Roger B. Myerson (2014) - Moral-Hazard Credit Cycles with Risk-Averse Agents

So, I think this is a theoretical model I’m going to present to you. It’s a theoretical model with an important purpose. I should say what the purpose is. We’ve had financial crisis, problems with moral hazard in the banking sector at a variety of levels in 2008 in America and Euro zone problems. So moral hazard in the financial sector seems to be central, an important problem in macroeconomics instability. And yet traditional macroeconomic models have often had little or no place for financial intermediation never mind the details of moral hazard. So my goal here is to show in a model how moral hazard in financial intermediation by itself can be driving macroeconomic instability as an attempt to help people to see more clearly that when we’re thinking about the problem of macroeconomics instability, other things like money supply, aggregate demand, all are important but that if banking seems important there’s no reason, theoretical reason not to think it actually is important. I’m going to talk today about a paper. Well, let me say this is so important I spoke about the same subject at the Lindau 2011 meetings, which only a few of you were here. That paper is now published in the Journal of Political Economy. So today I’ll talk about a paper that just came out a few weeks ago in the Journal of Economic Theory, that’s a sequel. If you find it interesting to read either is fine. And let me say dear students. This is an example of a speaker who has 30 minutes and 21 slides. That’s too many. I must skip 10 and I invite you to keep track of how many I skip. But I should have... whatever. Alright get to work. So the point, again to have it written in front is important, that moral hazard in financial intermediation by itself can be a fundamental driver of macroeconomic instability, that it’s often difficult when you write down the classical capital growth models to get them to be unstable. It can be done but one has to add some arcane bells and whistles perhaps. And what I want to show is when we think about moral hazard in financial instability, a simple model where you strip out almost everything else just works. I think I as a theorist am supposed to be showing you the simplest possible model not the most realistic. But what I am hoping is that students and colleagues, faculty colleagues in the audience might think that the principle that’s going to be illustrated in this model is robust and simple that you can put it into your more realistically applicable model in the future, that future research should take this kind of model and use it to build more realistic and applicable models for policy guidance. The basic insight is that because of moral hazard financial agents... A standard result in agency theory is that dynamic moral hazard problems with limited liability are optimally solved by having late career rewards. So because of moral hazard financial agents are going to need long term relationships with investors because they’re going to get late career rewards for good performance over a long career. But these relationships, long term relationships can create complex macroeconomic dynamics. Anything that’s long term becomes a dynamic state variable. The first paper, the one that came out in 2012 in the Journal of Political Economy, showed how credit cycles can be sustained over an infinite time horizon in a simple model where the bankers or financial agents were assumed to be risk neutral. As you know risk neutrality is always a convenient and natural simplifying assumption. It’s nice and linear. But linear models with risk neutrality tend to go to corner solutions. So the optimal contracts in the paper I presented here 3 years ago looked rather extreme with agents receiving incentive payments only at the end of their career if all their investments throughout their career were successful. That is an extreme result. So today I’m going to show you a model that has risk averse agents. They’re going to have nonlinear utility of consumption in every period. The analysis becomes more complicated because it’s nonlinear. So I’m going to simplify things relative to the other paper and have people have only 2 active careers. They work when they’re young and they work when they’re old and then there’s a third period when they retire. The analysis becomes more complicated but the resulting optimal contracts are going to look much more realistic. Risk averse financial agents are going to get substantial rewards in every period when they supervise investments. But remarkably we also find that when I add this complication, this was a total surprise to me between 2011 and now when I get to return to now in Lindau, we find that the economy can become in a sense even more unstable when the agents are risk averse. That’s from the last slide, forget it, we’re going on to the next. So here’s a slide with too much text on it. I’m trying to introduce the variables. So let me say a little bit about the variables. The parameters of the model are listed... let’s see on the bottom here are some base line numerical values for the variables. So let me introduce them one at a time. I need to say up here. Theta...We’re going to have people’s utility function is going to be in every period. The agents are going to live 3 periods. They’re going to work when they’re young, then work when they’re old and then agents who are young in period T, in T plus 1 they’re old and in T plus 2 they’re retired and in T plus 3 they’re gone making room for others in the world. So the consumption in period T plus 0 is C0 and C1, C2. So I get to consume when I’m a young worker, an old worker and in retirement. And in each period we’re going to raise it to the theta power. And I like to think about the square root. That’s a good one in your utility