The Solow Model of Economic growth, named for Nobel 1987 winner Robert Solow, helps us understand the dynamics of national economic growth and the pace of said growth. In this article, we’ll explore down the conceptual basis for the Solow Growth Model, understand the relationships it presents via graphs, derive the model mathematically, and finally explore the model’s implications.
The Solow model describes the relationship between output and capital. It says that the amount of capital determines the amount of output being produced, which in turn determines the amount of savings, and finally the amount of capital accumulated over time, thus cyclically compounding. Let’s break down these relationships:
- More output—goods and services being produced—means that consumers can save more money, since there are more goods and services to go around and they are wealthier versus peers in economies where output is smaller.
- A higher savings rate, or % of income saved/invested, means that the amount of capital each consumer owns—not to mention the national capital stocks—should compound over time, thus increasing the amount of capital compared to economies in which the savings rate is lower.
- More capital accumulated ultimately means that more goods and services will be produced, since more money and liquidity floating around means capital for businesses to invest and individuals to start new businesses.
- Thus, we arrive at the start of our cycle: more output, driven by more capital.
We can look at this relationship through a graph. We’ll take two assumptions for granted: first, the size of the population is constant and the number of workers is proportional to the population (therefore output per worker is proportional to output per person, and second, there is no technological progress, so we’re not shifting the production function due to changes in technology.
The three curves in the graph represent the following:
- Depreciation, e.g. the value of an asset decreasing over time due to natural use and degradation. For example, a machine in a factory will eventually break down or hit its shelf life, and thus depreciation represents the even distribution of its decline over its lifespan. So, it may only break and have to be taken offline in year 10, but it can thus be depreciated at 10% per year.
- Output simply represents the total production of goods and services in the economy described in the model. In the graph we’ll see below, the point of equilibrium (steady state) is a function of depreciation and savings & investment, and output is in turn affected by these variables, not the other way around (think exogenous versus endogenous variables). Simply put: savings and investment rates, as well as depreciation rates, are taken as exogenous assumptions.
- Savings & investment: the net rate of investment in the economy described in the model. As an assumption of the model, the amount that you save translates into investments. Investments refer to the creation of new, physical capital — not stocks or bonds. We assume that this savings → investments cycle happens through intermediary, so while you may buy a stock, that company then takes your money and invests it into physical capital of some sort.
Note the following relationships on the growth:
- As we increase capital, we get a slightly upward—fairly proportional—sloping increase in depreciation.
- Output per worker is a function of capital per worker. As you increase capital per worker (think machinery), you get more output per worker.
- Investment is a function of output per worker multiplied by the savings rate.
Now, here’s the critical point: the equilibrium (steady state) is the point at which savings and investment meets depreciation. At any point below this equilibrium, savings and investment will drive more output that creates more capital, savings, investment, etc. At any point above this equilibrium, depreciation will decrease the value of assets faster than new savings and investment can create it. Examine that relationship on these three graphs:
On the middle graph, savings and investment (capital) will drive more output, thus driving c to c* and o to o*. If for any reason capital and output continues to expand to c**/o**, then a natural retraction should occur that returns the model to its equilibrium, or steady state. In a moment we’ll derive the model, but let’s first examine some of the implications of the model as presented thus far.
Implications of The Solow Growth Model
The Solow model implies that a country can raise its standard of living in the long-run by increasing the savings rate; this ties back to the relationship between capital and output, via the savings rate, described earlier.
Additionally, a core conclusion is that growth slows down as an economy develops. Even growth from increasing the saving rate is temporary, and once an economy is in steady state, output per worker stays stagnant and instead technological progress drives growth.
Also, increasing the savings rate increases the output per worker, inferring that getting people to save more will ultimately increase the national output of goods and services and contribute to economic growth.
We’ll explore another big implication of the model at the end of this article—let’s first derive the model to further understand it.
Deriving the Solow Growth Model
You should understand the model and its implications by now—the fun’s just getting started, though. Here are our assumptions as we explore the math driving the model:
- Closed economy, meaning that total investment equals national savings (basically, people can’t invest money in other countries). So, I = S + (T — G), with S being private saving, (T — G) being public saving, and I being investment.
