2 Main assumptions

This is a particular version of the Solow model, one where we are making some special assumptions (more special than the ones Solow made). Some of them are technical assumptions meant to make the model easier to analyze. Throughout, I will try to be clear about what those special assumption are, and you should think about how things might change if those assumptions changed. As the opening of Solow’s article put it:

“All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive.”

2.1 Consumption and investment

To set things up, we’re going to make a few simplifying assuptions about the economic environment.

Assumption 2.1 (Closed economy without government) We assume that there is no trade, so net exports are zero. We also assume there is no government spending and taxes are zero.

This makes things very simple, because all GDP is either consumption or investment:

\[ Y_t = C_t + I_t \]

Important thing to remember: This is a statement about accounting, not about causal relationships! It will always be true, but it won’t be the case that we can make statements like “an increase in consumption increases GDP” in this model – that’s a more complicated question.

Along those lines, we will make the further assumption about how investment is related to the level of GDP.

Assumption 2.2 (Constant rate of saving) Assume a constant fraction of output, \(0<\gamma \leq 1\), is saved. (\(\gamma\) is the lower case Greek letter “gamma”). That is, \(\gamma Y_t\) is the amount of investment that occurs in a given period of time. We call \(\gamma\) the saving rate. \(\gamma\) is a parameter of the model – one that takes on a constant value by assumption.

It’s hopefully easy to see that if \(\gamma Y_t = I_t\), then as a matter of accounting, \[C_t = (1-\gamma)Y_t\]

2.2 The Aggregate Production Function

Now that we’ve set up some accounting facts, we need to figure out how much production there actually is. We’ll imagine a very stylized version of the economy where there are several different different “factors of production” – inputs – which are combined to create a single output. That output is real GDP.

We summarize the relationship between the quantity of inputs and the quantity of output using a mathematical relationship, which we’ll call the aggregate production function

\[ Y_t = f (A_t, L_t, K_t, H_t)\]

Where at a particular time \(t\) (say, a given year)

  • \(Y_t\) is real GDP in that year
  • \(L_t\) is the size of the labor force - the number of workers in the country
  • \(K_t\) is the stock of physical capital - the amount of equipment, physical structures, etc that are used to produce goods and services.
  • \(H_t\) is the stock of human capital - the knowledge, skills, and ability that individual workers have. They acquire human capital via education, training, experience, and other investments.
  • \(A_t\) is productivity. Productivity is the ability to combine factors of production into output. As \(A_t\) increases, a given amount of labor, physical, and human capital will produce more final output. We can think about this as improvements in technology, but also improvements in efficiency with which we use the factors of production.
  • \(f (A_t, L_t, K_t, H_t)\), the production function. This summarizes the mathematical relationship between the amount of labor, physical capital, human capital, productivity, and the quantity of real GDP produced.4

We will conventionally use upper-case letters for aggregate variables. Lower case letters will be for “per worker” variables. That is, \(Y_t\) is GDP in a country in year \(t\), and \(y_t \equiv Y_t/L_t\) is GDP per worker in year \(t\).

2.3 Assumptions about the aggregate production function

Assumption 2.3 (Constant Returns to Scale (CRS)) We assume that the production function is constant returns to scale (CRS) in the “reproducible factors of production” (labor, physical capital, and human capital). This means that if we increase each of these factors by the same proportion, output also increases proportionately. Mathematically, if \(z\) is a constant:

\[ z Y_t = f(A_t, z L_t, z H_t, z K_t) \]

The reasoning behind CRS is what’s often referred to as a “replication argument.” Suppose you have a factory with a certain amount of workers, and it produces a given amount of output. If you build an identical factory across the street, and put another identical set of workers in the new factory, you should be able to produce twice as much output as before.

So far, we’ve just made assumptions about the function \(f\) without saying what that function is. This is intentional, because if we know how the model works with a generic function (that satisfies some conditions), then we should know how it works when we make more restrictive assumptions. Sometimes, however, we want to be a little more specific, so we use particular functional forms.

The most common example of a constant returns to scale production function is something like the following

\[ Y_t = A_t K_t^\alpha \left( \frac{H_t}{L_t} \times L_t \right)^{1-\alpha} \]

This function may look a little strange. It’s an example of what’s called a “Cobb-Douglas” function.5 The part inside the parentheses looks like some terms should cancel out, and mathematically, they do. The reason we write it this way is to remember that human capital and physical labor are separate inputs. The term \(H_t/L_t\) is the average amount of human capital workers have. Multiplying that by the number of workers gives us the total quantity of human capital inputs. The way human capital works in this particular production function is by making it “as if’’ we have more workers than actually physically exist.

Exercise:. Show that the Cobb-Douglas production function above is constant returns to scale. (That is, show that \(Y_t\) is the same as the equation above, and if we multiply all the reproducible factors by \(z\), you’ll get \(z \times Y_t\) output).

The CRS assumption is a standard assumption. The next assumption is special: We don’t need to make it, the standard Solow model doesn’t make it, but it will make solving the model easier.

Assumption 2.4 (Special assumptions about the production function) We will assume that the level of productivity, the number of workers, and the amount of human capital is fixed over time.

Why make these assumptions? They allow us to go from the aggregate production function to one that is in per worker terms easily.

