Beginners Guide to Solow Model

Spring 2007 187

A Beginner’s Guide to the Solow Model

Sheldon H. Stein

Abstract: The Solow model is widely regarded as the workhorse model of the the-

ory of economic growth. Although at one point this model was first encountered

in graduate school, it has since filtered down to the intermediate and, occasion-

ally, to the principles of macroeconomics course. Many have commented on how

difficult it is to teach the Solow model to undergraduates, especially to students

in the principles of macroeconomics course. The author demonstrates that under

the assumption that the level of savings is autonomous, the essence of the stock-

flow adjustment of the Solow model becomes much easier to comprehend.

Keywords: depreciation, economic growth, investment, savings, steady state,

stock-flow adjustment

JEL code: A22, E10, 040

Within the past 30 years, the theory of economic growth, as represented by the

Solow growth model, has filtered down to introductory principles courses. Many

observers have commented on the difficulties associated with teaching the Solow

growth model to undergraduates. I present a simplified version of the Solow

model to make it more accessible to undergraduates, with the assumption that the

level of savings is autonomous. With this assumption, the adjustment of capital

toward a stock equilibrium is greatly simplified. Once the stock-flow equilibrium

process is understood, the transition toward the standard Solow framework fol-

lows naturally.

Sheldon H. Stein is an associate professor of economics at Cleveland State University (e-mail:[email protected]). Copyright © 2007 Heldref Publications

In this section, the Journal of Economic Education publishes articlesconcerned with substantive issues, new ideas, and research findings ineconomics that may influence or can be incorporated into the teaching ofeconomics.

HIRSCHEL KASPER, Section Editor

Content Articles in Economics

TEACHING THE SOLOW MODEL AT THE UNDERGRADUATE LEVEL

Taylor (2000, 90–91) was pessimistic about the ability of beginning students to

comprehend the notion of a “steady-state growth equilibrium” as found in the

Solow model. Colander (2000, 77) stated that the basic Solow model is too difficult

for principles students and perhaps a bit of a stretch for most students at the inter-

mediate level. DeLong, in an essay posted on his Web site, http://www.j-bradford-

delong.net/MHText/Older%20Files/Textbooks.html, states that two-thirds of the

undergraduates at Berkeley have difficulty coping with the notion of capital per

“effective worker.” Baretto (http://www.wabash.edu/EconMacro/home.htm), in a

commentary posted on his Web site, writes that presentations of the Solow growth

model in all of the major textbooks require “fantastic feats of imagination” by stu-

dents. He recommends that the properties of this model be presented with Excel

spreadsheets and Word documents. In the spirit of a comment attributed to Albert

Einstein, who said that physical theories should lend themselves to descriptions

so simple that even a child could understand them, I have discovered that a sim-

ple version of the Solow model enables the student to understand the essence of

the dynamic processes behind the Solow model. By making one unrealistic heroic

assumption, I present a version of the Solow model that I call the beginner’sSolow model or BSM. The benefit of learning BSM is that it enables the student

to understand the essence of the stock-flow equilibrium process found in the orig-

inal Solow model. Once BSM is mastered, and it generally does not take long, the

transition to a more standard description of the Solow model such as that found

in intermediate texts is relatively quick and painless. The level of mathematics

used in BSM makes it amenable to principles students.

THE BEGINNER’S SOLOW MODEL

In the beginner’s Solow model, I make the assumption that the level of savings

is completely autonomous, being independent of its usual determinants such as

disposable income, interest rates, wealth, and so forth. This assumption can eas-

ily be dispensed with once the student has the “Eureka moment” with regard to

the nature of stock-flow equilibrium. I begin by setting the level of autonomous

savings at $50.

The standard assumption that depreciation is equal to a fixed fraction of the

capital stock is retained. To make it possible for students to do the mathematical

calculations in their heads, without the need to use calculators or spreadsheets, I

assume that the depreciation rate is 10 percent. Suppose now that the initial value

of the capital stock is $300. With savings equal to a constant $50 and deprecia-

tion equal to 10 percent of $300 (i.e., $30), net investment, the difference between

saving (or gross investment) and depreciation in the first period, is $20. Adding

this $20 to the initial capital stock of $300 provides the capital stock for the

following period, which is $320. Surprisingly, this often turns out to be the major

hurdle that a student must leap before being able to comprehend the essence of

the Solow model. But the vast majority of them are able to do so.

