Economic Development
Name 1: Student ID 1:
Name 2: Student ID 2:
Name 3: Student ID 3:
Name 4: Student ID 4:
| ARE/ECN 115A | Fall 2019 |
| Problem Set 3: Asymmetric Information, Risk, and Financial Markets
Due: 3 December, 2019 11:59pm |
Instructions:
An excel template to this problem set can be found in the PS3 excel template on Canvas. Please fill out this
Excel file and copy each of the figures described below into this document as instructed. We will ask each member of the group to upload their own excel file to Canvas. Your group will turn in a PDF of this document on gradescope. Please fill in the questions below in their designated boxes. Please provide final equations along with numeric answers wherever necessary. Please fill in the answers below in their designated boxes.
Credit Market Equilibrium under Multiple Borrower Types
In problem 1, the lender is a monopolist who offers limited liability loans under symmetric information. In problem 2, the lender is a monopolist who offers limited liability loans under asymmetric information.
Problem 1 Limited Liability and Symmetric Information. Ram is a moneylender who lives in the village of Palampur in India. Half of the farmers in Palampur are SAFE farmers and the other halves are RISKY farmers. Both types of farmers need a loan of $200 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 $400 with 100% probability. They never have a bad harvest. RISKY farmers have a good harvest in which they earn revenues of $600 with 50% probability. They have a bad harvest in which they earn revenues of $0 with 50% probability. Ram has perfect information about the farmers, i.e. he knows who is a SAFE farmer and who is RISKY.
As a result, he can offer different contract terms to SAFE and RISKY types. Ram’s opportunity cost in money is 20%. Ram 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 𝑦s and 𝑦r denote the incomes of SAFE and RISKY farmers, respectively. Derive expressions for 𝐸(𝑦s) and 𝐸(𝑦r), the expected incomes of SAFE and RISKY farmers respectively. Report your expressions in intercept-slope format as in the questions above.
(b) Let 𝜋s and 𝜋r denote Ram’s profits from a loan to SAFE and RISKY farmers, respectively. Derive expressions for 𝐸(𝜋s) and 𝐸(𝜋r), the expected values of Ram’s profits from loans to SAFE and RISKY farmers respectively, as functions of the interest rate, i. Report your expressions in intercept-slope format as in the questions above
(c) Graph 𝐸(πs), 𝐸(πr) , 𝐸(𝑦s) and 𝐸(𝑦r) 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, with 0.1 as intervals). Label this “Figure1. Credit Market under Symmetric Information” and copy the figure in the box provided below.
(d) Using your equations and graph, answer the following questions:
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 Ram would be willing to charge on a loan to a SAFE farmer?
iv. What is the lowest interest rate Ram would be willing to charge on a loan to a RISKY farmer?
(e) Assume that Ram is a monopolist.
i. What is the equilibrium interest rate Ram would charge to a SAFE farmer?
ii. What is the expected profit that Ram, earns on this loan to SAFE farmers?
iii. What is the equilibrium interest rate Ram would charge to a RISKY farmer?
iv. What is the expected profit that Ram earns on this loan to RISKY farmers?
Problem 2 Limited Liability and Asymmetric Information: Ram has decided to retire. Ali is a lender from a neighboring village who decides to offer loans in Palampur. However, since he is from a different village, he does not know the farmers in Palampur. 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 Ram, Ali’s opportunity cost is also 20%.
(a) What type of asymmetric information problem does Ali face?
(b) What is the maximum interest rate Ali can charge so that both types of farmers would want to borrow?
(c) Let 𝜋 be Ali’s profit. Derive an expression for 𝐸(π), the expected value of Ali’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 (b). (Remember: Over this range of the interest rate Ali cannot tell to which type of farmer she has given the loan!).
(d) Explain what will happen if Ali increases the interest rate above the interest rate you identified in (b)?
(e) What is the maximum interest rate Ali can charge so that at least one type of farmer will want a loan?
(f) Derive an expression for Ali’s expected profit, 𝐸(π), as a function of the interest rate for values between the interest rates you identified in part (b) and part (e).
