The typical real business cycle model is based upon an economy populated by identical infinitely lived households and firms, so that economic choices are reflected in the decisions made by a single representative agent. It is assumed that both output and factor markets are characterized by perfect competition. Households sell capital, kt, to firms at the rental rate of capital, and sell labor, ht, at the real wage rate. Each period, firms choose capital and labor subject to a production function to maximize profits. Output is produced according to a constant-returns-to-scale production function that is subject to random technology shocks. Specifically yt=ztf(kt, ht), where yt is output and zt is the technology shock. (The price of output is normalized to one.) Households’ decisions are more complicated: given their initial capital stock, agents determine how much labor to supply and how much consumption and investment to purchase. These choices are made in order to maximize the expected value of lifetime utility. Households must forecast the future path of wages and the rental rate of capital. It is assumed that these forecasts are made rationally. A rational expectations equilibrium consists of sequences for consumption, capital, labor, output, wages, and the rental rate of capital such that factor and output markets clear.
While it is fairly straightforward to show that a competitive equilibrium exists, it is difficult to solve for the equilibrium sequences directly. Instead, an indirect approach is taken in which the Pareto optimum for this economy is determined (this will be unique given the assumption of representative agents). As shown by Debreu (1954), the Pareto optimum as characterized by the optimal sequences for consumption, labor, and capital in this environment will be identical to that in a competitive equilibrium. Furthermore, factor prices are determined by the marginal products of capital and labor evaluated at the equilibrium quantities. (For a detailed exposition of the connection between the competitive equilibrium and Pareto optimum in a real business cycle model, see Prescott, 1986 .) We now provide an example of solving such a model.