We build a one-period general equilibrium model with money. Equilibrium exists, and fiat money has positive value, as long as the ratio of outside money to inside money is less than the gains to trade available at autarky. We show that the nominal effects of government fiscal and monetary policy can be completely described by a diagram identical in form to the IS-LM curves introduced by Hicks to describe Keynes’ general theory. IS-LM analysis is thus not incompatible with full market clearing, multiple commodities, and heterogeneous households. We show that as the government deficit approaches a finite threshold, hyperinflation sets in (prices converge to infinity and real trade collapses). If the government surplus is too large, the economy enters a liquidity trap in which nominal GNP sinks and monetary policy is ineffectual.
We build a one-period general equilibrium model with money. Equilibrium exists, and fiat money has positive value, as long as the ratio of outside money to inside money is less than the gains to trade available at autarky. We show that the nominal effects of government fiscal and monetary policy can be completely described by a diagram identical in form to the IS-LM curves introduced by Hicks to describe Keynes’ general theory. IS-LM analysis is thus not incompatible with full market clearing, multiple commodities, and heterogeneous households. We show that as the government deficit approaches a finite threshold, hyperinflation sets in (prices converge to infinity and real trade collapses). If the government surplus is too large, the economy enters a liquidity trap in which nominal GNP sinks and monetary policy is ineffectual.
We extend the standard model of general equilibrium with incomplete markets (GEI) to allow for default. The equilibrating variables include aggregate default levels, as well as prices of assets and commodities. Default can be either strategic, or due to ill-fortune. It can be caused by events directly affecting the borrower, or indirectly as part of a chain reaction in which a borrower cannot repay because he himself has not been repaid.
Each asset is defined by its promises A, the penalties lambda for default, and the limitations Q on its sale. The model is thus named GE(A,λ,Q). Each asset is regarded as a pool of promises. Different sellers will often exercise their default options differently, while each buyer of an asset receives the same pro rata share of all deliveries. This model of assets represents for example the securitized mortgage market and the securitized credit card market.
Given any collection of assets, we prove that equilibrium exists under conditions similar to those necessary to guarantee the existence of GEI equilibrium. We argue that default is thus reasonably modeled as an equilibrium phenomenon. Moreover, we show that more lenient lambda which encourage default may be Pareto improving because they allow for better risk spreading.
Our definition of equilibrium includes a condition on expected deliveries for untraded assets that is similar to the trembling hand refinements used in game theory. Using this condition, we argue that the possibility of default is an important factor in explaining which assets are traded in equilibrium. Asset promises, default penalties, and quantity constraints can all be thought of as determined endogenously by the forces of supply and demand.
Our model encompasses a broad range of moral hazard, adverse selection, and signalling phenomena (including the Akerlof lemons model and Rothschild-Stiglitz insurance model) in a general equilibrium framework. Many authors (including Akerlof , Rothschild and Stiglitz) have suggested that equilibrium may not exist in the presence of adverse selection. But our existence theorem shows that it must. The problem is the inefficiency of the resulting equilibrium, not its nonexistence. The power of perfect competition simplifies many of the complications attending the finite player, game theoretic analyses of the same topics.
The Modigliani-Miller theorem typically fails to hold when there is the possibility that the firm or one of its investors might default.
One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent’s beliefs about the state of the world, his beliefs about other agents’ beliefs about the world, his beliefs about other agents’ beliefs about other agents’ beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the “richness” of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for “interesting” sets of events, and show that the answer depends on the definition of “interesting.”
Many advocates of social security privatization argue that rates of return under a defined contribution individual account system would be much higher for all than they are under the current social security system. This claim is false. The mistake comes from ignoring accrued benefits already promised based on past payroll taxes, and from underestimating the riskiness of stock investments.
Confusion arises because three distinct reforms are muddled. By privatization we mean creating individual accounts (which could, for example, be invested exclusively in bonds). By diversification we mean investing in stocks, and perhaps other assets, as well as bonds; diversification might be undertaken either by individuals in their private social security accounts, or by the social security trust fund. By prefunding we mean closing the gap between social security benefits promised to date and the assets on hand to pay for them. Any one of these reforms could be implemented without the other two.
If the system were completely privatized, with no prefunding or diversification, the social security system would need to raise taxes and/or issue new debt in order to pay benefits already accrued. If the burden were spread evenly across all future generations via a constant proportional tax, the added taxes would completely eliminate any rate of return advantage on the individual accounts. We estimate that the required new taxes would amount to about 3 percent of payroll, or about a quarter of all social security contributions, in perpetuity. Unlike privatization, prefunding would raise rates of return for later generations, but at the cost of lower returns for today’s workers.
