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Unformatted text preview: that you can produce with the last computer is very small. This is
what we call Diminishing marginal product of capital (MPK). Suppose again that we have a Cobb Douglas production
function. Then, if we double the amount of capital,
F (A, 2K, L) = A2α K α L1−α = 2α AK α L1−α < 2AK α L1−α
Note that the extra “GDP” that you produce with a new computer is always positive. That is, a new computer, will not
reduce the amount of portfolios that you have to manage. This amount will increase, but this increase will be smaller
and smaller as we increase the number of computers.
2 Figure 1: A production function that shows DMPK (holding labor constant) How can we explain this concept using Maths? Well, ﬁrst notice that once we introduce a new computer, we produce
some extra GDP. How can we explain this in mathematical terms ? The ﬁrst derivative is positive! That is, once we
increase one argument of the function, the function increases. So, our ﬁrst requirement for a production function is that
MPK = ∂ F (A, K, L)
∂K The second requirement is that the increase in GDP should be smaller and smaller once we introduce new computers.
This means the the second derivative is negative! (Also, remember that Calculus is a requirement for this class...). So, we
∂ 2 F (A, K, L)
• Make sure that you understand the difference between constant returns to scale and diminishing MPK. They might
look contradictory at ﬁrst sight. But you should realize that when we show CRS, we are augmenting both factors
(capital and labor)!!! To show DMPK, we are holding labor CONSTANT and we increase capital.
• We will also require that the marginal product of labor is positive but diminishing (you can do the same as before,
but you should take the derivatives with respect to labor instead of capital)
If a production function fulﬁlls these requirements, it looks like the one in Figure 1.
Suppose that the production function is Cobb-Douglas, Y = AK α L1−α . Which is the condition such that DMPK holds?
Finally, let’s say something about the amount of people in the economy. It might be very simple to assume that population
is constant. We observe in the real world that people are born, they die, they migrate...So, at every period of time, the
amount of people in a country is different. Remember that the growth rate of population can be expressed as:
nt = f ertilityt − mortalityt + net migrationt
And also, remember that these factors are strongly correlated with GDP per capita. That is, the higher is GDP per capita
in a country, the lower is the fertility rate, the lower is the mortality rate, and the higher is the inmigration rate. However,
we are going to simplify further the model, because we want to focus on capital, so
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- Spring '10