problem set 42
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In Question 1, the lender faces a single borrower who has a choice between two activities. In Question 1 (a)(e), the lender offers limited liability loans under symmetric information. In Question 1 (f)(k), the lender offers limited liability loans under asymmetric information.
The following assumptions describe the borrower and lender throughout Question 1.
Borrower: Michelle is an entrepreneur who is deciding between two investment projects. Both projects require an investment of $200. She does not have any money, so she needs a loan in order to undertake one of the projects, which have the following characteristics:
Project 1 consists of founding an economic consulting firm â€œExcellence with Michelle,â€ which is safe: with 100% probability it succeeds and generates $400 of revenues.
Project 2 is to open â€œJam with Michelleâ€, a ukulele class for children. Although Michelle is an outstanding musician, this project is much riskier: with 50% probability, it succeeds and generates $500 of revenues, and with 50% probability, it fails and generates no revenues.
Lender: Joanne is a banker who may offer Michelle a loan. Joanneâ€™s opportunity cost of money is 15%. In other words, she would earn a 15% interest rate if she invested the money in a bank instead of lending it to Michelle.
Limited Liability and Symmetric Information: We begin by assuming that Joanne offers a limited liability loan contract and faces symmetric information (i.e. asymmetric information is not a problem). Under the limited liability contract, if Michelleâ€™s project succeeds, she must repay 100% of the debt obligation (principal plus interest); however, if her project fails, she doesn’t have to repay anything. Symmetric information means that Joanne can specify which project Michelle must select, and she can enforce this selection. A credit contract thus specifies two terms: the Project and the interest rate.
(a)Derive expressions for and , the expected values of Michelleâ€™s income under the two projects, as functions of the interest rate i charged by Joanne. Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution. The solution should be written in interceptslope form. For example, for Project 1, report , where A and B are numbers that you calculate.
(b)Derive expressions for and , the expected values of Joanneâ€™s profits on a loan to Michelle when Michelle does Project 1 and 2 respectively, as functions of the interest rate i. Again, please report your â€œfirst stepâ€ and your solution. Report the solution in interceptslope form. For example, for a loan that finances Project 1, report , where A and B are numbers that you calculate.
(c)Using the Excel template available on Canvas, graph, , and as functions of the interest rate i. Title this graph â€œFigure 1: Credit Market under Limited Liabilityâ€. (Put i on the horizontal axis and graph over the range i = 0 to i = 3 with 0.1 intervals).
(d)If the credit market is characterized by perfect competition:
i.What is the equilibrium interest rate charged by Joanne?
ii.Which project does Joanne make Michelle do?
iii.How much expected profit does Joanne earn from the equilibrium contract?
iv.How much expected income does Michelle earn from the equilibrium contract?
(e)If the credit market is characterized by monopoly:
i.What is the equilibrium interest rate charged by Joanne?
ii.Which project does Joanne make Michelle do?
iii.How much expected profit does Joanne earn from the equilibrium contract?
iv.How much expected income does Michelle earn from the equilibrium contract?
Limited Liability and Asymmetric Information: Now, letâ€™s examine the impact of asymmetric information on the credit market equilibrium. Specifically, in this question, we assume that Joanne is not able to observe or enforce the project that the borrower chooses. As a result, the loan contract can only specify the interest rate (not the project). Assume that if Michelle is indifferent between the two projects, she chooses the one that makes Joanne better off. Everything else remains as it was in (a)(e).
(f)What type of asymmetric information problem does Joanne face? (Moral Hazard or Adverse Selection)
(g)Joanne now must consider how her choice of the interest rate affects Michelleâ€™s choice of project. For what interest rates will Michelle prefer Project 1 to Project 2? Please report your â€œfirst stepâ€ and your solution.
(h)For what interest rates will Michelle prefer Project 2 to Project 1? Please report your â€œfirst stepâ€ and your solution.
(i)For what interest rates will Michelle prefer not to borrow?
(j)If the credit market is characterized by monopoly:
i.What is the equilibrium interest rate?
ii.What project does Michelle choose?
iii.How much expected profit does Joanne earn?
iv.How much expected income does Michelle earn?
(k)If the credit market is characterized by perfect competition:
i.What is the equilibrium interest rate?
ii.What project does Michelle choose?
iii.How much expected profit does Joanne earn?
iv.How much expected income does Michelle earn?
Part 
First step or reasoning 
Solution 
(a) 

(b) 

(c) 

(d) 
(i) 

(ii) 

(iii) 

(iv) 

(e) 
(i) 

(ii) 

(iii) 

(iv) 

(f) 

(g) 

(h) 

(i) 

(j) 
(i) 

(ii) 

(iii) 

(iv) 
(k) 
(i) 

(ii) 

(iii) 

