Homework 2 Fall 2018 Please hand in your answers at the beginning of class on Oct 16. 1 Karlan and Zinman (2009) Karlan and Zinman (2009) conduct an experiment to distinguish hidden action from hidden information in a credit market. 1. Draw a diagram to illustrate the structure of the experiment. List the different comparisons the authors make and explain what they tell us about reasons for default. 2. Theory questions: (a) Show that ê(ro , Cb (r), θi ) is decreasing in θi . This means that the (anticipated) amount of effort is smaller for higher risk borrowers than lower risk borrowers. (b) The authors define θ(ro ) as the θi at which individuals are indifferent between accepting and rejecting the loan at the offered interest rate. What is the indifference condition that defines θ? (c) Set Cb (r) = 0 and assume there is no strategic (voluntary) default. Differentiate dθ implicitly to calculate dr o . In which direction does selection operate with respect to a marginal increase in the offer rate? 2 Credit—moral hazard in project choice Each borrower can choose either a risky (R) or a safe (S) project. Project i (i = R, S) has a return Yi with probability pi and 0 otherwise. We assume that the risky project has a higher return when successful, but that the expected return is higher for the safe project: YR > YS , pR YR < pS YS . Assume that borrowers maximize expected net return. 1. Consider first an individual loan contract with limited liability. The borrower pays a gross interest r if the project is successful, and nothing if it fails. For what range of interest rate r will the borrower choose the safe project? 2. Consider now a joint liability contract with two borrowers. Assume independent returns between the two projects. Each borrower pays r if his project is successful and an addition q if his project is successful and the project of his partner has failed. (a) 1 Show that the borrowers will choose to coordinate on the same type of projects. (You may want to consider a Nash equilibrium.) (b) In what range of r, q will the borrowers choose safe projects? 3. Now assume that the returns are positively correlated within the same type of projects,1 and that the borrowers still coordinate in choosing the same risk level. The correlation of returns can be modeled by increasing the probability of symmetric events (success or failure) by ρpi , and decreasing that of asymmetric evens (success for one borrower and failure for the other) by the same amount. In what range of r, q will the borrowers choose safe projects? 4. Interpret your results and give intuition for the above 1. 2. and 3. 3 Informal insurance—full risk sharing There are two individuals who have von Neumann Morgenstern utility and are strictly risk averse. Each individual i earns exogenous random income yi which follows distribution fi . Let ρ(y1 , y2 ) be individual 1’s consumption, so individual 2 consumes y1 + y2 − ρ(y1 , y2 ). Suppose that risk is fully pooled between them. 1. Show that ρ(y1 , y2 ) = ρ̃(y1 + y2 ). (In other words, show that the optimal consumption of 1 depends only on aggregate income.) −σ c −σ c 2. Assume that individuals 1 and 2 have CARA utility −e σ11 1 and −e σ22 2 , respectively. Show that ρ̃(y1 + y2 ) is linear. Calculate its slope as a function of risk aversion parameters σ1 and σ2 . How does individual 1’s weight affect ρ̃(·)? Give intuition for this relationship. 1 Events of success and failure remain independent between different type of projects. 2