188 JOURNAL OF ECONOMIC EDUCATION

Next, with savings assumed to remain at $50, depreciation is equal to 10 percent of

$320 or $32. Net investment is now $50 – $32 � $18. The capital stock at the begin-

ning of the next period is $320 plus $18 or $338. If one continues with this process,

the movement toward the steady-state level of the capital stock can be understood

without having to use electronic aids. Finally, students are asked what happens when

the capital stock reaches $500. There is always at least one student who responds that

no further changes in the level of the capital stock will occur because the $50 of sav-

ings is now absorbed entirely by $50 of depreciation, which is 10 percent of $500. For

some, arriving at this point is a struggle, but it does not take long for most if not all of

the students in the class to acknowledge that they understand. A short homework

assignment that I give at the end of the class verifies that this is the case.

In addition to the assumption of autonomous savings, I also assume that labor and

technology are fixed, which is fairly standard in introductory expositions of the

Solow model, such as that found in Mankiw (2003, 181). If one assumes that the cap-

ital-output ratio is constant, then output grows as the capital stock grows. At this

stage of the exposition, simplifying assumptions of this kind are quite welcome by

my students. Also, the whole notion of whether we are dealing with national savings,

private savings, or public savings is an issue that can be explored on another day.

At this point, I begin the whole process again, with an initial value of the cap-

ital stock above the steady state of $500, say $510. To those who understand how

this model works, it is obvious that the capital stock must initially fall to $509 if

the level of autonomous savings remains at $50 because depreciation is 10 per-

cent of $510, or $51, thus making net investment equal to –1.00. The capital stock

will continue to fall until it equals $500, where once again, depreciation will

absorb the entire $50 of exogenous savings and a steady state is reached.

I now illustrate these examples with a spreadsheet. In Table 1, autonomous sav-

ings is set equal to 50, the depreciation rate is set at 10 percent, and the initial

value of the capital stock is $300. In Table 2, I begin a process of capital decumu-

lation by setting the initial value of the capital stock above the steady state of

$500 at $510. Once the template is set up, a user can change the level of autono-

mous savings and the depreciation rate and observe the process of movement

toward the steady state. Thus, in Table 3, the depreciation rate is set at 20 percent

with savings set at $70. The steady-state capital stock now occurs at $350, where

$70 of savings is offset by $70 of depreciation.

Next, the lesson can be reinforced with the algebraic approach. The change in

the capital stock is

�K � S0 – �K, (1)

where S0 is autonomous savings, K is the capital stock, and � is the depreciation

rate. The steady state, where K is in a long-run equilibrium, occurs where �K is

equal to zero, or

S0 – �K � 0. (2)

This can be illustrated by referring to Tables 1, 2, and 3. Solving for K,

K* � (S0/�). (3)

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TABLE 2. Capital Accumulation when Savings Is $50, the Initial CapitalStock Is $510, and the Depreciation Rate Is 10 Percent

NetTime K ($) S ($) Depreciation ($) investment ($)

1 510.00 50.00 51.00 �1.002 509.00 50.00 50.90 �0.903 508.10 50.00 50.81 �0.814 507.29 50.00 50.73 �0.735 506.56 50.00 50.66 �0.666 505.90 50.00 50.59 �0.597 505.31 50.00 50.53 �0.538 504.78 50.00 50.48 �0.489 504.30 50.00 50.43 �0.43

10 503.87 50.00 50.39 �0.39...

74 500.00 50.00 50.00 0.00

Note. K � capital stock; S � savings. Autonomous savings � $50.00; Depreciation rate �0.10; Steady-state capital � $500.00.

TABLE 1. Capital Accumulation When Savings Is $50, the Initial CapitalStock Is $300, and the Depreciation Rate Is 10 Percent


1 300.00 50.00 30.00 20.002 320.00 50.00 32.00 18.003 338.00 50.00 33.80 16.204 354.20 50.00 35.42 14.585 368.78 50.00 36.88 13.126 381.90 50.00 38.19 11.817 393.71 50.00 39.37 10.638 404.34 50.00 40.43 9.579 413.91 50.00 41.39 8.61

10 422.52 50.00 42.25 7.75...

102 500.00 50.00 50.00 0.00


So the steady state for the numbers used in equation 1 is K* � (So/�) �$50/(0.10) or $500. By changing the numbers in this equation along with the

parameters of the template, as was done in Table 3, the student gets even more

reinforcement about the nature of the steady state and its relationship to savings

and depreciation.

Finally, a graph (Figure 1) can be drawn to illustrate the steady state visually.