(iv) 
Question 2: Credit Market Equilibrium under Multiple Borrower Types
Now we turn to a different problem; namely, what happens when lenders face borrowers of different types. In Question 2(a)(e), the lender is a monopolist who offers limited liability loans under symmetric information. In Question 2(f)(p), the lender is a monopolist who offers limited liability loans under asymmetric information.
Limited Liability and Symmetric Information. Oscar is a moneylender who lives in the village of Oboadaka in Ghana. Half of the farmers in Oboadaka 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 $350 with 100% probability. They never have a bad harvest. RISKY farmers have a good harvest in which they earn revenues of $500 with 70% probability. They have a bad harvest in which they earn revenues of $0 with 30% probability. Oscar has perfect information about the farmers, i.e. he knows who is a SAFE farmer and who is a RISKY one. As a result, he can offer different contract terms to SAFE and RISKY types. Oscarâ€™s opportunity cost of money is 30%. Oscar offers limited liability credit contracts in which the farmers must repay the full loan plus interest if harvest is good, but nothing if harvest is bad.
(a)Let and denote the incomes of SAFE and RISKY farmers, respectively. Derive expressions for and , the expected incomes of SAFE and RISKY farmers, respectively, as functions of the interest rate i charged by Oscar. Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution. Report your final expressions in interceptslope format as in the questions above.
(b)Let and denote Oscarâ€™s profits from a loan to SAFE and RISKY farmers, respectively. Derive expressions for and , the expected values of Oscarâ€™s profits from loans to SAFE and RISKY farmers, respectively, as functions of the interest rate, i. Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution. Report your final expressions in interceptslope format as in the questions above.
(c)Graph , , and ) 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). Title this graph â€œFigure 2: Credit Market under Symmetric Informationâ€.
(d)Using your equations and graph, answer the following questions. For each, please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution.
i.What is the highest interest rate a SAFE farmer would be willing to pay for a loan?
ii.What is the highest interest rate a RISKY farmer would be willing to pay for a loan?
iii.What is the lowest interest rate Oscar would be willing to charge on a loan to a SAFE farmer?
iv.What is the lowest interest rate Oscar would be willing to charge on a loan to a RISKY farmer?
(e)Assume that Oscar is a monopolist.
i.What is the equilibrium interest rate Oscar would charge to a SAFE farmer?
ii.What is the expected profit that Oscar earns on this loan to SAFE farmers?
iii.What is the equilibrium interest rate Oscar would charge to a RISKY farmer?
iv.What is the expected profit that Oscar earns on this loan to RISKY farmers?
Limited Liability and Asymmetric Information. Oscar has decided to retire. Isaac is a lender from a neighboring city, Konkonuru, who decides to offer loans in Oboadaka. However, since he is from an outside village, he does not know the farmers in Oboadaka. He only knows that half of the farmers are SAFE and half are RISKY. As a result, he must charge a single interest rate to everybody who wants a loan. Like Oscar, Isaacâ€™s opportunity cost is also 30%. Because Oscar retired, Isaac is also a monopolist.
(f)What type of asymmetric information problem does Isaac face?
(g)What is the maximum interest rate Isaac can charge so that both types of farmers would want to borrow? Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution.
(h)Let be Isaacâ€™s profit. Derive an expression for , the expected value of Isaacâ€™s profit from a loan, as a function of the interest rate when the interest rate is less than or equal to the value you identified in part (g). Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution. Please report your final expression in slope intercept form. (Remember: Over this range of the interest rate, Isaac cannot tell which type of farmer he has given the loan to!). Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution.
(i)Explain what will happen if Isaac increases the interest rate above the interest rate you identified in (g)?
(j)What is the maximum interest rate Isaac can charge so that at least one type of farmer will want a loan? Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution.
(k)Derive an expression for Isaacâ€™s expected profit, , as a function of the interest rate for values between the interest rates you identified in part (g) and part (j). Please report your â€œfirst stepâ€ (i.e., how you set up your calculation) and your solution.
(l)What will happen if Isaac increases the interest rate above the interest rate you identified in (j)?
(m) Use the expressions from parts (h) and (k) to graph Isaacâ€™s expected profit as a function of the interest rate for interest rates between 0 and 3. Title this graph â€œFigure 3: Lender Expected Profit under Asymmetric Informationâ€
(n)What is the equilibrium interest rate charged by Isaac?
(o)What is Isaacâ€™s expected profit?
(p)Which type or types of farmers take the loan?
Part 
First step or reasoning 
Solution 
(a) 

(b) 

(c) 

(d) 
(i) 

(ii) 

(iii) 

(iv) 

(e) 
(i) 

(ii) 

(iii) 

(iv) 

(f) 

(g) 

(h) 

(i) 

(j) 

(k) 

(l) 
(m) 

(n) 

(o) 

(p) 
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