- Public savings is zero, e.g., T = G = 0. Thus, I = S, or investment is equal to private savings. This basically assumes that private savings is proportional to income. That savings rate is exogenous (determined outside the model), and it’s between 0 and 1, so S = Ys, with 0 < s< 1.
- As stated prior, the size of the population is constant, the number of workers is proportional to the population (labor participation rates), and there is no technological progress.
- There are two inputs into the production function: K and N. K refers to capital, and N refers to labor.
- Constant returns to scale in both inputs, but decreasing returns in each input. This means that adding either capital or labor to the economy will individually get decreasing returns, but in combination you’ll get constant returns.
- The rate of capital depreciation is exogenous, and we’ll describe the depreciation rate as d.
With those assumptions taken, the model starts with these four equations:
- The production function.
- The resource constraint function.
- The resource allocation equation.
- The capital accumulation equation.
These equations are respectively as follows:
- Y = AKᵅNᵝ
- Y = I + C
- I = sYₜ
- Kₜ₊₁ = Iₜ + (1-d)Kₜ
Whereas:
- Y is the total quantity of output produced (total output).
- N is the quantity of labour (either measured in # of workers or hours).
- A is the level of technology (or productivity; generally a measure of efficiency).
- K is the quantity of capital.
- α (alpha) and β (beta) are positive constants (not fixed, but rather parameters) that represent the output elasticities of labour and capital, meaning how much output changes when labor or capital changes. These values typically sum to 1 for reasons we’ll see later.
- Since α and β sum to 1, α + β = 1, then β =1 — α. So, we can simplify our equation to: Y = A(Kᵅ)(N¹⁻ᵅ).
Now let’s break it down:
The production function states that output (real GDP) is equal to technology multiplied by capital and labor, which have (alpha) and (1- alpha) as exponents. The resource constraint says that output is the combination of investment and consumption. The resource allocation equation says that investment equals a share of output. Since the economy is closed, the money that consumers save is all investment - so “s” is really the savings rate, or what percent of output is saved, not consumed, by people. Finally, the capital accumulation equation just says that the capital of the next period (t + 1) equals the investment of the current period plus the capital of the current period that doesn’t depreciate. D is the depreciation rate, so if 90% of capital survives per period, we’re essentially just adding the 90% that survives to the new investment.
Now, in the Solow model we often consider the per capita, or per-person rates. Since N represents the number of workers, we can derive per capita equations from our base equations:
- Y/N = AKᵅN¹⁻ᵅ/N =AKᵅN⁻ᵅ = (A)(K/N)ᵅ
- Y/N = I/N + C/N
- I/N = sY/Nₜ
- Kₜ₊₁/N = I/Nₜ + (1-d)K/Nₜ
*note in the production function that the N on the bottom cancels out with the 1 in the 1 — alpha.
We’ll use lowercase to denote per-capita terms, so Y/N becomes y, C/N becomes c, I/N becomes i, and so on:
- y = Akᵅ
- y = i + c
- i = syₜ
- kₜ₊₁ = iₜ + (1-d)kₜ
We can see a few interesting things in these equations already: investment per worker in the economy equals savings per worker. So, more workers saving means more being invested and more capital being created. Furthermore, output itself reflect technology multiplied by capital per worker raised to the output elasticity.
Those are all the generation equations and formulas you need to know!
Now, taking a step back: we know that output is a function of capital and labour, so we can say that Y = f(K,N). We can rewrite this as xY = f(xK,xN), since constant returns to scale means that multiplying capital and labor by a fixed input will affect output similarly.
Taking xY = f(xK,xN), we’ll divide by N. We get:
1/N(Y) = (1/N(K), 1/N(N)). 1/N(N) cancels out, so we get:
(Y/N) = (K/N). Since we’re dividing output by labor, we’re essentially getting the output per person as a function of capital per person. Substituing our lowercase letters to represent per capita equations, this gives us one function:
y = f(k)
Well, y = f(k) is the output line on a graph labeled with output per worker and capital per worker. Here we can see the decreasing returns to scale from both inputs.
