Notes on the Solow Model

“Once one starts to think about [economic growth], it is hard to think about anything else”
— Robert Lucas (1988)

The bulk of undergraduate macroeconomics revolves around understanding the models used to analyze growth and sustained long term growth. One of the very first economic models we’re introduced to is the Solow model of growth. In this post I aim to recap and hopefully, in a concise manner summarize Romer’s advanced macroeconomics¹ sections on the Solow model to explain the dynamics underlying growth. Of course there are other advanced models that better explain certain theories and key ideas of growth but are still best understood in comparison to the Solow model.

What is it that allows countries to be vastly richer than they were a hundred, even fifty years ago? How do we characterize the empirical evidence of wealthier nations slowing down in growth while developing nations continue to see increased growth? Would we one day see convergence between all nations?

One remarkable example given in the opening pages of Advanced Macroeconomics is the growth of real income per person in India compared to the United States. During the average recession in the United States, real income per person often falls by a few percent relative to the usual path. In contrast, the productivity slowdown (as we’ll come to see is quite important) — despite that the average annual productivity growth since the 1970s has been roughly 1 percentage point below its previous level — has reduced real income per person in the United States by 20 percent relative to what it otherwise would have been².

Let’s compare India and the United States. If India continues its growth at its recent average rate of approximately 5.5 percent per year, it will take roughly fifty years for Indian real incomes to reach the current U.S levels. And if growth slows to 3 percent, the process will take closer to a hundred years. What if it slips down to just 1.3 percent — the average growth rate of India’s pre-reform era? This wait stretches to nearly two hundred years. Quite the reminder of how important the growth rate matters in the long run³.

We begin with the fundamental and probably simplest macroeconomic model. The base Solow model with no augmentations. Then we move to relaxing the fixed savings assumption to become endogenous instead of treating it as exogenous. Then we leave the Solow model to discuss the following augmentations: treating technological progress as an endogenous variable, and the determinants of the decision to invest in knowledge accumulation (empirically we see countries with strong public and private research institutions tend to be wealthier).

Will technological progress save us all?

The Model

The Solow model starts with a closed economy.

In this economy, denote all output and sum total of economic as , a function of capital, , and labour at time with a multiplicative factor , known as “knowledge” or effectiveness of labour or technological level.

Thus we obtain the production function⁴

It seems also that natural resources isn’t quite a heavy constraint on growth. It is reasonable to assume that there are constant returns to and . The intensive form of the production function is . is capital per unit of labour⁵, and rewrite , . In other words, we can write output per unit of labour as a function of capital per unit of labour.

For mathematical niceness, the intensive-form production function satisfies , , . Additionally, satisfies the Inada⁶ conditions

Next, I’m going to rattle off a few assumptions, and derive a discrete time version of the Solow model (before moving to the continuous time version in the Romer book). Assume a closed economy, investment is a fraction of disposable income, labour is paid a wage and capital is paid by interest or rental price . Capital depreciates at rate and to obtain the next period’s capital stock, whatever’s left of the remaining capital in this period must be included with investment for next period’s capital. We call this the law of motion for capital accumulation. Thus we end up with the following basic equations of the Solow model:

Dynamics of inputs to production

To put the Solow model into use, we must concern ourselves with how each of the inputs to production vary over time. The model here is set in discrete time for simplicity, though for continuous time it is much nicer to know that these variables are defined at every point in time. The initial levels of capital, technical progress, and labour are taken as given.

Dynamics of K

It’s convenient to focus on capital stock per unit of labour rather than the unadjusted capital stock . Then we have . Or, equivalently we have . What this tells us is that the change in capital stock from one period to the next time period is equivalent to the investment per unit of labour minus depreciation of capital stock at the current time period.

In other words, can be viewed as the break-even investment, the amount needed to keep at its existing level at time .

The Inada conditions also imply the following about :

• When , so it is large, meaning the investment line is much steeper than the depreciation line. Thus for small , actual investment is much larger than break-even investment.
• Conversely, when becomes large, falls to zero. At some point, this slope of the actual investment line falls below the slope of the break-even investment (depreciation) line.
• Finally, implies that for , these two lines and must cross only once.

This is known as the steady state. Some properties include, when is initially less than , actual investment, > , exceeds required investment, and so . Thus k is rising. If exceeds , then . Finally, if , . Thus, regardless of the initial value of , it will eventually converge to .

We come to three observations about the exogenous savings rate .

• The saving rate has no long-run effect on the growth of output per unit of labour since the steady state is characterized by a constant level of output per worker.
• Nevertheless, the saving rate determines the level of output per unit of labour in the long run. All else being equal, countries with a higher savings rate will also have a higher level of output per unit of labour in the long run.
• An increase in the savings rate will lead to higher growth of output per unit of labour for a period of time, known as a positive shock, but not indefinitely. This period of growth will end when the economy transitions to a new steady state.

Golden-rule consumption

We turn to the golden-rule or golden-rule level of capital, a description that I try not to use but in this case I will indulge, which describes the level of capital or the savings rate that produces the highest level of steady-state consumption.

