A Simplified Version of the Stochastic Growth Model

We first consider a special case of the stochastic growth model, without government expenditure and with a depreciation rate of 100%. The equations that describe the accumulation of capital and the determination of the real interest rate then simplify to

Because of the assumption of competitive markets and the absence of externalities, the equilibrium of the model is Pareto optimal.

Let us focus on two variables: labor supply per person l and the savings rate s. Defining the savings rate, we also determine aggregate consumption as C = (1 − s)Y.

We focus on both behavioral equations of the representative household (13.21) and (13.23). Once we determine labor supply and the savings rate, the rest follows automatically from either equilibrium conditions or definitions.

The Euler equation for consumption (13.21) can be written as

We can substitute for consumption and the real interest rate in the Euler equation by using the relations

This allows us to rewrite the Euler equation as

Because the left-hand side is constant, the right-hand side must also be constant. Thus, the savings rate is constant and is determined by

The savings rate is independent of the real interest rate and constant in this simplified model because of the assumption of logarithmic preferences.

From (13.23), and by noting that c_t = (1 − )Y_t/N_t and w_t = (1 −α)Y_t/(l_tN_t), we arrive at the conclusion that labor supply per household member is also constant in this simplified model. It is given by

Labor supply is constant in this simplified model, because the impact of the shocks in technology (labor efficiency) on the real wage and the real interest rate cancel each other out, so that no intertemporal substitution in labor supply is induced by such shocks.

This result is due to the specific assumptions that we made to simplify the model, and, as one can see from the analysis of the full model in section 13.4, is not a general feature of the model. In fact, the opposite is true. In general versions of the model, intertemporal substitution in labor supply induced by temporary real shocks is the key driver of fluctuations in aggregate employment.

13.2.1 Fluctuations of Output in the Simplified Stochastic Growth Model

We can now determine fluctuations in total output. Taking logs on both sides of the production function (13.1), we get

We know that K_t = Y_t−1 and that L_t = lN_t. Therefore,

We can substitute the logarithm of A and N from equations (13.8) and (13.9). This implies

We can express (13.30) as

where y_t = ln Y_t − ln denotes the percentage deviation of output from its long-run trend.

The logarithm of trend output in this model is defined by

Trend output grows at the exogenously given rate of n + g, as happens with steady state output in the Ramsey model of growth. From (13.10) and (13.31), we end up with

Equation (13.32) suggests that the percentage deviations of aggregate real output from trend follow a second-order autoregressive process (AR(2)), depending on the share of capital in the production process and the degree of persistence of productivity shocks. Because α is low (about 1/3), the dynamic behavior of total real output depends primarily on the degree of persistence of productivity shocks. If the persistence of productivity shocks η_A is high, then we have considerable persistence in the fluctuations of output. Otherwise, the persistence of output fluctuations around trend is relatively low.

Note that if the degree of persistence of productivity shocks were zero, then (13.32) would simplify to

Equation (13.33) suggests that if productivity shocks did not persist, deviations of output around trend would follow a first-order autoregressive process, with a degree of persistence equal to α.

13.2.2 The Simplified Stochastic Growth Model and the Evidence on Aggregate Fluctuations

Empirical estimates suggest that percentage deviations of output around trend do indeed follow an AR(2) process. For example, running a regression of the log of US real GDP on a linear trend, and its two lags, using annual data for the period 1890–2014, one gets a fairly good statistical fit. The coefficient on the first lag of output is equal to 1.211, with a standard error of estimate of 0.086, and the coefficient on the second lag of output is equal to −0.319, with a standard error of estimate of 0.087.

Hence, the econometric estimates of the model parameters are quite sensible, suggesting that, as a first approximation, this simplified model can account for both the long-run growth rate and the aggregate fluctuations of US output.

However, many other features of aggregate fluctuations are not adequately described by this simplified version of the stochastic growth model. The first such feature is the constant savings ratio. This essentially means that consumption will display the same degree of variability as output and investment. As shown by the stylized facts presented by Hodrik and Prescott [1997] and others since the seminal Hodrik-Prescott paper first appeared in 1980, this does not seem to be a property of the data. Empirical estimates suggest that fluctuations in aggregate consumption are much smoother than output fluctuations.

A second counterfactual feature of the simplified model is the constant employment rate. In reality, the employment rate is not constant over the business cycle, whether one looks at total hours or the number of employees. As shown by Hodrik and Prescott [1997], employment and hours worked are pro-cyclical and are very strongly correlated with output.

Finally the behavior of real wages over the business cycle is not as predicted by the simplified stochastic growth model. In the model, real wages would be pro-cyclical and as volatile as GDP per capita, which is not the case in the data, as suggested by, among others, Neftci [1978].

One must thus consider the more general version of the model to assess its empirical significance.

13.3