect test in Part (c) if the expected benefit from the imperfect test outweighs the cost c. The decision to take the test will depend on the specific values involved in the inequality.