function. You could have a negative theta like minus 1 but then we better divide by a theta there to make sure that we have... the derivative always has a positive slope. So theta is the risk aversion parameter, the relative risk aversion parameter. An agent can make an investment of different sizes, can invest a large amount of money or a small amount of money on behalf of investors. You can only invest in our island through a local banker. But the agent can steal a gamma fraction of it without anybody knowing the difference. They will not know the difference now but the result will be... If they steal up to gamma, they can get... In the next period we’ll discover that the investment fails. But otherwise the investment, if they don’t steal, the investment will succeed. And it says the returns from investment are going to be such that if they invest HS now next period they’re going to have HS plus, 1 plus some profit rate divided by delta. The previous slide I forgot to emphasise said this is a stationary non stochastic model. And yet I have an expected profit rate. I’ve divided by delta because I want to say when we invest per unit invested. The expected present discounted value of returns if it’s managed properly is the amount invested times 1 plus the net profit rate. But it has a T subscript. Amount invested by an agent who was young in T, appeared T plus S, gets a profit of T plus S. So that sounds like I’ve just cheated, invested profit rates on the island depend on what period we’re in. But no, now the macro part of the model. There’s going to be a given function, a nice linear, perhaps downward sloping demand curve, investment demand curve that says the more the aggregate investment on our island the lower our profit rates in any given period. And that’s the model. So first the micro part. Here’s one slide about the micro model in depth. I must show you something rigorous. I can’t... the time is too short to give it proper. But I must show you a rigorous model. First the micro and then the macro in the next slide and then the results on the slide after that. Ok get to work. So, I’m going to try to prejudice. I’m going to try to stack the deck against having recessions by saying, by having no financial institutions. Investors, there are global investors who also discount the future. I should say the previous slide the investors are discounted, the financial agents are discounting the future by delta, discount factor delta per period. They only work in 2 periods. Period is I guess 20 years. So delta equals 0.5 as a reasonable discount factor for discounting over 20 years. And they are global investors who also are willing to perfectly elastically supply investment funds at this. And I’m going to assume that those global investors can form a consortium and go and hire a new banker at any time. And that on the island there is a very large supply, an ample supply of new young MBAs ready to invest billions of other people’s money on the island. And there’s no problem of the investors finding them and hiring them, ok. So this is the optimal incentive problem for the investors over here. This is the key. We have to go through this. H0 is how much is invested in the first period, H1 is how much is invested in the second period. It’s discounted by delta in the returns to the investors. Each amount invested is multiplied by the net profit rate, discounted to that period. We’re going to pay the agent C0 or C1, C1 in her second period and C2 in her third period. But of course the future expenses are discounted. Everything is homogeneous in this model. So I’m going to linearise things. I’m going to normalise things by assuming whatever the unit is. That an agent is handling one. Call it 100 million unit of funds. All those numbers have to be non-negative. And now the incentive constraints. I’m going to assume that when the agent cheats, what she can do is she can take her. She’s got C1. Let’s talk about in the 0 period when she’s young. She’s offered C0. She could steal gamma H0, the gamma fraction of her investment amount. And now she’s going to run away with it. She knows she wants to now live on it. So she’s going to spend some part. How much does she spend? She spends it uniformly over the 3 years of her life. So one plus delta, plus delta squared per period. And you raise that to the theta power. That’s the consumption applied in the utility function. And that utility in the first period, second period and third period just with appropriate discount factors will be her utility if she steals. And that should be less than or equal to what she gets on the equilibrium path which is her C0 in the first period, delta C0, C1 to the theta, delta squared, C2 to the theta power. Similarly in the second period of her life, in period T plus 1 the start of the period T could run away with C1, is expecting to get C1 to the theta this period utility and delta time C2 to the theta in the second period divided by theta just to make sure it’s got the right derivative, sign of derivative. But she could run away, take her C1 and what she steals and parcel it out over 2 periods and get the utility of that. So that’s the incentive constraint period. That’s our incentive consumption. It has a solution. I wanted to find 3 important functions. One is for any... since H0 is 1 in the objective function, P T is 1. What we’re going to find is I want to ask: What’s the profit rate now such that the optimal value of this problem is zero? Why zero? Because this is the net profit that investors get from hiring a new young banker. I said in the big world global investment funds are perfectly elastically supplied. We investors, there are lots of us wanting to invest in this island. Our investment is limited by the need to find bankers. But there are lots of bankers. So if our present discounted value relative