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neoclassical model with diminishing returns to capital

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Is there convergence across economies?

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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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