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    neoclassical model with diminishing returns to capital

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    1 - The production function is:
    <br>
    <br>F(K) = A*K
    <br>
    <br>Since this functions is linear in K (that is, it is a constant -A- multiplied by K), then the first derivative is simply:
    <br>
    <br>F'(K) = A
    <br>
    <br>Now, to find the 2nd derivative, we must derive F'(K). This one is easy, because F'(K) is a constant. The derivative of a constant is zero. Therefore,
    <br>
    <br>F''(K) = 0
    <br>
    <br>
    <br>2 - We have that y=Y/N, and k=K/N. In order to find y as a function of k, we must divide both sides of the production function by N:
    <br>
    <br>Y = F(K) = A*K
    <br>Y/N = (A*K)/N
    <br> y = A*(K/N)
    <br> y = A*k
    <br>
    <br>So that's y as a function of k
    <br>
    <br>3 - Since I can't write points above letters in text mode, I'll call k' to the k with a point above it (that is, the derivative of k with respect to time). We know from the data that:
    <br>
    <br>K' = sY - delta*K
    <br>
    <br>But, since delta=0,
    <br>
    <br>K' = sY
    <br>
    <br>What we want to find here is k' (the derivative of k with respect to time) and not K', but since k=K/N, then k'=(K/N)'. So we will calculate this expression:
    <br>
    <br>k' = (K/N)' = (K'*N - K*N')/N^2
    <br>(N^2 means N squared)
    <br>
    <br>This equality comes from the fact that this is the derivative of a division. Now, since we know the expressions ...

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