Lecture 5 Crisis: Sustainable Debt, Public Debt Crisis, and Bank Runs

Lecture 5 Crisis: Sustainable Debt, Public Debt Crisis, and Bank Runs Last few years have been tumultuous for advanced countries. The United States and many European countries have been facing major economic, financial and fiscal crises resulting in steep decline in output and rise in unemployment and public debt. The effects of these crises are still being felt. Many countries such Greece and Ireland are not able to meet their debt obligations and have to be bailed out by other countries. These events have brought to the fore the issue of sustainability of public debt and financial system. The recent crisis originated in the financial sector. Banks and other financial institutions have to be rescued by governments and central banks. This required massive infusion of public resources in financial institutions and in many cases outright government takeover. Central banks had to resort to unconventional monetary measures (e.g. quantitative easing, forward guidance) in order to avoid depression and shore up public confidence. Financial crisis severely reduced real activities and tax revenues and substantially increased public expenditure. This led to rapid increase in government debt with many countries facing fiscal deficit running in double-digits. In 2009, the debt-gdp ratio for the U.S. was 87.5%, for Britain 80.6%, Japan 214%, and Greece 125%. This massive run up in debt has raised the question of sustainability of public debt. In this lecture, we analyze twin issues of fiscal and financial sustainability. We begin with the issue of sustainability of public debt. 1. Dynamics of Public Debt We first examine the dynamics of public debt in a partial equilibrium deterministic model. This model is quite useful in understanding inter-linkages among fiscal deficit, public debt, rate of growth of GDP and interest rate. The question we ask is whether a government can run a persistent fiscal deficit and whether public debt can grow forever without bankrupting a country. Denote D(t) = Debt at time t, y(t) = GDP at time t, r = Interest rate on debt and rd(t) = Interest payment at time t. Suppose that at the initial period D(0) = 0 and y(0) = 1. For simplicity, we assume that r is exogenous and constant. By making r exogenous, we are essentially looking at a partial equilibrium model. We are assuming that the rate of interest is independent of the public debt and the government can always borrow at this rate. Later on we will relax these assumptions. Let the time path of national debt and GDP be as follows. Assume that the government borrows a constant fraction, b, of GDP every period. In other words, the budget deficit is a constant proportion b of GDP. Also assume that GDP grows at a constant rate g. Then the time paths of debt and GDP are governed by following differential equations: and Ḋ = by(t); b > 0 (1) ẏ = gy(t); g > 0. (2) 1

Now we ask the question whether a country can bankrupt itself. For simplicity, assume bankruptcy means whether the ratio of interest payment to GDP will exceed one i.e. 1 at some point in time. rd(t) y(t) Given the set up, we want to know: (i) The time path of rd(t) y(t) z(t) and (ii) What is the steady-state value of z(t)? Does it exceed one? To answer these questions, we first derive the expression for z(t) using (1) and (2). (2) implies that the time path of y(t) is given by Then (1) and (3) imply that y(t) = y(0) exp gt. (3) D(t) = y(0)b expgt g + C (4) where C is an unknown constant. Since D(0) = 0 and y(0) = 1, (4) implies that Then from (3) and (5) we have D(t) = b g (expgt 1). (5) z(t) rd(t) y(t) = r b g (1 exp gt ). (6) (6) gives the time path of the ratio of interest payment and GDP. We want to find out (i) steady state value of z(t) and (ii) whether it exceeds one. Taking limit of (6) we have lim z(t) = r b t g. (7) (7) gives the steady state value of z(t). Interest payment on the debt converges to a constant proportion of GDP equal to r b g in the long run. If r b g < 1, then the government can run persistent deficit. On the other hand, if r b g > 1, then a country cannot run persistent deficit without bankrupting itself. The intuition for these results is as follows. (1) and (2) show that Ḋ ẏ = b g. Thus, for every dollar increase in GDP, debt increases by b g. Suppose that b g = 0.5. Then this implies that for every dollar increase in GDP, debt increases by 50 cents. Clearly, GDP is growing faster than the debt and thus the ratio of public debt to GDP will always be less than unity. Since, rate of interest r is typically less than one, in this case the ratio of interest payment to GDP will always be less than unity. On the other hand, if b g = 1.5, for every dollar increase in GDP, debt increases by 1.5 dollars. Debt is growing faster than GDP and thus the ratio of public debt to GDP will eventually exceed unity. In this case, if r is high enough then interest payment may exceed GDP. 2