to the competitive delta discounting and global investment markets was strictly positive, money would flow in. We’d all want to hire bankers on this island. And money would flow in. But then of course there would be more investment in the island. So the key equilibrium, it’s going to be in the next slide but let me say it right now, is going to be that an equilibrium, global investors will only hire young bankers at zero net expected gain because if we had positive... because you can’t have strictly, perfectly elastically supplied market. The market cannot make strictly positive returns. So given next period's profit rate. What profit rate this period would just get, is the lowest profit rate. This period that would get global investors willing to hire new young bankers. That’s why P T plus 1 and given an optimal solution it’s got a simple formula right there. The next important one. I have G for growth. Is the growth in responsibilities in the optimal solution per unit, well H0 is actually 1 in this problem, but per unit invested in the first period. How much will the optimal contract have the banker handling in the second period? And one other. I’m going to let the... V1 stands for the constant equivalent consumption. The amount of consumption that would give, in both periods, that would give the agent the same utility. And at the beginning of time we might worry about how much in the past the old bankers were promised some expected utility from their contracts. And per unit of constant equivalent consumption that’s generated in the utility. This is the measure, C1 is measuring the utility of their promise in the contract. How much are they actually handling as old agents? So H1 is the old agent’s responsibilities in their second period per unit of constant equivalent consumption promised to them. I use constant equivalent consumption instead of utility because utility is a nonlinear. Everything is linear in consumption and investments, not in utility because utility is a nonlinear function of investments. Ok, macro. This is the one macro slide. Let JT denote the total investments that are managed by all young agents in any period T. The one I was talking about last time. That was in 100 million... 1 maybe stood for 100 million euro or 100 million dollars or whatever. This is going to be perhaps in trillions. It’s a much bigger number. It’s a macro number. This is the macro slide. On the island in any period T JT stands for the total amount of investment that’s being handled by new young bankers in that period. Now at period zero... we’re going to start the economy at some period and call it zero. But there was a past and there are presumably, maybe not, there are presumably some older bankers in the second period of contracts with global investors. Whatever they did in the first period is past. We don’t care about it. But V0 is an aggregate measure of how many old bankers we have in the sense of what is the constant equivalent payoff, this amount this period plus this amount next period, when their retirement would be just enough to give those old bankers in aggregate the utility that they were promised under their contracts. That’s going to be our initial condition. Now, the elasticity of supply of global funds means that we can never have the profit rate be greater than the... Remember YPT is PT plus 1, is the amount of profits in period T, the profit rate in period T that would just enable investors to break even when they expect PT plus 1 next period to be the profit rate when they’re hiring new young bankers. So we can never have... we global investors can never expect positive profits. So an equilibrium. PT can never be bigger than YPT plus 1. But if we’re actually hiring them, if JT is strictly positive, then we must be breaking even. We’re not going to hire at a loss. So here is the definition of equilibrium and we’re done with the model. An equilibrium, given the initial condition V0, an equilibrium is a sequence of profit rates forever. And a sequence of young investment responsibilities such that, let’s say, in the first period the investment in the first period must be equal to what the old guys were promised times in the optimal solution of their problem how much investment do they handle per unit promised plus J1 which is the amount handled by the young guys. Now the young guys in their second period in optimal contract their responsibilities are going to grow by the growth factor. So they’re going to be handling in period T plus 1. And now JT plus 1 is what the young guys come in. And that must be equal to what the profit rate is in the next period run through the investment demand curve. I of PT plus 1 is how much aggregate investment in our island would be on the demand curve for that profit rate. Of course the JTs have to be non-negative. The profit rates have to be between 0 and YPT plus 1. And complementary slackness when JT is positive, PT must be equal to the Y of PT plus 1. That’s it. Ok, results. One. There exists a unique steady state P* will be the steady state such that P* equals Y of P*. If the G is less than 1, that means as the agents get older they handle less responsibilities. Then it turns out the steady state is stable because it looks just like a capital model. That’s not going to be very likely. I’ll show you in a picture in a little bit. What I want you to see is, remember, the profits that the investors are earning is really the amount of investment handled by their young guys times the profit rate in that period plus the profit rate next period multiplied by 2 things. They’re going to discount the future delta because that’s next period. And the investors... On the other hand if G is bigger than 1, which it typically is going to be, then the guy who we hired to handle H0 is going to handle more than that next period. So is the future