Lets make fun of Obama while he is still relevant.

[Warren Dew]

(12-06-2016, 04:06 PM)Mikebert Wrote:

Warren Dew Wrote:d(population)/dt / population > 0 (percentage population increases as old people have retired but not died yet)
 
d(GDP/population)/dt = population * d(GDP)/dt + GDP * d(population)/dt

Using more compact notation you are saying:
 
d(GDP/N)dt = N * dGDP/dt + GDP dN/dt
 
this is the differentiation of the product of GDP and N, done correctly you would have
 
d[GDP * (1/N)]/dt = (1/N) dGDP/dt - GDP (1/N)^2 (dN/dt)
 
I'm still confused

Sorry, yes, messed up on that intermediate line.  The conclusion is still the same, though.  Derivation with the corrected equation below.

d(GDP/worker)/dt / (GDP/worker) > 0 (percentage individual worker productivity increases)

d(workers)/dt / workers < - d(GDP/worker)/dt / (GDP/worker)
    (percentage worker population decreases faster than percentage worker productivity increases)

d(GDP)/dt / GDP = d(workers*GDP/worker)/dt / (workers*GDP/worker)
    = (GDP/worker * d(workers)/dt + workers * d(GDP/worker)/dt) / (workers*GDP/worker)
    = d(workers)dt / workers + d(GDP/worker)/dt / GDP/worker < 0 (GDP decreases)

d(population)/dt / population > 0 (percentage population increases as old people have retired but not died yet)

d(GDP/population)/dt = d(GDP)/dt / population - GDP * d(population)/dt / population^2
    = (GDP/population) * (d(GDP)/dt / GDP - d(population)/dt / population) < 0
    (average living standard for the economy cannot keep up)