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The aggregate production function describes the relationship between aggregate output and the production inputs that drive said output.
“Output” is real GDP, or total real income. This essentially describes on a national level the value of goods and services that are produced once adjusted for inflation or deflation.
For inputs, we’ll consider capital and labour. Capital, or physical capital, is equivalent to all the machines, factories, office buildings, and other infrastructure that drives productive output in the economy. Labour is simply the number of workers in the economy.
We’ll use Y to denote output, K to denote capital (not C, which often denotes consumption), and N to denote labour).
Thus, let’s form our aggregate production function:
Y = f(K,N)
This says that output is a function of capital and labor. We don’t necessarily quantify that relationship specifically, but rather generally denote the inputs driving the output. Also note that we can alter the function as follows:
2Y = f(2K,2N)
This means that the function has constant returns to scale (CRTS), e.g. if we double the labor and capital, we’ll also double the output. We can generalize this to say that for any positive constant x, we get:
xY = f(xK,xN)
However, despite constant returns to scale, we get decreasing returns to individual inputs. This means that increasing just capital leads to smaller and smaller increases in output, and vice versa with labor. We call the extra output from one extra “unit” of capital the marginal production of capital, or MPK. We can say that the MPK equals the change in Y:
MPK = ΔY = Yₜ – Yₜ ₋ ₁
This means that the total output in the current period equals the output in the current period minus the output of last period. We can substitute each Y for f(K,N). Thus:
MPK = ΔY = Yₜ — Yₜ ₋ ₁ = f(K +1,N) – f(K,N).
MPK = f(K +1,N) — f(K,N).
We can use K + 1 to denote the increase (the ΔY) as being one unit of capital. In that equation, note that the MPK is declining as K increases if labour is held as a constant. N + f(K + 1) — f(K). MPL, or marginal product of labour, is similarly as follows, and diminishes as N increases:
MPL = ΔY = Yₜ — Yₜ ₋ ₁ = f(K,N + 1) — f(K,N).
Returning to our aggregate production function, we can find the output per worker, which generally correlates with productivity and standard of living:
xY = f(xK,xN)
Output per worker is simply Y/N, or output divided by the number of workers in the economy. Given constant returns to scale, we’ll set x to 1/N:
1/N(Y) = f((1/N)K,(1/N)N) = Y/N = f(K/N,1)
We can get rid of 1, so:
Y/N = f(K/N)
This means that output per worker equals capital per worker.
Now, here’s the relationship we’ve figured out in graphical form:
We can see that increases in capital per worker leads to those decreasing returns to capital as the curve levels off.
Overall, when thinking about the two inputs (capital and labor), we can keep in mind that increasing the capital stock (amount of capital) by itself cannot sustain growth because of the diminishing marginal product of capital. The supply of available workers will likely also level off over time due to population constraints and birth rates dropping as countries develop.
So, instead, sustained growth really required that x factor of technological progress that increases per-worker productivity, and that’s a core takeaway of the aggregate production function. There is theoretically no limit to how far the production function can shift given increases in technological progress.
Hope you found the article 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.