more important or less important than this period? The product of delta times GPT plus 1 is the critical point. What the interesting case is going to be... Let me say in the linear case this G was exactly 1 over delta. The first paper where everybody was risk neutral and discounted the same G was exactly 1 over delta. And this was exactly 1. The interesting case we’re going to get now relative to the risk averse... Why is it 1 over delta? Because as the agent... all the rewards come in the last period. As the agent gets closer to... in retirement. In the linear case everything is... We go to a corner. All rewards are at the end. And consequently in the second period, that end of period has one less delta discounting of it relative to when we’re young and consequently it’s that much more valuable by a factor of 1 over delta. And therefore you can trust me against current temptations to steal by a factor of 1 over delta. And therefore you can give me 1 over delta more responsibilities. As I get closer to retirement I become... the end of career rewards that have been motivating me throughout my career are coming closer and closer and therefore you can trust me more with greater responsibilities. They’re more valuable simply because I discount the future. But if you discount the future at the same amount, then those exactly cancel out. But with risk aversion or nonlinear consumption more properly... There’s no risk at all in this model. But with square root utility function for example my marginal utility of consumption at 0 is infinite. So I really am very hungry. And I’m very tempted to steal to assuage that terrible hunger. So you should expect that you’ll pay me something at the beginning so as to have me less tempted to steal and move consumption forward. And that means that the growth in responsibilities will be less than it was otherwise. So G is typically less than 1 over delta. But it’s still typically positive because as I go forward. So let me just say that’s the normal case. It doesn’t always happen. There’s a theorem. I don’t have time to say... it’s in front of you. There’s a regular case. Either that or something else. But the something else has never been found. So now let’s just say we’re in the regular case. That’s the only... I’ve looked at several thousand randomly generated cases and I never found it. Let’s go straight to the main result at the bottom. When G is bigger than 1 and less than 1 over delta which is going to be the normal... when the regular case applies, from any initial condition other than the one that gives us the steady state. There’s a unique initial condition that corresponds to the steady state. If we’re not there, then there exists an equilibrium that enters an extreme cycle where new bankers are only hired every other period and it will enter that cycle within finitely many periods. So we need to see pictures. First of all the cases. Basically there are 3 parameters in the model but the only ones that count are the utility exponent. This is risk neutrality at 1. Square root utility function is here. These are getting progressively more risk averse. And I suggest that delta equals 0.5 was a good discount factor. So that’s a nice line to look at. Above this line things are stable when people are very risk averse or very patient. So waiting 40 years for payment doesn’t mean very much to you. Then everything is stable. I don’t have an unstable model. But in this normal case and I think these are... Here is logarithmic utility at zero. At these normal parameter values it is quite... we’re going to get the G bigger than 1 but less than 1 over delta. In this zone here G gets so bit that it’s bigger than 1 over delta. And I’ll show you what happens. This is one of the slides I’m skipping. But I kept wanting to say: Ah look, here is the consumption in the first period. Here is the consumption in the second period and here’s how much money you’re being paid in retirement. Anyway let’s get on, move on. Skip numerical examples. Skip pictures, skip. Here is the important picture. This is for square root utility function, a reasonable delta for 20 year discount factor, a linear downward sloping investment demand curve. And this is the steady state. What am I showing you? I’m showing you the total investment... I’ve set up this model so that... I’ve parameterised it so that one unit of aggregate investment is the steady state. That means that’s a unit investment that generates a profit rate such that over... when you have that profit rate every period, the amount of... this is surplus profits over the discount factor, over the normal discount rate that investors require, that the surplus over that is just enough to cover the payments to the banker that are needed to prevent the banker... to keep the banker from not wanting to steal the investment. In that steady state the green part is the investments managed by young bankers in the steady state and the brown is the investments managed by old agents. And there’s a growth of responsibility. So the brown part is a larger part of it. But suppose we start out with just a few too many old bankers in the first period. Just a little bit away. Then of course they crowd out some young guys. But then when those young guys get old, they become... we don’t have enough old guys. So we’re going to have to hire more young guys. And there’s an aggregate fluctuation. And you can see its increasing and increasing. Until we get to. It stops increasing, this is. Here we’ve hit the limit cycle where young guys are hired only in periods 28 and 30. And in periods 27 and 29 there are lots of old bankers. But when they retire then suddenly we go into recession. Here’s the bifurcation diagram that shows the magnitude of these cycles. What really is happening, there’s something strange going on. The reason why we have