MFE Macroeconomics Week 8 Exercises

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1 MFE Macroeconomics Week 8 Exercises 1 Liquidity shocks over a unit interval A representative consumer in a Diamond-Dybvig model has wealth 1 at date 0. They will need liquidity to consume at a random time in the interval t [0, 1], where the liquidity shock has cumulative distribution function F (t) with F (0) = 0 and F (1) = 1. 1 agent s expected utility is therefore: U = The associated density function is f(t). The representative 1 0 u(c t )f(t)dt where C t is the consumption they attain if the liquidity shock hits at time t. discounting. For simplicity there is no There is no storage technology so the representative agent has to invest all its wealth in an investment technology. The return on the investment depends on the date at which the investment is liquidated, with the returns increasing the longer the investment is allowed to run. An investment liquidated at time t provides return R t, i.e. it depends on how long passes before liquidation. We assume R 0 > 0, Ṙ = dr dt > 0 and Ṙ/R is increasing in t so the earlier an investment is liquidated the lower return it pays. The return profile of all investments is the same. A bank offers a deposit contract in which a depositor can choose when to withdraw their deposit. If the depositor withdraws at time t [0, 1] they receive consumption C t. Liquidity shocks are i.i.d. across depositors, and assumed to be publicly observable. The bank has to ensure that suffi cient projects are liquidated at each point in time to honour the requests of depositors wanting to withdraw, i.e. they face a flow budget constraint. 1. The number of depositors wanting to withdraw between time t and t + dt is f(t)dt. Given that these depositors have been promised a consumption level of C t, what proportion of investments x t have to be liquidated between time t and t + dt to satisfy the flow budget constraint of the bank? 2. Why must 1 0 x tdt 1? 3. Set up the optimisation problem solved by the first best deposit contract. (Hint: it will maximise utility subject to the flow budget constraint and the condition in part (2)). Solve the problem and show that u (C t )R t is independent of t under the optimal contract. Interpret this result. 1 In the model in the lectures the agent had to consumer in either period 1 or period 2. Here there is a continuum of possible consumption dates along the unit interval. 1

2 4. Assume that the utility function is of the CRRA type with coeffi cient of relative risk aversion greater than 1. Manipulate your answer to part (3) to show that the optimal contract implies: Ċ t C t < Ṙt R t, i.e. consumption rises slower over time than the return on the investment technology. Prove that this implies the optimal insurance contract is front-loaded, with C t > R t for t < t and C t < R t for t > t for some t [0, 1]. Plot the time paths of C t and R t on a diagram. 5. Show that the first best outcome in part (4) is not incentive compatible, because depositors may want to withdraw their deposits early and reinvest their deposits in the investment technology themselves. 2

3 2 Diamond-Dybvig with random investment returns 2 Consider a model in which consumers are of the Diamond-Dybvig type. They have unit resources available for investment in period 0, and learn at date 1 whether they will be patient or impatient consumers. They are patient with probability t, in which case their utility is u(c 1 ), and impatient with probability 1 t, in which case their utility is u(c 2 ). Discounting is ignored for simplicity. There is a storage technology that returns 1 between periods 0 and 1 and between periods 1 and 2. The investment technology differs from that in the lecture note in two respects. Firstly, it provides no return if the investment project is liquidated in period 1. Secondly, if it is held until period 2 is pays a random return R. The random return is the same for all projects and is observed at the start of period 1 after resources have been allocated to the storage and investment technologies. It is assumed that R has a well-behaved distribution such that E(R) > 1. All consumers report their type truthfully. 1. The optimal insurance contract specifies levels of consumption C 1 (R) and C 2 (R) to be paid respectively to impatient and patient consumers when then random investment return observed is R. Write down the problem solved by the optimal contract and derive the first order conditions with respect to C 1 (R) and C 2 (R). Use the notation i 1 to denote the quantity of resources placed in the storage technology in period 0, so 1 i 1 is the quantity of resources placed in the investment technology in period What happens if R is small? 3. What happens if R is large? 4. Plot the optimal allocations C 1 (R) and C 2 (R) as a function of R, paying particular attention to any special points of interest. 5. A bank is interested in implementing the optimal allocation using a variant of the suspension contract. They offer to pay C 1 (R) to depositors who want to withdraw in period 1, and will pay in full if they can. Note that C 1 (R) is conditional on R. If the total amount of withdrawals requested in period 1 exceeds the available funds i 1 then each consumer wanting to withdraw receives a pro rata share of i 1. The income available in period 2 is shared between all agents who did not try and withdraw in period 1. Long run assets are never liquidated as doing so would yield no return. Denote the fraction of patient consumers who demand their deposits in period 1 by x(r) [0, 1]. Explain how it is possible to choose C 1 (R) so as to implement the social optimum. 2 This is intentionally a diffi cult question. It shows the scope to extent the Diamond-Dybvig model. 3

