Assignment
Problem 1: Credit Market Equilibrium
Javier is an entrepreneur who is deciding between two investment projects. Both projects are risky and require an investment of $100. The following are the details on the two projects:
Project 1 consists of foundingan economic consulting firm "Excelencia con Javier," which is relatively safe: with 80% probability it succeeds and generates $600 of revenues, and with 20% probability it fails and generates only $200 of revenues.
Project 2 is to open "Kickin' it with Javier", a soccer clinic for children. Although Javier is an outstanding soccer player, this project is much riskier: with 20% probability, it succeeds and generates $1200 of revenues, and with 80% probability it fails and generates only $150 of revenues.
Oscar is a banker who may offer Javier a loan.Oscar's opportunity cost of money is 10%. In other words, he would earn a 10% interest rate if he invested the money in a bank instead of lending it to Javier.
First, consider that Oscar offers unlimited liability contracts, i.e. Javier must repay the loan (full principal and interest) whether his project fails or succeeds. Oscar can specify which project Javier must select and enforce this selection. A credit contract thus specifies two terms: the Project and the interest rate. Let denote the contract. For example, the contract means that Javier must do Project 1 and the interest rate is 10%. Based on the information provided, answer the following:
Let y_1 and y_2 denote Javier's incomes and π_1 and π_2 denote Oscar's profits from Javier's Project 1 and 2, respectively. Derive the expressions for ?E(y?_1) and E(y_2), the expected values of Javier's incomes under the two projects; and, E(π_1) and E(π_2), the expected values of Oscar's profits from loans that finance Projects 1 and 2 respectively, as functions of the interest rate, i.
Using Excel template available on Canvas, graph ?E(y?_1), E(y_2), E(π_1) and E(π_2) as functions of the interest rate, i (i.e., put i on the horizontal axis and graph over the range i = 0 to i = 5 with 0.1 intervals). Title this graph "Figure 1: Credit Market under Unlimited liability Contract".
Which of the following will be the equilibrium contract if Oscar is a monopolist?
What is Oscar's expected profit under the above equilibrium contract?
What is Javier's expected income under the above equilibrium contract?
Which of the following will be the equilibrium contract be if perfect competition instead characterizes the credit market? (Under perfect competition, the equilibrium contract will make the borrower as well off as possible while allowing the lender to earn at least zero expected profit)
What is Oscar's expected profit under the above equilibrium contract under perfect competition?
What is Javier's expected income under the above equilibrium contract under perfect competition?
Now, consider that Oscaroffers a limited liability contract which means that Javier does not have to repay the loan in full if his project fails.However, if Javier's project succeeds he must repay the full loan (principal plus interest). If his project fails, he only has to repay 70% of the total debt obligation (for example, if the interest rate is 20%, he would have to repay 0.7*(1+0.2)*100 if his project fails).
Under alimited liability contract as described above, derive expressions for ?E(y?_1) and E(y_2), the expected values of Javier's incomes under the two projects; and, E(π_1) and E(π_2), the expected values of Oscar's profits from loans that finance Projects 1 and 2 respectively, as functions of the interest rate, i.
Using the Excel template available on Canvas, graph?E(y?_1), E(y_2), E(π_1) and E(π_2) as functions of the interest, i.Title this graph "Figure 2: Credit Market under Limited Liability".
What will the equilibrium interest rate be if perfect competition characterizes the credit market?
What is Oscar's expected profit under the above equilibrium contract?
What is Javier's expected income under the above equilibrium contract?
Explain intuitively what happens when Oscar is a monopolist and cannot observe/enforce Javier's choice of project. Would she still be able to extract all surplus from Javier given this information asymmetry?
Problem 2: Adverse Selection & Moral Hazard
Ed is a moneylender who lives in the village of Chibuto in Mozambique. Half of the farmers in Chibuto are SAFE farmers and the other half are RISKY farmers.Both types of farmers need a loan of $150 in order to farm. Farmers will take a loan as long as they can earn at least zero expected income. SAFE farmers have a good harvest in which they earn revenues of $300 with 100% probability. They never have a bad harvest. RISKY farmers have a good harvest in which they earn revenues of $450 with 50% probability. They have a bad harvest in which they earn revenues of $0 with 50% probability. Ed has perfect information about the farmers, i.e. he knows who is a SAFE farmer and who is RISKY.Ed's opportunity cost is money is 10%. The farmers must repay the full loan plus interest if harvest is good, but nothing if harvest is bad.