Again, we express everything in terms of per unit of labour for convenience. In the steady state, the level of consumption is

And the level of actual investment equals the level of break-even investment which implies that

The maximization problem for is

Which gives us the first order conditions , giving . Stated plainly, the golden-rule savings rate is the one that supports in steady-state via .

If initially the savings rate is less than the golden-rule savings rate, then a small increase in will result in a long run increase in consumption when in the new steady-state. Conversely, if the initial savings rate is greater than the golden-rule savings rate, then an increase in will result in a long run decrease in consumption in the new steady-state. One can think of it as an increase in output is offset by the increase in depreciation because of the larger capital stock.

Law of motion when population is varying

Relaxing the assumption of a constant population (i.e number of workers), we find that a growth in the number of workers causes capital per unit of labour to fall.

Suppose that the number of units of labour is growing over time with growth rate .

Starting with the law of motion for capital accumulation,

Giving us

Which becomes the new law of motion for capital accumulation per unit of labour when population is allowed to grow at rate .

Varying population at the steady-state

Since the level of capital does not change at the steady state,

And solving for yields

Which gives the relation that the level of actual investment should be equal to the break-even investment, which happens to be the sum of the population growth rate and depreciation rate.

When technical progress is labour augmenting

In this case, let us define a new variable which we can think of as a measure of productivity which is assumed to be constant.

Here, our production function will look like . It is still natural to assume constant returns to scale for and .

Since technology is ever changing and constantly evolving rapidly, we can model the evolution of with a technological growth rate .

Thus our new set of equations characterizing the augmented Solow model is

Law of motion when technical progress and population are varying

The following derivation needs no explanation. Though I should reference that in the footnotes we say is capital per unit of effective labour, which we can denote here as .

Thus, we have that

Steady state analysis

Here, an increase in leads to an increase in , thus decreasing the capital stock per effective unit of labour. Therefore we require a proportional increase in to maintain the current level of capital per effective worker.

In the steady state, .

Since both growth rates are less than zero, we can approximately say .

Thus

Or

In this case, the actual investment is equal to the break-even investment where the depreciation rate is modified by the sum of population and technical progress growth rates.

The sum is also called the growth rate of effective labour. Again, we observe that the marginal product of capital, must equal for the optimal amount of capital stock per unit of effective labour in the steady-state.

Kaldor’s facts in relation to the augmented Solow model

• In the steady-state, output per effective worker grows at a constant rate given by the growth rate of technology
• In the steady-state, output and capital per effective worker will grow at constant rates, equal to . So the capital-output ratio will be constant in the steady state.
• In the steady-state, the change in rental rate of capital is constant. which is constant so is constant.
• In the steady-state, . Therefore implying that . In other words, the real wage will grow at a constant rate.
• Labour’s share of income is constant because and both grow at rate .

Relating productivity output and unemployment

Naturally, we may have some questions about employment from the given production function. If we rewrite as , we have the relation that as technological progress grows or labour productivity grows, the lower the level of employment. How much then, should output increase to offset the unemployment caused by increases in per unit of effective labour productivity?

Recall the IS-LM model, though here we let the interest rate be exogenous and therefore reduces the LM curve to just a bank policy rate .

We can write the level of output determined by the IS and LM relation as

Where is a risk premium and . If we denote the total labour force as , then the unemployment rate is simply

We can now turn to what the labour market implies about real wages.

Price setting

By the production function, each unit of output requires workers. Firms set prices as a markup over the per unit of labour costs. Therefore the unit labour cost is , giving the price . From this we can see that an increase in productivity decreases costs, which decreases the price level given the nominal wage. A higher productivity raises the real wages that firms can afford.

Wage setting

We begin with a simple intuition. Workers and firms are always in a bargaining state over wages taking into account expected productivity, unemployment rate, and the broader market conditions captured by exogenous variable (things like unemployment benefits, unions, hiring firing costs etc). In short, wages are typically set to reflect the increase in productivity over time.

From the price setting equation we have . Now, if the expectation for prices is equal to the current level of prices, and likewise for productivity, the previous equation yields

Thus the natural rate should depend on neither the level of productivity nor on the rate of productivity growth.

Infinite growth in a finite world?

Not sure who this quote is attributed to but it goes along the lines of, “whoever believes infinite growth can be achieved in a finite world is either a madman or an economist”.

It’s a great thing that I’m heading down either road.

The last hundred years were nothing short of an anthropological miracle. The Solow model, albeit simple and reductionist, gives us a lens to begin to understand the mechanics of that miracle. The last six years, driven by breakthroughs in machine learning has only reinforced the conclusions drawn from the simple assumptions of the Solow model, that technology is the binding constraint on long-run growth. It is just as difficult to imagine what the world looks like a hundred years from now as it would have been for someone in the 18th century to envision the 20th. Whether this quip turns out to be wisdom or irony may depend entirely on our will to continue exercising dominion over life.