instability, a quick intuition as to why we have instability. Giving me an end of career reward, promise me an end of career reward which is the standard result in dynamic agency theory, that agents are going to be given a big end of career reward conditional on good performance throughout their career. Because that can motivate good behaviour throughout a long career whereas when you pay me earlier that can’t motivate. Anything you pay me when I’m young won’t motivate good behaviour when I’m old. The fact... that looks a lot like investment, a capital investment. Good old fashioned capital theory. A onetime payment that earns returns over an interval. However it comes at the end. And instead of... More importantly its value, the productivity appreciates over time. That looks like we’ve reversed the standard assumption of depreciation of capital to appreciation of capital. That looks like reversal of time. And reversal of time, changing depreciation of capital to appreciation of the relational capital that comes from the backload of moral hazard rents looks like a reversal of time. And a reversal of time makes unstable systems unstable. However in some sense... Let me just back up and say the rate at which we accelerate away from the steady state towards... If we’re near but not at the steady state, the rate at which these oscillations amplify as the agent gets more risk averse and G gets closer to 1 that rate gets faster. However the final amplitude gets smaller. And that’s what’s shown in this picture. As we look at more and more risk averse agents, the amplitude of the largest cycle gets smaller. In this section if people are just a little bit risk averse... this is the oscillation for the same parameters but with a risk neutral model. If people are just a little bit risk averse then actually the economy is stable. But I can actually get an even larger amplitude if I started with no bankers at all. This is the worst amplitude. This is for theta equals 0.9. It’s almost risk neutral. Ok, I think I have to skip everything else. Lots of other nice results. Conclusions. Look, employers commit... Moral hazard means that employers have to commit to long term incentives to backload. And that’s a non-trivial problem. Employers have to promise in an efficient solution of moral hazard in banks as in any other corporation. Responsible managers or responsible investing financial agents in this case have to be motivated by, among other things, late career rewards which means employers have to commit to long term incentives for agents who are in responsible positions. That means that new hiring in banks as in any other sector must take account of expected future returns. The dynamic economy then is going to include commitments to mid-career agents. So now you see why just looking at standard solutions to agency problems gives us new state variables that might create interesting dynamics. In the recessions of our model... We found recessions when productive investment was reduced by a scarcity of trusted mid-career or old agents for financial intermediation. Competitive recruitment of new agents could not fully remedy the problem because if you hired enough young bankers to get back to the steady state full, what we call full employment in one period, it would be in a contract where their responsibilities were going to increase over time. And that would mean next period we would have too many young bankers and therefore returns to banking would be too small to pay the costs of the bank itself. So a large adjustment to reach steady state financial capacity in one period would create over supply in the next period. For new agents to be hired low profit rates in one period must be followed by higher expected profit rates in the next period which in turn then have to be followed by lower expected profit rates. Early payments... Once I had risk aversion early payments to risk averse agents reduced their growth of responsibilities making their employers, here’s the key, making their employers profits less sensitive to future profit rates. So for any deviation of this period's profit rate, epsilon in one direction, we needed more than epsilon deviation next period if the growth rate is... if delta times the growth rate is less than 1. And so deviations have to keep increasing and amplifying until something gives. And what gives is the zero profit condition. Instead we get to periods where the profit on hiring new bankers is negative which means that nobody hires new bankers. And the result is if there is no intervention in the economy for macro stabilisation, the private sector... In this model if we’re not exactly at the right number of bankers for the steady state, then spontaneously generational inequalities will amplify over time until we reach a limit cycle in which new young bankers are hired only every other period and the investment and the economy suffers booms and busts. Thank you. Applause.

Roger B. Myerson (2014)

Moral-Hazard Credit Cycles with Risk-Averse Agents

Roger B. Myerson (2014)

Moral-Hazard Credit Cycles with Risk-Averse Agents

Abstract

We consider a simple overlapping-generations model with risk-averse financial agents subject to moral hazard. Efficient contracts for such financial intermediaries involve back-loaded late-career rewards. Compared to the analogous model with risk-neutral agents, risk aversion tends to reduce the growth of agents' responsibilities over their careers. This moderation of career growth rates can reduce the amplitude of the widest credit cycles, but it also can cause small deviations from steady state to amplify over time in rational-expectations equilibria. We find equilibria in which fluctuations increase until the economy enters a boom/bust cycle where no financial agents are hired in booms.

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