Supplement 3:  The General Solow Balanced Growth Model

	Most intermediate Macroeconomics textbooks offer a version of the Solow (1956) model of balanced growth, and some explanation of Solow’s proof that the economy converges to the balanced growth rate.  However, students at this level not only often have difficulty comprehending this proof, but also struggle to understand how changes in the saving rate, population growth rate, technological progress rate, and depreciation rate affect the long run growth path.  An Excel spreadsheet simulation of the Solow model gives students visual and numerical illustrations of these comparative static and convergence results.  Without the spreadsheet program, it is very tedious, if not impossible, to provide numerical examples.

Deriving the Balanced Growth Path
	Consider the following variant of the Solow (1956) model.  We will first derive the balanced growth condition in a way identical to section 5 of the paper.  Solow showed that an economy is in a steady state when it is in a state of balanced growth.  Balanced growth occurs when capital (K) grows at the same rate of the effective (adjusted for productivity) labor supply (LEt).  Since the labor force grows at rate n, and labor productivity grows at rate g, the effective labor supply grows at rate (n + g).  Labor supply growth is defined in equation (S3.1); balanced growth capital growth is defined in equation (S3.2).
					(LEt) = (n+g)(LEt-1)					(S3.1)
					Kt = (n+g)Kt-1						(S3.2)
	Income is either consumed or saved, where the saving rate is s.  This defines consumption, Ct in terms of the saving rate and income (Y):
					Ct = Yt – sYt						(S3.3)
All saved output becomes capital, and is put into production the following period.  Thus, for each period, total saving equals gross investment (It):
					St = sYt = It						(S3.4)
Capital depreciates at rate d.  Net investment (Kt) is thus gross investment (It) less depreciation (dKt).
	For capital to grow at the balanced growth rate of (n + g), gross investment must be equal to:						I t* = (n + g + d)Kt*					(S3.5)
An asterix is used to denote balanced growth variables.  Equation (S3.5) can be rewritten by using equation (S3.4) to form:
					sYt* = (n + g + d)Kt*					(S3.5)
When equation (S3.5) is solved for the balanced growth capital to output ratio (Kt*/Yt*), we see that Kt*/Yt* is a constant:
					Kt*/Yt*= s/(n + g + d)					(S3.5)
As noted in section 5 of the paper, the balanced growth capital to output ratio defines steady state capital as a function of output, and vice versa.  When paired with a production function, steady state output and GDP are jointly defined.
	The aggregate production function assumes that output depends on a technology parameter (A), capital (Kt), and effective (adjusted for productivity) labor (LEt).  A Cobb-Douglas form is used, with parameter a<1:
					Yt* = AKt*a(LEt)1-a					(S3.6)
Equations (S3.5) and (S3.6) together define the steady state levels of output and capital, which are a function of the effective labor supply (LEt), total factor productivity (A), and parameters.  Depending on the value for a, solving (S3.5) and (S3.6) simultaneously will likely require a mathematical method.  

Deriving the actual GDP path
	As noted in the introduction to this supplement, Solow showed that an economy will converge to its balanced growth path.  To exhibit this property on a numerical spreadsheet model, we must construct a model of a simple economy, based on all of the assumptions of the Solow model, except the assumption of balanced growth.
	By ignoring the balanced growth condition, capital evolves according to the level of investment.  Gross investment is again equal to savings, as in equation (S3.4).  We rewrite equation (S3.4) as (S3.7) for expositional purposes.
					St = sYt = It						(S3.7)
Capital depreciates at rate d:
					Depreciationt = dKt					(S3.8)
Which leaves net investment, Kt to be:
					Kt = sYt – dKt						(S3.9)
Thus, the capital stock evolves according to the following equation:
					Kt+1 = Kt + Kt						(S3.10)
Next, define a general production function which is identical to (S3.6) except no assumption is made about the level of capital.  Capital is as defined in equation (S3.10), which need not be consistent with balanced growth.
 					Yt = AKta(LEt)1-a						(S3.11)

