advanced microeconomics, principal agent Question:Consider the following principal agent problem with adverse selection. A firm facesa worker who may be one of two types, with equal probabilities. The firm's profitsfrom a type i worker are given by πį = ¤¿liSi, i1, 2, where e; is the effortsupplied by a type i worker and s; is the payment to a type i worker. The costfunction of the more productive worker (type 1) is given by c₁ = e² and the costfunction of the less productive worker (type 2) is given by c₂ = 2e2. The utilityfunction of a worker of type i is given by: uį = siSi C₂ and his opportunity costutility is u 0. Find the solution to the firm's problem (assuming that effort isobservable and contractible).1/2=Use below Methods to solve above question:4 Principal-Agent - Hidden Information. Consider the following principal-agent problem.• Effort (e) is observable, so we can contract on it.● But, the agent has some private information.Agent:●• For example, suppose the agent can be either one of two "types": type 1 or 2.● Each agent knows her type, but the principal does not.• The principal's probability that the agent is type i is pi.. An alternative interpretation is as follows.● There are many agents in the population, some of type 1, others of type 2, where the proportion of type iagents is pi.• Now, assume that the two types differ in terms of their "productivity."• Specifically, their effort cost functions are different.. Let the cost functions of the two types be given by:c²whereand 0² =0² 6(e) >● thus, marginal costs for type 2 are also higher:Let s; be the firm's payment to a type i agent.• The utility function of a type i agent is given by:=. The firm is, therefore, also risk-neutral.u² : = si − 0¹ þ(ei)oi p(ei)).• Assume that the opportunity cost utility for a type i agent is u? (we can take it as u = 0).The Firm:● Let the firm's profits from a type i agent be given by:The agent is, therefore, risk-neutral (his utility is linear in s; –●0² (ei)oʻ(ei) > 0, ø″(ei) > 0Oh > 0² = 0¹ ⇒0¹ (e) for all e,0² þ′(e) > 0¹ þ′(e) for all e$1Ti = ei Si-4.3 Case II: Hidden Information● Again, remember that effort is observable, but whereas agents know their type, the firm does not.• The firm's problem is to choose "two packages," (e₁, 8₁) and (e2, 82), each designed to be attractive only to one(the correct one) type, such that expected profits are maximized:-max (e₁ - 81)p1 + (e₂ - $2)p2 :ei, siparticipation constraints● This can be re-written as:$₁0¹0(e₁) $2 - 0¹0(e₂)≥52-0² (е₂) ≥ 8₁-0² 0(e₁)S2$18₁ - 0¹6(e₁) ≥ 0pc1 = pclS20² (₂) ≥ 0pc2 = pchincentive compatibility constraintsS2 =0² 0(e₂)ICC1= ICCIICC2 = ICCh. Note that this implies that type 1 will also participate.• Why?● Since type 1 can always "pretend" to be type 2 and then receive:$2 − 0¹ 0(e₂) > s2 – 0² (€₂) = 0-● The constraint ICC1 ensures that a type 1 agent is better off accepting the package designed for her (ratherthan pretending to be a type 2 agent).● The constraint ICC2 ensures that a type 2 agent is better off accepting the package designed for her (ratherthan pretending to be a type 1 agent).Considerand pc2:pc1• For a type 2 to accept, we need s2 ≥ 0² 0(e₂).• But, since the firm wants to make the payment as low as possible, it will set:=(32)$₁0¹0(e₁) ≥ s₂ - 0¹¹ (е₂):S20² 0(e2) — 0¹ 0(e₂) (0² - 0¹) (e₂) > 0since 0² >01(33)● Again, note that from the above, we see that type 1 extracts a surplus:$1$₁ - 0¹¹ (е₁) > 0• But, to prevent type 1 from acting as type 2, it is sufficient to set:$₁ - 0¹6(e₁) = (02² - 0¹) (e₂)(43)(44)i.e., if she acts like a type 2, she will get a strictly positive surplus.• This says that if type 2 participates, type 1 will also participate.Next, consider the two incentive compatibility constraints ICC1 andICC2:(45)(46)(47)=(48)(49)• Primarily, we are interested in revealing type 1 (low-cost type).• We will, therefore, concentrate on Icci rather than IcC2.Later, we will show that if Icc₁ holds, so does ICC2.• Hence, plugging pc2 (50) into ICC1(48), we get that in order to prevent type 1 from acting as type 2, we haveto satisfy the ICC1:(50)(51)(52)(53)(54)(55)(56)$₁ = 0¹ 6(e₁) + (0² – 0¹) (e2)● Condition (57) will make type 1 (just) indifferent between acting her type or pretending to be a type 2.● In other words, it does not leave any extra surplus over and above what is necessary to prevent type 1 frompretending to be type 2.(57)

An agent may act in a way that is contrary to the best interests of the principal. Thus,…

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 - -