Exercise:. Exercise: Starting from \[ Y_t = f (A_t, L_t, K_t, H_t)\] Apply assumption 2.4 and then 2.3 to derive the “per worker” production function (lower case letters referring to per-worker quantities): \[ y_t = f(A, k_t, h) \]

We make one more, critical assumption about the production function:

Assumption 2.5 (Diminishing marginal product of capital) The amount of output per worker always increases when we add additional capital per worker, but at a diminishing rate. Each additional unit of capital per worker adds less and less output, holding everything else fixed.

This mean if you have a very small amount of capital, adding an additional machine makes a big difference in the amount you produce, all else equal. If you have lots of machines, and the same number of workers, one additional machine won’t help as much.6

Exercise:. Explain the difference between assuming constant returns to scale and diminishing marginal product of capital (why don’t these assumptions conflict?)

Exercise:. Draw a production function (with capital per worker on the horizontal axis, and output per worker on the vertical axis) that is consistent with 2.5. Then, draw a function for investemnt per worker \(i_t = \gamma y_t\) that is consistent with the production function you drew. Label output, investment, and consumption at a particular level of capital per worker.

2.4 The evolution of physical capital

Given the assumptions we’ve made, the only factor of production that changes over time is the amount of physical capital (the capital stock) per worker. The next set of assumptions is about describing how the capital stock changes over time.

To think about this, we’ll focus on two special features about physical capital. First, it is created by the act of investing. Investing is the decision to use resources today to create more physical capital that will be available for use in the future. The other feature is is that physical capital depreciates (wears out) over time. For instance, a car will break down over time as its parts wear out, and those parts will have to be replaced.

In other words, the amount of capital available at a given time could be generally described as

\[\begin{equation} \begin{aligned} \text{Capital this year } = & \text{ Capital last year }&\\ & - \text{Capital that "wears out" (depreciates) after last year} &\\ & + \text{New capital built (investment) last year }& \end{aligned} \end{equation}\]

We need to make an assumption about how much capital wears out in each period.

Assumption 2.6 (Constant rate of depreciation) Assume that the rate of depreciation is fixed over time at a number \(0<\delta\leq 1\). (\(\delta\) is the lower case Greek letter “delta”.) That is, \(\delta K_t\) is the amount of physical capital that wears out in a period of time.

Exercise:. Graphically depict how the level of depreciation of capital per worker changes as the amount of capital per worker changes. That is, draw a “depreciation function” with capital per worker on the horizontal axis, and depreciation per worker on the vertical axis that is consistent with 2.6.

Mathematically, we can represent the evolution of the capital stock in the aggregate as

\[ K_t = K_{t-1} - \delta K_{t-1} + I_{t-1} \]

where \(I_{t-1}\) is the aggregate amount of investment in year \(t-1\).

By dividing by the number of workers, we can come up with the per-worker capital evolution equation

\[ k_t = k_{t-1} - \delta k_{t-1} + i_{t-1} \]

(Hopefully, it’s easy to intuit how this would change if we assumped population was growing over time: the amount of machines per worker would depend on how many machines were wearing out, how many new machines we were building, and how quickly new workers were arriving).

2.5 Summarizing the main assumptions

In this section we’ve outlined the crucial mathematical assumptions behind our version of the Solow model:

  • Assumption 2.1 Closed economy without government, so, (in per worker terms) \[ y_t = c_t + i_t \]

  • Assumption 2.2 Constant rate of saving and investment, so that (in per worker terms) \[i_t = \gamma y_t \]

  • Assumption 2.3 The aggregate production function is CRS:

\[ Y_t = f(A_t, L_t, K_t, H_t) \]

  • Assumption 2.3 For simplicity, assume productivity, labor, and human capital do not change over time:

\[ Y_t = f(A, L, K_t, H) \]

  • Assumption 2.4 Combining the previous two assumptions,

\[ y_t = f(A, k_t, h) \]

  • Assumption 2.5 Diminishing marginal product of capital

  • Assumption 2.6 Constant rate of depreciation, \[0 < \delta \leq 1\]

In the next section, we’ll start combining these assumptions to figure out what they imply for the predictions of the model.

  1. To keep all of these straight, it may be helpful to think about production of a particular good, and pretend that good is the only good produced in the country. The economist Paul Romer (who won the 2018 Economics Nobel Prize for his contributions to growth theory) uses the analogy of producing a chair. The chair is the final good (real GDP). The worker who produces the chair is part of the labor force. The tools she uses to produce the chair are physical capital. Her training as a carpenter is her human capital; nobody else can “use” her human capital without hiring her to use it. Productivity, on the other hand, is like the Pythagorean theorem or other facts about geometry that anyone can use to make a chair once they’re discovered. Physical and human capital, clearly, take resources to produce; we have to spend resources to create tools or to train carpenters. However, in general, we do not need to expend resources to re-invent geometry once we’ve discovered it. And once we know how many workers we have, their individual skill as carpenters, the tools they work with, and how productive they are using their physical and human capital, we can know how many chairs they produce.↩︎

  2. Cobb-Douglas functions are functions that are the products of exponential terms. If you’re feeling rusty with exponents, there are some reminders in the Appendix @ref(#mathappendix).

    A constant returns to scale Cobb-Douglas function is one where all the exponents sum to 1. You could also have decreasing returns or increasing returns to scale with this function, by changing the exponential terms.

    You often see Cobb Douglas production functions, but sometimes you see them pop up in other contexts in micro- and macroeconomics. They’re very convenient to work with and have some nice properties.↩︎

  3. With a littl ebit of calculus, it’s easy to verify this assumption holds for the Cobb-Douglas case.↩︎