What is the expected profit that jonathan earns/ Managerial Economics
We begin with a story of Moral Hazard. Felipe is an entrepreneur who is considering between two investment projects – each requires an investment of $120. Project 1 is opening a Filipino Fusion Taco Truck. This project is risky. With probability 1/2the Taco Truck is successful and generates $240 of revenues. However, with probability 1/2the Taco Truck fails and generates zero revenues. Project 2 is becoming an economics tutor. This project has no risk and always generates $200 of revenues. If Felipe chooses not to undertake any investment, he earns zero income.
Cristina is a banker. She is considering offering Felipe a limited liability credit contract. Under this contract, Felipe only has to repay the loan when he is able to. In other words, he repays the loan plus interest when his project is successful, but he doesn’t have to repay anything if his project fails. Cristina’s opportunity cost of money is 10%. In other words, she would earn a 10% interest rate if she invested the money in a bank instead of lending it to Felipe.
1. Let y1 and y2 denote Felipe’s income from Projects 1 and 2 respectively under a limited liability credit contract.
a. Derive expressions for E(y1) and E(y2), the expected value of Felipe’s income under the two projects, as functions of the interest rate, i. Your expressions should take the form y = A + Bi where you have to find A and B.
b. In Excel, graph E(y1) and E(y2) as functions of the interest rate, i (i.e., put i on the horizontal axis).
c. Looking at your graph, how does the interest rate affect Felipe’s project choice?
2. Let Π1 and Π2 denote Cristina’s profit from a limited liability loan which finances Felipe’s Project 1 and 2 respectively.
a. Derive expressions for E(Π1) and E(Π2), the expected value of Cristina’s profits from loans that finance Projects 1 and 2 respectively. Similar to part (1a), you should express these two expected profit functions as functions of the interest rate.
b. Graph E(Π1) and E(Π2) on the same graph you made for part (1b).
3. Assume that Cristina is from the same village as Felipe, so she can easily observe and enforce Felipe’s choice of project (i.e., Cristina does not suffer from asymmetric information) so that a contract specifies two things: In other words, the contract specifies which project Felipe must undertake and the interest rate. For example, the contract would require Felipe to open the Taco Truck and would charge Felipe a 3% interest rate.
a. Find the equilibrium contract assuming that Cristina is a monopolist. (Find the contract that maximizes the Cristina’s expected profit while allowing Felipe to earn at least zero expected income.) Make sure your contract specifies both the interest rate AND the project that the borrower will do.
b. What is the value ofFelipe’s expected income and Cristina’s expected profit under this monopoly contract?(You should report dollar values here).
c. Find the equilibrium contract if instead the Cristina is part of a perfectly competitive loan market. (Find the contract that maximizes Felipe’s expected income while letting Cristina earn at least zero expected profit).
d. What is the value of Felipe’s expected income and Cristina’s expected profit under this competitive equilibrium contract?
4. Now let’s change our assumption about information. Let’s assume that Cristina is not from the same village as Felipe so that she cannot observe Felipe’s project choice. As a result, now a loan contract specifies only the interest rate. Felipe will look at the interest rate and choose whichever project is best for him.
a. In a separate figure, graph the Cristina’s expected profit as a function of the interest rate. Discuss the shape of the your graph. (Make sure that you – like Cristina – consider how the interest rate affects Felipe’s choice of project!)
b. Find the equilibrium interest rate under if Cristina is a monopolist.
c. Which project does Felipe choose?
d. What is the Felipe’s expected income and Cristina’s expected profit under this monopoly contract with asymmetric information?
ADVERSE SELECTION
We now turn to a story of Adverse Selection. In the village of Keystone, half of the farmers are SAFE farmers and half are RISKY farmers. Both types of farmers need a loan of $100 in order to farm. Farmers will take a loan as long as they can earn at least zero expected income. The only difference between SAFE and RISKY farmers is as follows:
Jonathan is a moneylender who lives in Keystone. His opportunity cost of money is 0.25 (i.e., he would earn a 25% interest rate if he invested the money in a bank instead of lending it to farmers). Jonathan offers limited liability loans, so a farmer does not have to repay the loan if he has a bad harvest. Since Jonathan lives in Keystone, he has perfect information about farmers. Specifically, he knows who is a SAFE farmer and who is a RISKY farmer.
5. Let ys be the income of a SAFE farmer and yR be the income of a RISKY farmer.
a. Derive expressions for E(ys) and E(yR) – the expected value of income for SAFE and RISKY farmers, as functions of the interest rate, i. As above, your expressions should take the form E(y) = A + Bi where you have to find A and B.
b. Graph E(ys) and E(yR) as functions of the interest rate, i (i.e., put i on the horizontal axis).
6. Let Πs be Jonathan’s profit on a loan to a SAFE farmer and ΠR his profit on a loan to a RISKY farmer.
a. Derive expressions for E(Πs) and E (ΠR) as functions of the interest rate, i.
b. Graph E(Πs) and E(ΠR)on the same graph you made for question 9.
7. Using your equations and graph, answer the following questions:
a. What is the highest interest rate a SAFE farmer would be willing to pay for a loan from Jonathan?
b. What is the highest interest rate a RISKY farmer would be willing to pay for a loan from Jonathan?
c. What is the lowest interest rate Jonathan would be willing to charge on a loan to a SAFE farmer?
d. What is the lowest interest rate Jonathan would be willing to charge on a loan to a RISKY farmer?
8. Assuming that Jonathan is a monopolist:
a. What is the equilibrium interest rate Jonathan would charge to a SAFE farmer?
b. What is the expected profit that Jonathan earns on this loan to SAFE farmers?
c. What is the equilibrium interest rate Jonathan would charge to a RISKY farmer?
d. What is the expected profit that Jonathan earns on this loan to RISKY farmers?
Now assume Jonathan retires and no longer offers loans in the village. Juan is a loan officer from a bank in a nearby city. He is considering offering limited liability loans to farmers in Keystone. Like Jonathan, Juan’s opportunity cost of money is 0.25, and he is a monopolist. Juan, however, does not know the farmers in Keystone, so he cannot tell who is a SAFE farmer and who is a RISKY farmer. All he knows is that half are SAFE and half are RISKY farmers. As a result, he has to charge a single interest rate to everybody who wants a loan.
9. In contrast to Jonathan, Juansuffers from asymmetric information. So when Juan thinks about the interest rate he will charge, he must think about who will want the loan.
a. What is the maximum interest rate Juan can charge so that both types of farmers would want to borrow?
b. Let Π be Juan’s profit. Derive an expression for E(Π), the expected value of Juan’s profit from a loan, as a function of the interest rate for interest rates below the interest rate you identified in 9a. (Remember: Over this interest rate range, Juan can’t tell to which type of farmer he has given the loan!).
c. What will happen if Juan increases the interest rate above the interest rate you identified in 9a?
d. What is the maximum interest rate Juan can charge so that at least one type of farmer will want a loan?
e. Derive an expression for E(Π) as a function of the interest rate for values between the interest rates you identified in 9a and 9d.
f. Graph E(Π) as a function of the interest rate for interest rates between 0 and 5 (0% to 500%).
g. What will be the equilibrium interest rate charged by Juan?
h. What will Juan’s expected profit be?
i. Which type or types of farmers will take the loan?