These results show that whether a country can run a persistent fiscal deficit depends on the rate of interest and the rates of growth of GDP and public debt. Fiscal deficit is likely to be unsustainable in countries with low GDP growth. The level of fiscal deficit itself plays a crucial role. Higher the fiscal deficit less likely the deficit is going to be sustainable. Next, we study the case in which a country is caught in a debt trap. This is a situation in which the government has to borrow even in order to pay interests on the public debt. To analyze the issue of debt trap, we modify the environment slightly. Suppose that the public debt evolves as follows: Combining (3) and (8), we have Ḋ = by(t) + rd(t). (8) Ḋ = b exp gt +rd(t). (9) In order to find path of public debt, we need to solve (9). (9) is an example of first order non-autonomous differential equation. The general form of the first order non-autonomous differential equation is ẋ + a(t)x(t) = e(t). (10) The solution of (10) is given by x(t) = exp A(t) [ exp A(t) e(t)dt + C] (11) A(t) = a(t)dt and is known as the integrating factor. Using (11) we can derive the solution for (9) which is given by In this case, z(t) is given by D(t) = b g r expgt +C exp rt. (12) b z(t) = r g r [1 exp(r g)t ]. (13) From (13) it is immediately clear that z(t) converges to a finite limit only when g > r. Growth rate of GDP must exceed rate of interest. In the case, r > g, debt-servicing ratio, z(t) goes to infinity in the long run. In the case, g > r, we have lim z(t) = r b t g r. (14) The condition that g > r puts severe restriction on the ability of countries to run fiscal deficit on the long term basis. Most likely, countries caught in debt trap will either have to default or run budget surplus (b < 0). 3

So far, we have analyzed a deterministic model of public debt with fixed interest rate. However, in real world the rate of interest paid by a government depends on the magnitude of public debt and fiscal deficit. It has been observed that countries with higher debt level and fiscal deficit pay higher rate of interest and are more likely to default. Also, fiscal crisis and debt default happen suddenly. Next we develop a model to analyze the interaction among debt level, rate of interest and default probability. 2. Debt Crisis and Default Suppose that the government has D amount of debt maturing in the current period. But the government does not have revenue to pay the debt and it wants to roll over the debt to the next period. Suppose that the gross rate of interest is R(= 1 + r). Thus total amount payable next period will be RD. It will be obtaining tax revenue T next period. But tax revenue is random and its cumulative distribution function, F (), with support (T min, T max ) is continuous. Suppose that the government pays to the debt holders if T RD. It repudiates the debt if T < RD. Let π be the probability of default. Suppose that investors are risk-neutral and they can obtain a risk-free rate ˆR. Since, investors can buy both risky and risk-less bonds, equilibrium requires that they would be indifferent between the two at margin. Thus (15) implies that (1 π)r = ˆR. (15) π = R ˆR R. (16) It is easy to see that (16) traces an upward sloping concave curve between R and π. Higher the default probability, higher will be the rate of interest demanded by investors. When π = 0, the government debt is risk free and thus it will have to pay risk free rate. On the other hand, π = 1, no investor will be willing to buy the debt and the rate of interest demanded will approach infinity. Whether government defaults depends on the realized tax revenue relative to the amount due to bondholders. The government defaults if and only if T < RD. Thus the probability of default is given by π = F (RD). (17) (17) shows that higher is the debt, D, and higher is R, higher is the default probability. Assume that the density function of the probability distribution is Bell-shaped (or inverted U shape). Thus, the probability of occurring very high or low taxes is low. With this assumption, (17) traces a S-shaped relationship between π and R. The intersections of these two curves determine equilibrium values of π and R. Combining, (16) and (17) we have F (RD) 1 + ˆR R 4 = 0. (18)

Since (18) is a non-linear equation, there can be more than one equilibrium. First, note that R = is always an equilibrium. At this equilibrium, default probability, π = 1, and no investor buys government bond. For reasonable shapes of functions, there is also two other equilibria one with low R l and low π l and other with high R h and high π h. One can show that equilibrium with low R l and low π l is stable, while equilibrium with high R h and high π h is unstable. There are number of implications of this model. Firstly, small differences in fundatmentals can lead to large differences in interest rates and default probabilities. Secondly, the model suggests that when default occurs it may occur quite unexpectedly. Finally, default depends not only on self-fulfilling beliefs, but also on fundamentals. In particular, an increase in the amount the government wants to borrow, an increase in the safe interest rate, and a leftward shift in the distribution of potential revenue all make default more likely. Next, we study the issue of bank runs and financial crisis. As mentioned earlier, the recent crisis originated in the financial sector. Many countries witnessed severe erosion in the credibility of their financial sector. Many financial institutions failed and closed down. In the U.S., more than 250 banks failed in last two years. Governments and central banks have to take extraordinary measures such as government take over, quantitative easing, government guarantees to save the financial system. In the next section, we develop a model of deposit banking to analyze the causes of bank runs and financial crisis. This model is known as Diamond-Dybvig model. 3. Demand Deposit Banking and Bank Runs One of the most important roles of banks is that they convert assets of shorter maturity into assets of longer maturity. This intermediation role usually leads to mismatch in maturity duration of assets and liabilities of banks. Banks have liabilities payable on demand, but assets that are not. This mismatch of assets and liabilities raises the possibility of a bank run. A bank may fail because the assets it owns or the loans it has made may realize such unexpectedly low returns that the bank no longer has the resources to pay depositors. It may also fail if a sudden rush of withdrawals forces it to sell off assets at a loss. To understand bank runs, we first develop a model of demand deposit banking. Assume that there are N three-period lived individuals. Each individual receives y units of goods as endowment in the first period. These individuals are of two types. Type-I individuals want to consume in the second period and Type-II individuals want to consume in the third period. For simplicity assume that nobody consumes in the first period. Suppose that half of the population is of each type. Individuals face the problem of financing their future consumption. Suppose that individuals can save in two ways: (i) by storing their endowments and (ii) and capital investment. Storing the endowment gives the return of one. Capital investment gives superior return, but only after two periods. Suppose that one unit capital investment in the first period returns X > 1 units of goods in the third period. Though, capital does not produce anything in the second period, an investor can choose to sell his capital in the 5