For households able to invest in the stock market on their own, diversification would not raise rates of return, correctly adjusted to recognize risk. Households that are constrained from holding stock, due to lack of wealth outside of social security or to fixed costs from holding stocks, would gain higher risk-adjusted returns and would benefit from diversification. If this group is large, diversification would raise stock values, thus helping current stockholders, but it would lower future stock returns, thus hurting young unconstrained households. Overall, since the number of truly constrained household is probably not that large, privatization and diversification would have a much smaller effect on returns than reformers typically claim.
This paper describes how three money’s worth measures — the benefit-to-tax ratio, the internal rate of return, and the net present value — are calculated and used in analyses of social security reforms, including systems with privately managed individual accounts invested in equities. Declining returns from the U.S. social security system prove to be the inevitable result of having instituted an unfunded (pay-as-you-go) retirement system that delivered $7.9 trillion of net transfers (in 1997 present value dollars) to people born before 1917, and will deliver another $1.8 trillion to people born between 1918 and 1937. But young and future workers cannot necessarily do better by investing their payroll taxes in capital markets. If the old system were closed down, massive unfunded liabilities of $9–10 trillion would still have to be paid unless already accrued benefits were cut. Alternative methods of calculating these accrued benefits yield somewhat different numbers: the straight line calculation is $800 billion less than the constant benefit calculation we propose as the benchmark. Using this benchmark in a world with no uncertainty, we show that privatization without prefunding would not increase returns at all, net of the new taxes needed to pay for unfunded liabilities. These new taxes would amount to 3.6 percent of payroll, or about 29 percent of social security contributions. Prefunding implemented by reducing accrued benefits or by raising taxes, would eventually increase money’s worth for later generations, but at the cost of lower money’s worth for today’s workers and/or retirees.
Computing money’s worth when there is uncertainty is much more difficult unless four conditions hold, namely optimization, time homogeneity, stable prices, and spanning. Under these conditions, the diversification of social security investments into stocks and out of bonds has no effect whatsoever on money’s worth when it is properly adjusted for risk: a dollar of stock is worth no more than a dollar of bonds. When spanning fails, diversification can raise welfare for constrained households, but the exact money’s worth must depend on specific assumptions about household attitudes toward risk. Calculations like those of the Social Security Advisory Council that attribute over $2.85 of net present value gain to each $1 shifted from bonds to stocks completely overlook the disutility of risk. By contrast, we estimate that a 2 percent of payroll equity fund carved out of social security would increase net present value by about 59 cents per dollar of bonds switched into equities, instead of $2.85. When the likely reductions in income and longevity insurance are factored in, the net advantage of privatization and diversification is substantially less than popularly perceived.
Many advocates of social security privatization argue that rates of return under a defined contribution individual account system would be much higher for all than they are under the current social security system. This claim is false. The mistake comes from ignoring accrued benefits already promised based on past payroll taxes, and from underestimating the riskiness of stock investments.
Confusion arises because three distinct reforms are muddled. By privatization we mean creating individual accounts (which could, for example, be invested exclusively in bonds). By diversification we mean investing in stocks, and perhaps other assets, as well as bonds; diversification might be undertaken either by individuals in their private social security accounts, or by the social security trust fund. By prefunding we mean closing the gap between social security benefits promised to date and the assets on hand to pay for them. Any one of these reforms could be implemented without the other two.
If the system were completely privatized, with no prefunding or diversification, the social security system would need to raise taxes and/or issue new debt in order to pay benefits already accrued. If the burden were spread evenly across all future generations via a constant proportional tax, the added taxes would completely eliminate any rate of return advantage on the individual accounts. We estimate that the required new taxes would amount to about 3 percent of payroll, or about a quarter of all social security contributions, in perpetuity. Unlike privatization, prefunding would raise rates of return for later generations, but at the cost of lower returns for today’s workers.
For households able to invest in the stock market on their own, diversification would not raise rates of return, correctly adjusted to recognize risk. Households that are constrained from holding stock, due to lack of wealth outside of social security or to fixed costs from holding stocks, would gain higher risk-adjusted returns and would benefit from diversification. If this group is large, diversification would raise stock values, thus helping current stockholders, but it would lower future stock returns, thus hurting young unconstrained households. Overall, since the number of truly constrained household is probably not that large, privatization and diversification would have a much smaller effect on returns than reformers typically claim.