4 3 Liquidity shocks over a unit interval 1. The flow budget constraint is: R t x t dt = C t f(t)dt 2. The initial total quantity of investments in the economy is normalised to 1, so the total quantity of investments liquidated over t [0, 1] must similarly sum to 1. Note that optimality requires the integral to be equal to one, otherwise some investments would be left unliquidated at the end of time. The bank could then improve welfare by liquidating these investments earlier. 3. The optimal contract solves: for which the first order condition is: max {C t} u(c t )f(t)dt s.t. C t f(t) R t dt 1 [u (C t ) µrt ] f(t) = 0 and u (C t )R t = µ is independent of t. This result is due to the optimal contract trading off risk sharing against allowing as many investments as possible to run close to maturity (so the rate of return is maximised). If the return was independent of t then R t = R and we have the standard perfect risk sharing result u (C t )R = µ. However, here the fact that Ṙ/R is increasing in t means it is optimal to tilt the consumption stream more towards depositors who withdraw later. The cost in welfare terms of lost insurance is less than the welfare gain of having more resources available to distribute to late withdrawers. 4. Totally differentiating the first order condition from part (3) gives: Hence: u (C t )ĊtR t + u (C t )Ṙt = 0 Ċ t u (C t ) u (C t ) C t + Ṙt = 0 C t R t u (C t) u (C C t) t is the coeffi cient of relative risk aversion (sometimes defined as the minus of this). Since it is greater than 1 it follows that: Ċ t < Ṙt C t R t and consumption under the optimal contract increases at a rate below the return on the investment technology. It must be that consumption is front-loaded, as anything else contradicts either optimality or feasibility. If C 0 < R 0 then the level of consumption is below the return on investment in every period, which does not distribute all available resources to consumption so by definition cannot be optimal. If C 1 > R 1 then consumption exceeds the return on investment in every period, which clearly is not feasible. 4

5 It follows that C 0 > R 0 and C 1 < R 1 and there must be a critical t at which the consumption and return profiles cross. R 1 C 1 C 0 R 0 t* 5. Suppose a depositor withdraws their deposits from the bank time τ, soon after period 0 so τ is small. They obtain C τ under the optimal contract, which they can invest themselves in the investment technology. If they do and have to withdraw in some future period t they will obtain C τ R t C 0 R 0 R t because τ is small. The question to ask when thinking about incentive compatibility is whether this is greater or less than the C t promised if they leave their deposits in the bank. In other words, is C 0 R 0 R t C t when C t satisfies u (C t )R t = µ? This condition can be re-written as: We know already that C0R0Rt C t with respect to t is: C 0 R 0 R t C t 1 = 1 when t = 0. The derivative of the left hand side of the expression d C 0 R 0 R t = Ṙt Ċt, dt C t R t C t which is positive from part (4). It follows that the left hand side is increasing in t and must therefore exceed 1 for all t [0, 1]. We have: C 0 R 0 R t > C t so there is always an incentive for the agent to withdraw at time τ. The optimal deposit contract is not incentive compatible. By immediately withdrawing their deposits, the depositor can achieve the higher dashed line of consumption in the following figure. R 1 C 1 C 0 R 0 t* 5

6 4 Diamond-Dybvig with random investment returns 1. The optimal insurance contract solves: The first order conditions are: max [tu(c 1 (R)) + (1 t)u(c 2 (R))] s.t. tc 1 (R) i 1 (1 t)c 2 (R) (i 1 tc 1 (R)) + R(1 i 1 ) u (C 1 (R)) = µ + λ u (C 2 (R)) = λ where µ is the Lagrange multiplier on the period 1 budget constraint and λ is the Lagrange multiplier on the period 2 budget constraint. 2. The period 2 budget constraint can be written as (1 t)c 2 (R) R(1 i 1 ) + tc 1 (R) i 1, so for small R it must be that tc 1 i 1 and the period 1 budget constraint does not bind. We have that µ = 0 and consumption is equalised across periods under the optimal contract. In this case the impatient consumers are insuring the patient consumers. When R is low the optimal contract rolls over some of the resources in the storage technology so as to boost the consumption of the patient consumers. 3. When R is large the first period budget constraint binds so u (C 1 (R)) > u (C 2 (R)), C 1 (R) < C 2 (R) and patient consumers receive more than impatient consumers. Insurance no longer operates as there is no mechanism by which patient consumers can transfer resources to impatient consumers (remember that liquidating long-run investments provides no return). This means that the benefits of a large R accrue solely to the patient consumers. 4. The special point of interest is the R at which the first period budget constraint starts to bind. At this point tc 1 (R) = i 1, (1 t)c 2 (R) = R(1 i 1 ) and C 1 (R) = C 2 (R), so R is defined by: R = 1 t t i 1 1 i 1 When R < R there is perfect risk sharing and C 1 (R) = C 2 (R) = i + R(1 i). When R > R the 6

7 impatient consumers receive C 1 (R) = i1 t and patient consumers C2 (R) = R(1 i1). Graphically: Period 2 consumption Period 1 consumption R* R 5. The deposit contract promises: { } i 1 min C 1 (R), t + (1 t)x For R > R it is clear from the figure in part (4) that the patient consumers will not want to withdraw in period 1 - if they wait they obtain R(1 i1) which by definition is greater than R (1 i 1) = i1 t which is what they get if they withdraw early. The optimal allocation is therefore supported for R > R.To see the support for R < R, start by noting that it cannot be that x(r) = 1 in equilibrium, because R > 0 would otherwise imply that a patient consumers could gain infinite utility by delaying consumption until period 2. This is because there are always some resources available for distribution in period 2 under the suspension contract. Similarly, it cannot be that x(r) = 0 in equilibrium in this case because that would imply an incentive for a patient consumer to withdraw in period 1. A patient consumer waiting would receive R(1 i1) which by definition is now less than R (1 i 1) = i1 t from withdrawing. With x(r) = 1 and x(r) = 0 ruled out as equilibrium behaviour, it must be that a fraction x(r) of patient agents withdraw for R < R. If a only a fraction of patient agents withdraw then they must be indifferent between withdrawing or not, i.e. we require C 1 = C 2 and the socially optimal risk sharing equilibrium is the only one supported. The full characterisation of the equilibrium is: [t + (1 t)x(r)] C 1 (R) = i 1 Ri 2 (1 t)(1 x(r)) = C 2 (R) C 1 (R) = C 2 (R) 7