Working Paper 2004-39 December 2004
Abstract: Recent work by Greenwood, Hercowitz, and Krusell (1997 and 2000) and Fisher (2003) has emphasized the importance of investment-specific technological change as a main driving force behind long-run growth and the business cycle. This paper shows how the growth model with investment-specific technological change has a closed-form solution if capital fully depreciates. This solution furthers our understanding of the model, and it constitutes a useful benchmark to check the accuracy of numerical procedures to solve dynamic macroeconomic models in cases with several state variables.
JEL classification: E10, E32, D90
Key words: growth model with investment-specific technological change, closed-form solution, long-run growth, business cycle fluctuations
The recent work of Greenwood, Hercowitz, and Krusell (1997 and 2000) has focused the attention of economists on the role of investment-specific technological change as a main driving force behind economic growth and business cycle fluctuations. Fisher (1999) documents two key empirical observations that support these conclusions. First, the relative price of business equipment in terms of consumption goods has fallen in nearly every year since the 1950s. Second, the fall in the relative price of capital is faster during expansions than during recessions.
Models of investment-specific technological change have also being successfully used to account for the evolution of the skill premium in the U.S. since the Second World War (Krusell et al., 2000) or the cyclical behavior of hours and productivity (Fisher, 2003), among several other applications.
Unfortunately, the standard growth model with investment-specific technological change, as presented in Fisher (2003), does not have a known analytic solution. Therefore, researchers have employed computational methods to solve the model.
In this paper, we show how this standard model has a closed-form solution when there is full depreciation of capital. We derive the exact solution in the case where there is a labor/leisure choice and long-run growth in the economy. The solution has a simple backward representation that allows to gauge the importance of each parameter on the behavior of the model.
There are, at least, two reasons that make our result important. First, the closed-form solution improves our understanding of the dynamics of the model beyond the findings provided by numerical computations. The law of motions for variables uncover the main driving forces in the model and develop intuition that is difficult to obtain from the computer output. In particular, we illustrate how shocks propagate over time and which factors determine the persistence of the model. This exercise highlights the importance of the capital participation share as a determinant of propagation.
Second, the closed-form solution is an excellent test case to check the behavior of numerical procedures like solution methods for a dynamic macroeconomic models. The approximated solutions generated by those algorithms in the case of full depreciation can be compared against the closed-form solution. In that way, we can evaluate the accuracy of the solution method. The model with investment-specific technological change is a more interesting test case than the neoclassical growth model because the presence of two shocks increases the dimensionality of the problem and, consequently, makes it more representative of interesting macro applications.
2. A Growth Model with Two Shocks
We present a simple growth model with two shocks, one to the general technology and one to investment as described in Fisher (2003).
There is a representative household in the economy, whose preferences over stochastic sequences of consumption [c.sub.t] and leisure [l.sub.t] can be represented by the utility function: