Assume that the production function in an economy is given by y=k1/2, where y and k are the per-worker levels of output and capital, respectively. The savings rate is given by s=0.2 and the rate of depreciation is 0.05. What is the optimal savings rate to achieve the golden-rule steady state level of k?
Steady state occurs when change in k = 0 and as change in k = sy - dk = 0 where d = depreciation rate = 0.05 => sy = dk
Formula : c = y - i where i = investment per worker = sy and c = consumption per worker
We have to find that steady state level of k which will maximize c(That level of k is what we called golden rule level of k)
change in k = sy - dk = 0 where d = depreciation rate = 0.05 => sy = dk = 0.05k
At steady state we sy = dk and also i = sy(discussed above)
=> c = y - i = y - dk = k1/2 - 0.05k
Maximize : c
First order condition(FOC) :
dc/dk = 0 => 0.5k-1/2 - 0.05 = 0
Solving this we get k = 100
so sy = dk => sk0.5 = 0.05k
=> s*1000.5 = 0.05*100 => 10s = 5 => s = 0.5 = 50%
Hence, Golden rule level of saving rate = 50%
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