General equilibrium models investigate the pricing of real and financial assets resulting from the balance of supply and demand in an economy. The participants in the economy (often called agents) make their investment and consumption decisions to optimize their individual objectives, typically the maximum expected utility of end-of-period wealth, or the maximum expected utility of lifetime consumption. This creates a demand and supply for transactions, whose pricing is then set by the equality of supply and demand.

Equilibrium is not a stationary state. It changes at every moment, depending on the stochastic nature of the flow of capital and goods, of investment results, and of technology changes.

One result obtained from the solution of a general equilibrium model is the relationship of interest rates to economic variables. Interest rates are determined by economic forces through the equilibrium of supply and demand. Term structure models describe the behavior of interest rates of different maturities as joint stochastic processes; these models do not relate interest rates to economic variables. General equilibrium models explain why interest rates behave the way they do, not just how they behave.

Most of the modern general equilibrium models fall into two broad categories: pure exchange models and production models. Pure exchange models assume that each participant receives some endowment (such as income from labor) during his lifetime, which he can trade with other participants to maximize his expected utility of consumption. Thus, a participant who assigns large utility to immediate consumption will borrow from those participants who assign higher utility to consumption at a later date. The mechanism of supply and demand will determine the pricing of such contracts, resulting in a description of the term structure of interest rates and the pricing of bonds. For this kind of a model, see, for instance, Karatzas and Shreve (1998).

Models of production economies often start with an initial endowment assigned to each participant. The economy contains production opportunities, which consist of production processes with stochastic rates of return on investment. The production processes can be viewed as exogenously given assets that are available for investment in any amount. The amount of investment in the production, however, is determined endogenously. The parameters of the production process can themselves be stochastic. This can be interpreted as representing uncertain changes in production technology.

It is assumed that investors can issue and buy any derivatives of any of the assets and securities in the economy. The investors can lend and borrow among themselves, either at a floating short rate or by issuing and buying term bonds. The resultant market is complete. It is further assumed that there are no transaction costs and no taxes or other forms of redistribution of social wealth. The investment wealth and asset values are measured in terms of a medium of exchange that cannot be stored unless invested in the production process. For instance, this wealth unit could be a perishable consumption good. A model of production economy with these characteristics is described in Cox, Ingersoll, and Ross (1985a). They assume that the investors have identical preferences.

For a meaningful economic analysis, it is essential that a general equilibrium model allows heterogeneous participants. If all participants have the same preferences, they will all hold the same portfolio. Since there is no borrowing and lending in the aggregate, there is no net holding of debt securities by any participant, and no investor is exposed to interest rate risk. Moreover, if the utility functions are the same, it does not allow for study of how asset pricing and interest rates depend on differences in investors' preferences.

The main difficulty in developing a general equilibrium model of production economies with heterogeneous participants had been the need to carry the individual wealth levels as state variables, because the equilibrium depends on the distribution of wealth across the participants. It is shown in the 2005 paper “The Economics of Interest Rates” (Chapter 11) that the individual wealth levels can be represented as functions of a single process, which is jointly Markov with the technology state variable. This allows construction of equilibrium models with just two state variables, regardless of the number of participants in the economy.

The papers in Part Three investigate an economy in continuous time with production subject to uncertain technological changes described by a state variable. Each investor maximizes the expected utility from lifetime consumption. The participants have different utility functions and different time preferences.

The economy contains a production process whose rate of return dA/A on investment is

where is a Wiener process. The process represents a constant return-to-scale production opportunity.

The parameters of the production process are stochastic, reflecting the fact that production technology evolves in an unpredictable manner. It is assumed that their behavior is driven by a Markov state variable . The state variable can be interpreted as representing the state of the production technology. The process , which can be a vector, may be correlated with the production process .

In equilibrium, the total wealth must be invested in the production process (which justifies referring to the production process as the market portfolio). Any lending and borrowing (including lending and borrowing implicit in issuing and buying contingent claims) is among the participants in the economy, and its sum must be zero.

It may seem more realistic to have a model of the economy with multiple production processes: factories for different goods, farming of different commodities, and so on. It may be noted, however, that the equilibrium conditions would simply determine in which proportion these production processes are held by the aggregate of the economy participants. Now, this total is actually known and observed: It is the market portfolio. Rather than specifying the vectors of expected returns for each production and the covariance matrix of their risks (and perhaps arriving at a market portfolio different from the observed one due to misspecification of the inputs for the individual productions), it serves the purpose of investigating an economic equilibrium better to model the properties of the market portfolio directly.

An economy cannot be in equilibrium if arbitrage opportunities exist. A necessary and sufficient condition for absence of arbitrage is that there exists a process , called the state price density process, such that the price P of any asset in the economy satisfies the equation

Equilibrium is fully described by specification of the process , which determines the pricing of all assets in the economy, such as bonds and derivative contracts, by means of the previous equation. Bond prices in turn determine the term structure of interest rates. The state price density process also determines each participant's optimum investment strategy. Solving for the equilibrium means solving for the process .

In “The Economics of Interest Rates” (Chapter 11), it is assumed that consumption takes place continuously at rates based on the investor's optimal investment and consumption strategy. The equilibrium conditions are used to derive a nonlinear partial differential equation whose solution determines the state price density process and consequently the term structure of interest rates. (The results are stated in terms of the so-called numeraire portfolio ). While the solution to the equation can be approximated by numerical methods, the nonlinearity of the equation could present some difficulties.

The 2013 article “General Equilibrium with Heterogeneous Participants and Discrete Consumption Times” (Chapter 12) provides the exact solution for the case that consumption takes place at a finite number of discrete times. If the time points are chosen to be dense enough, the discrete case will approximate the continuous case with the desired precision. This solution does not require solving partial differential equations, and explicit computational procedure is provided. The algorithm requires no more complicated mathematical tools than finding the root of a monotone function.

In many applications, the technology risk is independent of the production risk. For instance, if the production is farming, the progress in development of new agricultural methods, hybrids, fertilizers, and so on is independent of weather. The unpublished 2013 memorandum “Independence of Production and Technology Risks” (Chapter 13) provides an intriguing formula for the equilibrium value of the short rate in the case that each participant maximizes the expected utility of end-of-period wealth.

The paper “Risk-Neutral Economy and Zero Price of Risk” (Chapter 14), written in 2014, investigates the equilibrium in an economy in which all participants are indifferent to risk. The mechanism of asset and derivative pricing in such an economy is identified. It is shown that no economy in equilibrium with stochastic interest rates can be simultaneously risk-neutral and have zero market price of risk. On the other hand, there exist equilibrium economies with risk-averse participants and zero prices of risk. The paper explains the paradox: In a risk-neutral economy in equilibrium, the expected returns are the same on all assets, regardless of their riskiness, over the one period that is relevant to the investors, namely to the point of consumption. Due to the nonlinearity of compounding, however, this precludes the expected instantaneous returns to be the same, unless they are deterministic. The market price of risk will not be zero.