Usually we write yt f at kt ht where ht is the amount

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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 Cobb-Douglas production function, Y = AK α L1−α . Now, multiply both RIVAL factors by 2. A(2K )α (2L)1−α = A2α 21-α K α L1−α = 2AK α L1-α = 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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