Modeling in Excel
	We can now use a spreadsheet to set up a numerical example of the dynamic growth model just developed.  To show Solow’s convergence proof, both the balanced growth path and actual GDP path economies are modeled.  These models are shown in Figure S3.1, below.  Figure S3.1 is set up much the same way as Figure 5.1 (the balanced growth model) in the paper, with a few exceptions.  First, the top row is used to track time (in years).  Second, effective labor (LEt) is be allowed to evolve over time.  In time 0, an initial effective labor supply is assumed; in time 1 and after, the labor supply grows according to equation (S3.1).  In Figure S3.1, this is accomplished in rows 10 and 11 by finding the growth in the labor force every year (n + g)(LEt), which is added to the labor force the following year (LEt+1 = (n+g)(LEt)).  The balanced growth path is calculated as before, relying on equations (S3.5) and (S3.6) (solved using circular references and the Iteration calculation option described in section 5 of the paper).  Additional rows are inserted to track the growth rate of steady state real GDP and the balanced growth capital to output ratio.  The actual GDP path is determined by equations (S3.7) – (S3.11).  Rows 19-22 are defined by equations (S3.7-S3.9), and row 24 (capital) is defined by equation (S3.10), where an initial endowment of capital is provided in year 0.  Output (row 26) is determined by equation (S3.11).  Note that it is not a circular reference with the equation for capital (row 24).  Additional lines were inserted to track the actual GDP growth rate and actual capital to output ratio.  As a side note, cells with constants are in bold face, and cells with circular references are shaded.  Also note that once the year 1 column has been set up, it is easily copied to as many other columns as desired, where each column represents an additional year.  As a side note, the Excel file corresponding to Figure S3.1 (available on this web page) exhibits forty one years, where Figure S31 displays four years.
Figure S3.1:  A Dynamic Solow Growth Model

	Figure S3.1 depicts an economy that is initially on its balanced growth path.  That is, the initial endowment of capital is set at the balanced growth level in year 0.  Note that in years 1, 2, and 3, the economy stays on the balanced growth path.  That is, capital (and output) naturally grows at the same rate as the labor supply.  This is an example showing part of Solow’s proof of the stability of the balanced growth path.  With the spreadsheet model, it is relatively simple to examine an economy that starts off its growth path.  For example, if capital is initially set above the balanced growth level, output growth will be below balanced growth level, and the capital to output ratio will fall to the balanced growth level.  That is, the economy converges to the balanced growth path.  Graphs of such examples are particularly instructive.
	Aside from the variables mentioned above, it may be desirable to add additional lines to the spreadsheet to calculate other variables.  For example, the capital to labor ratio (Kt/Lt), capital to effective labor ratio (Kt/LEt), consumption (Ct=Yt‑St), per capita income (Yt/Lt) and income per effective unit of labor (Yt/LEt) may be helpful to explore certain topics.  Graphs of these variables may illustrate some central results of this growth model.  As a side note, on some graphs (e.g., Y over time), it may be desirable to change the scale of the Y axis to a log scale. 
	This simulation can also be used to conduct comparative static analyses.  Changing a parameter value at a certain date will show the appropriate change in the growth (or actual) path, and will show the economy converging to the new path.  Such examples can illustrate that a change in the saving rate will change the position, but not growth rate of the long run (potential) GDP path, and changes in n, g, and d change both the slope and position.  Figure S3.2 displays an example of an economy that is initially on its long run path, but suddenly suffers a decrease in the saving rate (in period 5).  Note that the balanced growth path shifts down, but grows at the same rate, and the growth of actual GDP slows in order to converge back to the long run path.
Figure S3.2:  Comparative static example:  A decrease in the saving rate

	Without a computer, numerical examples such as the one in this supplement are all but impossible, a fact that distances intermediate level students from the economics. Yet, when using a spreadsheet to solve the model, all rigor is not lost.  Students must set up the spreadsheet and input the equations properly in order for it to work; they must understand the model in order to get the spreadsheet to solve it.  Once set up, the spreadsheet can be used for a wide variety of applications and examples.