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Unformatted text preview: Usually
we write Yt = F (At , Kt , Ht ) where Ht is the amount of human capital.
Let’s start with the assumptions!
Assumption 1. Total human capital in the economy is given by Ht = ht Lt , where Lt is the amount of people in the country at
time t. However, we are going to assume that everybody has the same amount of human capital, ht , so, we can just normalize it to
ht = 1. Then, Ht = Lt .
Also, we cannot use all production functions. We need a production function that has some nice properties. That is,
Assumption 2. Our production function will be a neoclassical one. That is, it will show Constant Returns to Scale and also,
Diminishing Marginal Returns to Capital and Labor (but we will focus on capital).
Constant Returns to scale
Suppose that we have a Ford factory in Michigan. And, we copy exactly the same factory, with the same amount of
machines, and the same people, and we install the new factory in Iowa. The production of cars of the new factory should
be exactly the same. Hence, we double the amount of capital and the amount of labor, and we double also output, USING
THE SAME RECIPE. Suppose a CobbDouglas production function, Y = AK α L1−α . Now, multiply both RIVAL factors
by 2. A(2K )α (2L)1−α = A2α 21α K α L1−α = 2AK α L1α = 2Y . Thus, we observe CRS. Notice that A does not change,
since we are assuming that we use the same recipe. Formally, we have Constant Returns to Scale if,
F (At , λKt , λLt ) = λF (At , Kt , Lt )
where λ is a constant (2, 3, one milion...).
Diminishing (but positive) marginal product of capital
Suppose that you are a broker. You have a cubicle in an amazing 50th ﬂoor of a NYC building, and you have a computer
in front of you, where you take care of the portfolios of your clients. However, your boss decides to provide you with
an extra computer. Now, you have to split your time with both computers. Yeah, maybe you can do it. You will use the
new computer a bit less than the other one, but, you can still manage that. That is, the extra “GDP” that you produce
with the second computer is still high, but smaller than with the ﬁrst computer. You are taking care of many more
clients at the same time. Your boss observes that you are still a young boy, with a lot of energy, and he decides to buy
another computer for you. Now, you have to deal with even more portfolios in three different computers, every day.
Ok...you can!! You cannot manage so many portfolios with the third computer, but, you can still do it... And so on,
and so on...When your boss gives you the 50th computer, obviously you cannot manage more portfolios with the extra
computer, or maybe you can take care of only one or two extra portfolios (if you can do more, maybe you should think
if you come from Mars...). Therefore, the extra “GDP”...
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 Spring '10
 Xaviersalaimartin

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