Q1. Let a firm producing widgets have an owner-manager with a utility function C+ BC2, where is the owner's discount factor. The firm can invest I at date 1, which yields total revenue Of(I) at date 2, where 0 is the price of widgets (the firm is a price taker) and f(I)is the quantity of output. The firm has initial resources at date 1 of E, and it has contracted outstanding debt of D with a bank, payable at date 2. Finally, assume that the owner manager consumes anything that is not invested at date 1, and it is not available for repayment to the lender at date 2. If, however, revenue is generated at date 2, the minimum of that revenue and the debt payment is paid out to the bank. Assume that the owner invests after 0 is realized (i) How much will the owner-manager invest if he does invest? Is this socially optimal? What is the maximum value of D such that he will invest (hint: he can consume his endowment instead of investing)? How does this maximum debt level vary with O and B? (Derive expressions) (ii) The bank wants to maximize the repayment it gets. It will want to forgive debt in order to get the owner to invest. Suppose the bank has to agree to forgiveness before o is realized. Let the distribution of O be g(0) with range [[0,7) (a) If the bank forgives debt down to a fixed amount, what will that amount be? (derive expressions) (b) Will the owner-manager make the first best level of investment if the bank links repayment to the amount of output produced at date 2 (that is, D=D(X))? () Will the owner-manager make the first best level of investment if the bank links repayment to the price of the widget at date 2 (that is, D=D(0))? What is the expression for the debt level the banker will set? Based on (a). (b). and (c), what is the best form of debt forgiveness from the banker's perspective? Why? (iii)Now assume that the banker does not know the owner-manager's discount factor . It could either be " with probability p, or with probability (1p), with BH> B'. The owner-manager knows his discount factor (so we have a situation of asymmetric information). The banker wants to maximize his expected repayment. Let us assume he can offer the owner-manager the following choices: Either invest Iy and repay Dy or invest I and repay D. Assume he chooses the menu of contracts so that (1) he induces the high type owner to choose the first contract and the low type to choose the other, and (2) he maximizes expected repayment. (a) What would the banker set investment and debt repayment at if he had full information and could offer each type the contract appropriate to it? Call the contracts {1\,D*,},{11,D";}. (b) Now turn to the case where he is asymmetrically informed. Write down the banker's optimization problem, assuming the optimal menu intends to separate the types, with the Incentive Compatibility constraint for each type (high type should prefer his contract to the low types contract rice versa) the Individual Rationality constraint for each type (each type should prefer to invest than to consume immediately). (C) Why would both types not choose the contract with high debt repayment? Explain the single crossing property here. (d) Intuitively, which type should invest at the socially optimal level? It turns out that the binding constraints are that the high type should not want to pick the low" type's contract, and the low type should just want to invest rather than consume (as you will learn in your course on information economics, this pattern is characteristic of these problems). Substitute these constraints into the objective function and show that 1. =/;D B'. The owner-manager knows his discount factor (so we have a situation of asymmetric information). The banker wants to maximize his expected repayment. Let us assume he can offer the owner-manager the following choices: Either invest Iy and repay Dy or invest I and repay D. Assume he chooses the menu of contracts so that (1) he induces the high type owner to choose the first contract and the low type to choose the other, and (2) he maximizes expected repayment. (a) What would the banker set investment and debt repayment at if he had full information and could offer each type the contract appropriate to it? Call the contracts {1\,D*,},{11,D";}. (b) Now turn to the case where he is asymmetrically informed. Write down the banker's optimization problem, assuming the optimal menu intends to separate the types, with the Incentive Compatibility constraint for each type (high type should prefer his contract to the low types contract rice versa) the Individual Rationality constraint for each type (each type should prefer to invest than to consume immediately). (C) Why would both types not choose the contract with high debt repayment? Explain the single crossing property here. (d) Intuitively, which type should invest at the socially optimal level? It turns out that the binding constraints are that the high type should not want to pick the low" type's contract, and the low type should just want to invest rather than consume (as you will learn in your course on information economics, this pattern is characteristic of these problems). Substitute these constraints into the objective function and show that 1. =/;D