# The Endogenous Growth Model

Let us investigate the effect of fiscal policy using an endogenous growth model. In this model, the conventional neoclassical model is modified to have an endogenous growth rate. The endogenous growth model generally incorporates externalities of human capital accumulation and/or research development so as to determine economic growth endogenously. There are many formulations of endogenous economic growth (e.g., Romer 2011).

## The Optimal Growth Model

We formulate a very simple version of the endogenous growth model in order to analyze the effect of fiscal policy in Sect. 4.2. Before doing this, it is useful to formulate the optimal growth model in this section. In the optimal growth model, households optimize consumption/saving behavior over time. The optimal allocation of consumption over time is determined by the relationship between the time preference and the interest rate, as explained in Chap. 3. If the rate of interest r is greater than the rate of time preference p, the agent greatly prefers future consumption to present consumption. Thus, the growth rate of consumption, ra_{c}, increases with the rate of interest and decreases with the time preference rate.

We may formulate this as follows:

P is a positive constant. We may also regard ra_{c}as the economic growth rate because it corresponds to the growth rate of GDP and/or capital. In the optimal growth model, saving/consumption behavior is endogenously determined in order to maximize lifetime utility. In this sense, the model is a sophisticated version of the neoclassical growth model and is called the optimal growth model.

As explained in Sect. 2, economic growth normally enhances capital intensity and reduces the marginal product of capital. In the competitive market, the rate of interest is equal to the marginal product of capital. Thus, economic growth reduces the rate of interest over time. In the long run, the rate of interest declines so that it is equal to the exogenously given time preference rate.

Then, Eq. (5.20) means that economic growth stops even in the optimal growth model. This corresponds to long-run equilibrium in the neoclassical growth model, as shown in Fig. 5.4. If labor grows exogenously, consumption and capital grow at the rate of n in the long run, while per capita consumption and per capita capital stock do not grow in the long run. This property is a key outcome of the conventional neoclassical growth model. As long as labor supply is exogenously given, we cannot derive the endogenous growth rate in the long run.