Solow model golden rule with my exact answer

Question

In a perfectly competitive Solow economy with physical capital accumulation, population growth and a Cobb-Douglas production function, show that the “golden rule” steady-state would be reached if at every period aggregate consumption coincides with the aggregate labour income.

Answer 1

Let production function $$F(K_t,L_t)=K_t^aL_t^{1-a}$$

$$max[c^*=f(k^*)-(δ+n)k^*]$$ with respect to $k^*$

Then $MPK=a(k^*)^{a-1}$

So $k_G= (\frac{a}{(δ+n)})^{1/1-a}$ which is golden rule level of capital stock per capita.

Now calculate consumption at golden rule level

$$c_G=f(k_G)-(δ+n)k_G$$ $$c_G=(k_G)^a-(δ+n)k_G$$

$$c_G=(\frac{a}{(δ+n)})^{a/1-a}-(δ+n)(\frac{a}{(δ+n)})^{1/1-a}$$

$$c_G=(\frac{a}{(δ+n)})^{a/1-a}[1-(δ+n)(\frac{a}{(δ+n)})]$$

$$c_G=(\frac{a}{(δ+n)})^{a/1-a}[1-a]$$ $$c_G=k_G^a[1-a]$$

Ow consider to calculate wage

$w=f(k)-kf’(k)=(1-a)f(k)$

Then,

$$c_G=k_G^a[1-a]=w_G$$

$$c_G=w_G$$

Now consider aggregate version,

$C_G/L=W_G/L$

so,

$C_G=W_G$

I am confused at the aggregate transformation.

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