The steady-state level of the capital stock occurs where the horizontal S curve,

which represents the autonomous level of savings, crosses the D curve, which rep-

resents the level of depreciation and whose slope is the depreciation rate. On the

basis of the previous discussion, one can see that whenever the initial capital

stock is below K*, the capital stock will rise during the next period by the verti-

cal difference between the curves. Whenever the initial capital stock is greater

than K*, the capital stock will fall similarly by the difference. Once the economy

arrives at K*, it is in a steady-state equilibrium because all of the savings is

absorbed by depreciation. The student can raise or lower the autonomous savings

curve to observe the effects of savings on capital accumulation. The student can

also rotate the depreciation curve by changing its slope, which is the depreciation

rate. These effects will carry over once I introduce the original Solow model.

CONCLUSION

One might object to the lack of pedagogical purity of the BSM exercise with

the charge that the flow-savings equilibrium component of this model is imposed

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TABLE 3. Capital Accumulation when Savings Is $70, the Initial CapitalStock Is $300, and the Depreciation Rate Is 20 Percent


1 300.00 70.00 60.00 10.002 310.00 70.00 62.00 8.003 318.00 70.00 63.60 6.404 324.40 70.00 64.88 5.125 329.52 70.00 65.90 4.106 333.62 70.00 66.72 3.287 336.89 70.00 67.38 2.628 339.51 70.00 67.90 2.109 341.61 70.00 68.32 1.68

10 343.29 70.00 68.66 1.34...

43 350.00 70.00 70.00 0.00


by assumption. For this, I offer no apology because this is one case where the end

really does justify the means. Once students understand this abbreviated model,

it becomes easier to move to more advanced versions of the Solow model where

savings is determined by the technology available to the firm. We thus can replace

the horizontal savings curve with an upward sloping savings function as was done

by O’Sullivan and Sheffrin (2006, 190–91) in their principles text. They assumed

that the level of labor and technology is fixed, and appealed to the upward slop-

ing total product of capital curve and a fixed average propensity to save to obtain

a savings curve that is an upward sloping function of the capital stock. Other than

that, the rationale for this intersection being a steady state is exactly the same as

for the BSM. Although I feel uncomfortable using a model in which savings are

assumed to be autonomous, if I want students to learn the standard Solow model,

I believe that I have no choice but to start out this way.

In the analysis presented in intermediate texts, BSM equation (1) is replaced

with equation

�K � sY � �K , (4)

where S0 is replaced by sY where s is the savings rate. This equation can be found

in Delong (2002, 99) and Mankiw (2003, 185). The steady state is obtained by

setting �K equal to zero to obtain

K/Y � s/�, (5)


FIGURE 1. The steady state of the beginner’s Solow model.

which is similar to equation (3). This expression for the steady-state, capital-

output ratio is a special case of the following found in DeLong (2002, 76)

K/Y � s/(n � g � �), (6)

where n represents the proportional growth rate of the labor force, and g represents

the proportional rate of growth in the efficiency of the labor force (EL) where the

production function is

Y � K�(EL)1–�. (7)

It is interesting that the BSM diagram in Figure 1 is visually identical to one

found in an essay on DeLong’s Web site (http://econ161.berkeley.edu/

macro_online/gt.primer.pdf) titled “Growth: An Introduction,” as a primer to

growth theory. The only difference is in the interpretation of the two curves. In

DeLong’s graph, the horizontal line represents the savings rate, not autonomous

savings. There is also a line through the origin that represents the effect not only

of depreciation but also of growth in the labor force and the growth of technical

change.

Also, once students understand the process of capital accumulation in the BSM

model, it really is not difficult to run through the exercise in Table 7-2 of Mankiw

(2003, 188) in which capital and labor are inserted into a production function to

compute output, which is then multiplied by the savings rate to get the level of

savings. So, once students master the BSM, they are not too many steps removed

from a conventional Solow model with labor force growth and technological

progress.

REFERENCES

Colander, D. 2000. Telling better stories in introductory macro. American Economic Review 90(May): 76–80.

DeLong, J. B. 2002. Macroeconomics. New York: Irwin McGraw-Hill.Mankiw, N. G. 2003. Macroeconomics. New York: Worth.O’Sullivan, A., and S. Sheffrin. 2006. Macroeconomics: Principles and tools. Englewood Cliffs, NJ:

Prentice-Hall.Taylor, J. 2000. Teaching macroeconomics at the principles level. American Economic Review 90

(March): 90–94.

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Beginners Guide to Solow Model

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