Week 7 Problem Set

Question 1:

Financial market imperfections

Machines are used to produce a single output good in a two period economy. The stock of machines in period t=\{1,2\} is denoted K_{t} and produces output goods A K_{t} in period t with A>1. The stock of machines in period 1, K_1, is given and machines never depreciate.

At the end of period 1 a one-for-one technology is available which converts output into machines, so the representative agent chooses how to allocate output AK_{1} between consumption C_{1} in period 1 and an investment to increase the stock of machines in period 2. The utility of the representative agent is \ln(C_{1})+\ln (C_{2}), so there is no discounting.

The representative agent has no initial debt, but can borrow (from abroad) at the rate R \in (1,A) in period 1 to finance either consumption in period 1 or additional investment in machines for period 2. Denote the amount borrowed by the representative agent in period 1 as B, to be repaid in period 2 with interest. Crucially, there is a financial market imperfection in the economy: the representative agent cannot borrow more than a fraction \psi<1 of the quantity of their machines in the second period. That is, the machines serve as collateral for the loan.

a) Set up the maximisation problem of the representative agent.

b) Derive the agent’s optimality condition(s). How does the Euler equation for consumption look in this case? Interpret.

Hint: You could solve this problem by setting up the Lagrangian and writing down the Kuhn-Tucker first-order conditions. Alternatively, argue why all constraints faced by the agent, including the constraint on borrowing, must bind at the optimum, and then solve the problem by substituting the constraints into the objective function.

c) Solve for C_{1},\ C_{2},\ K_{2} and B as functions of exogenous parameters A,\ \psi ,\ R and K_{1}.

d) Describe the effects of an increase in \psi, which can be interpreted as an increase in the degree of financial openness of the economy.

Question 2:

Bankers cannot abscond with government money

This question builds on the Gertler-Kiyotaki model we studied in the lecture by incorporating the government as an additional agent. A representative household has an endowment y in period 1 of a two-period economy. They have preferences \sqrt{c_{1}}+\beta \sqrt{c_{2}} over consumption in the two periods, where \beta is the discount factor. The government imposes a lump sum tax on the household in period 1. The government then deposits the tax revenue T at the bank in period 1 and distributes the return RT as a lump sum tax rebate to households in period 2.

The representative bank takes in deposits from both the consumer (d) and the government (T) in period 1, adds its own net wealth N, and invests the total with entrepreneurs for return R^{k}. The bank’s revenue in period 2 is therefore R^{k}(d+T+N). The banker can default in period 2 before repaying consumer deposits, but if he does so he can only abscond with a fraction \theta of the funds he owes to households and owns himself. He cannot abscond with any of the government deposits.

a) Define and solve the utility maximisation problem of a representative household, assuming they take as given the deposit rate R, the lump-sum tax T, and the banker’s profit \pi. How do lump-sum taxes affect deposits d and the household’s consumption choices? Why do you think this is the case?

b) Write down a sufficient condition that stops the banker from defaulting. Assuming this is satisfied in equilibrium, define the profit maximisation problem of the bank, and discuss its implications. How do lump-sum taxes affect the behaviour of the banker?

c) What is the socially optimal (first-best) consumption allocation in the absence of financial frictions?

d) Discuss whether the government can use lump sum taxation to implement the first-best consumption allocation in part (c). How realistic do you think the underlying assumptions are?

Question 3:

Modifying assumptions in the Diamond-Dybvig model [Romer 10.11]

Consider the Diamond Dybvig model we studied in the lecture, but suppose that \rho R < 1.

a) In this case, what are c^{a*}_1 and c^{b*}_2? Is c^{b*}_2 still larger than c^{a*}_1?

b) Suppose the bank offers the contract described in the lecture: anyone who deposits one unit in period 0 can withdraw c^{a*}_1 in period 1, subject to the availability of funds, with any assets remaining in period 2 divided equally among the depositors who did not withdraw in period 1. Explain why it is not an equilibrium for the type-a’s to withdraw in period 1 and the type-b’s to withdraw in period 2.

c) Is there some other arrangement the bank can offer that improves on the autarky outcome?

Question 4:

Deposit insurance in the Diamond-Dybvig model [Romer 10.12]

Consider deposit insurance in the Diamond Dybvig model as studied in the lecture.

  1. Suppose fraction \phi > \theta of depositors withdraw in period 1. How large a (lump-sum) tax must the government levy on each agent withdrawing in period 1 to be able to increase consumption of those agents who wait to withdraw until period 2 to c^{b*}_2? Explain why your answer should simplify to zero when \phi = \theta, and check that it does.

    For concreteness, suppose that in period 1 the government collects taxes after agents withdraw funds from the bank. Since the projects are already liquidated at that point, the government does not earn return R on the collected resources.

  2. Suppose the tax is marginally less than the amount you found in part (a). Would the type-b’s still prefer to wait until period 2 rather than try to withdraw in period 1?