Let y_S and y_R denote the incomes of SAFE and RISKY farmers, respectively andπ_S and π_R denote Ed's profits from loans to SAFE and RISKY farmers, respectively. Derive the expressions for E(y_S) and E(y_R), the expected incomes of SAFE and RISKY farmers; and, E(π_S) and E(π_R), the expected values of Ed's profits from loans to SAFE and RISKY, respectively, as functions of the interest rate, i.
Graph E(π_S), E(π_R) , E(y_S) and E(y_R) as functions of the interest rate, i (i.e., put i on the horizontal axis and graph over the range i = 0 to i = 3). Label this "Figure 3".
Using your equations and graph, answer the following questions:
What is the highest interest rate a SAFE farmer would be willing to pay for a loan from Ed?
What is the highest interest rate a RISKY farmer would be willing to pay for a loan from Ed?
What is the lowest interest rate Ed would be willing to charge on a loan to a SAFE farmer?
What is the lowest interest rate Ed would be willing to charge on a loan to a RISKY farmer?
Assume that Ed is a monopolist.
What is the equilibrium interest rate Ed would charge to a SAFE farmer?
What is the expected profit that Ed earns on this loan to SAFE farmers?
What is the equilibrium interest rate Ed would charge to a RISKY farmer?
What is the expected profit that Ed earns on this loan to RISKY farmers?
Now, Anubhab is from a neighboring village and does not know the farmers in Chibuto. He only knows that half of the farmers are SAFE and half are RISKY. As a result, he has to charge a single interest rate to everybody who wants a loan. Like Ed, Anubhab's opportunity cost is also 10%.
What is the maximum interest rate Anubhab can charge so that both types of farmers would want to borrow?
Let π be Anubhab's profit. Which of the following is an expression for E(π), the expected value of Anubhab's profit from a loan, as a function of the interest rate, i when both types of farmers want to borrow?
Explain what will happen if Anubhab increases the interest rate above the interest rate you identified in (e)?
What is the maximum interest rate Anubhab can charge so that at least one type of farmer will want a loan?
Problem 3: Risk
Anubhab, Ed and Xinda are cotton farmers in the village of Qacha's Nek in rural Lesotho. They each have zero wealth (W=0), so their consumption is equal to the income they earn from their economic activity. Each of them must choose one (and only one) of the following three activities:
Activity 1: Full time farming with UKM08 cotton seeds: A farmer should work full time (7 days per week) on their farm if they are cultivating UKM08 cotton. Working full time a farmer has a 50% probability of having a GOOD harvest and earning income of $420 and a 50% chance of having a BAD harvest and earning only $60.
Activity 2: Full time farming with UK92 cotton seeds: UK92 variety is very well adapted to local weather conditions and thus has no risk. If a farmer works full time she will earn $210 with certainty.
Activity 3: Part-time farming with UKM08 cotton seeds: In this third activity, a farmer plants UKM08 variety cotton seeds and works Monday through Thursday on his farm and he works Friday through Sunday as a construction worker in the nearby city of Geita. Since he is not able to work full-time on the farm, it is more likely that he suffers damages from pests or bad weather. Specifically, the probability of having a GOOD cotton harvest and earning $420 drops to 25%, while the probability of having a BAD harvest and earning only $60 increases to 75%. The individual also earns $60 with certainty as a construction worker (i.e., he earns $60 in addition to his farm income under both a GOOD and a BAD harvest).
What is the expected value of consumption for each activity?
Anubhab, Ed andXinda view risk differently. This is reflected in the differences in their utility functions, which are listed below. Using those utility functions, compute the certainty equivalent (CE), the risk premium (RP) and expected utility (EU) associated with each of the three activities for each individual.
Anubhab: U(C)=2C
Ed: U(C)=40C-0.05C^2
Xinda: U(C)=0.1C^2
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Table 1. Certainty Equivalent, Risk Premium and Expected Utility for 3 Activities
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Activity 1
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Activity 2
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Activity 3
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EU Anubhab
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CE Anubhab
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RP Anubhab
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EU Ed
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CE Ed
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RP Ed
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EU Xinda
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CE Xinda
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RP Xinda
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c. Match each individual with the activity that they will choose.
Anubhab Activity 1
Ed Activity 2
Xinda Activity 3
d. Match the following based on the type of risk preferences that describe each individual.
Anubhab Risk Neutral
Ed Risk Averse
Xinda Risk Loving
e. Assume that the three farmers start with the same level of wealth. In this environment would you expect wealth inequality to remain the same, increase or decrease over time? Explain your answer in a few sentences.
Attachment:- Excel-Template.rar