Inspecting the Mechanism: closed‐form Solutions for Asset Prices in Real Business Cycle Models

Abstract

We derive closed-form solutions for asset prices in an RBC economy. The equations are based on a log-linear solution of the RBC model and allow a clearer understanding of the determination of risk premia in models with production. We demonstrate not only why the premium of equity over the risk-free rate is small but also why the premium of equity over a real long-term bond is small and often negative. In particular, risk premia for equity and long real bonds are negative when technology shocks are permanent. Recently, a growing literature has explored the asset pricing implications of real business cycle (RBC) models. Examples are Rouwenhorst (1995), Jermann (1998) and Boldrin et al. (1995). 1 From a methodological point of view, models with production allow a more realistic modelling of consumption and dividends than do the pure exchange economies of Lucas (1978). However, explaining the behaviour of asset prices in production economies is also more challenging. For example, in an exchange economy, an increase in risk premia can be obtained by increasing risk aversion. This is not necessarily true in production economies, since agents can choose a smoother consumption path by substituting between work and leisure in response to productivity shocks. The papers just cited find that RBC models tend to generate counterfactual asset pricing implications unless some extreme form of rigidities is introduced. This paper attempts to provide a clearer understanding of the challenges posed by asset pricing behaviour for RBC models. Instead of solving a standard model numerically and simulating it over a large number of parameter permutations, we argue here that this understanding can be better developed by analysing approximate closed-form solutions for prices of a variety of financial assets. Although there are now many accurate solution algorithms available for solving RBC models (e.g., see the overview in Taylor and Uhlig (1990)), the approximate closed-form expressions obtained here make the relationship between asset prices and exogenous technology shocks particularly transparent by providing simple analytical expressions that decompose the effect of technology shocks on asset prices into (i) a direct effect due to the shock itself and (ii) an indirect effect stemming from capital accumulation.