I spent a few days mulling about Chapter 5 on capital to income ratio β. Thus β is C/I where C is capital and I is income. Note that his definitions may be slightly different from the usual ones.I tried to do an exercise to help understand it better. He says that in the long run, it is s/g where s is the savings rate and g the growth rate. He also has an identity on the return from capital

α = r*β where * denotes multiplication and r is the rate of return on capital. To consider the long term changes, I will denote these quantities at a given time by subscripts β_t etc and after an year by β_(t+1) and growth rate for that year by g_t etc.

So C_(t+1)=C_t+r_t*C_t . Now divide both sides by I_(t+1). the right hand side is (C_t/I_t)*(I_t/I_(t+1)) +r_t*(C_t/I(t+1)). Now divide the numerator and denominator of the second factors on the right hand side by I_t. I assume that the population growth rate is zero when writing I_(t+1) in terms of I_t and g_t. Simplifying, we get the equation β_(t+1)= (β_t/(1+g_t))* (1+r_t).

Now assume that the growth rate also stabilizes. Since we know that β_t stabilizes, we must also have r_t stabilizing. Denoting the limits by β, r, t etc and taking limits on both sides and assuming that none goes to infinity, we have β=β(1+r)/(1+g) or we should have r=g in the long run, if we assume stability.

All this may be nonsense. If anybody who knows some economics (I do not) has corrections and suggestions, it will help me to understand the book a little better.

α = r*β where * denotes multiplication and r is the rate of return on capital. To consider the long term changes, I will denote these quantities at a given time by subscripts β_t etc and after an year by β_(t+1) and growth rate for that year by g_t etc.

So C_(t+1)=C_t+r_t*C_t . Now divide both sides by I_(t+1). the right hand side is (C_t/I_t)*(I_t/I_(t+1)) +r_t*(C_t/I(t+1)). Now divide the numerator and denominator of the second factors on the right hand side by I_t. I assume that the population growth rate is zero when writing I_(t+1) in terms of I_t and g_t. Simplifying, we get the equation β_(t+1)= (β_t/(1+g_t))* (1+r_t).

Now assume that the growth rate also stabilizes. Since we know that β_t stabilizes, we must also have r_t stabilizing. Denoting the limits by β, r, t etc and taking limits on both sides and assuming that none goes to infinity, we have β=β(1+r)/(1+g) or we should have r=g in the long run, if we assume stability.

All this may be nonsense. If anybody who knows some economics (I do not) has corrections and suggestions, it will help me to understand the book a little better.

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