Now consider a standard aggregate demand function of Y = C + C¹Y + I + G. We’ll assume that government spending is zero and consumption independent of output is zero, so we get Y = C¹Y + I, or consumption as related to output plus investment equals national output. Thus:
Y = C¹Y + I
Y - C¹Y = I
Y(1 - C¹) = I
C¹ is the amount of consumption you consume for each dollar you’re given/earn—thus 1 - C¹ is functionally the savings rate, or how much you don’t consume for each dollar you get. We can thus substitute 1 - C¹ for the savings rate, or s:
Ys = I, meaning output multiplied by the savings rate is equal to investment. To find the investment per person (lowercase i), we’ll swap in Y for y, or output per person. So:
ys = i (investment per person = output per person multiplied by the savings rate)
However, we found prior that y = f(k), and thus we can make the substitution:
i = f(k)*s (investment per person equals capital per person multiplied by the savings rate)
y = f(k) must always be larger than i = f(k)*s unless s = 1, which is highly unlikely since that assumes a savings rate of 100% and zero consumption.
Well, remember that the investment & savings curve on our graph was always below the output line. f(k)*s is why, so let’s add that curve to the graph:
The depreciation line is equivalent to the depreciation rate, which we’re calling d, multiplied by the volume of capital. Since we want the per-person rate, we can say that the depreciation function is d(K/N). We’re using k for capital per person, so we can swap out K/N to produce kd. Let’s add it to the graph:
Also consider the following:
- Capital stock decreases by Kdₜ (t being time period) naturally over time.
- Capital stock increasing via new capital being created through investment takes the form of Iₜ = sYₜ
- To find the capital stock of the next period, we can consider the capital stock of this period (t) plus what we’re adding through investment minus the depreciation. So: Kₜ₊₁ = Kₜ + Iₜ - Kdₜ (Kₜ₊₁ is the total capital stock at time t).
- If we combine that relationship with the investment function (just swap out I for sYₜ and express it in per-worker terms, we get: (Kₜ₊₁/N) = Kₜ/N + sYₜ /N— Kdₜ/N
- If the equilibrium or steady state is reached, then the change in capital per person, or ΔK/N = 0.
- To change to the growth rate version of the Solow equation, we’ll divide both sides of the standard equation, ΔK/N = s(Y/N) - kd, by K/N or capital stock per person. Note that s(Y/N) is substituted for sf(K/N) since y = f(k). That gives us ΔK/N/(K/N) = sf(K/N)/(K/N) — d (since k = K/N, dK/N/K/N cancels out to d). To interpret that growth rate equation, note that sf(K/N) is really just the investment per worker or I/N, so the term represents the average investment per each unit of capital. The second term is just the depreciation rate. So, the average investment per unit of capital minus depreciation equals the change in capital per worker divided by the capital per worker (basically, the difference between the new and old capital per worker).
- We know that the savings rate increases the level of output per worker, but what happens to consumption? If savings rate (s) = 0, output is zero and so is consumption over the long-run. At s = 1, consumption is also zero since s + c = 1. This infers that steady state consumption is zero for s = 0,1 and that consumption will be positive at any other value. The steady state, or equilibrium point, that yields the highest long-run consumption per worker is called the Golden Rule steady state. The level of capital per worker associated with this steady state is creatively called the Golden-Rule level of capital per worker. To find this level, we can return to our prior Y = C + I (given our assumption that G and exports/imports = 0). So, C = Y - I. Thus C/N = Y/N - I/N. Since I = Ys, C/N = Y/N — s(Y/N), or C/N = (1-s)(Y/N). Thus the golden rule level of capital per worker is equivalent to the consumption rate (1-s) multiplied by the output per worker.
Based on the functions we’ve looked at, I want to emphasize two more implications that tie into the dynamics of national comparative growth:
Given that economies in this model slow down in growth as they approach the steady state equilibrium, poorer economies should grow faster than richer economies if they have the same steady state. This is the concept of convergence and can explain higher growth rates in developing countries.
However, countries that have different savings rates, different production functions, or different steady states won’t necessarily converge. Here we still get faster and then slower convergence to the steady state, but the end target is different. This can explain many developing countries that still lag behind richer countries.
So, that’s all from me on the Solow growth model!
Hope you found it helpful! Here’s another article detailing nominal and real GDP (here), and one more about how to calculate inflation using multiple GDP deflators (here). Also:
- Read about the Income Approach to GDP here.
- Read more economics stories here.
- To learn more about the oil market, consider reading about PADD Districts, the Why WTI and Brent are Crude Oils, and Why There are Price Differences Among Crude Oils, and my Oil & Gas Terms Guide.
