1.1 Deriving the basic Solow equation
The Solow model is one of the simplest models one can use to explain economic growth. Represent a country’s aggregate production funciton by = ( ) ( )1− (1)
where for each time period is gross domestic product (GDP), is the capital stock level and is the size of the labor force. To derive the Solow model, we begin with a very simple expression: +1 = − + (2a)
This equation "says:" The level of the capital stock tomorrow (+1 ) is equal to the level of the capital stock today ( ), minus the capital stock lost through depreciation during period ( ), plus investment in period ( ). Note that and are called variables, while and are called parameters. To derive the Solow equation we make two assumptions. First, we assume that investment today is exactly equal to savings today. Another (shorthand) way of writing this is = This equation simply says investment in period is exactly equal to savings in period The second assumption is that the rate of savings is constant over time. Is this a reasonable assumption? Figures 1 and 2 show that savings rates are relatively stable for some countries and not for others. In ﬁgure 1, Australia and Belgium have relatively constant savings rates from the 1990s through 2008, while the savings rate for Costa Rica is stable from the mid 80s through the mid 90s, then increases between 1995 and 2000 and drops again after 2000. In ﬁgure 2, the U.S. and U.K. have savings rates that appear to have drifted downwards over time. Savings rates in the U.S. fell from around 20% in 1980 to 13% in 2007.
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Figure 1. Savings rates
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Figure 2. Savings rates
One can conclude that the constant savings rate assumption can be quite reasonable for some countries, but problematic for others. In practical, applied work, one would likely use the average savings rate for the last ten or twenty years. Returning to the Solow model, with the constant savings assumption, investment in period is equal to = = where is a constant savings rate. Substituting (1) into equation (3) gives = ( ) ( )1− which, substituted into equation (2a) yields +1 = − + = (1 − ) + ( ) ( )1− (4) (3)
For the moment, assume the labor force is constant. We can represent this by writing = for all Then equation (4) can be written as +1 = (1 − ) + = (1 − ) + ( ) 1− To get the Solow equation, we now divide the above expression by to get µ ¶ µ ¶1− +1 ( ) 1− = (1 − ) + = (1 − ) + or +1 = (1 − ) + ( ) where =
is capital stock per unit of labor.
Working with the Solow equation
Equation (5) is the basic Solow model — i.e., the Solow model without labor force growth and technical change. It has the exact same meaning as equation (2a): i.e., "the capital stock in period + 1 is equal to the capital stock in period , less the capital stock lost through depreciation, plus new investment. 1.2.1 The steady state
In class we observed that (5) is a diﬀerence equation in A diﬀerence equation (for our purposes) tells us how the level of a variable in time depends on the level of that variable in period − 1 3
Here, the Solow equation gives us the level of the capital stock in period as a function of the capital stock level in period − 1 An important concept associated with diﬀerence equations is the steady state. A steady state is a level of a variable such that the level of that variable in period is exactly equal to the level of that variable in period − 1 In other words, for the basic Solow