second period at price v k. Thus a buyer of capital in the second period will get return X in the third period. To create the need for the liquidity and emergence of demand deposit banks, we make two assumptions. Firstly, in the first period no individual knows his type. Each individual has an equal chance of being either Type-I or Type-II. Individuals come to know their type only in the second period. They have to choose their mode of savings in the first period before knowing their type. Secondly, there is informational imperfections in the asset market. It is possible to issue fake titles to capital. One can verify genuineness of titles only through costly effort. Suppose that verification costs θ goods per unit of capital. Assume that θ > X 1. With the possibility of creating fake titles on capital, a buyer of such title will always verify the genuineness of the title. Thus, the price paid by a buyer to a seller of the title in the second period will be v k θ. Let us now look at the relative rate of return of two methods of saving. One period rate of return on storage is one. On the other hand, one period rate of return on capital investment if it is sold in the second period is v k θ. What will be v k? Since, one unit of capital in the second period pays X units of good in the third period, individuals will pay at most X goods in the second period for capital i.e. v k X. Then the rate of return on capital in the second period v k θ X θ. Since, θ > X 1, it implies that θ > v k 1 and thus 1 > v k θ. This ensures that the one period rate of return on storage is higher than the one period rate of return on capital sold after first period. Given the returns on storage and capital investment, if individuals know their type in the first period, type-i individuals would store their endowments, while type-ii individuals would invest in capital. But individuals do not know their type in the first period. This uncertainty induces them to diversify their portfolio. They will store part of their endowment and put the other part in capital investment. Thus, individuals will get lower return/utility in the uncertainty case compared to the certainty case. Now we show that banks can improve the rate of return which individuals get. In fact, in the presence of banks individuals will get as high a return as in the certainty case. Imagine there is a bank which accepts deposits from individuals. It offers rate of return of 1 on one period deposits and X on two period deposits. Individuals can withdraw their deposits in any period they like. Question then is whether individuals would deposit their endowments in the bank and whether the bank would be able to honor its commitment. Answer to both questions is yes! Individuals of both types will get better rate of return by depositing with banks. To see this, suppose that individuals put fraction s of their endowments in capital investment. Then type-ii individuals will get return of X from deposits with the bank. But they will get only sx + (1 s) which is less than X, if they do not deposit. On the other hand, type-i individuals will get only s(v k θ) + (1 s), which is less than one if they do not deposit with the bank. But, they will get return of one, if they deposit with the bank. Thus, it is in the interest of individuals to put their endowments as deposits with the bank rather than save on their own. The other question is whether the bank will be able to honor its commitment. In the economy, half of the population is of one type. If all individuals put their endowments with the bank, then total deposit with the bank will be Ny. Thus, the bank knows that 6

total obligation in the second period is Ny/2 and in the third period NyX/2. The bank can put Ny/2 in storage and the rest in capital investment and meet its commitment. In the model, banks have social role. The economy achieves better outcome than otherwise. The role of banks is to create liquid liabilities (demand deposits) from illiquid assets. They allow individuals flexibility in their timing of spending even when assets are not flexible in their returns. They do so by taking advantage of the fact that there is more randomness for an individual than for the aggregate economy. An individual does not know when he wants to consume and thus cannot select the asset with the best return for his situation. By pooling the resources of many individuals, however, a bank can be confident that it knows the fraction of its depositors who will withdraw after one period, and thus it can hold the correct fraction of goods in long-run assets (here, capital) and good short-run assets (storage). Bank Runs By converting illiquid assets in liquid assets, banks enhance social welfare. But this very function also makes banks susceptible to bank runs. In the model, bank run can occur when type-ii individuals start withdrawing their deposits in the second period. In the second period, the bank has enough in storage to pay the promised return of y good to N/2 individuals. But if more then N/2 individuals start withdrawing their deposits the bank will have to start selling its capital at price v k θ, which is less than one. For every y units of withdrawal, the bank will have to sell more than y units of capital investment. In this case, if every type-ii individual starts withdrawing his deposit, the bank will not have enough resources to meet its obligations and it will become insolvent. When can type-ii individuals withdraw their deposits in the second period? Suppose that you are a type-ii person and hear a rumor that every other type-ii is going to withdraw their deposit in the second period. Then the question is whether you will wait till the third period or you will join the crowd and also withdraw your deposit in the second period. You know that if every other individual withdraws his deposit, then bank will become insolvent and you will get nothing in the third period. But if you withdraw in the second period, you will get something though less than y. Thus, it will be optimal for you to withdraw your deposit in the second period. Since, every type-ii individual is like to you, it will be optimal for them to withdraw their deposits. Result will be run on the bank. All are worse off if such panic occurs. However, each individual is rational in withdrawing early, given that the others are also withdrawing early. Measures to Prevent Bank Runs There are number of measures that can be taken to prevent bank runs. The key here is to inspire confidence in type-ii individuals that the bank can meet its obligations in the third period. We briefly discuss some of the most important measures: (i) Interbank Lending: A run on a bank makes that bank insolvent by forcing it to sell its assets at a loss. Suppose that the bank can borrow from other banks to meet its obligations instead of selling its assets. These loans can be repaid back next period. 7

