2 Main assumptions
This is a particular version of the Solow model, one where we are making some special assumptions (even 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, and we have to dig in beyond the accounting identity.
Along those lines, we will make the further assumption about how investment is related to the level of GDP:
Exercise: Show that
\[ C_t = (1- \gamma) Y_t\]
(This should be a pretty quick line of algebra)
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. We can think of a unit of that output as a unit of 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 way we quantify the our 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, 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 amout 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.
The most common example of a constrant returns to scale production function is the “Cobb-Douglas” production function:
\[ Y_t = A_t K_t^\alpha \left( \frac{H_t}{L_t} \times L_t \right)^{1-\alpha} \]
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 (and important) assumption in the Solow model. 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: 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.
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.
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
\[ k_t = k_{t-1} - \delta k_{t-1} + i_{t-1} \]
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 these 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.
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.↩