(g) What will happen if Ali increases the interest rate above the interest rate you identified in (e)?
(h) Use the expressions from parts (c) and (f) to graph Ali’s expected profit as a function of the interest rate for interest rates between 0 and 3. Label this “Figure 2: Lender’s Expected Profit under Asymmetric Information” and copy it in the box provided below.
(i) What is the equilibrium interest rate charged by Ali?
(j) What is Ali’s expected profit?
(k) Which type of farmers takes the loan?
Problem 3 Risk Preferences and Insurance
Rachel, Phoebe and Monica are sunflower farmers in the village of Girasol. They each have zero wealth, 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. Sunflower farming is risky because of a combination of weather and pests. Under full time farming, the farmer works 7 days per week on their farm. There is a 50% probability of having a GOOD harvest and a 50% chance of having a BAD harvest. If the harvest is GOOD, the farmer earns an income of $200. If the harvest is BAD, the farmer earns an income of only $40.
Activity 2: Full time construction work. This activity has no risk. An individual who decides to work full time in construction earns $80 with certainty.
Activity 3: Part-time farming. In this third activity, the farmer works during the week as a sunflower farmer and works in construction during the weekend. Since she is not able to work full time on the farm, the probability of having a GOOD harvest and earning $200 drops to 25%, and the probability of having a BAD harvest and earning only $40 increases to 75%. The individual also earns $30 with certainty as a construction worker (the person earns this $30 from construction in addition to her farm income under both a GOOD and BAD harvest).
(a) What is the expected value of consumption for each activity?
A. Activity 1: Full time farming:
B. Activity 2: Full time construction work:
C. Activity 3: Part time farming:
Rachel, Phoebe and Monica 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. Report your answers in Table 1 below.
Rachel: 𝑈(𝐶) = 0.05𝐶2
Monica: 𝑈(𝐶) = 20𝐶 − 0.05𝐶2
Phoebe: 𝑈(𝐶) = 0.5C
(b) Table 1. Certainty Equivalent, Risk Premium and Expected Utility for 3 Activities
Note: Please put in your final answers as whole numbers or upto one decimal point wherever necessary
| Full time farming | Full Time Construction Work | Part Time Work | |
| CE Rachel | |||
| CE Monica | |||
| CE Phoebe | |||
| RP Rachel | |||
| RP Monica | |||
| RP Phoebe | |||
| EU Rachel | |||
| EU Monica | |||
| EU Phoebe |
(c) Which activity each individual will choose?
A. Rachel
B. Monica
C. Phoebe
(d) Which type of risk preferences describes each individual? (Risk Neutral, Risk Averse, or Risk
Loving?)
A. Rachel
B. Monica
C. Phoebe
Joey is an insurance agent who offers conventional crop insurance contracts only to full time farmers.
He is not interested in offering insurance to part time farmers. The contracts are straightforward. At the beginning of the season, farmers pay a premium of $50. At the end of the season, Joey pays farmers an indemnity payment of $100 if the farmer had a BAD harvest. If the farmer had a GOOD harvest, Joey doesn’t pay the farmer anything. For questions e-f, assume that Joey has perfect information about the farmer’s activity choice. In other words, he can write and enforce a contract that requires the farmer to choose full time farming.
(e) What is Joey’s expected profit from this contract? (Joey’s profit is just the premium he collects from the farmer minus the indemnity payment he makes to the farmer).
(f) What is the expected consumption for an individual who chooses full-time farming with Joey’s insurance contract?
(g) What is the expected utility associated with full-time farming with an insurance contract for Rachel, Monica and Phoebe?
A. Rachel:
B. Monica:
C. Phoebe:
(h) Now assume that each individual can choose between the four available activities: Full Time Farming without Insurance (Activity 1 above), Full time construction work (Activity 2 above), Part Time Farming without insurance (Activity 3 above) and Full Time Farming with Joey’s insurance contract (Activity 4). Which activity will each individual choose?
A. Rachel:
B. Monica:
C. Phoebe:
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