In this way, the bank can maintain enough capital to meet its obligation to every type-ii person who do not withdraw early. Once type-ii individuals know that the bank can meet their obligations in the third period, no type-ii individual will withdraw early. (ii) Government Deposit Insurance: The government can also help prevent bank runs by guaranteeing type-ii individuals that they will receive their promised return even if the bank becomes insolvent. If the guarantee is credible, there will be no reason for type-ii individuals to panic. However, such guarantees by the government can have perverse effects on the bank. If a bank is not insured, it must choose its assets carefully, weighing the risks and returns of assets in order to attract shareholders and depositors. But if the government insures depositors against all losses, then depositors will not care about their bank s exposure to risk. They will only care about high rate of return. Banks in order to attract depositors will increase their average return by holding riskier portfolio. This is the moral hazard problem of insurance: insuring people against losses removes the incentives for the insured to act to reduce the risk of these losses. There are number of ways in which the government can reduce moral hazard problems: (i) Capital requirement: A capital requirement forces banks to maintain a net worth no less than some fraction of their assets. This provides a larger cushion to absorb asset losses before depositors or the insurer of the depositors suffer any losses. (ii) Regulations: Regulations can be used to limit the kinds of assets which banks can hold. This way the government can reduce the riskiness of asset mix of banks. (iii) Closing insolvent or near insolvent banks: If a bank is insolvent or near insolvent, it gives shareholders incentive to invest in very risky assets. This happens because they have nothing to lose and the potential payoff can be very high. By closing such banks the government can limit excessive risk-taking behavior. (iii) Temporary Suspension/Curtailment of Withdrawals: One way a bank might structure itself to prevent panics is by temporarily closing its doors once its reserves of the liquid short-term assets (storage) have been used up. Then bank can reopen in the next period when its long-term capital pays it return. In this case, type-ii individuals will know that the bank will be able to meet its obligations in the third period and bank run will not occur. 4. Financial Crisis In the last section, we analyzed causes of bank run and measures to prevent them. Many of the measures such as facility for interbank loan, deposit insurance etc. are effective if bank runs are isolated events. There is a run on a bank, but other banks and financial institutions are not affected. However, if there is run on many banks at the same time these measures may not work. There may not be enough solvent banks to lend to banks who are facing crisis. Also the government may not have enough resources and thus its guarantees may not be credible enough. This may result in collapse of financial system and the financial crisis. The bank runs can be widespread for many reasons. Banks in a particular country share common macroeconomic fundamentals. Adverse productivity shocks can affect all 8

the banks. Similarly, they may be equal participants in housing bubbles, capital inflow bonanzas, increasing private and public leveraging etc. There also may be cross holding of assets by banks. One bank may have investment in other bank and vice versa. Thus, if the asset quality of one bank deteriorates, it may adversely affect the balance-sheet of other banks and financial institutions. 9

Aftermath of Financial Crisis Financial crises are protracted affair. Their adverse effects are quite severe and long lasting. K. Rogoff and C. Reinhart after surveying the causes and consequences of financial crisis over 800 years in their book This Time is Different find that the consequences of severe financial crisis share following three characteristics: (i) Collapse in asset markets: Asset market collapses are deep and prolonged. Real housing prices decline on average by 35 percent and the decline persists over six years. The equity prices collapses average 56 percent over a downturn of about three and half years. (ii) Severe decline in output and rise in unemployment: The unemployment rate rises an average of 7 percentage points during the down phase of the cycle, which lasts on average more than four years. Output falls more than 9 percent during the down phase of the cycle, which lasts on average for two years. (iii) Large increase in government debt: The government debt increases quite substantially. On average government debt rises by 86 percent in real terms compared to the pre-crisis level. Since output falls, government debt-gdp ratio tend to explode. 10