We construct stationary Markov equilibria for an economy with fiat money, one non-durable commodity, countably-many time periods, and a continuum of agents. The total production of commodity remains constant, but individual agents’ endowments fluctuate in a random fashion, from period to period. In order to hedge against these random fluctuations, agents find it useful to hold fiat money which they can borrow or deposit at appropriate rates of interest; such activity may take place either at a central bank (which fixes interest rates judiciously) or through a money-market (in which interest rates are determined endogenously).
We carry out an equilibrium analysis, based on a careful study of Dynamic Programming equations and on properties of the Invariant Measures for associated optimally-controlled Markov chains. This analysis yields the stationary distribution of wealth across agents, as well as the stationary price (for the commodity) and interest rates (for the borrowing and lending of fiat money).
A distinctive feature of our analysis is the incorporation of bankruptcy, both as a real possibility in an individual agent’s optimization problem, as well as a determinant of interest rates through appropriate balance equations. These allow a central bank (respectively, a money-market) to announce (respectively, to determine endogenously) interest rates in a way that conserves the total money-supply and controls inflation.
General results are provided for the existence of such stationary equilibria, and several explicitly solvable examples are treated in detail.
We construct stationary Markov equilibria for an economy with fiat money, one non-durable commodity, countably-many time periods, and a continuum of agents. The total production of commodity remains constant, but individual agents’ endowments fluctuate in a random fashion, from period to period. In order to hedge against these random fluctuations, agents find it useful to hold fiat money which they can borrow or deposit at appropriate rates of interest; such activity may take place either at a central bank (which fixes interest rates judiciously) or through a money-market (in which interest rates are determined endogenously).
We carry out an equilibrium analysis, based on a careful study of Dynamic Programming equations and on properties of the Invariant Measures for associated optimally-controlled Markov chains. This analysis yields the stationary distribution of wealth across agents, as well as the stationary price (for the commodity) and interest rates (for the borrowing and lending of fiat money).
A distinctive feature of our analysis is the incorporation of bankruptcy, both as a real possibility in an individual agent’s optimization problem, as well as a determinant of interest rates through appropriate balance equations. These allow a central bank (respectively, a money-market) to announce (respectively, to determine endogenously) interest rates in a way that conserves the total money-supply and controls inflation.
General results are provided for the existence of such stationary equilibria, and several explicitly solvable examples are treated in detail.
In the classical general equilibrium model, agents keep all their promises, every good is traded, and competition prevents any agent from earning superior returns on investments in financial markets. In this paper I introduce the age-old problem of broken promises into the general equilibrium model, and I find that a new market dynamic emerges.
Given the legal system and institutions, market forces of supply and demand will establish the collateral levels which are required to secure promises. Since physical collateral will typically be scarce, these collateral levels will be set so low that there is bound to be some default. Many kinds of promises will not be traded, because that also economizes on collateral. Scarce collateral thus creates a mechanism for determining endogenously which assets will be traded, thereby helping to resolve a long standing puzzle in general equilibrium theory. Finally, I shall show that under suitable conditions, in rational expectations equilibrium, some investors will be able to earn higher than normal returns on their investments.
The legal system, in conjunction with the market, will be under constant pressure to expand the potential sources of collateral. This will lead to market innovation.
I illustrate the theoretical points in this paper with some of my experiences on Wall Street as director of fixed income research at the firm of Kidder Peabody
In the classical general equilibrium model, agents keep all their promises, every good is traded, and competition prevents any agent from earning superior returns on investments in financial markets. In this paper I introduce the age-old problem of broken promises into the general equilibrium model, and I find that a new market dynamic emerges.
Given the legal system and institutions, market forces of supply and demand will establish the collateral levels which are required to secure promises. Since physical collateral will typically be scarce, these collateral levels will be set so low that there is bound to be some default. Many kinds of promises will not be traded, because that also economizes on collateral. Scarce collateral thus creates a mechanism for determining endogenously which assets will be traded, thereby helping to resolve a long standing puzzle in general equilibrium theory. Finally, I shall show that under suitable conditions, in rational expectations equilibrium, some investors will be able to earn higher than normal returns on their investments.
The legal system, in conjunction with the market, will be under constant pressure to expand the potential sources of collateral. This will lead to market innovation.
I illustrate the theoretical points in this paper with some of my experiences on Wall Street as director of fixed income research at the firm of Kidder Peabody.
The existence of Nash and Walras equilibrium is proved via Brouwer’s Fixed Point Theorem, without recourse to Kakutani’s Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.”